Elementary Preservice Teachers? Conceptions of
Common Approaches to Teaching Science and Mathematics
by
Kimberly Nunes-Bufford
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 12, 2011
Copyright 2011 by Kimberly Nunes-Bufford
Approved by
Charles Eick, Chair, Associate Professor of Curriculum and Teaching
Sara Wolf, Associate Professor of Educational Foundations, Leadership and Technology
Deborah Morowski, Assistant Professor of Curriculum and Teaching
ii
Abstract
This study investigated preservice teachers? thinking about common approaches to math
and science education for elementary children in grades K-6. Specifically, this study focused on
preservice teachers' thinking on the interpretation of tools for conceptual development,
consideration of processes for meaningful learning, and conceptions of pedagogical approaches
between mathematics and science. The study occurred while elementary preservice teachers
where in a jointly enrolled science and mathematics methods classes and subsequent internship.
The learning cycle was a common approach used in the methods courses and used by elementary
preservice teachers in the field. The nature of the students? understandings was examined
through several data sources: open-ended pre and post-course tests and weekly blogs. Results
indicated varied conceptions of tools, processes, and approaches in science and math teaching at
the beginning of the methods courses. Many of the participants initially thought of science and
math as being approached in differing ways. Initial views of science ranged from preservice
teachers thinking of science in terms of teaching out of the textbook, watching videos, or
conducting experiments. Initial views of mathematics teaching ranged from teachers
demonstrating and students practicing, teaching real-world mathematics, or teaching
mathematics with hands-on learning. All of the participants expressed broadened ideas about
teaching mathematics and science at the end of both methods courses. At the end of the semester
82 percent of preservice teachers recognized commonalities in teaching approaches for math and
science, including use of inquiry-based teaching, as well as the use of the learning cycle. Follow-
iii
up observations were conducted from a portion of the participants during their student teaching
experience. Case studies were presented of two of the preservice teachers? conceptions of use of
tools, processes, and approaches to mathematics while in their internship. Jane wanted students
to understand mathematics beyond memorizing formulas. With some mentoring, Jane developed
and implemented lessons that used tools for conceptual development, processes for reasoning,
and a learning cycle approach. Kate believed that mathematics needed to be approached with real
life mathematics, such as time and money, in order to keep students engaged. With mentoring,
Kate also implemented a few lessons following standards based math curricula that involved
tools to promote understanding and processes for reasoning. However, by the end of the
observation time she reverted back to a style of teaching that focused on learning facts in
isolation.
iv
Acknowledgements
First and foremost, thank you to my Heavenly Father. I am grateful to my family for all
of their support as I traveled on this academic journey. I could not have done this without the
love and support from my husband. Many thanks for all of the support and encouragement from
the faculty and staff in the College of Education. Special thanks to Dr. Charles Eick for all of the
guidance during this process. Special thanks to the preservice teachers for letting me be a part of
their learning process.
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Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgments.............................................................................................................. iv
List of Tables .................................................................................................................... xii
Chapter 1. Introduction ........................................................................................................1
Overview and Rationale of Study ............................................................................1
Brief Literature Review ...........................................................................................2
National Standards .........................................................................................2
Science Framework for Mathematics Teaching ............................................4
The Learning Cycle in Elementary Teacher Preparation ...............................6
Research Questions ..................................................................................................9
Participants, Research Methods, and Organization of this Dissertation ................11
Overview of Results .....................................................................................12
Chapter 2. Review of Literature.?????????????????????? 14
Efforts to Reform Science and Mathematics Education ?????????? 14
National Standards??????????????????????15
Reform Movement ..? ???????????????????? 17
Curriculum??????????????? ??????? ???20
Knowledge .??????????????????????????? 22
Content Knowledge .?????????????????????22
vi
Pedagogical Content Knowledge .????????????????25
Curricular Knowledge ..????????????????????27
Constructivist Perspective on Learning .................................................................28
Processes for Meaningful Learning in Science and Mathematics ...???29
Problem Solving.................................................................................31
Reasoning and Proof ..........................................................................32
Communication ..................................................................................33
Connections........................................................................................34
Representations ..................................................................................35
Common Teaching Approaches in Mathematics and Science ...............................37
Reflection .....................................................................................................37
Inquiry ..........................................................................................................38
Assessment ...................................................................................................39
Discourse......................................................................................................40
Tools for Conceptual Development .............................................................41
Preparation of Elementary Teachers ......................................................................44
Improving Teacher Education Programs .....................................................45
Science and Mathematics in Elementary Teacher Education Programs ......46
Disparities in Teacher Education Programs .................................................47
Mathematics Courses for Elementary teacher Programs .............................49
Science Courses for Elementary Teacher Programs ....................................50
Teacher Beliefs and the Impact on Practice ...........................................................51
Student Teaching ...................................................................................................55
vii
Conceptual Change ................................................................................................57
The Learning Cycle for Mathematics and Science Teaching ................................61
Historical Context of the Learning Cycle ....................................................61
5E Instructional Model of the Learning Cycle ............................................65
The Learning Cycle: Elementary School through Higher Education ..........69
Conclusion .............................................................................................................74
Chapter 3. Methodology ....................................................................................................75
Introduction to the Study .......................................................................................75
Research Paradigm.................................................................................................76
Constructivist Grounded Theory Methodology .....................................................77
Researcher as Part of the Research Process ...........................................................79
Design and Research Methods ...............................................................................82
Case Study Approach ...................................................................................82
Participants ...................................................................................................83
Data Sources ................................................................................................87
Data Management ........................................................................................91
Data Analysis: Phase I .................................................................................92
Data Analysis: Phase II ................................................................................97
Ethical Considerations ...............................................................................100
Data Triangulation .....................................................................................101
Chapter 4: Elementary Preservice Teachers? Conceptions of Common Approaches to
Mathematics and Science during Methods Course ..............................................103
Abstract ................................................................................................................103
viii
Introduction ..........................................................................................................104
Background ..........................................................................................................107
Standards Documents.................................................................................107
Commonalities in Standards Documents .........................................108
Learning Theory and the Learning Cycle ..................................................109
Research on the Learning Cycle ................................................................111
Methods................................................................................................................113
Research Design.........................................................................................113
Context of the Study ..................................................................................113
Data Collection ..........................................................................................115
Data Analysis .............................................................................................115
Categorical Themes .........................................................................117
Results ..................................................................................................................118
Conceptions of Tools .................................................................................118
Science .............................................................................................118
Mathematics .....................................................................................120
Conceptions of Processes ...........................................................................123
Traditional View ..............................................................................123
Expanded View ................................................................................125
Progressive View .............................................................................126
Conceptions of Pedagogical Approaches ...................................................127
Traditional View ..............................................................................127
Expanded View ................................................................................129
ix
Progressive View .............................................................................129
Common Approaches between Mathematics and Science ........................131
Discussion ............................................................................................................134
Impact of Beliefs and Prior Experiences ...................................................137
Issues with Inquiry .....................................................................................138
Implications................................................................................................139
References ............................................................................................................141
Appendix A: Pre-test............................................................................................153
Appendix B: Post-test ..........................................................................................154
Appendix C: Weekly Blogs .................................................................................155
Chapter 5: Elementary Student Teachers? Conceptions and Use of Tools, Processes, and
Approaches for Mathematics Teaching ...............................................................158
Abstract ................................................................................................................158
Introduction ..........................................................................................................159
Issue as Conceived in Study ......................................................................161
Research Questions ....................................................................................163
Literature Review.................................................................................................164
Common Processes ....................................................................................164
Problem Solving...............................................................................164
Reasoning and Proof ........................................................................165
Communication ................................................................................166
Connections......................................................................................166
Representations ................................................................................167
x
Common Use of Tools for Conceptual Development......................167
Common Pedagogical Approaches ............................................................169
Student Teaching and Mathematics ...........................................................170
Methodology ........................................................................................................172
Case Selection ............................................................................................172
Context of the Study ..................................................................................173
Data Collection ..........................................................................................175
Data Analysis .............................................................................................177
Jane and Kate: Methods Class ...................................................................179
Results ..................................................................................................................181
Jane: Student Teaching ..............................................................................181
Kate: Student Teaching ..............................................................................186
Discussion ............................................................................................................190
Implications................................................................................................193
References ............................................................................................................196
Appendix A: Follow-up Discussion Questions....................................................208
Appendix B: Final Reflection ..............................................................................209
Chapter 6: Conclusions ....................................................................................................210
Summary ..............................................................................................................210
Tools for Conceptual Development .....................................................................212
Methods Course Cases ...............................................................................212
Student Teaching Cases .............................................................................214
Processes for Meaningful Learning .....................................................................217
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Methods Course Cases ...............................................................................217
Student Teaching Cases .............................................................................219
Pedagogical Approaches ......................................................................................222
Methods Course Cases ...............................................................................222
Interpretations of Hands-on Approach .............................................222
Interpretations of the Learning Cycle Approach .............................224
Student Teaching Cases .............................................................................226
Final Note.............................................................................................................230
References ........................................................................................................................235
Overall Appendix .............................................................................................................268
Appendix A: Pre-test........................................................................................................269
Appendix B: Post-test ......................................................................................................270
Appendix C: Weekly Blogs .............................................................................................271
Appendix D: Follow-up Discussion Questions................................................................275
Appendix E: Final Reflection ..........................................................................................276
Appendix F: Coding Guide ..............................................................................................277
Appendix G: Sample Coding Level 1 Phase I .................................................................280
Appendix H: Sample Coding Level 2 Phase I .................................................................283
Appendix I: Sample Coding Level 3 Phase I ...................................................................284
Appendix J: Sample Coding Phase II Jane .....................................................................285
xii
List of Tables
Table 1: Thematic Categories for Conceptions of Tools .................................................119
Table 2: Thematic Categories for Conceptions of Processes ...........................................123
Table 3: Thematic Categories for Conceptions of Pedagogical Approaches ..................127
Table 4: Thematic Categories for Approaches between Mathematics and Science ........131
Table 5: Jane- Methods Class ..........................................................................................180
Table 6: Kate-Methods Class ...........................................................................................180
Table 7: Themes: Combined Chart ..................................................................................211
1
CHAPTER 1: INTRODUCTION
Overview and Rationale of Study
The purpose of this study was to investigate how preservice elementary teachers relate,
understand, and use the common or shared teaching practices for effective mathematics and
science learning. The study examined preservice teachers? concepts of the learning cycle, a
teaching approach for mathematics and science with embedded processes and tools for
conceptual understanding. The study took place with a class of preservice teachers (N=22) who
were in their science and mathematics methods courses and subsequent student teaching.
Conceptions of pedagogy, tools for conceptual development, and processes for meaningful
learning were examined for mathematics and science teaching during the science and
mathematics methods courses from a constructivist grounded theory perspective. A small
number of participants (N=2) were observed as case studies during their full-time student
teaching in order to ascertain if preservice teachers continued to use common approaches in
mathematics teaching. During student teaching, the learning cycle approach, conceptions of
pedagogy, tools for conceptual development, and processes for meaningful learning were
examined for mathematics teaching from a qualitative case study perspective.
Solving problems with unknown solutions and the thinking processes involved in such
tasks were foundational principles of the science and mathematics standards (National Council
for Teachers of Mathematics (NCTM), 2000; National Research Council (NRC), 1996).
Additionally, Partnership for 21st Century Skills (2006) identified problem solving and critical
2
thinking as minimal skills required by employers of their employees in the workplace. In order to
produce a work force with problem solving and critical thinking skills, teacher preparation
programs needed to promote problem solving and critical thinking across the disciplines.
Fostering interdisciplinary approaches in science and mathematics in preservice teacher
education programs supported the development of common pedagogical approaches that focused
on inquiry and problem solving, as found in the national science education and mathematics
education standards (NCTM, 2000; NRC, 1996).
Brief Literature Review
National Standards
National standards came about in response to criticisms made about the state of education
in America (Wong, Guthrie, & Harris, 2003). National standards in science and mathematics for
students in the K-12 setting have been a focus of the National Council for Teachers of
Mathematics (NCTM) (1989, 2000), National Research Council (NRC) (1996), and the
Partnership for 21st Century Skills (Partnership for 21st Century Skills, 2009). Similarly,
standards have been developed for teacher preparation programs in science (National Science
Teachers Association (NSTA), 2003) and mathematics (Conference Board of the Mathematical
Sciences (CBMS), 2001; Senk, Keller, & Ferrini-Mundy, 2000). Interstate New Teacher
Assessment and Support Consortium (INTASC) (n.d.) developed general standards for new
teacher candidates as well. All of the documents called for teachers to be able to design,
implement, and integrate teaching that makes content meaningful through inquiry and problem
solving processes. This shift from teaching that focuses on a transmission model of teaching was
known as reform-based education movement.
3
The reforms in science and mathematics education recognized the richness and
complexities of the subjects that can not be conveyed by teachers or learned by students in a
teaching environment that focuses on skills in isolation (Akerson & Donnelly, 2008; Molina,
Hull, Schielack, & Education, 1997). In order for students to develop an understanding of
science and mathematics, conceptual understanding must be the goal for the subjects (Molina et
al., 1997). Best practices for reform-based classrooms designed instruction to allow students to
generate their own questions and seek answers, use tools to investigate concepts, and use
processes to reason, conjecture, and demonstrate proof of their conclusions (Beeth, Hennesey, &
Zeitsman, 1996; NCTM, 2000; NRC, 1996; Williams & Baxter, 1996). Posing problems with
unknown solutions became a means to support exploration of ideas, helps students make
connections, supports communication among students and teacher, and expand students? new
knowledge to other areas (NCTM, 2000; NRC, 1996). In mathematics when students were
working on a problem to determine the relationship between area of squares and area of
triangles, they worked with tools, such as geoboards, to explore the concept. This exploration
often created a discussion among students of their findings. Students may have used their
geoboards to demonstrate why the area of a triangle is half of the area of a square and use the
geoboard to support their claims. Similarly in science, students may have wondered about the
role of earthworms in nature. After studying the earthworms in their natural habitat, they created
miniecosystems in small groups to study the earthworms. The students within the small groups
communicated, recorded their observations, and then presented their findings to the class based
on the data they collected. In both cases, the lessons were designed to foster and promote finding
out about the unknown. Elementary preservice teachers learned to development and implement
such lessons as a part of their science and mathematics methods courses.
4
Mathematics education has been considered a critical needs area. Teachers have been
charged with the task of teaching students beyond basic computations, but to complete complex
mathematical tasks as well (U.S. Department of Education, 2011). In order to improve student
achievement in mathematics, the quality of teaching needed to be improved (U.S. Department of
Education, 2011). The use of the learning cycle in a mathematics methods course for elementary
preservice teachers offered a unique way to help beginning teachers develop the skills necessary
for quality teaching. Furthermore, it provided a framework for teaching mathematics that focused
the mathematics on higher order thinking skills in a practical hands-on method of instruction.
Science Framework for Mathematics Teaching
Reformed based science classrooms involved students in concept development prior to
learning definitions or algorithms (Marek & Cavallo, 1997). This is distinctly different from
traditional classrooms, which were typically characterized by the teacher giving information and
the students completing practice on the concept (Marek & Cavallo, 1997). In science, this type of
traditional instruction may have included an experiment or demonstration to support the content
provided by the teacher (Marek & Cavallo, 1997). However, proponents of concept development
prior to explanation for science teaching have been around since the 1960s, prior to the national
science standards (Atkin & Karplus, 1962; Bybee, 1997). Atkin and Karplus (1962) recognized
that children entered schools with preexisting understandings and interpretations of the world
around them, and teachers must provide additional experiences to help students connect their
prior knowledge with the new knowledge. Engaging students in experiences helped them make
connections to their existing understandings, developed questions for further study, and
expanded the understanding of concepts in deeper ways. Atkin and Karplus (1962) concluded
5
that students should explore concepts, define and develop the concept from the experience, and
apply the newly learned concept to a new situation.
This model of exploration, concept development, and concept application has become the
framework for science teaching and has been modified and expanded since the 1960s (Bybee,
1997). It often was referred to as the learning cycle since it mimics the way people naturally
learn (Lawson, Abraham, & Renner, 1989; Schmidt, 2008). People began with consciousness of
a new idea, then investigation of the new idea, and finally use of the new idea (Schmidt, 2008).
As part of a learning cycle, inquiry-based teaching involved exploration around a central idea,
formulation of questions, investigations to answer the questions, and reflection of learned ideas
(Bell, 2001; Morrison, 2008; Tracy, 1999). Inquiry, therefore, was embedded within the learning
cycle (Gee, Boberg, & Gabel 1996; Tracy, 1999; Withee & Lindell, 2006).
Although science education has included the learning cycle for several decades, the
inform-verify-practice method of instruction still persists (Marek & Cavallo, 1997). Yet even
within the inform-verify-practice model, students in science worked with materials and
conducted cook-book experiments (Marek & Cavallo, 1997). The same could not be said in
mathematics education. National standards called for a focus on mathematics for conceptual
understanding rather than skill and drill (NCTM, 2000). However, the expository mode of
teaching in mathematics still prevailed (Taylor, 2009). Within this system students learned
isolated math skills and then practiced those math skills (Alsup, 2003). Learning skills in
isolation and without context inhibited students making connections within mathematics and to
other areas (U.S. Department of Education, 2011).
Conceptual understanding, as promoted in the science and mathematics standards, often
was not the school experience of elementary preservice teachers (CBMS, 2001). Teacher
6
candidates were often products of a traditional science and mathematics classroom where the
teacher delivered the knowledge (Taylor, 2009). This pattern in the traditional mathematics
classroom somewhat mirrored the traditional science classroom. In both cases students were
being informed of concepts rather than learning concepts for themselves through a scaffolded
approach (Marek & Cavallo, 1997). Approaching teacher candidate preparation from a
perspective of inquiry and problem solving processes aligned the program with teacher
preparation standards for science and mathematics teaching. The learning cycle in mathematics
offered a framework in which students can explore mathematic concepts, investigate new ideas,
and apply new ideas to other situations (Marek & Cavallo, 1997).
The Learning Cycle in Teacher Preparation
Elementary preservice teachers entered the undergraduate program with beliefs about
science and mathematics that have been shaped by their mathematics and science experiences
(Manouchehri, 1997). They often found mathematics dull and uninteresting (Jong & Brinkman,
1999) and filled with procedures to follow (Manouchehri, 1999). Elementary preservice teachers
took methods courses prior to student teaching with the purpose to ?help students discard some
knowledge, beliefs, and dispositions about mathematics and pedagogy they bring to the
preservice program? (Manouchehri, 1999, p. 199). The preservice methods course typically
focused on the pedagogy for the applicable subject area (Manouchehri, 1999). The use of the
learning cycle in both the mathematics and science methods courses was one way for preservice
elementary teachers to experience science and mathematics through exploration, concept
development, and extension of ideas (Cathcart, Pothier, Vance, & Bezuk, 2006; Marek, 2008;
Reys, Lindquist, Lambdin, Suydam, & Smith, 2003). The learning cycle also offered a way of
incorporating similar teaching methods found in science and mathematics in using tools for
7
conceptual development (Chick, 2007; Fuller, 1996; Hill, 1997; Marek & Cavallo, 1997),
discourse (Akerson, 2005; Heywood, 2007; Williams & Baxter, 1996), assessing student
knowledge (Manouchehri, 1997; Marek & Cavallo, 1997; NCTM, 2000; NRC, 1996) inquiry-
based teaching (Morrison, 2008; Manouchehri, 1997; Marek & Cavallo, 1997; NRC, 1996; Weld
& Funk, 2005), and reflection (Bleicher, 2006; Hill, 1997; Manouchehri, 1997). Using the
learning cycle in the science methods course and continuing it into the mathematics methods
course offered a means of bridging and connecting standards-based approaches and would
potentially help preservice teachers to be more effective science and mathematics teachers with
the use of an inquiry-embedded approach for both subjects.
Although the learning cycle, as an inquiry embedded approach, had its roots in the
science field (Bybee, 1997; Atkin & Karplus, 1962; Marek & Cavallo, 1997), the tenets align
closely with the national mathematics standards. The mathematics standards promoted students
working together (NCTM, 2000), as recommended in the exploration and elaboration phases
(Bybee, 1997). Using concrete tools and solving problems to develop a deep understanding of
the concepts was core to the mathematics standards (NCTM, 2000) and aligned with the
exploration, explanation, and elaboration phases of the learning cycle (Bybee, 1997). According
to the mathematics standards, students generated their own mathematical questions (NCTM,
2000), a part of the engagement phase (Bybee, 1997), and students generated conclusions
(NCTM, 2000), a part of the explanation phase (Bybee, 1997). The exploration, explanation, and
elaboration phases supported a focus on thinking and reasoning skills that leads to conjectures or
arguments about the mathematics being examined (Bybee, 1997; NCTM, 2000).
The research that exists on preservice teachers and the learning cycle focused primarily
on science (Lindgren & Bleicher, 2005; Marek & Cavallo, 1997; Marek, Maier, & McCann,
8
2008; Urey & Calik, 2008). Researchers have concluded that K-12 students using the learning
cycle performed better on content questions than students in a traditional science classroom
(Cardak, Dikmenli, & Saritas, 2008; Ergin, Kanli, & Unsal, 2008; ?ren & Tezcan, 2009;
Soomor, Qaisrani, Rawat & Mughal, 2010). Similarly, direct teaching of the learning cycle
approach positively influenced elementary preservice teacher?s efficacy of teaching science
using the learning cycle (Settlage, 2000). Hanuscin and Lee (2008) used the learning cycle as an
effective approach with their elementary preservice teachers during the science methods course.
They found the learning cycle was an effective approach with elementary preservice teachers.
After experiencing science with the learning cycle, the preservice teachers successfully
developed a sequence of activities to use with students following the learning cycle approach.
The results from learning cycle assessments indicated that elementary preservice teachers had
greater understanding of some aspects of the learning cycle, but that their conceptions of
implementation of the learning cycle did not always follow its intended use (Hampton, Odom, &
Settlage, 1995; Marek, Laubach, & Pedersen, 2003). Gee et al. (1996) similarly noticed
preservice teachers implemented the learning cycle in their field placements but left out the
extend phase. Understanding of the learning cycle improved with the continued exposure of the
learning cycle in methods class, but some had a difficult time conceptually changing their ideas
about teaching science (Hanuscin & Lee, 2008; Lindgren & Bleicher, 2008).
In addition to examining preservice teachers understanding of the learning cycle,
researchers noticed the acceptance of teaching science with the learning cycle approach. Gee et
al. (1996) were disappointed to find that preservice teachers were not fully committed to the
learning cycle as an approach for science. They believed that it was difficult for the preservice
teachers to accept a different way to approach teaching science. Lindgren and Bleicher (2008)
9
drew similar conclusions when some of the strongest science students were reluctant to use the
learning cycle as an approach for teaching science. However, students who had negative science
experiences or were dissatisfied with the way they were taught, embraced the learning cycle
approach.
There was a scarcity of research on the learning cycle in mathematics education.
However, researchers teaching preservice teachers have provided examples of the learning cycle
and mathematics in science lessons (Marek & Cavallo, 1997; Marek et al., 2008; Simon, 1992).
Marek (2008) described a science lesson in which students measured the circumference of
objects to make a conclusion about the relationship of the circumference and diameter of circles.
His description illustrated how the learning cycle could be used in mathematics teaching.
Although his example was for a science lesson, it very easily could have been for a mathematics
lesson on understanding the area of circles. Similarly, Marek and Cavallo (1997) explained that
the learning cycle could be used in mathematics for problem solving and provided examples of
learning cycle lessons to teach measurement and geometry concepts. Simon (1992) recognized
the use of the learning cycle in mathematics methods courses. He described a mathematics
methods class in which teacher candidates solved a problem situation, discussed solutions, and
extended new ideas into other problem situations.
Research Questions
Preservice teachers in this study were in science and mathematics methods classes which
were taught with the learning cycle approach. Research needed to be conducted to analyze the
bridge between mathematics and science methods courses and student teaching experiences in
mathematics for elementary preservice teachers (Jong & Brinkman, 1999). In order to ascertain
how preservice teachers carried over the learning cycle approach into their mathematics
10
teaching, preservice teachers needed to be studied as they moved through the teacher preparation
process. Although numerous studies existed on various levels of content knowledge (Ball, 1988;
Davis & Petish, 2005; Hill, 1997; McLeod & Huinker, 2007; Rice, 2005; Summers, Kruger, &
Childs, 2001; Turnuklu & Yesildere, 2007), beliefs (Ball, 1996; Fuller, 1996; Manouchehri,
1997), and other factors (Chick, 2007; Even, Tirosh, & Markovits, 1996; Manouchehri, 1997)
influencing preservice teachers within elementary education programs, very little data existed on
the study of elementary preservice teachers in relation to their understanding of the learning
cycle and subsequent commonalities in practice between mathematics and science teaching
(Offer & Mireles, 2009; Stinson, Harkness, Meyer, & Stallworth, 2009) as recommended by the
National Research Council (1996), National Science Teachers Association (2003), and the
Partnership for 21st Century Skills (2009).
This research offered an opportunity to examine preservice teachers dually enrolled in
science and mathematics methods courses and as they moved from methods courses through
student teaching. Furthermore, it provided insight into the ways in which elementary preservice
teachers considered tools for conceptual development, processes for learning, and approaches for
learning mathematics during a student teaching experience. In order to examine the response
preservice teachers had with processes that occurred similarly in science and mathematics, the
participants? science and mathematics methods courses were approached from a similar
pedagogical perspective with use of the learning cycle. Preservice teachers then were expected to
use the learning cycle approach in their laboratory teaching experiences. In order to determine if
teacher candidates continued with the learning cycle approach in mathematics, a sub group of
these participants were studied during their student teaching experience. The research questions
for this study were: (1) What are preservice teachers? conceptions of tools for conceptual
11
development, processes for meaningful learning, and pedagogical approaches to mathematics
and science prior to and after taking the mathematics-science methods courses? (2) What
changes, if any, in development of knowledge and understanding of the common approaches
between mathematics and science occurred while taking the mathematics-science methods
courses? (3) How did preservice teachers put into practice during student teaching their thinking
from the methods courses on tools for conceptual development, processes for meaningful
learning, and pedagogical approaches to mathematics teaching?
Participants, Research Methods, and Organization of this Dissertation
The research design was a qualitative study conducted in two phases. The study was
conducted in two phases due to the nature of the teacher preparation process. During Phase I,
students were participating in science and mathematics methods courses. The purpose of the
methods courses was to provide preservice teachers with the opportunity to learn the pedagogy
and put the pedagogy into practice on a small scale. During the methods courses, participants
spent part of their time on campus and part of their time in field experiences. The second phase
of the study occurred during the student teaching experience. The student teaching experience
provided more time for the preservice teachers to develop as teachers and practice teaching. In
the student teaching experience preservice teachers spent every day of the semester in a
classroom gradually accepting all of the responsibilities of the classroom teacher. They were
expected to take over all of the responsibilities for a two-week time period. During that time the
student teacher was responsible for planning lessons, implementing, lessons, and the day-to-day
business of a classroom. Conducting the study in two phases allowed for examination and
delineation of growth of thinking and practice in mathematics teaching as preservice teachers
progressed through the latter part of their program.
12
The first phase consisted of 22 elementary pre-service teachers who completed pre-tests,
post-tests, and on-line blogs during their jointly enrolled mathematics and science methods
courses. The differing modes of data collection were used to reveal preservice teachers?
thoughts and knowledge of common science and mathematics practices as they developed
throughout the methods courses. Phase II of the study consisted of studying two of the Phase I
participants during their student teaching experience. Data sources for Phase II consisted of
lesson plans, observations, follow-up interviews, and a final reflection on their mathematics
teaching experiences. The two cases were selected for several reasons. First they were placed in
a school that participated in science and mathematics reform initiatives and had resources for
their mathematics teaching. Secondly, they both were given the freedom by their cooperating
teachers to design and implement mathematics lessons as long as they taught the assigned
objectives. Thirdly, the cases represented two perspectives on approaching mathematics
teaching. The two cases were selected due to the richness of data they provided. The various
types of data collected during student teaching were used to indicate how preservice teachers?
thinking and practice changed as they moved along the teacher development continuum.
Following Phase II data collection, the cases studies were examined for emerging themes and
compared with themes from Phase I. Following data analysis of Phase I, the cases were
examined for emerging themes within each case and across cases.
Overview of Results
Data collected during the methods portion of the study revealed varying conceptions of
tools, processes, and approaches for science and mathematics teaching. Participants overall
entered the methods courses with the perceptions of science and mathematics as being two
separate subjects with two different methods of teaching. By the end of the semester, through the
13
use of the learning cycle, most of preservice teachers recognized that science and mathematics
could be approached similarly. Participants who held traditional views, in which the teacher
disseminated the information and students practiced, initially were more likely to hold onto those
beliefs. However, preservice teachers who entered the methods courses with more expanded
notions of science and mathematics teaching broadened their understandings.
The two cases studied offered additional insight into the development of elementary
preservice teachers. The cases presented two very different student teachers with similar and
differing pedagogical issues. Teaching practices that were exhibited during the methods class
were maintained during student teaching. In the case of Jane, she exhibited expanded notions of
teaching science and mathematics through hands-on teaching during the methods course. During
student teaching, after some guidance, she was able to implement hands-on lessons in
mathematics, although not on the level of full inquiry. Kate, on the other hand, demonstrated
more traditional views of science and limited views of mathematics in terms of real world
mathematics. During her student teaching, in mathematics she taught investigation type lessons
but under a great amount of teacher control which limited the richness of the learning for the
students.
This dissertation consists of a review of the literature, a methodology section, two
articles, and a final chapter. The first article focuses on the themes that emerged from the
findings of the study conducted while pre-service teachers were jointly enrolled in mathematics
and science methods courses. The second article consists of the themes from practice in
mathematics that emerged from the two case studies during the student teaching. The final
chapter of this dissertation examines the overall findings and conclusions.
14
CHAPTER 2: REVIEW OF LITERATURE
The review of literature focused on elementary preservice teachers and common
pedagogical approaches to mathematics and science and is presented in the following sections:
Efforts to Reform Science and Mathematics Education
Since the launching of Sputnik in 1957 (Garber, 2007), cries for reform in science and
mathematics education have been made throughout the educational arena (Bybee, 1997; Gates,
2005). The National Science Foundation (NSF), established in 1950, became the only federal
agency to promote science and engineering research and education (NSF, 2009). With the
establishment of NSF came the National Science Board (NSB), the people appointed by the
president to provide input to the foundation (NSF, 2009). Beginning in 1954, a summer program
was funded through the National Science Foundation to support the mathematical knowledge of
teachers (NSF, 2009). Since that time NSF grants have continued to provide funding for
research, education, and other programs directed towards the improvement of science and
mathematics (NSF, 2009).
Despite the efforts of the National Science Foundation and the research efforts that have
been funded as a direct effort of the agency, concerns with the state of mathematics and science
education persisted (NSB, 2004). In examining the data on science and technology in the United
States, the National Science Board (2004) concluded that the number of jobs requiring
knowledge of science and engineering would increase without a prepared job force to fill those
positions if things continue in the United State with science and mathematics. The National
Science Board Report (2004) also indicated that fewer American school children and women
15
were choosing science and engineering fields. Furthermore, the percentages of foreign born
individuals in areas of science and engineering with graduate degrees increased over the last few
years from 24 to 38 percent (NSB, 2004).
Concerns for the state of education persisted in the United States with other stakeholders,
including interest groups, businesses (Gates, 2005), and parents (Kadlec & Friedman, 2007).
The Public Agenda conducted a survey among students and parents to determine the level of
satisfaction of science and mathematics education (Kadlec & Freidman, 2007). Of parents and
students surveyed by Kadlec and Freidman (2007), both groups believed that America was
behind other countries in mathematics and science, yet they were pleased with the mathematics
and science education in the local schools. African American and Hispanic parents surveyed,
however, indicated that their schools were below where they needed to be in the teaching of
mathematics and science (Kadlec & Freidman, 2007). The disparity of satisfaction and
complacency with American schools presented additional problems for a nation that was
operating high schools as they did fifty years ago (Gates, 2005).
National Standards
Like the launching of Sputnik, the release of A Nation at Risk by the National
Commission on Excellence in Education (1983) sparked renewed concerns with the education
system of the United States (Wong, Guthrie, & Harris, 2003). Recommendations by the eighteen
member commission included increasing high school graduation requirements, raising standards
and expectations for four year colleges, increasing the time students spent in school, improving
the preparation of teachers, and expecting the public to hold the system accountable for making
these improvements (National Commission on Excellence in Education, 1983). At the national
16
level, standards were developed as a means of improving and reforming the American
educational system to improve student achievement (Wong et al., 2003).
Those standards for science and mathematics came about from the efforts of the National
Research Council (1989, 1996) and the National Council of Teachers of Mathematics (1989,
1991, 2000). Science standards documents included the National Science Education Standards
(NSES) (National Research Council, 1996) and Benchmarks for Science Literacy (American
Association for the Advancement of Science, 1993). Mathematics standards began with
Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), Assessment
Standards for School Mathematics (NCTM, 1995), and Professional Standards for Teaching
Mathematics (NCTM, 1991). From the preceding documents, the areas of curriculum,
assessment and professional standards were updated and combined in Principles and Standards
for School Mathematics (NCTM, 2000). More recently NCTM published Curriculum Focal
Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2006) and
Focus in High School Mathematics: Reasoning and Sense Making (2009). These documents
focused on more specific grade level standards for pupils. The National Research Council (2001)
published Adding it Up, a report that described proficiency in mathematics as conceptual
understanding, procedural fluency, strategic competence, adaptive reasoning, and productive
disposition. Most recently the Common Core State Standards Initiative (2010), headed by the
National Governors Association and Council for Chief State School Officers, developed
standards for language arts and mathematics that focused on reasoning, critical thinking, and
problem solving for all states to adopt.
Standards have been developed for teachers and teacher preparation programs as well.
Interstate Teacher Assessment and Support Consortium (INTASC) (n.d.) developed general
17
standards for new teacher candidates. The standards included development, planning, and
implementation of lessons that incorporate a variety of strategies, meeting the needs of diverse
learners, and working within the school community. The Conference Board of the Mathematical
Sciences (2001), in The Mathematical Education of Teachers, focused their standards and
recommendations on what teachers and teacher preparation programs needed to have in order to
promote successful mathematics programs. They described the challenges elementary teacher
programs have in the preparation of teachers in teaching mathematics. They also recommended
that preparation programs work with mathematics faculty to design content courses that provides
a deep examination of mathematics.
Similarly, the National Science Teachers Association (2003) developed Standards for
Science Teacher Preparation. The Standards for Science Teacher Preparation (NSTA, 2003)
described the content knowledge elementary and secondary teachers need in their preparation
program, including earth, space, and life science. Accredited teacher preparation institutions
follow standards to maintain the accreditation of their programs. The Association for Childhood
Education International (ACEI) in association with the National Council for Accreditation of
Teacher Education (NCATE) developed standards for teacher preparation institutions to receive
accreditation of their programs (ACEI, 2007).The resulting documents called for teacher
candidates to be able to design, implement, and integrate teaching that makes content meaningful
through inquiry and problem solving processes.
Reform Movement
Despite the efforts of the NCTM and NRC over the last two decades, preservice teachers
were often products of a traditional science and mathematics classroom where the teacher
delivers the knowledge (Taylor, 2009). In the science classroom this traditional approach was
18
known as inform-verify-practice (Marek & Cavallo, 1997). With this approach the teacher
informed the student about a science concept, the student verified the concept, and then
completed additional practice problems (Marek & Cavallo, 1997). Sometimes these verification
activities were called experiments (Marek & Cavallo, 1997). However, they were not true
experiments because the students already knew what to expect (Marek & Cavallo, 1997).
Furthermore, since students were informed and told the information, they weren?t involved in
true science learning (Marek & Cavallo, 1997). The traditional mathematics classroom often
began with the teacher introducing the new material, students worked on problems often
algorithmic in nature, and the teacher answered questions (Alsup, 2003).
However, the inform-verify-practice approach for traditional science classrooms and the
tell-practice approach of traditional mathematics classrooms did not take into account the
richness and complexities of these two fields. A complex framework of generalizations, ideas,
and relationships composed the nature of mathematics (Molina et al., 1997) and the nature of
science (NOS) (Akerson & Donnelly, 2008). The recognition of this complex framework
created a different schema for the teaching and learning of these two fields. In order to develop
the skills to make generalizations, to understand ideas, and to examine relationships, conceptual
learning must be developed (Molina et al., 1997). The shift in thinking about the nature of
teaching mathematics and science created a change in curriculum (NCTM, 1989, 1991, 2000;
NRC, 1989. 1996) from the teaching of isolated skills to a development of skills in context
(Molina et al., 1997) and became known as the reform movement (Williams & Baxter, 1996).
Constructivist philosophy has influenced educational efforts greatly since the 1980s when the
reform movement began (Vernette, Foote, Bird, Mesibov, Harris-Ewing, & Battaglia, 2001).
Within the standards documents constructivist philosophies could be seen as the teaching and
19
learning process is described as active involvement to develop understanding for each student
(NCTM, 2000; NRC, 1996).
Reforms in science and mathematics education were characterized with a move away
from lower order thinking skills to higher order thinking skills (Tal, Dori, Keiny, & Zoller,
2001). In a reformed mathematics classroom it was accepted that students generate their own
mathematical conclusions and questions (Beeth, Hennesey, & Zeitsman, 1996; NCTM, 2000;
Williams & Baxter, 1996). A reformed science classroom indicated a shift from learning
isolated science information to a development of understanding about science phenomena and
the nature of science (Beyer & Davis, 2008). Both science and mathematics reformed
classrooms were characterized by a focus on thinking and reasoning skills that leads to
conjectures, or arguments, about the science or mathematics being examined (NCTM, 2000;
NRC, 1996; Williams & Baxter, 1996). Reform classrooms involved students working together,
using concrete tools, and solving problems to develop a deep understanding of the concepts
(Williams & Baxter, 1996).
Integral to the enactment of the reform movement were teacher education programs
(Manouchehri, 1997; McGinnis, Kramer, & Watanabe, 1998). Teacher education programs
provided the knowledge base for future teachers that include content and pedagogy
(Manouchehri, 1997). Teachers in reform classrooms acted as a facilitator, helping guide
students to mathematical or science conceptual understanding, rather than serving as the giver of
knowledge (Beyer & Davis, 2008; Williams & Baxter, 1996). Teacher education programs have
to establish a process that provided preservice teachers the opportunities to develop those skills
as facilitators in context (Manouchehri, 1997; Noori, 1994; Speilman & Lloyd, 2004).
20
Curriculum
The reform movement also impacted the development of curricular materials different
from traditional textbooks (Klein, 2003). Battista (1999) noted that the traditional textbook
curriculum developers were not fully apprised of research on student learning, and concluded
that ?sound curricula must include clear long-range goals for ensuring that students become
fluent in employing those abstract concepts and mathematical perspective that our culture has
found most useful? (p. 424). Li and Fuson (2002) identified two concerns of existing curricular
materials including how the materials aid in student achievement and the embedded ideas in the
presentation of material. Educators have to be mindful of how concepts are presented in
curricular materials and how they use such materials in teaching (Renner, Abraham,
Grzybowski, & Marek, 1990).
Curricular materials were often in the form of textbooks, and textbooks played a role in
the teaching and learning in the classroom (Stamp & O?Brien, 2005). Traditional textbooks in
science and mathematics often included a large number of concepts that focus on examples and
definitions of concepts (Stamp & O?Brien, 2005). De Villiers (2004) argued that ?just knowing
the definition of a concept does not at all guarantee understanding of the concept? (p. 708). Just
because students could recite the definition of a rectangle doesn?t mean that the students could
recognize that a square is a specialized rectangle. Furthermore, Cannizzaro and Menghini (2006)
felt that definitions needed to arise from students? experiences with concepts. From their
research with students and textbook problems, Renner et al. (1990) concluded that science
teachers that focus on the textbook create a classroom in which students are not experiencing
science but rather just reading about science.
21
Reform-based curricular materials differed from traditional textbooks (Lloyd &
Frykholm, 2000). Reform curricula emphasized problem solving tasks, real world applications,
and exploration of topics (Goodnough & Nolan, 2008; Lloyd & Frykholm, 2000). Reform-based
materials focused on students making sense of the concepts rather than isolation of facts. ?In
sense-making curricula, on the other hand, because students retain learned ideas for long periods
of time, and because a natural part of sense-making is to interrelate ideas, students accumulate an
ever-increasing store of well-integrated knowledge? (Battista, 1999, p. 432). Reform curricula
often included examples of student thinking, information about the concept, discussion
questions, and various means of assessing student learning (Lloyd & Frykhom, 2000).
All of these factors influenced the enacted curriculum in the classroom. The enacted
curriculum was the curriculum the teacher actually teaches. For traditional science classrooms
this often entailed reading of texts, vocabulary lists, worksheets, and lecture (Weiss, 1994).
Renner et al. (1990) found that when teachers followed textbooks that their role was then to
support the textbook content. Furthermore, as students moved from primary to intermediate
grades the number of hands-on activities in science often diminishes (Fulp, 2002). Similarly, a
traditional mathematics classroom often encompassed learning specific skills in isolation and the
practice of those skills (Alsup, 2003).
However, curricular materials alone did not ensure a reform based classroom (Battista,
1999; Huang, 2000). Battista (1999) found that even with reform based materials that teachers
potentially could distort the ideas in the materials if the teacher had misconceptions about the
content. The variety of responses from students when given the same curriculum ??provides
evidence of the interplay between curriculum as designed and curriculum as ?wrapped? around
the ongoing action and interaction of students and teacher? (Cannizzaro and Menghini, p. 376,
22
2006). Similarly, Huang (2000) expressed concerns for teachers who followed curricula
materials closely without taking into account the learning needs of the students.
Knowledge
Content Knowledge
There are different types of knowledge needed by teachers (Stotsky, 2006). Shulman
(1986) identified three types of knowledge needed by teachers: content knowledge, pedagogical
content knowledge, and curricular knowledge. Content knowledge of teachers, sometimes
referred to as subject matter knowledge, involved the actual content that teachers themselves
know (Ball, 1996; Manouchehri, 1997). Fact, concepts, and principles of a subject comprised
content knowledge (Manouchehri, 1997). Knowledge of facts was also known as declarative
knowledge (Jacobs & Paris, 1987). Knowledge of processes was called procedural knowledge
(Jacobs & Paris, 1987). Content knowledge included both common knowledge and specialized
knowledge (McLeod & Huinker, 2007).
Elementary teachers were expected to have sufficient content knowledge (McLeod &
Huinker, 2007; Shulman, 1986). Elementary teachers also were expected to have content
knowledge of all subjects including, but not limited to, language arts, social studies, science, and
mathematics (Davis & Petish, 2005). Content knowledge of mathematics included knowledge of
numbers and operations, algebra, data and statistics, and geometry (NCTM, 2000). Content
knowledge of science included life science, physical science, space science, and earth science
(Davis & Petish, 2005). Stotsky (2003) argued that ?a deep knowledge of the academic content
supporting the field of teacher?s license is the sine qua non for defining teacher quality? (p. 257).
In examining the knowledge of environmental science content with preservice teachers and in-
service teachers, data indicated some misconceptions in in-service teachers but greater
23
misconceptions in preservice teachers (Summers et al., 2001). Teachers can not teach effectively
what they do not understand themselves (Ball, 1996; Summers et al., 2001).
Teachers were expected to not only know the what of mathematics and science
knowledge and the why for teaching the mathematics and science (Even et al., 1996; Kelly
2000). Teachers also were expected to know common algorithms and science facts, examples of
common content knowledge (McLeod & Huinker, 2007). Knowing why and when strategies
should be applied is known as conditional knowledge (Jacobs & Paris, 1987). Teachers were also
expected to know why algorithms work or why scientific tenets hold true (Even et al., 1996), an
example of specialized content knowledge (McLeod & Huinker, 2007). Specialized content
knowledge enabled teachers to examine students? solutions and students? misconceptions (Even
et al., 1996). Teachers recognized their content knowledge as an influencing factor in their
ability to teach a subject (Goodnough & Nolan, 2008). Knowledge about mathematics and
science, including mathematics and science origins, history of mathematics and science, and how
truths are determined, also comprised knowledge of mathematics and science (Ball, 1996; Justi
& Gilbert, 2000; Kistler, 1997). Teachers expressed implicitly and explicitly the knowledge they
have about a subject through the types of assigned tasks, ways unpredicted student responses are
handled, and the role of curricular materials in the lessons (Ball, 1988).
Furthermore, content knowledge involved teachers not only knowing the content but
having the ability to incorporate new knowledge into the previously learned knowledge
(Anderson & Hoffmeister, 2007). The authors concluded that ??a definition of teachers? c ontent
knowledge should include not only the capacity to learn, but also the concomitant recognition
that learning involves attention to connections and concepts? (p. 201). Furthermore, it was not
just content knowledge that was important, but it was also important how teachers learned the
24
content (Jones, 2000). If teachers learned content for facts, then their knowledge base was
limited to facts. If teachers learned concepts and the relationships to other concepts, then their
knowledge base was deeper. Zull (2002) determined that understanding the prior knowledge and
experiences of the students was critical for teachers to build new knowledge. Teacher knowledge
affected the decisions of the day-to-day classroom from the interpretation of the materials, the
presentation of materials, and interaction with the students (Paton, Fry, & Klages, 2008;
Shulman, 1986). Teacher knowledge was important for interpreting students? unpredicted
mathematics and science responses (Ball, 1996; Summers et al., 2001)
If teachers were expected to be effective, they must have an understanding of the
concepts, rules, and principles of the content (Beeth et al., 1996; Manouchehri, 1997). They were
expected to be able to explain and interrelate the concepts to students (Ball, 1996; Even et al.,
1996; NCTM, 2000; Shulman, 1986). Skemp (1976) described knowing the what and the why of
mathematics as relational understanding. An instrumental view was a view of mathematics as a
set of memorized formulas and rules (Skemp, 1976). Hill (1997) found that elementary
preservice teachers initially held an instrumental view of mathematics. Similarly, Ball (1988)
found that elementary preservice teachers had fragmented knowledge of mathematics and Rice
(2005) found preservice teachers lacked an understanding of basic science concepts. This
fragmented knowledge inhibited teachers in analyzing students? misconceptions and determining
alternate means of teaching a concept (Rice, 2005).
In addition to knowledge of content, Hill et al. (2008) identified specialized knowledge of
mathematics as mathematical knowledge of teaching (MKT). Mathematical knowledge of
teaching was quite complex. It included a respect of mathematics and a level of content
knowledge that was different from what a mathematician might need (Hill & Ball, 2009).
25
Furthermore, teachers were to be able to ?unpack? ideas encompassed in mathematical concepts
and to be able to understand mathematical justifications of the end result (Hill et al., 2008). For
example, teachers may have memorized a rule for division of fractions to ?flip and multiply? but
this did not take into account ?the mathematical knowledge to teach this topic? (Hill & Ball,
2009, p. 68). Finally, teachers with MKT knew where students were in their mathematical
thinking and where the students should grow and develop in mathematical thinking (Hill et al.,
2008).
Pedagogical Content Knowledge
Subject matter or content knowledge alone was not sufficient to be an effective teacher
(Davis & Petish, 2005; Manouchehri, 1997; Turnuklu & Yesildere, 2007). Teachers needed
knowledge of students and how to teach (Even et al., 1996; Heywood, 2007; Hovey, Hazelwood,
& Svedkausite, 2005; NCTM, 2000; Turnuklu & Yesildere, 2007). Shulman (1986) named this
type of knowledge as pedagogical content knowledge. This knowledge included teachers
knowing the most useful representation of a concept, understanding interrelationships of
concepts, and obstacles learners may have with the representations or interrelationships of
concepts (Davis & Petish, 2005; Even et al., 1996; Heywood, 2007; Manouchehri, 1997;
Summers et al., 2001). Daehler and Shinohara (2001) considered pedagogical content knowledge
as knowledge of the level of difficulty of the concept, as well as knowledge of how to present the
concept to make it understandable to students. Both knowledge of the content and knowledge of
the best way to teach the content were necessary for effective teaching and learning and must
take into account students? prior knowledge and experiences (Daehler & Shinohara, 2001; Davis
& Petish, 2005; Manouchehri, 1997).
26
New teachers often exhibited limited pedagogical content knowledge (Davis & Petish,
2005; Manouchehri, 1997). For both science and mathematics this type of knowledge involved
an in-depth understanding of the concept along with proficiency in being able to identify student
thinking (Daehler & Shinohara, 2001; Davis & Petish, 2005; Heywood, 2007; NCTM, 2000).
Once the teacher had identified and assessed where the students were conceptually, then the
teacher created learning experiences that helped students build a depth of understanding based on
the students? level of comprehension (Heywood, 2007). Turnuklu and Yesildere (2007) found
that elementary preservice teachers did not have sufficient knowledge of assessment to develop
and elicit understanding of student conceptions.
Pedagogical content knowledge also included decisions teachers make with regard to
methods of teaching and knowledge about students? abilities (Davis & Petish, 2005; Even et al.,
1996; Fuller, 1996). Teachers? content knowledge influenced pedagogical choices made by the
teachers (Even et al., 1996). Pedagogical choices teachers made influence learning opportunities
for students (Davis & Petish, 2005; Martin, McCrone, Bower, & Dindyal, 2005). In inquiry-
based classrooms, teachers made pedagogical decisions between letting students arrive at their
own conclusions and making sure students know science terminology (Harlow & Otero, 2006).
Martin et al. (2005) found that ?the teachers choice to pose open-ended tasks (tasks which are
not limited to one specific solution or solution strategy), engage in dialogue that places
responsibility for reasoning on the students, analyze student arguments, and coach students as
they reason, creates an environment? (p.95) for students to actively participate in the proof and
reasoning process. In addition to pedagogical choices made by teachers, the manner in which
teachers dealt with unexpected responses and questions from students reflected the pedagogical
knowledge of that teacher (Even et al., 1996; Fuller, 1996; Martin et. al, 2005).
27
Preservice teacher education programs needed to provide experiences that allowed
preservice teachers to move from learning content to thinking about content and students in the
teaching/learning environment (Even et al., 1996; Manouchehri, 1997). Preservice teachers
needed knowledge of educational theory as well as knowledge of day-to-day decision making in
the classroom (Manouchehri, 1997). New teachers were more concerned with their teaching
performance, when they first begin to practice teach, than with student learning (Manouchehri,
1997). This transition from self to students was found to occur as the new teachers developed
pedagogical reasoning defined as the ability to examine the pedagogical choice, recognize
potential solutions, and devise a solution that best fits the learning situation (Davis & Petish,
2005; Manouchehri, 1997).
Curricular Knowledge
Shulman (1986) identified curricular knowledge as another type of knowledge needed by
teachers. The manner in which teachers converted curricular materials into learning experiences
demonstrated their curricular knowledge (Davis & Petish, 2005; Shulman, 1986). Furthermore,
Shulman (1986) criticized teacher education programs for their lack of curricular knowledge
preparation. Textbooks often were considered lacking by teachers in the presentation or approach
of a topic (Shulman, 1986; Stamp & O?Brien, 2005). Not only did teachers need to be familiar
with the curriculum they were expected to teach, but they also needed knowledge of alternative
curriculum materials (Shulman, 1986; Stamp & O?Brien, 2005). Shulman (1986) identified that
teachers with strong curricular knowledge were able to relate what students were learning in
other classes as well. Shulman (1986) called this lateral curricular knowledge.
Teachers needed to be able to evaluate the appropriateness of the curricular materials,
determine how to present or alter the information presented, and perhaps determine alternative
28
curriculum materials (Davis & Petish, 2005; Fuller, 1996; Shulman, 1986). Curricular materials
played a powerful role in the classroom (Renner et al., 1990; Spielman & Lloyd, 2004).
Preservice teachers had a disadvantage when using curricular materials due to their lack of
experience and may not interpret the materials appropriately (Davis & Petish, 2005).Teacher
actions may have resulted from reactions to classroom events that occur spontaneously and not
from carefully considered pedagogical choices (Martin et. al., 2005), or from other beliefs about
the curriculum (Chick, 2007). A teacher whose goal was to make sure students write proofs
similar to the textbook has different goals than a teacher who wanted to aid students to make
sense of the proofs and the reasoning behind the mathematics. Davis and Petish (2005)
recommended that teacher education programs should provide opportunities for preservice
teachers to learn how to critically examine curricular materials.
Constructivist Perspective on Learning
Constructivism was found in the research and conceptions of learning based on the works
of John Dewey (1933), Jean Piaget (1976), Lev Vygotsky (1962), Howard Gardner (1983), and
Jerome Bruner (1968) (Beeth et al., 1996; Lever Duffy & McDonald, 2011; Vernette et al.,
2001). Knowledge occurred at two levels, with the first level as the actual experience (Beeth et
al., 1996; Zull, 2002). At the second level was how the person categorized the experience with
other experiences (Beeth et al., 1996; Zull, 2002). The constructivist perspective recognized that
learning is unique for each person and is based on each person?s experiences (Beeth et al., 1996;
Lever-Duffy & McDonald, 2011; Marek & Cavallo, 1997). Therefore, teachers with a
constructivist philosophy viewed learning as a unique experience for each student as each
student is unique (Beeth et al., 1996).
29
The manner in which a teacher teaches based on a constructivist philosophy was called
constructivist pedagogy (Beeth et al., 1996). A constructivist classroom was characterized as a
move away from a transmission model of teaching (Jaeger & Lauritzen, 1992; Vernette et al.,
2001). Active learning was a key component in a constructivist classroom which encompasses
students making connections, scaffolding of learning, options in learning, learning for
understanding of concepts, using a variety of assessment methods, inquiry, and a student
centered environment (Vernette et al., 2001). Social interaction was a key element of social
constructivism involving negotiation of meaning with classroom peers to foster learning (Kim &
Darling, 2009). This negotiation and meaning making was different from a traditional teacher
centered classroom. Ka-Ming and Kit-Tai (2006) found the students learning in a constructivist
classroom, in comparison to students in a teacher-centered classroom, had deeper understanding
of the material.
In addition to content knowledge, curricular knowledge, and pedagogical content
knowledge, teachers needed to have the ability to foster knowledge development, reflection, and
communication of learning (Beeth et al., 1996). Since constructivist philosophy placed more
demands cognitively on students to go beyond regurgitation of facts, teachers had to be able to
assess student cognition (Beeth et al., 1996). Furthermore, processes must be facilitated by the
teacher in order for students to ?make meaning of their world by logically linking pieces of their
knowledge, communication and experiences? (Jaeger & Lauritzen, 1992, p.1).
Processes for Meaningful Learning in Science and Mathematics
Processes for meaningful learning were based on the processes as described in the
Principles and Standards for School Mathematics (NCTM, 2000) and the National Science
Education Standards (NRC, 1996). Process skills were also considered thinking skills (Sambs,
30
1991). These processes included problem solving, reasoning and proof, communication,
connections, and representations (Goodnough & Nolan, 2008; Justi & van Driel, 2005; NCTM,
2000). In mathematics, students were asked to reason about their answers and demonstrate proof
(NCTM, 2000). Similarly in science, students were expected to be able to validate findings and
be able to evaluate the conclusions of others (NRC, 1996). In mathematics, students were
expected to communicate their findings (NCTM, 2000). Likewise, in science students were
expected to be able to explain their findings (NRC, 1996). In mathematics, students were
expected to solve problems (NCTM, 2000). In science, students were expected to ask questions
and then find the answers to those questions (NRC, 1996). In mathematics, students were
expected to be able to represent their work (NCTM, 2000). In science, students were expected to
present their scientific knowledge (NRC, 1996). In mathematics and science, students were
expected to make connections or identify relationships between and among concepts and areas of
study (NCTM, 2000; NRC, 1996).
Manouchehri (1997) recommended that preservice teachers must ?explore, analyze,
construct models, collect and represent data, present arguments, and solve problems? in
mathematics (para.11). Cady, Meier, and Lubinski (2006) found that when using practices
aligned with NCTM recommendations in the methods course and in the field placement that
those preservice teachers continued to implement problem solving, classroom discourse, and
multiple forms of assessment. Likewise, Banchoff (2000) determined that, ?given opportunities
to act as mathematicians do and to share their thinking with classmates, students will develop the
skills, habits, and dispositions of young mathematicians? (p. 350). After examining knowledge of
preservice teachers, Akerson and Donnelly (2008) concluded that the National Science
Education Standards (NRC, 1996) needed to be emphasized in teacher preparation programs in
31
order to help preservice teachers more fully develop their understanding of the nature of science
and how science works.
Problem Solving. Problem solving was one of the processes common to science and
mathematics teaching. ?Problem solving means engaging in a task for which the solution is not
known in advance? (NCTM, 2000, p. 52). For science, the problem solving process often
occurred during scientific inquiry in which students are finding out the unknown (NRC, 1996).
Whether called problem solving, as in mathematics, or inquiry, as in science, students were
expected to learn how to gather information, record collected data, and offer answers and
explanations of those answers (NCTM, 2000; NRC, 1996). Since not all problems are simple,
problem solving caused the learner to wrestle with alternative solutions and take risks in thinking
(Manouchehri, 1997, NCTM, 2000). Thinking about alternatives in solutions allowed the learner
to expand problem solving skills and gain insight into the content that can be reapplied in later
situations (Manouchehri, 1997). Swartz (1982) expounded on the importance of students to
problem solve using a variety of strategies to gain understanding. The problem solving process,
therefore, promoted the development of thinking skills (NCTM, 2000).
Problem solving often began, for both science and mathematics, in problem posing
(NCTM, 2000, NRC, 1996). Children were noted to have a natural curiosity about the world in
which they live (NCTM, 2000). These natural curiosities led them to question and explore the
world around them. Students built knowledge through the natural problem solving process
(Goodnough & Nolan, 2008; NCTM, 2000). With problem solving, students were expected to
find an answer to a question which is not readily known (NCTM, 2000). The processes students
go through to solve the problem helped them build an understanding (NCTM, 2000). Unlike rote
memorization, the processes of problem solving also provided a means for teachers to create a
32
mathematics and science classroom that fostered teaching through conceptual learning (Molina
et al., 1997, NCTM, 2000).
Polya was a mathematician who greatly influenced problem solving strategies in
mathematics (NCTM, 2000). Some of the most common strategies he described include ?using
diagrams, looking for patterns, listing all possibilities, trying special values or cases, working
backward, guessing and checking, creating an equivalent problem, and creating a simpler
problem? (NCTM, 2000, p. 54). These strategies were used as students solve problems within a
context (NCTM, 2000). For example, children in a science classroom may have used a
magnifying glass to observe organisms or objects to answer a question. Their observations then
led them to look for patterns or to list common features, and this was very similar to students in a
mathematics class examining and sorting pattern blocks to determine which ones fill a space. In
both scenarios students were actively involved in problem solving.
Reasoning and Proof. Reasoning and proof helped students develop logical thinking to
decide if an answer makes sense. Students developed guesses or conjectures about concepts,
experiments, or observations (NCTM, 2000; NRC, 1996). In mathematics, students noted
patterns. They used reasoning and proof processes to determine if the ?patterns are accidental or
if they occur for a reason? (NCTM, 2000, p. 56). Reasoning and proof included the ability of
students to use counterexamples to disprove a conjecture (Chick, 2007; NCTM, 2000). Similarly
in science, students used reasoning and proof processes in experiments and tests to determine if
the results are consistent (NRC, 1996). Reasoning and proof were especially critical in science
due to the nature of science requiring evidence of conclusions based on experiments and
observations (NRC, 1996).
33
Tools became a helpful resource in science and mathematics for students to be able to
reason and provide proof of their solution (NCTM, 2000; NRC, 1996). Students were expected
to use concrete materials as a means of investigating conjectures held about concepts (NCTM,
2000; NRC, 1996). Tools for mathematical calculations often were used in science as a means of
providing evidence for the conjecture: measurement devices for time, length, capacity,
temperature, and weight. Tools in mathematics for reasoning and proof may have involved
measurement devices but they also might be strings or numbers or calculations or diagrams.
Communication. As students were involved in problem solving and reasoning and proof
of a situation, communication became another common process that emerges. Communication
involved talking, writing, and listening, (NCTM, 2000) about the mathematics or science concept
being learned. Communication allowed students to refine their thinking and cement ideas
(Goodnough & Nolan, 2008; NCTM, 2000; Zull, 2002). Classroom discourse became a critical
arena for students to share, question, and revisit ideas (NCTM, 2000). Students benefited from
learning about the way other students in the class thought about and solved problems (NCTM,
2000). Communication also aided students in the development of more formal mathematical
language (NCTM, 2000). Since science was based on experiments and observations,
communication in science was important for students to express their understanding, make
predictions, and develop conclusions of experiments and observations (Goodnough & Nolan,
2008; NRC, 1996). In mathematics communication became important as students presented their
proofs. The other students in the classroom needed to be able to understand the reasoning behind
the proof (NCTM, 2000).
This process of students and teachers discussing the science and mathematics was called
discourse. Teachers played a critical role in the form and function of the classroom discourse
34
(Goodnough & Nolan, 2008; NCTM, 2000; Newton & Newton, 2001; Zull, 2002). Furthermore,
subject matter knowledge of the teacher could influence the type of discourse (Newton &
Newton, 2001). Teachers with limited subject matter knowledge would have difficulty fostering
discourse for ideas they themselves did not know or understand. Teachers may have found it
difficult to relinquish control of the learning context to allow for students to fully communicate
their learning (Goodnough & Nolan, 2008). In order for preservice teachers to develop
communication skills in students and promote discourse in the classroom, Kistler (1997)
recommended that preservice teachers also needed to learn how to communicate about
mathematics.
A specialized means of communication was through the use of technology and more
specifically, computers. Computer mediated communication [CMC] provided a means for
communication of ideas and concepts that can move potentially beyond the classroom (Spears,
Postmes, Lea, & Wolbert, 2002). Computer mediated communication included electronic mail
[e-mail], weblogs [blogs], or discussion boards (Bull, Bull, & Kajder, 2003; Repman, Zinskie, &
Carlson, 2005; Wiegel & Bell, 1996). Wiegel and Bell (1996) found that students who used
electronic communication shared information more than students who hand wrote their
reflections. Computer mediated communication may also have acted as an equalizer in the sense
that everyone with access had equal opportunity to contribute (Spears et al, 2002). Bull et al.
(2003) used blogs as a forum for preservice teachers to share thoughts and ideas on texts and
current topics in the news.
Connections. Connections allowed students to have a deeper understanding of a concept
(NCTM, 2000; Zull, 2002). Connections could be made within topics, to other subjects, or to
experiences (NCTM, 2000). Furthermore, connections helped students to link the concepts rather
35
than learn isolated facts (NCTM, 2000). In mathematics, students may have worked on a task
determining the volume of pairs of cones and cylinders with the same height and base. They
could then make connections between the two volumes to make a generalization about the
volume of cones and cylinders with the same height and base. In science, students may have
taken their knowledge of what makes a simple circuit to make connections to series or parallel
circuits.
Mathematics was an integral part of science, commerce, and the everyday world (NCTM,
2000). As such students in a science class were expected to be making connections to the
mathematics involved in the science (NCTM, 2000). In studying the way the brain operates, Zull
(2002) determined that students make connections when they use language to express their
learning and test ideas. He found that when children use language, they use it to connect old
ideas with new ideas. Therefore, making connections and communicating those connections
became critical to student learning (Zull, 2002). Journal writing could provide a way for students
to make connections between the hands-on experiences and the concept (Bleicher, 2006).
Representations. As students solved problems in mathematics and science they relied on
representations to express ideas (Davis & Petish, 2005; Justi & van Driel, 2005; NCTM, 2000).
Representations were not only what occurs on paper, but what also occurs in the mind (NCTM,
2000; Zull, 2002). Symbols, diagrams, graphs, and images were examples of representations
(NCTM, 2000; Posner, Strike, Hewson, & Gertzog, 1982). In science as students tested a simple
circuit consisting of a battery, a light bulb, and connecting wire, students represented on paper
their various tests. Similarly, in mathematics, as students determined combinations of outfits to
wear, they represented the combinations through colored squares or drawings labeled by their
own invention. Concrete tools allowed students to begin to develop more complex forms of
36
representations (NCTM, 2000). However, there were situations in which representations may
serve as an obstacle if the learner does not know how to interpret the representation (Heywood,
2007). For example, if students in a classroom did not understand how base 10 blocks
represented the number system, this would be an obstacle in using the representations.
Furthermore, representations allowed students to relate concepts to the real world and are
important in communication, reasoning and proof, and problem solving (NCTM, 2000).
Different types of representations were used in teaching (Davis & Petish, 2005). Mental
representations differed from instructional representations (Davis & Petish, 2005, Justi &
Gilbert, 2000; Zull, 2002). Instructional representations included real world applications,
physical demonstrations, and visual aids (Davis & Petish, 2005). Mental representations included
mental images and ideas created in the mind (Zull, 2002). In Ball?s (1988) study of classroom
teachers, she found that teachers who used a mixture of symbolic forms with drawings enabled
the students to develop meaning through the representations. Zull (2002) also found that images
played a key role in students processing new information to existing neural networks. Problems
may arise when teachers selected representations that did not align with the learning goals (Davis
& Petish, 2005). Furthermore, choice in representations conveyed a teacher?s knowledge and
beliefs about mathematics and impacts student learning of the content (Ball, 1996; Davis &
Petish, 2005). It was important that students develop clear and accurate mental representations of
concepts. This indicated that teachers must also carefully choose the instructional representation
for a concept, in order to help students make connections with the representation to help develop
mental representations.
37
Common Teaching Approaches in Mathematics and Science
Along with common processes in mathematics and science were common approaches in
teaching mathematics and science. Reflection (Bleicher, 2006; Cannizzaro & Menghini, 2006;
Heywood, 2007; Hill, 1997; Manouchehri, 1997), inquiry (Manouchehri, 1997; Morrison, 2008;
NRC, 1996; Weld & Funk, 2005), assessment (Manouchehri, 1997; NCTM, 2000; NRC, 1996),
discourse (Akerson, 2005; Goodnough & Nolan, 2008; Williams & Baxter; 1996; Woodruff &
Meyer, 1997), and tools for conceptual development (Bleicher, 2006; Justi & Gilbert, 2000;
NRC, 1996; Sriraman & Lesh, 2007 ) were common to science and mathematics teaching.
Reflection
Reflection of thinking allowed students to examine past experiences and merge the new
information with the old (Manouchehri, 1997). Cannizzaro and Menghini (2006) used the
process of reflection to aid preservice teachers to expand on conceptions of definitions other than
the traditional definition. Hill (1997) found that reflection allowed preservice teachers to present
and clarify their understanding of mathematics. Through the reflection process preservice
teachers also realized the significance of relational learning (Hill, 1997). Reflection provided a
means for learners to make connections between experiences and concepts (Bleicher, 2006).
Reflection also helped in the conceptual change process by providing time to examine
conflicting ideas and examine new ideas (Heywood, 2007). Philipp, Thanheiser, and Clement
(2002), in a course that integrated content and children?s mathematical thinking, used reflection
as a means for preservice teachers to make connections between the content and pedagogy.
For teachers, reflection was considered a part of the teaching process (Heywood, 2007).
Teachers reflected on their own knowledge, their own teaching practice, and the learning of the
students (Heywood, 2007; Howitt, 2007; Kelly, 2000). Teachers were expected to reflect on
38
student reasoning when giving an explanation in order to provide an appropriate counter
response (Beyer & Davis, 2008). Therefore, reflection could promote pedagogical content
knowledge through the examination of practice, learning, and knowledge of students (Davis &
Petish, 2005; Heywood, 2007; Kelly, 2000).
Inquiry
Inquiry and inquiry-based instruction have been promoted in the National Education
Science Standards (NRC, 1996), by researchers of mathematics education (Manouchehri, 1997),
and in science education (Morrison, 2008; Weld & Funk, 2005). Inquiry involved authentic
questions, often based on learners? experiences, followed by investigations to answer the
questions, and a reflection on the results of the investigations (Morrison, 2008; NRC, 1996).
Inquiry-based teaching has been credited for students developing a deeper understanding of the
nature of science (Morrison, 2008; Weld & Funk, 2005). Students may sometimes have
experienced frustration when examining science through inquiry (Morrison, 2008). This
frustration may be due to the fact that the teacher was not giving them the answers, and the
students have to work through the unknown to find answers. Teacher education programs should
have aided preservice teachers in developing schemes of inquiry and provide inquiry
explorations (Manouchehri, 1997; Morrison, 2008; Weld & Funk, 2005). Developing schemes of
inquiry in teachers helped them understand the inquiry process and experience lessons
approached with inquiry. For teacher training programs schemes of inquiry could be developed
and built upon throughout the program in content courses (Sanger, 2006; Weld & Funk, 2005)
and methods courses (Akerson & Donnelly, 2008; Morrison, 2008).
The National Science Education Standards (NRC, 1996) called for science teachers to
base their teaching on using inquiry. Morrison (2008) found that many teachers, due to lack of
39
inquiry-based science experiences, needed opportunities to experience inquiry activities and
reflect on their learning. Appleton and Kindt (1999) found that teachers? use of ?cook-book?
lessons limited inquiry due to the predictable nature of such lessons. A key component of
inquiry-based teaching revolved around authentic questions. Morrison (2008) found that most of
his preservice teachers in a science methods course were able to generate authentic questions;
however, some had difficulty generating good questions. In a study comparing elementary
preservice teachers in an inquiry-based chemistry course and secondary teachers in a traditional
course, Sanger (2006) found that the elementary preservice teachers performed as well or better
on chemistry content questions than the students in the traditional class. Experiences with
inquiry-based teaching and learning have been found to impact elementary preservice teachers?
conceptions of science, address misconceptions, and build confidence in science skills
(Bhattacharyya, Volk, & Lumpe, 2009; Sanger, 2006; Weld &Funk, 2005). After taking a
chemistry course for preservice elementary teachers using inquiry-based methods, the
participants? felt confident in their science knowledge (Sanger, 2006). Weld and Funk (2005)
noticed similar results in a life science course for elementary preservice teachers.
Assessment
Teacher education programs needed to provide preservice teachers methods in analyzing
students? cognitive abilities; otherwise known as assessment (Manouchehri, 1997). Assessment
was a necessary part of teaching. The standards documents recommend that assessments operate
throughout a teaching program and not just at the end (NCTM, 2000; NRC, 1996). Assessments
provided insight to teachers as to what students know or don?t know about mathematics and
science (NCTM, 2000; NRC, 1996). However, assessments only measured certain aspects of
student knowledge and have to be interpreted in the scope of what the assessment was designed
40
to measure (NCTM, 2000). Furthermore, the national standards recommend that teachers assess
student knowledge throughout a lesson including when students are discussing, conducting
experiments and trials, or completing journal or open-ended responses (NCTM, 2000; NRC,
1996). Teachers assess students through questions, written modes, and performance (Wang,
Kao, & Lin, 2010).
Discourse
Discourse was another aspect of reformed based mathematics (Williams & Baxter, 1996)
and science classrooms (Akerson, 2005). Discourse helped facilitate development of new
knowledge (Williams & Baxter, 1996). In order for discourse to promote this development of
new knowledge, certain factors were to be in place (Williams & Baxter, 1996). First of all, the
classroom environment had to be one that encouraged students to share their thinking (Williams
& Baxter, 1996). Teachers were to determine expectations for discourse among the class
members (Williams & Baxter, 1996) and to provide modeling (Goodnough & Nolan, 2008).
Discourse did not mean that students just have conversations with one another (Williams &
Baxter, 1996). Students would experience difficulties in group interactions in which some
students dominated the conversation over other students (Goodnough & Nolan, 2008). The
classroom teacher had to know how to facilitate the discourse to ensure that students are gaining
knowledge from the conjectures, explanations, or contradictions that are presented (Williams &
Baxter, 1996; Woodruff & Meyer, 1997). Kistler (1997) recommended that elementary
preservice teachers should experience discourse in their preparation program.
Discourse provided a venue for students to learn more deeply about science (Akerson,
2005). Akerson (2005) observed teachers starting with questions and encouraging discussion in
order for students to gain a better understanding of the science concepts. In order to bridge the
41
gap from the prior knowledge and experiences of the students, discourse in the form of group
discussion and inquiry about a concept is critical to promote connection of concepts, vocabulary,
and experiments with the old knowledge (Woodruff & Meyer, 1997). Discourse provides an
opportunity for students to reason and construct information together which promotes reasoning
(Akerson, 2005). As students reason through the discourse process, teachers make decisions of
what to say, questions to ask, and information to provide (Heywood, 2007; Williams & Baxter,
1996). Kelly (2000) identified that discourse among preservice teachers helped them clarify
misconceptions, discuss new ideas, and test new concepts. Furthermore, discourse aids students
in making connections between their prior knowledge and new concepts (Kelly, 2000).
Tools for Conceptual Development
Experiences impacted the way students learn and understand (Hoffer, 1993; Speilman &
Lloyd, 2004; Zull, 2002). Concrete tools offered opportunities for direct experiences and added
opportunities for learning (Hoffer, 1993; Marek & Cavallo, 1997; Zull, 2002) as well as
supporting real world mathematical and science situations (Jurdak & Shanin, 2001; NRC, 1996).
After studying the way the brain works, Zull (2002) concluded that students must have physical
interactions to connect ideas in the brain with the actual actions of objects as well as connecting
the concrete ideas to abstract ideas.
The National Research Council (1996) recommended that students must have tools in
order to be able to directly investigate scientific phenomena. Tools for developing concepts are
often in the form of models (Justi & Gilbert, 2000). ?A model can be taken to be a
representation of an idea, object, event, process, or system? (Gilbert & Boulter, 1995 as cited in
Justi & Gilbert, 2000, p. 994). Learning aids in mathematics are known as manipulatives
(Sriraman & Lesh, 2007). Manipulatives should precede symbolic notation (Sriraman & Lesh,
42
2007). The use of concrete tools in mathematics and science lessons is often referred to as
?hands-on? (Bleicher, 2006). Hands-on teaching can be used to introduce or develop a concept
(Bleicher, 2006).
Concrete tools played a role in linking school to real world activities, connecting subject
matter, and building conceptual understanding (Jurdak & Shanin, 2001). After studying how
plumbers and students solved the same given problems, Jurdak and Shanin (2001) found there to
be a paradox between the type of mathematics completed in school and the type used in the
workplace. They recommended that ?one possible approach is for the mathematics curriculum to
build bridges between conceptual tools and concrete tools? (Jurdak & Shanin, 2001, p. 314).
However, building bridges between conceptual tools and concrete tools was a complex teaching
process that required teachers? pedagogical content knowledge (Chick, 2007; Davis & Petish,
2005).
Merely giving students concrete materials did not ensure that connections will be made or
deep levels of understanding will be attained (Justi & Gilbert, 2000). When teachers plan
lessons, choices were made as to the representations or models to be used in the lesson (Chick,
2007; Goodnough & Nolan, 2008). Teachers discarded some options and choose other options
based on strengths and weaknesses each option offers (Chick, 2007). Teachers analyzed the
representation, model, or illustration in relation to how closely the tool will aid students in
obtaining the new knowledge based on students? prior knowledge and level of thinking (Chick,
2007). Once a tool had been selected, teachers then decided how the tool would be used in the
learning experience (Chick, 2007). Teachers determined questions, explanations, and tasks
involving the tool while keeping in mind the learning objective for the lesson (Chick, 2007;
43
Goodnough & Nolan, 2008). Teachers were expected to provide a means for students to make
connections from the experiences with concrete tools (Bleicher, 2006)
Examples of situations or problems also served as a tool for building understanding
(Chick, 2007; Zull, 2002). Chick (2007) recommended that teacher education programs aid
future teachers in discriminating between fruitful and fruitless examples. Counter-examples
were useful tools in helping students think about extensions and depth in problems (Chick,
2007). In order for teachers to produce counter examples, they needed experience in examining
the nuances of examples in order to determine which examples would be appropriate (Chick,
2007).
Manipulatives were often thought of as a tool for developing concrete experiences in
mathematics. Fuller (1996) found that teachers use manipulatives in various ways. Teachers
who choose to demonstrate with the manipulative while the students look on represent a content-
focused approach (Fuller, 1996). Whereas teachers who had students use the manipulative
themselves indicate a learner center approach (Fuller, 1996). Zull (2002), in considering the way
young children naturally examine what is in front of them, recommended that teachers put
objects in front of the students to allow the students to explore.
Software applications served as another tool for conceptual development. Hoffer (1993)
used three-dimensional software to investigate geometric properties of polyhedra. The software
allowed students to conjecture and make proofs about the polyhedra (Hoffer, 1993). Students
moved between physical models of polyhedra to software generated models of polyhedra to test
mathematical ideas (Hoffer, 1993). After further investigations, Hoffer (1993) suggested that the
experiences allowed the students to apply the scientific method of exploration, research, and
determining conclusions.
44
Use of concrete tools played a powerful role in teacher training programs as well (Hill,
1997). Since preservice teachers experiences with mathematics often consist of rules and
postulates, the use of manipulatives, models, and alternative approaches allowed the preservice
teacher to confront past experiences and develop relational understanding of mathematics (Hill,
1997). Working with children in schools allowed preservice teachers to put into practice the use
of same or similar manipulatives, models and alternative approaches experienced in teacher
preparation classes (Hill, 1997). Furthermore, Hill (1997) speculated that work with children
may allow new teachers to recognize that the ?rule and postulates? view of mathematics that they
experienced is not the most effective way of teaching mathematics (Hill, 1997). Moreover, Hill
(1997) found that when preservice teachers used concrete experiences in courses and with
students in field placements, it set the stage for preservice teachers to achieve conceptual change
due to bolstering self-confidence in teaching, gaining a sense of accomplishment, and deepening
of mathematical understanding.
Preparation of Elementary Teachers
Teacher education programs established a process that allows preservice teachers
opportunities to develop the recommended skills of designing, planning, and implementing
lessons using standards-based practices to meet the needs of the students (Manouchehri, 1997;
Noori, 1994; Speilman & Lloyd, 2004) and to develop an understanding of the complex task of
teaching (Grossman et al., 2009). This preparation program typically occurred in three phases:
subject matter, theory of education, and pedagogy (NCATE, 2010). During the first phase,
preservice teachers took general content courses in various departments within the institution.
Preservice teachers then move into the teacher preparation portion of their program. Institutions
partnered with nearby school systems to provide clinical teaching experiences (ACEI, 2011).
45
Methods courses often included part-time clinical practice. Teacher candidates experienced an
increase in responsibilities as they complete various field experiences (ACEI, 2011). Methods
courses were often taught by subject specific professors and utilize a combination of on campus
and clinical practice in the field. Student teaching, meant to solidify the preservice teacher?s
knowledge of teaching (Fennel, 1993), was the final phase that takes place after preservice
teachers have completed their methods courses. It consisted of a full-time immersion in a school
setting where the preservice teacher assumes the role of the classroom teacher.
Improving Teacher Education Programs
With the call for improvement of the preparation of elementary teachers (National
Commission on Excellence in Education, 1983; U. S. Department of Education, 2011), efforts
had been made to improve elementary teacher education programs. The federal government
recently established initiatives for top teacher preparation programs based on a linking process of
student test scores and the teacher?s preparation institute. In 1997, Texas organized a
commission of K-16 educators to design guidelines for elementary teacher programs in
mathematics (Molina et al., 1997). The purpose of the commission was to design an elementary
teacher program which would strengthen elementary teachers in the teaching of mathematics
(Molina et al., 1997). Furthermore, the commission recognized the importance of aligning
teacher education programs with the standards as advocated by the National Research Council
(1989) and the National Council of Teachers of Mathematics (1989, 1991) (Molina et al., 1997).
The commission recognized that mathematics is more complex than a series of formulas, that
technology has affected the way mathematics is used in the real world, and that mathematics
requires sense-making and the development of mathematical sophistication (Molina et al., 1997).
With the recognized changes in mathematics and society, changes are necessary in teacher
46
education programs (Kistler, 1997; Molina et al., 1997). Furthermore, a lack of performance in
students could be attributed to the lack of mathematical preparation of teachers (Molina et al.,
1997).
In order to improve the education of elementary teachers, faculty were expected to model
the instructional practices as advocated as well (Kistler, 1997; Molina et al., 1997; Olgun, 2009).
Preservice teachers needed experiences designed by the teacher educators to promote
connections between the content and the pedagogy (Bleicher, 2006; Kelly, 2000; Manouchehri,
1997; Molina et al., 1997; Tal et al., 2001). Teacher candidates entered the program with varied
personal feelings about mathematics (Harkness, D?ambrosio, & Morrone, 2007). In particular,
elementary preservice teachers may have had anxiety about mathematics (Gresham, 2007;
Trujillo & Hadfield, 1999). ?Mathematics anxiety is a feeling of helplessness, tension, or panic
when asked to perform mathematics operations or problems? (Gresham, 2007, p. 182). Education
programs had to take into account students? prior experiences and provided experiences that
allowed the preservice teachers to shift or alter their belief system to envelop the ideologies of
the program (Ball, 1988; Heywood, 2007; Kelly, 2000).
Science and Mathematics in Elementary Teacher Education Programs
Teacher education programs came under criticism as well for the lack of preparation for
teaching science and mathematics (De Villiers, 2004, Tal et al., 2001). De Villiers (2004)
recommended teacher education programs needed to allow ?sufficient opportunity for
exploration, conjecturing and explaining? (p. 704). The author also suggested that if preservice
teachers have not been exposed to mathematics that involves problem-posing, conjecturing,
refuting, and reformulating, then how can they adequately stimulate this in their classrooms?
Similarly, Tal et al. (2001) determined that teacher education programs must provide experiences
47
that allow new teachers to understand Science-Technology-Society (STS) concepts with a focus
on the process of gaining scientific literacy. Providing students with opportunities to explore,
investigate, and reason would also aid preservice teachers in challenging deeply held beliefs
about mathematics (Manouchehri,1997; Philipp et al., 2002) and science (Heywood, 2007;
Kelly, 2000), and replace those beliefs with reform-minded ones. Spielman and Lloyd (2004)
found that the use of reform curricula with preservice teachers had a positive impact on their
beliefs about mathematics. Similarly, researchers expressed that preservice teachers?
involvement in courses aligned with Principles and Standards for School Mathematics (NCTM,
2000) would produce more effective elementary teachers (Alsup, 2003; Kistler, 1997).
Disparities in Teacher Education Programs
Consensus had not been reached as to the best manner for preparing elementary teachers
to teach (Even et al., 1996). Shulman (1986) argued that content knowledge alone was not
sufficient. Furthermore, he found that the curricular knowledge was most neglected in teacher
education programs (Shulman, 1986). Similarly, Newton and Newton (2001) determined that all
of the content preparation could not possibly prepare preservice teachers for classroom events
and must include pedagogical training as well. Even et al. (1996), on the other hand, concluded
that teacher education programs should focus on empowering teachers in decision making rather
than employ curricula developed by ?experts?. After examining teacher education programs from
six countries, Schmidt et al. (2007) found that countries with strong student test scores produced
strong teacher education programs that focused on extensive content teaching along with
practical pedagogical experiences.
Another disparity in teacher education had been attributed to the complexity of teacher
education programs (Bales & Mueller, 2008). Teacher education programs involved education
48
professionals from four distinct settings (Bales & Mueller, 2008). Preservice teachers worked
with faculty in science and mathematics classes, faculty in education courses, practicing teachers
in field experiences, and other school personnel (Bales & Mueller, 2008). Complications could
occur in the development of the preservice teachers with such distinct units each with different
agendas for the preservice teacher (Bales & Mueller, 2008).
Another factor that influenced teacher development programs was the content knowledge
of the preservice teachers (Ball, 1988; Cannizzaro & Menghini, 2006; DeVilliers, 2004; Even et
al., 1996; Matthews & Seaman, 2007). Preservice teachers had misconceptions and gaps of
mathematical knowledge some of which has been attributed to poor or deficient mathematics
experiences (Cannizzaro & Menghini, 2006; De Villiers, 2004). Similarly, studies revealed
elementary preservice teachers have fragmented knowledge of mathematics (Ball, 1988; Even et
al., 1996; Quinn, 1997). Performance by U.S. future educators, when compared to other
countries? future educators, indicated weaker levels of knowledge in all areas of mathematics
(Schmidt et al., 2007).
Deficiencies in science content knowledge have also been found (Davis & Petish, 2005;
Harlow & Otero, 2006; Kelly, 2000). Kelly (2000) found the lecture and text based cookbook
style science experiences were a limiting factor in the way preservice teachers thought about the
nature of teaching science. Teachers themselves admitted to their own lack of content knowledge
in teaching science (Hatton, 2008; Harlow & Otero, 2006). Justi and van Driel (2005)
determined that teachers? content knowledge was the most important factor in how students
learned science. Furthermore, a connection existed between the level of science knowledge and
the pedagogical content knowledge in science teachers (Appleton, 2008; Davis & Petish, 2005).
49
Mathematics Courses for Elementary Teacher Programs
In addition to efforts made by states to improve elementary teacher education programs,
The No Child Left Behind Act (NCLB) (2002) created additional mathematics course
requirements for elementary teacher education programs. Unfortunately, mathematics courses for
elementary preservice teachers were often taught by faculty in the mathematics department with
a traditional lecture based teaching format (Alsup, 2003; Manouchehri, 1997). This form of
teaching continued the fragmented knowledge base preservice teachers already have in
mathematics (Ball, 1988). In an effort to provide a reform model of mathematics, efforts were
made on the part of multiple universities to establish mathematics content courses that relied on
problem solving, conjecturing, conceptual development, and reasoning (Alsup, 2003; Emenaker,
1996; Even et al., 1996; Gresham, 2007; McLeod & Huinker, 2007). Problem solving allowed
students to construct their own knowledge in an active learning environment (Alsup, 2003).
Reform models in the science and mathematics content courses provided a continuum of reform-
based teaching as preservice teachers moved from content courses to methods class that also
employed reform models of teaching (Cady et al., 2006). Even et al. (1996) found that when
preservice teachers participated in activities that challenged their previous notions of
mathematics that they were able to clarify and reevaluate the meaning of mathematical ideas and
mathematical instruction.
McLeod and Huinker (2007) incorporated the four main principles of mathematical
knowledge of teaching (MKT), as determined by Hill et al. (2008), in a mathematics course for
preservice teachers. One course focused on problem solving, communication, and reflection.
Other courses focused specifically on geometry, discrete mathematics and statistics, and algebra.
Students indicated improved confidence in their mathematical abilities. The researchers
50
concluded that the problem solving course was the course that had the greatest impact on the
teachers? mathematical knowledge of teaching (McLeod & Huinker, 2007).
Science Courses for Elementary Teacher Programs
Science departments similarly made attempts to improve courses for elementary
preservice teachers (Harlow & Otero, 2006; Heywood, 2007; Sanger, 2006). As with
mathematics, elementary teachers were found to be deficient in content knowledge of science
(Heywood, 2007; Kelly, 2000; Rice, 2005; Summers et al., 2001). Elementary teachers often
avoided teaching science (Appleton & Kindt, 1999; Howitt, 2007; Sanger, 2006) which has been
attributed to their lack of confidence in their knowledge of science (Heywood, 2007; Kelly,
2000; Olgun, 2009; Weld & Funk, 2005). Just as with mathematics, teachers held negative views
about teaching science (Fulp, 2002; Kelly, 2000). In comparison to other subjects, elementary
teachers reported they have less confidence to teach science (Fulp, 2002; Weiss, 1994) and as a
result may not teach science or may postpone the teaching of science (Appleton & Kindt, 1999).
Furthermore, elementary preservice teachers attributed their prior negative experiences with
science as a reason for a lack of confidence in teaching the subject (Weld & Funk, 2005).
Content and methods courses, conducted in a non-traditional manner with a focus on inquiry-
based teaching and learning, provided an opportunity for preservice teachers to experience
science class in the manner that followed national standards on science teaching, as well as a
means to boost confidence in the subject (Heywood, 2007; Howitt, 2007; Kelly, 2000; Olgun,
2009; Sanger, 2006).
Studies with preservice teachers on science content have occurred in content courses
(Sanger, 2006; Weld & Funk, 2005) and methods courses (Davis & Petish, 2005; Dawkins,
Dickerson, McKinney, & Butler, 2008; Gee et al., 1996; Heywood, 2007; Howitt, 2007; Kelly,
51
2000; Olgun, 2009; Rice, 2005; Wang et al. 2009). Elementary preservice teachers often lacked
content knowledge in science (Rice, 2005; Summers et al., 2001; Weld & Funk, 2005) and held
na?ve views about science (Heywood, 2007). Content courses in science, just as in mathematics,
were often presented as lecture based classes with cook-book labs (NRC, 1996). As a means of
developing content and pedagogical knowledge, researchers focused on active learning (Olgun,
2009), inquiry-based teaching (Harlow & Otero, 2006; Morrison, 2008; Sanger, 2006), and
misconceptions about science (Dana, Campbell, & Lunetta, 1997; Dawkins et al., 2008;
Heywood, 2007). Heywood (2007) found that addressing misconceptions with elementary
preservice teachers was effective in building their science knowledge for teaching. Justi and van
Driel (2005) recommended that teachers should be taught about scientific models and how to use
models. Wang et al. (2009) recommended that preservice teachers? concepts of assessment in
pedagogical practices should be addressed in the methods class as well. Kelly (2000) found that
preservice teachers? experiences in a constructivist-based methods class, along with field
experiences, to be effective in helping the preservice teachers gain an understanding of what it
means to teach science.
Teacher Beliefs and the Impact on Practice
Teachers made decisions about teaching practices based on underlying beliefs and past
experiences (Kelly, 2000; Manouchehri, 1997). Beliefs and knowledge were knitted together
and cannot be separated from one another (Manouchehri, 1997). Hancock and Gallard (2004)
defined beliefs ?as an understanding held by an individual that guides that individual?s intentions
for actions? (p. 281). Shaw and Cronin-Jones (1989) shared that researchers could not look at
knowledge alone; beliefs had to be taken into account. Similarly, Poole (1995) argued that
beliefs and values could not be ignored and were embedded within all aspects of education.
52
Beliefs were developed through childhood (Poole, 1995; Waters-Adams, 2006). New
experiences caused an individual to examine the experience in relation to their belief system
(Pehkonen, 2001). Negotiations were then made with the new experiences and existing beliefs
(Pehkonen, 2001). People could change their beliefs and choose differing belief systems from
youth (Pehkonen, 2001; Poole, 1995).
Preservice teachers entered their teacher training programs with existing notions and
ideas about teaching and learning based on their experiences and ultimately beliefs (Pajares,
1992; Shaw & Cronin-Jones, 1989). These belief systems positively or negatively impacted
preservice teachers? attitudes (Tosun, 2000). However, field experiences and methods classes
could modify or reinforce belief systems (Hancock & Gallard, 2004). Reform based preparation
programs could also positively impact preservice teachers? beliefs and attitudes (McGinnis et al.,
1998). However, Waters-Adams (2006) observed teachers, whose experiences in school were
different from the methods of their teacher training program, which led to a struggle to teach
according to the espoused methods.
Researchers examined how belief systems impact various aspects of teaching
mathematics (Ball, 1988; Cannizzaro & Menghini, 2006; Fuller, 1996; Martin et.al., 2005). A
teacher?s view of mathematics included ?beliefs about mathematics, beliefs about oneself as a
learner and user of mathematics, beliefs about mathematics teaching, and beliefs about
mathematics learning? (Pehkonen, 2001, p.14). Teachers? beliefs about mathematics affected the
way in which the teachers designed the learning environment (Pehkonen, 2001). A teacher
whose goal was to make sure students write proofs similar to the textbook had different goals
than a teacher who wanted to aid students to make sense of the proofs and the reasoning process
of determining proofs (Martin et.al., 2005). Furthermore, teachers who focused on making tasks
53
?fun? held different beliefs from teachers who wanted students to complete tasks that were
worthwhile and significant to learning mathematics (Ball, 1988). Mathematics classes in which
the teacher believes mathematics was about following procedures and rules looks completely
different from a class in which the teacher believed students should develop understanding and
reflective thought towards mathematics (Ball, 1988). Most significantly was that teacher beliefs
about what it means to do mathematics was conveyed in the teaching of the mathematics and
directly impacted how students thought about mathematics (Ball, 1988; Pehkonen, 2001).
Cannizzaro and Menghini (2006) designed research to improve the reflective practice of
teachers in their teaching ?towards the thinking of others and towards changes in one?s own
knowledge, beliefs, and didactical practice? (p. 370). The teachers in the study realized the
student responses were different in different classes depending on how the teacher approached
the problem and the beliefs the teacher had about the ability of her students. Similarly, Fuller
(1996) found after studying novice and experienced teachers that all of the teachers believed that
showing and telling students how to solve problems exemplified a good teacher. This indicated
that teachers needed to challenge their thinking about the dynamics of mathematics and the role
of the teacher in the mathematics classroom (Fuller, 1996).
Similarly, researchers examined teacher beliefs about science as well (Poole, 1995; Shaw
& Cronin-Jones, 1989; Waters-Adams, 2006). Teacher beliefs about science were influenced by
childhood experiences as well as teacher training programs (Water-Adams, 2006). Poole (1995)
concluded that within science education not only do individual beliefs systems affect the way
science is taught but that science was also affected by society. Society?s beliefs about science
were indicated by the importance on science in the education system, the resources that were
provided to teach the subject, and the decisions about what was taught in science (Poole, 1995).
54
Shaw and Cronin-Jones (1989) examined beliefs of elementary and secondary preservice
teachers. They recommended that methods courses help elementary preservice teachers develop
their belief systems. Water-Adams (2006) determined that pedagogical decisions were based on
a teacher?s beliefs about education. Teacher actions may result from reactions to classroom
events that occur spontaneously, from habit, or from carefully considered pedagogical choices
(Martin et. al., 2005; Waters-Adams, 2006). Decision making was affected by a person?s
emotions, cognition, and will (Poole, 1995).
It was recommended that teacher education programs take into consideration preservice
teachers? underlying beliefs when making decisions about the program (Manouchehri, 1997;
Philipp et al., 2002). Beliefs were the filter through which teachers translate the knowledge of
mathematics and science and the pedagogy of teaching (Manouchehri, 1997). By the time
preservice teachers entered teacher education programs, they had been enculturated through their
own school experiences (Kelly, 2000; Manouchehri, 1997). Those school experiences were often
comprised of isolated facts, disconnected concepts, and a surface level of understanding
(Heywood, 2007; Kelly, 2000; Manouchehri, 1997) in which students only learned information
for short term purposes (Heywood, 2007). In a survey of preservice teachers at the beginning of
a science methods course, Kelly (2000) found that over the half of the students held negative
beliefs about science and considered it to be too difficult to learn, boring, and required too much
rote learning. Due to the deep level of enculturation, teacher education faculty were expected to
undo what preservice teachers learned in the K-12 setting (Ball, 1988) as well as in content
college classes (Kelly, 2000).
Teacher education programs, in order to make a shift in beliefs, had take into
consideration what preservice teachers know and think about mathematics (Ball, 1988; Green,
55
Piel, & Flowers, 2008) and science (Heywood, 2007). Manouchehri (1997) concluded that due to
the enculturation of a traditional pedagogy, preservice teachers may not see the value of the
reform pedagogy nor focus their attention on the methods being used in the education classes.
Preservice teachers were unlikely to change or alter their existing beliefs unless they are
challenged to closely examine the beliefs they already hold (Feiman-Nemser & Buchmann,
1986). Even with challenges to existing beliefs, some teachers held on to misconceptions and
some were resistant to change (Gee et al., 1996; Heywood, 2007). Monhardt (2009) found that
elementary preservice teachers believe that they have sufficient science knowledge when in fact
they often do not. College experiences that allowed preservice teachers to solve problems,
explore, analyze, and present reasoning and proof of solutions created opportunities for
preservice teachers to challenge and change beliefs (Kelly, 2000; Manouchehri, 1997). New
teachers also often taught in na?ve ways (Heywood, 2007). They often taught through telling
rather than through insightful experiences (Heywood, 2007). Preservice teachers were also
concerned with their own teaching rather than student learning (Akerson, 2005). Furthermore, in
schools, literacy and mathematics often took precedence over science teaching (Heywood, 2007).
New teachers did not spend as much time teaching science due to the pressures for students to
perform well in reading and mathematics (Heywood, 2007).
Student Teaching
Teacher candidates practice taught on a small scale during their methods classes. When
preservice teachers entered full-time student teaching, it was expected that they would apply
what they learned during methods courses to student teaching (Grossman et al., 2009). Preservice
teachers were provided this opportunity during student teaching. Student teaching was the
pinnacle of the teacher preparation program and some considered the most influential
56
experience. However, the college or university had little control over the student teaching
experience. Preservice teachers often had the ?luck of the draw? when being assigned to a
cooperating teacher (Bolton, 1997).
The cooperating teacher played a powerful role in the development of the student teacher
(Putnam, 2009). Preservice teachers often were unfamiliar with the school and practices of the
cooperating teacher. Bianchini and Cavazos (2007) suggested that beginning teachers need help
negotiating the school culture and the culture of the students. Student teachers were required to
teach in a classroom that is designed and controlled by the cooperating teacher (Bolton, 1997).
This gave the cooperating teacher considerable power during the student teaching experience.
This meant the student teacher and the cooperating teacher had to establish and maintain clear
lines of communication (Bolton, 1997). The student teacher had to deal with the stress and
anxiety of simultaneously being a student and a teacher (Bolton, 1997). Although novice
teachers were enculturated by the cooperating teacher, both learned from one another through the
mentor-mentee relationship that develops during student teaching (Crawford, 2007). This
relationship that developed between cooperating teacher and student teacher often caused student
teachers to place more relevance on the role of the cooperating teacher than experiences from the
teacher preparation program (Securro, 1994).
Preservice teachers faced difficulties during their student teaching experience (Bolton,
1997; Fennell, 1993). Some struggled with student teaching despite doing well in university
courses (Bolton, 1997; Tracy, Follo, Gibson, & Eckart, 1998). Student teachers entered into the
student teaching experience feeling unprepared (Dana, 1992). Student teachers had concerns for
survival, teaching situations, and student concerns and issues outside of student teaching (Smith
& Sanche, 1993). Classroom management was a concern as well (Bolton, 1997).
57
Student teachers often adopted the teaching practices of the cooperating teacher (Putnam,
2009). This sometimes meant a student teacher shifted to more teacher oriented practices
(Putnam, 2009) that were at odds with the pedagogical teaching from their university courses.
Student teachers may have been taught reform methods of teaching in university courses but this
was at odds with the type of science and mathematics teaching they experienced in student
teaching (Crawford, 2007; Leonard, Boakes, & Moore, 2009). Additionally, cooperating teachers
implied that methods taught at the university were not the real-world of teaching (Batesky,
2001). Interaction with the cooperating teacher impacted and influenced the new teachers?
beliefs about their teaching abilities (Philippou, Charalambos, & Leonidas, 2003). A preservice
teacher in a study by Philippou et al. (2003) worked with a traditional teacher and successfully
taught a different way. However, another preservice teacher tried to teach differently from the
traditional style of her cooperating teacher and it was not successful (Philippou et al., 2003). She
felt dissatisfaction from her cooperating teacher and this affected how she felt about teaching
mathematics. Crawford (2007) noticed that skepticism of reform practices develop during
student teaching due to the culture clash of the assigned classroom and the university philosophy.
Conceptual Change
Teachers made decisions about teaching practices based on underlying beliefs and past
experiences (Kelly, 2000; Manouchehri, 1997) and were unlikely to change or alter their existing
beliefs unless they were challenged to closely examine the beliefs they already hold (Feiman-
Nemser & Buchmann, 1986). The process of altering beliefs was known as conceptual change
(Posner et al., 1982). Conceptual change began with learning. Learning occurred when new ideas
interacted with old ideas and the new ideas were seen as ideas that made sense to the learner
(Posner et al., 1982). People made judgments about new ideas based on data, verification, or
58
confirmation (Posner et al., 1982). In this sense, learning was inquiry that involves the
progression towards conceptual change (Posner et al., 1982). Assimilation was the first type of
conceptual change (Posner et al., 1982). With assimilation students modified old ideas with new
ideas (Posner et al., 1982). However, in some cases old ideas had to be completely replaced with
new ideas (Posner et al., 1982). This dramatic change was called accommodation (Posner et al.,
1982). Conceptual change often referred to this process of accommodation (Posner et al., 1982).
Brown and Clement (1989) concluded that conceptual change must be the goal of teaching.
With recognition of the lack of strong mathematical skills (Ball, 1988; Manouchehri,
1997) and science skills (Heywood, 2007; Rice, 2005) and changes in teacher education
programs (Gitomer, 2007; Molina et al., 1997), researchers examined the conceptual change of
teachers (Heywood, 2007; Hill, 1997; Tal, 2001). Examining conceptual change of preservice
teachers was necessary due to the different experiences that occur in the K-12 setting as
compared to the reform teaching experiences in the teacher education program setting (Gee et al.,
1996; Hill, 1997). Teacher education programs required student teachers to use teaching
practices different from what they themselves experienced (Hill, 1997; Kelly, 2000). Conceptual
change began with the learner losing faith in the original understanding of a concept (Posner et
al., 1982). Cognitive conflict occurred when the learner became discontented or displeased with
the existing notions (Heywood, 2007). This would not occur until the student encountered
problems with the old ideas and became willing to accept new ideas (Posner et al., 1982). The
process of reflecting on new ideas in the context of preexisting notions provided a means for
learners to become discontent or displeased with their old ideas (Howitt, 2007). Zull (2002)
argued that reflection becomes an important component to give the learner time to accept new
ideas, since the learner holds deep emotions about their existing knowledge.
59
Experiences and teaching methods in teacher education programs provided an
opportunity for preservice teachers to become dissatisfied with their existing notions and be
willing to accept different ideas (Heywood, 2007; Kelly, 2000). Conflicting experiences could
occur through the means of direct exploration and observation of a situation or incident (Kelly,
2000). Deliberations on readings also could provide an avenue for cognitive conflict (Heywood,
2007). Debates of contrasting views similarly allowed learners to become dissatisfied with the
current view and more amenable to a new view (Heywood, 2007). Whatever the strategy
employed, direct observation, readings of texts, or debates of contrasting views, preservice
teachers? views were expected to be made explicit and the experience provide a cognitive
dissonance (Heywood, 2007). However, the experience should not have made the learner feel
discouraged about their present level of knowledge (Heywood, 2007). Furthermore, the new
idea generated through the cognitive conflict experience had to be seen initially as being credible
or able to fit into the learner?s current schema (Posner et al., 1982).
The second phase of conceptual change was identified as the learner being able to
understand new concepts (Posner et al., 1982). For new conceptions to be learned, the student
had to be able to make sense of the ideas and representations (Posner et al., 1982). Learners then
had to believe that the new concepts will solve the problem (Posner et al., 1982). Finally,
learners had to feel that it was of value to put time and attention into the new concepts (Posner et
al., 1982).
For elementary education majors who often held negative attitudes towards mathematics,
the question begins with whether or not the preservice teachers believed themselves to be
capable of learning mathematics (Hill, 1997) or science (Kelly, 2000). Therefore, the ability of
preservice teachers to achieve conceptual change was further hindered by their lack of
60
confidence in their own mathematical abilities (Hill, 1997) or science abilities (Kelly, 2000).
Similarly, those with negative attitudes may have been more resistant to different methods of
teaching (Freeman & Smith, 1997). In an effort to combat negative attitudes or lack of
confidence, methods classes placed preservice teachers in classrooms to gain experience in
teaching (Hill, 1997). Interaction with children was a very valuable experience for preservice
teachers in the process of putting new ideas and concepts into practice (Heywood, 2007; Hill,
1997; Kelly, 2000). Work with students in schools was found beneficial for preservice teachers
to fuse subject matter knowledge with pedagogical knowledge (Heywood, 2007; Kelly, 2000).
Furthermore, children?s positive response to preservice teachers provided encouragement for
preservice teachers to teach in meaningful ways and may have helped preservice teachers
recognize that change of thinking was necessary (Hill, 1997).
Conceptual change in thinking about science was also necessary (Heywood, 2007; Tal et
al., 2001). In order for conceptual change in science to occur, teachers needed to assimilate new
methods for teaching science (Tal et al., 2001). Heywood (2007) determined that science classes
for preservice teachers should focus on misconceptions as a means of supporting conceptual
change. Misconceptions could be used to promote reflection of science concepts. Reflection
played a critical role in teachers replacing old ideas with new ideas (Heywood, 2007; Philipp et
al., 2002; Zull, 2002).
Whether it was science or mathematics, teacher education programs had to be in
alignment to help promote conceptual change (Gee et al., 1996; Heywood, 2007). A systematic
process had to be in place in teacher education programs so that the experiences from content
courses to methods courses to internship were in alignment (Heywood, 2007). Those experiences
61
should have moved the new teachers from content specific knowledge to pedagogical content
knowledge (Appleton, 2008; Heywood, 2007).
The Learning Cycle for Mathematics and Science Teaching
Just as mathematics and science content were intertwined so were the common
approaches to teaching science and mathematics (NCTM, 2000; NRC, 1996). The learning cycle
offered a way of incorporating similar teaching methods and use of tools for conceptual
development (Chick, 2007; Fuller, 1996; Hill, 1997), discourse (Akerson, 2005; Heywood, 2007;
William & Baxter, 1996), assessing student knowledge (Manouchehri, 1997; NCTM, 2000;
NRC, 1996) inquiry-based teaching (Morrison, 2008; Manouchehri, 1997; NRC, 1996; Weld &
Funk, 2005), and reflection (Bleicher, 2006; Hill, 1997; Manouchehri, 1997). Furthermore, the
learning cycle offered a way for preservice teachers to confront their notions about mathematics
and science and expanded their belief systems to undergo conceptual change.
Historical Context of the Learning Cycle
The learning cycle was not a new way to approach teaching (Lawson, Abraham, &
Renner, 1989). Rather, in the learning cycle existed a natural form of learning that has persisted
with child and adult (Lawson, Abraham, & Renner, 1989; Schmidt, 2008). On a physiological
level, Zull (2002) found that sensory experiences entered in the sensory cortex of the brain were
processed and actualized in the integrative cortex, and new ideas were tested in the motor cortex.
From a sociological perspective, adults have observed children tasting, touching, smelling, and
feeling the world around them, incorporating and reflecting the new ideas, and actively testing
the new ideas (Schmidt, 2008; Zambo & Zambo, 2007). Similarly adults undergo the same
processes when learning new material: consciousness of a new idea, investigation of the new
62
idea, and finally use of the new idea (Schmidt, 2008). Schmidt (2008) described the natural
learning cycle as consisting of ?awareness, exploration, inquiry and action? (p.12).
Teaching consisted of a set of methods and procedures, some more effective than others
(Lawson, Abraham, & Renner, 1989). Lack of subject matter knowledge, failure to implement
effective teaching practices and routines, inability to gauge students and adjust instruction for
individual needs, difficulty identifying what is going wrong in the teaching and learning
environment, and surface level reflection of teaching experiences all were identified as
ineffective teaching practices (Reynolds & Elias, 1991; Tracy et al., 1998). Adams, Cooper,
Johnson, and Wojtysiak (1996) found that ineffective teaching practices created a learning
environment that allowed students to be passive learners and less engaged in the teaching and
learning process. In contrast to passive learning environments, Adams et al. (1996) found that
solving problems and exploring real life situations, tenets of the learning cycle, were key
components of a meaningful curriculum which promoted active learning, engaged students, and
enacted more effective teaching practices.
The idea of appropriately designed learning situations was not a new one. Neal (1962)
proposed that teaching techniques should focus on the development of critical thinking and this
development of thinking depended largely on the design of the learning situation. It was during
the late 1950s and early 1960s, in which educational reforms were in demand, that Atkin and
Karplus (1962) developed their model for teaching and learning science which focused on the
natural active learning of children and later became known as the learning cycle.
The development of the learning cycle began when Professor Robert Karplus spoke with
his daughter?s second grade class (Lawson, Abraham, &Renner, 1989). Karplus began to think
about the development of science in the elementary grades. In the late 1950?s and early 1960?s
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Karplus developed learning units for the elementary grades that focused on the way students
naturally approach and learn new phenomena. Karplus began to work with J. Myron Atkin on
?discovery learning?.
In their research from the early 1960?s, Atkin and Karplus (1962) described a series of
science experiments in which students came to understand the concept of magnetic field. In the
series of experiments students were not led directly to the concept of magnetic field, but rather
were led in a circuitous route to understand the concept from several points (Atkin & Karplus,
1962). Through the series of lesson observations Atkin and Karplus (1962) confirmed the idea
that students should explore concepts, define and develop the concept from the experience, and
apply the newly learned concept to a new situation. Furthermore, Atkin and Karplus (1962)
offered that the teacher must never present scientific ideas in dictatorial ways due to the ever
changing nature and knowledge of science. Initially, Karplus and Atkins (1962) only discussed
invention and discovery (Lawson et al., 1989). They considered invention to be the development
of a concept and discovery to be the verification of the concept in a new situation. The
Karplus/Atkin model was later revised to include exploration (Lawson et al., 1989). At the time
of the initial research by Karplus and Atkin, Professor Chester Lawson was developing a similar
model of instruction (Lawson et al., 1989). Karplus and Lawson began working together on the
Science Curriculum Improvement Study program in the 1960?s and 1970?s (Lawson et al., 1989).
The three phases of the Atkin/Karplus model, exploration, invention, and discovery, were
incorporated in the units developed for the Science Curriculum Improvement Study program
(Kratochvil & Crawford, 1971). Lawson and others incorporated the same concepts in the
Biological Sciences Curriculum Study (Withee & Lindell, 2006).
64
The phases of the learning cycle in the Atkin/Karplus model aligned closely with Piaget?s
theory of learning and in particular the processes of assimilation, accommodation, and
organization (Abraham & Renner, 1983; Bybee, 1997; Kratochvil & Crawford, 1971; Marek &
Cavallo, 1997; Renner & Lawson, 1973). Piaget concluded that children interact with the
environment and encounter new concepts which create contradictions in thinking (Renner &
Lawson, 1973). Piaget called this state of contradiction disequilibrium (Marek & Cavallo, 1997;
Renner & Lawson, 1973). Children then assimilated the new information with the old
information through exploration or adult guidance (Renner & Lawson, 1973). This followed
similarly in the exploration phase of the learning cycle in which students were gathering and
assimilating new information in a nondirected manner (Bybee, 1997). It was important that
students in the exploration phase were allowed open-ended exploration with materials that
explored the concept (Renner & Lawson, 1973) in order for students to examine differing
avenues of thought (Lawson et al., 1989). Furthermore, directions given by the teacher should
guide the students without telling them the concept they are to learn (Marek & Cavallo, 1997).
The invention phase followed the exploration phase (Bybee, 1997; Kratochivil &
Crawford, 1971; Lawson et al., 1989). In the invention phase students were introduced to a new
concept either by the teacher or another student (Kratochivil & Crawford, 1971). The process of
accommodation in which new information replaced old information began at this point (Bybee,
1997). The discovery phase allowed for the learners to apply the newly learned concept to a new
situation (Atkin & Karplus, 1962) and expand their understanding of the concept (Lawson,
Abraham, & Renner, 1989). This completed the accommodation process and moved the learner
from the initial state of disequilibrium to equilibrium (Bybee, 1997) and what Piaget named
organization (Renner & Lawson, 1973).
65
The three phases have since been modified and interpreted by other researchers.
However, the initial tenets still remained (Abraham & Renner, 1983). Karplus continued to call
the phases of the learning cycle exploration, invention, and discovery until the 1970?s (Lawson,
Abraham, & Renner, 1989). In 1976, Karplus renamed the phases as exploration, concept
introduction, and concept application (Karplus, 1976). Abraham and Renner (1983) expressed
the phases as exploration, conceptual invention, and expansion of the idea. Marek and Cavallo
(1997) thought of the phases as exploration, term introduction, and concept application.
5E Instructional Model of the Learning Cycle
The three phases were expanded to five phases, still incorporating the initial phases of the
Karplus/Atkin model (Bybee, 1997), by Lawson and others on the Biological Sciences
Curriculum Study (Withee & Lindell, 2006). The five phases included engagement, exploration,
explanation, elaboration, and evaluation (Bybee, 1997). Chessin and Moore (2004) considered
the model in terms of six E?s: engage, explore, explain, expand, evaluate, and e-search. E-search
was the use of electronic media, Internet research, presentation software, spreadsheet software,
anywhere within the 5 E?s (Chessin & Moore, 2004).
The engagement phase was what it sounds like, engaging the students (Bybee, 1997).
During the engagement phase, teachers had students focus on an event or problem (Bybee,
1997). This may be accomplished through a question, a situation or a problem (Bybee, 1997;
Marek et al., 2008). Teachers may even present a discrepant event as a means of engaging
students (Marek et al., 2008; Tracy, 2003). Bircher (2009) even used juvenile literature to
engage students in the concept. Students revealed their prior knowledge of the situation
presented and raised questions during the engagement phase (Bybee, 1997; Withee & Lindell,
2006). As a result of the engagement phase students were curious and actively interested in the
66
new concept, beginning the stage of disequilibrium (Bybee, 1997). This part of the learning
cycle was typically short in duration (Bybee, 1997).
The exploration phase carried the same connotations for exploration as originally
designed by Karplus and Atkin (Bybee, 1997). Karplus and Atkin (1962) concluded that the
exploration phase was critical to provide common knowledge to all students since all students
have different background knowledge and life experiences. Student exploration should be with
concrete materials and hands-on. The goal of the exploration phase was to provide a common
experience that the students and teacher could discuss to further scientific understanding in later
phases (Atkin & Karplus, 1962). Teachers had the responsibility of providing the materials,
observing and ensuring students were conducting the experiment correctly, and interacting with
students while students were collecting data (Marek, 2008).
Physical experiences in the exploration phase were necessary (Bybee, 1997) and allowed
the learner to move beyond initial observations to generalizations (Renner & Lawson, 1973).
This generalization allowed the learner to think about the concept in other situations (Renner
&Lawson, 1973). Piaget named this mental structure as logical-mathematical (Renner
&Lawson, 1973). Furthermore, Renner and Lawson (1973) deduced from the work of Piaget that
the exploratory experience must occur before abstract concepts are introduced. In studying the
brain, Zull (2002) found that beginning with concrete experiences and examples was important
in learning because it engages students? senses and allows for new neural networks to connect
from existing networks. The exploration phase allowed learners, through the interactions of
materials, to be able to understand and make abstract generalizations (Renner &Lawson, 1973).
Another key component of the exploration phase was the interaction with others (Bybee,
1997; Renner & Lawson, 1973). Cooperative groups, or cooperative learning, provided students
67
with opportunities to recognize the perspective of others (Kelly, 2000). Students learned to listen
to others, asked questions, and shared ideas when working together (Kelly, 2000; Renner &
Lawson, 1973). The exploratory phase encouraged students to discuss with one another and built
communication skills (Renner & Lawson, 1973). While students were discussing and sharing
ideas in the exploration phase, the teacher circulated, asked questions, and guided students but
not in a direct way (Atkin & Karplus, 1962; Bybee, 1997). It was also important that teachers
provide students adequate time in the exploration phase before moving on to another phase
(Withee & Lindell, 2006).
In the explanation phase, students were explaining what they discovered in the
exploration phase (Withee & Lindell, 2006). Terms and definitions were expected to be
discussed and clarified at this time (Bybee, 1997; Withee & Lindell, 2006). The teacher used
this time to bring a connection with the students? thoughts from the explore phase to the focus
concept (Bybee, 1997). Questioning by the teacher was a key part of the explanation phase and
required caution on the part of the teacher not to tell the students the science concept (Marek,
2008). Questioning on the part of the teacher during explanation construction allowed the
students to clarify and support their thinking as well as addressed any misconceptions presented
in the initial explanation (Beyer & Davis, 2008). This phase closely aligned with the phase
Karplus originally called invention and later renamed concept introduction (Bybee, 1997;
Karplus 1976).
Once the students had an explanation for their experiences in the explore phase, the
learning moved to the elaboration phase (Bybee, 1997). This phase followed what Karplus
originally named discovery and later renamed concept application (Bybee, 1997; Karplus, 1976).
In this phase students were applying or extending what they learned to a new situation (Atkin &
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Karplus, 1962; Marek, 2008). Group interaction and group discussion were also important at
this point of learning (Bybee, 1997). The goal of this phase was to move students from the
physical experience to the logical-mathematical operation of thinking in which students can
make generalizations about the concept (Bybee, 1997; Renner &Lawson, 1973). Likewise, Zull
(2002) found that when learners tested, expanded, and manipulated ideas that true understanding
of the concept developed.
Evaluation was the final phase but does not have to occur last (Bybee, 1997; Marek,
2008; NCTM, 2000). Evaluation could occur throughout the lesson (Bybee, 1997; Marek, 2008;
NCTM, 2000). Evaluation provided feedback to teacher and student (Bybee, 1997; NCTM,
2000). The evaluation could be a way for teachers to determine students? level of conceptual
knowledge, for students to assess their group knowledge, and for students to determine their own
learning (Bybee, 1997; Withee & Lindell, 2006).
Although the learning cycle has its roots in the science field (Bybee, 1997; Atkin &
Karplus, 1962), the tenets aligned closely with the national mathematics standards. Researchers
provided examples of the learning cycle with mathematics (Marek & Cavallo, 1997; Marek et
al., 2008; Simon, 1992). Marek (2008) described a science lesson in which students measured the
circumference of objects to make a conclusion about the relationship of the circumference and
diameter of circles. His description illustrated how the learning cycle could be used in
mathematics teaching. Although his example was for a science lesson, it very easily could have
been for a mathematics lesson on understanding area of circles. Similarly, Marek and Cavallo
(1997) explained that the learning cycle could be used in mathematics for problem solving and
provided examples of learning cycle lessons to teach measurement and geometry concepts.
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The mathematics standards promoted students working together (NCTM, 2000), as
recommended in the exploration and elaboration phases (Bybee, 1997). Using concrete tools and
solving problems to develop a deep understanding of the concepts was core to the mathematics
standards (NCTM, 2000) and aligned with the exploration, explanation, and elaboration phases
of the learning cycle (Bybee, 1997). According to the mathematics standards, students generated
their own mathematical questions (NCTM, 2000), a part of the engagement phase (Bybee, 1997),
and students generated conclusions (NCTM, 2000), a part of the explanation phase (Bybee,
1997). The exploration, explanation, and elaboration phase supported a focus on thinking and
reasoning skills that led to conjectures or arguments about the mathematics being examined
(Bybee, 1997; NCTM, 2000).
The Learning Cycle: Elementary School through Higher Education
Swartz (1982) found that students should experience diverse methodologies that allowed
students and teachers to explore and test scientific concepts. Components of inquiry were
embedded in the learning cycle (Gee et al., 1996; Tracy, 1999; Withee & Lindell, 2006). Inquiry-
based teaching involved the exploration of students around a central idea, formulation of
questions, investigations to answer the questions, and reflection of learned ideas (Morrison,
2008; Tracy, 1999). Therefore, researchers recognized that the learning cycle provided a method
for learners in any grade to explore and conduct investigations in mathematics or science (Gee et
al., 1996; Tracy, 1999; Withee & Lindell, 2006).
Research was conducted with the learning cycle in the K-12 setting (Boddy, Watson, &
Aubusson, 2003; Cardak, Dikmenli, & Saritas, 2008; Liu, Peng, Wu, & Lin, 2009). Researchers
concluded that students using the learning cycle performed better than students in a traditional
science classroom (Cardak et al., 2008; Ergin et al., 2008). Furthermore, Boddy et al. (2003)
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found that the use of the 5E learning cycle promoted higher order thinking skills in primary age
students. When working with a primary teacher in using inquiry-based teaching within the 5E
model, the researcher and teacher showed students were more involved and more on task than
they would have been in a traditional science lesson (Clark, 2003). Tracy (2003) designed a unit
on volume that incorporated science and mathematics concepts using the learning cycle. She
designed the unit to demonstrate how science and mathematics could be taught within the
learning cycle framework. Schlenker, Blanke, and Mecca (2007) incorporated science and
mathematics in an introductory chemistry unit with eighth graders that utilized the learning
cycle. They found their students were excited about learning chemistry with the learning cycle
approach. They also noted that the benefit of the learning cycle was that ?the cycle could be
entered at any point, and it is possible to loop back or ahead to another part of the cycle?
(Schlenker et al., 2007, p. 86). Similarly, the 5E model had been used with high school students
(Brown, Freidrichsen, & Mongler, 2008; Ksiazek et al., 2009). Brown et al. (2008) had high
school students design miniecosystems based on the 5E model. The authors were surprised at the
level of complexity in students? designs of the miniecosystems. In the culminating activity,
students presented their findings of what occurred in the miniecosystems and asked and
answered questions from other students. Ksiazek et al. (2009) designed a unit on seagrass for
high school biology students that used the learning cycle. Students designed and conducted their
own experiments to answer questions about seagrass and the effects of humans on seagrass.
Their initial experiments led them to additional questions to answer about environmental issues.
The teachers found that students were highly engaged, developed questions beyond the initial
query, and demonstrated complexity and depth of scientific understanding of the topics (Brown
et al., 2008; Ksiazek et al., 2009; Schlenker et al., 2007).
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Research was conducted at the higher education level with the use of the learning cycle
as well (Illinois Central College, 1979; Walker, McGill, Buikema, & Stevens, 2008). In response
to students? low level of reasoning abilities, Illinois Central College designed freshman courses,
English, mathematics, physics, history, social science and sociology, around the Karplus model
of the learning cycle (1979) (Illinois Central College, 1979). Students who participated in the
Development of Operational Reasoning Skills (DOORS) Project at Illinois Central indicated that
the classes held more meaning for them (Illinois Central College, 1979). Walker et al. (2008)
compared college sophomore students in a traditional microbiology laboratory and a laboratory
that used the 5E model with embedded inquiry. They found that the students in the inquiry-based
lab were better able to answer test questions on the lab content.
Researchers examined preservice teachers in regard to inquiry-based teaching (Haefner &
Zembal-Saul, 2004; Morrison, 2008; Park Rogers & Abel, 2008). Haefner and Zembal-Saul
(2004) studied eleven elementary preservice teachers? development in a life sciences course.
They found that the inquiry-based course helped preservice teachers confront misconceptions of
science teaching and supported preservice teachers? understanding of science. Stamp and
O?Brien (2005) worked with graduate students and in-service elementary teachers to develop
science curricular units using the 5E learning cycle model that aligned with the state standards.
They found that the teachers in the study became more skillful in their science teaching having
received professional development on teaching with the 5E model. At the college level, Withee
and Lindell (2006) studied five methods instructors. They found that the instructors supported
inquiry based teaching but found some difficulties in using the 5E model. Difficulties included
students reluctant to move out of the explore stage, difficulty in separating the stages, and that
one model does not always meet the instructional needs of the lesson.
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Researchers developed tests to determine preservice teachers? understandings of the
learning cycle during methods class (Marek, et al., 2008; Odom & Settlage, 1996). Odom and
Settlage (1996) developed a test to determine preservice teachers understanding of the learning
cycle called the Learning Cycle Test (LCT). Lindgren and Bleicher (2005) examined how
preservice teachers learned the learning cycle. In the science methods class, the researchers
conducted experiments using the three historical phases of the learning cycle (Karplus, 1976):
exploration, concept introduction, and concept application. The LCT (Odom & Settlage, 1996)
was administered and the beginning and end of the course. Results indicated varied conceptions
of the learning cycle even after receiving instruction on the learning cycle and lessons modeled
with the learning cycle. To provide insight into students? thinking about the learning cycle,
Lindgren and Bleicher (2005), in addition to the LCT, also used reflective journals kept by the
students throughout the course and focus group discussions provided. Data indicated that
students had a difficult time conceptually changing the ideas about teaching science (Lindgren &
Bleicher, 2008). Some students embraced the learning cycle due to dissatisfaction of the way
they experienced science in school (Lindgren & Bleicher, 2005). Preservice teachers?
understanding of the learning cycle improved with the continued exposure of the learning cycle
in methods class (Lindgren & Bleicher, 2008). Marek et al. (2008) modified the Learning Cycle
Test (LCT) by adding questions that focused on the teacher?s role. He named the two-tiered test
Understanding the Learning Cycle (ULC). Instructors who gave the ULC indicated they liked it
better than the LCT and felt it provided a more accurate indication of preservice teachers?
conceptions.
The learning cycle was an inquiry-embedded model that supported conceptual change of
thinking (Withee & Lindell, 2006) in which the learners replaced old ideas with new ideas
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(Posner et al., 1982). Similar to the findings of Lindgren & Bleicher (2005), Gee et al. (1996)
found that elementary education majors, after experiencing the learning cycle in content courses
and methods courses, were still not completely committed to the use of the learning cycle in their
own teaching. They speculated that preservice teachers may have difficulty translating the theory
and experiences of the methods class into their teaching practice (Gee et al., 1996). However,
Lindgren and Bleicher (2008) noticed students, who had negative science experiences or were
dissatisfied with the way they were taught, embraced the learning cycle approach. This indicated
multiple factors methods instructors have to consider when using the learning cycle approach:
preservice teachers? prior experiences in science, flexibility in thinking about teaching science in
a different way, and learning modules to help breech the barriers.
Little research has been conducted with elementary preservice teachers with
mathematics and the use of the learning cycle (Lindgren & Bleicher, 2005; Marek et al., 2008).
The research that exists on preservice teachers and the learning cycle focused primarily on
science (Lindgren & Bleicher, 2005; Marek et al., 2008; Urey & Calik, 2008). Urey and Calik
(2008) determined that the use of the learning cycle promoted conceptual change in pre-service
science teachers? understanding of cells. Additionally, Marek (2008) provided examples of the
learning cycle in science and mathematics. He provided detailed descriptions of how teachers
could develop and implement the learning cycle. He also gave examples of a learning cycle
lesson on understanding diameter and circumference of circles. Simon (1992) proposed that
learning cycles existed in mathematics teacher education programs as preservice teachers learned
new ways to think about mathematics and applied that knowledge to the teaching of students.
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Conclusion
As previously mentioned, elementary preservice teacher education programs often were
compartmentalized based on subject and further divided into content courses and methods
courses (Quinn, 1997). Other researchers focused on the integration of science and mathematics
as subjects (Berlin & White, 1991; Frykholm & Glassen, 2005; Lonning & DeFranco, 1994,
1997; Lonning, DeFranco, & Weinland, 1998; Stuessy, 1993; Steussy & Naizer, 1996).
However, this research presented an opportunity to examine conceptions of common approaches
between science and mathematics teaching, and how preservice teachers put their knowledge
into practice. Since many of today?s preservice teachers lacked a model of standards-based
reform teaching (CBMS, 2001), approaching teacher development from a perspective of
commonalties in mathematics and science teaching could help elementary preservice teachers
better understand how to teach mathematics and science. Researchers called for more research
that uses new innovative teaching techniques (Manouchehri, 1997). Furthermore, Manouchehri
(1997) called for researchers to conduct long term studies on change of preservice teachers and
how this change was exhibited in mathematics teaching practices.
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CHAPTER 3: METHODOLOGY
Introduction to the Study
The purpose of this research was to examine elementary preservice teachers?
mathematics and science thinking and practice in relation to conceptions of pedagogy, tools for
conceptual development, and processes for meaningful learning while in a joint science and
mathematics methods class and subsequent internship experience. The research questions of this
study were: (1) What are preservice teachers? conceptions of tools for conceptual development,
processes for meaningful learning, and pedagogical approaches prior to and after taking the
mathematics-science methods courses? (2) What changes, if any, in development of knowledge
and understanding of the common approaches between mathematics and science occurred while
taking the mathematics-science methods courses? (3) How did preservice teachers put into
practice during student teaching their thinking from the methods courses on tools for conceptual
development, processes for meaningful learning, and pedagogical approaches to mathematics
teaching?
The purpose of this project was to examine elementary preservice teachers? thinking and
practice on common approaches mathematics and science as they moved through co-requisite
science and mathematics methods courses to student teaching. This study used a qualitative
design (Creswell, 1988; Merriam, 1998). The study took place in two phases. Phase I occurred
during the participants? science and mathematics methods classes. Phase II occurred during the
participants? student teaching experience. Data sources for Phase I included an open-ended pre-
test (see Appendix A in overall appendix section), open-ended post-test (see Appendix B in
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overall appendix section), and weekly blogs (see Appendix C in overall appendix section). Data
sources for Phase II included written observations of preservice teachers teaching mathematics
taken as field notes during 5 to 6 observations, follow-up interviews with preservice teachers
after the observation (see Appendix D in overall appendix section), and a final reflection (see
Appendix E in overall appendix section). Information from observations, interviews, and blog
responses were linked through a coding process.
Research Paradigm
This research is based on a constructivist paradigm. At the heart of constructivism is the
notion that a learner constructs knowledge (Beeth, Hennesey, & Zeitsman, 1996; Piaget, 1976).
Since each individual experiences the world in different ways, then realities are different for each
person (Beeth et al. 1996; Denzin & Lincoln, 2003). Therefore, researchers with a constructivist
paradigm recognize relativist ontology: that ?there are multiple realities? (Denzin & Lincoln,
2003, p. 35). With this recognition of multiple realities comes the acknowledgement of a person
only knowing their own reality and researchers attempting to understand someone else?s reality
from the view of their own (Beeth et al., 1996). Constructivism recognizes multiple ways of
thinking and freedom in thinking and understanding in different ways (Beeth et al., 1996).
A constructivist paradigm also assumes a subjectivist epistemology: that the ?knower and
responder cocreate understandings? (Denzin & Lincoln, 2003, p. 35). With this cocreation of
understanding is the supposition that the teacher will create a learning environment conducive to
learning. Methodological procedures focus on the natural world. Pattern theories or grounded
theories are used to communicate the results of a constructivist paradigm (Denzin & Lincoln,
2003). This product is often in the form of case studies or narratives (Hatch, 2002). The voice of
the participant becomes an important part of the written account (Hatch, 2002).
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Constructivist Grounded Theory Methodology
This research used a constructivist grounded theory methodology (Charmaz, 2005).
Glaser and Strauss (1967) are credited with the seminal work on grounded theory. A theory
?explains or predicts something? (p. 31). In their work they define grounded theory as ?the
discovery of theory from data? rather than using data to verify a theory (p. 1). They concluded
that since the theory was determined based on the data that the theory would be long lasting.
Furthermore, they argued that the process for generating the theory is just as important as the
theory itself.
Glaser and Strauss (1967) described comparative analysis as a means for generating
theory. Comparative analysis can be used to determine accuracy of evidence, make empirical
generalizations, specify a concept of analysis, verify theory, or generate theory. When
determining accuracy of evidence, comparative analysis begins with a conceptual category being
assigned to a piece of data. This conceptual category is derived from a set of data and can be
applied to other similar data sets. When comparing to other data sets, limits to the concept are
determined to make empirical generalizations (Glaser & Strauss, 1967). Negative cases that
stand out or positive instances that support the research are used to verify the theory that is being
generated. With grounded theory, concepts emerge from the data to generate substantive theory.
Substantive theory leads to formal grounded theory (Glaser & Strauss, 1967).
There are two elements of theory: categories and properties, and hypotheses. Glaser and
Strauss (1967) defined a category as ?a conceptual element of the theory? (p. 36). Within each
category are properties. ?A property, in turn, is a conceptual aspect or element of a category?
(Glaser & Strauss, 1967, p. 36). Although there are different approaches to analyzing qualitative
data, Glaser and Strauss (1967) recommended the constant comparative method. The constant
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comparative method combines coding and analyses of the codes together (Glaser & Strauss,
1967). This allows the researcher to stay close to the data to ascertain the dimensions and
properties of the data (Glaser & Strauss, 1967).
The constant comparative method involves four stages (Glaser & Strauss, 1967).
Although one stage leads into the next, the previous stages are still taken into account (Glaser &
Strauss, 1967). When reading through the information, the researcher designated a word or
phrase that represents the meaning of each part of the data (Saldana, 2009). This process of
designation is called coding and is the first stage of the constant comparative method (Glaser &
Strauss, 1967). ?To codify is to arrange things in a systematic order, to make something part of a
system or classification, to categorize? (Saldana, 2009, p. 8). Codes were derived from a phrase
from the participants in the study or as an explanation of the phenomena by the researcher
(Glaser & Strauss, 1967; Saldana, 2009). As the coding continued, comparisons had to be made
to same and different groups within the data (Glaser & Strauss, 1967). Coding continued until
saturation had been reached (Glaser & Strauss, 1967). Once saturation had been reached the
second stage of the constant comparative method began (Glaser & Strauss, 1967). In this stage
comparisons were no longer made from event to event but from event to properties of events
(Glaser & Strauss, 1967). Comparisons of events and properties often led to a beginning
conception of the theory (Glaser &Strauss, 1967).
Now that a theory was developing the researcher began to narrow the information (Glaser
& Strauss, 1967). This is the third phase of the constant comparative method (Glaser & Strauss,
1967). This narrowing process included eliminating irrelevant properties as well as unifying
concepts to create a reduction of concepts that helps to clarify the theory (Glaser & Strauss,
1967). Glaser and Strauss (1967) term this part of the process as theoretical saturation. At this
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point if new categories emerged, then the researcher continued with the new categories and
returned back to the beginning of the data with the new categories, if necessary (Glaser &
Strauss, 1967). The final stage of the constant comparative method involved actually writing the
theory (Glaser & Strauss, 1967). Using the data, codes, and developed theory, the researcher
wrote a framework outlining the theory. The researcher used specific incidents to give
illustration to the points being made about the theory (Glaser & Strauss, 1967).
Charmaz (2005) determined that grounded theories were being used beyond the original
scope of Glaser and Strauss (1967). Barney G. Glaser had an extensive background in
quantitative methods (Charmaz, 2006). Anselm L. Strauss, on the other hand, had experiences
with qualitative work in the Chicago School (Charmaz, 2006; Denzin & Lincoln, 2005). Unlike
grounded theory portrayed by Glaser and Strauss (1967), constructivist grounded theory does not
take an objective stance (Charmaz, 2005). It assumes the researcher brings experiences and bias
into the data collection and analysis and cannot be separated from that methodological
perspective (Charmaz, 2005). Therefore, interpretations of the data are not objective but rather
interpreted findings (Charmaz, 2005).
Researcher as Part of the Research Process
The researcher brought a framework of beliefs that could not be separated from the
research process (Denzin & Lincoln, 2003). ?All research is interpretive; it is guided by a set of
beliefs and feelings about the world and how it should be understood and studied? (Denzin &
Lincoln, 2003, p. 33). The researcher was a former elementary classroom teacher. As a
classroom teacher, the researcher mentored preservice teachers in methods lab placements as
well as supervised interns. The researcher also served as the school teacher leader and district
teacher leader for mathematics. Working under a multi-year NSF grant, TEAM-Math, the
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researcher had experience training inservice elementary teachers on research-based methods for
teaching mathematics. Similarly, the researcher also had experience teaching previous cohorts of
preservice teachers the mathematics methods course using researched based materials. However,
the researcher used the learning cycle in the mathematics methods course for the first time with
the cohort in this study.
The researcher?s perspective impacted the way observations were made, interpretations
of findings, and how concepts were integrated (Denzin & Lincoln, 2003; Merriam, 1998). Those
experiences provided the researcher with the facility to interpret the phenomenon and allowed
the researcher to adapt to situations being studied (Merriam, 1998). For example, the researcher
became familiar with the investigation materials for teaching mathematics as a classroom teacher
and school teacher leader. When the researcher observed lessons in which the preservice teachers
used those materials, she was aware of instances in which they altered the intent of the lessons or
expanded the lessons. The researcher was unaware of the type of teaching she would see when
going into the classrooms of the student teachers. However, she was aware of the types of
materials and the training the cooperating teachers in the schools had received. The researcher
was able to bring those resources to the attention of the student teachers to help them improve
their practice. Therefore, the researcher?s experiences, culture, and community played a part of
the research process (Denzin & Lincoln, 2003).
Phase I of the study took place in the classroom and in an on-line environment.
Participants completed the open-ended pre-test and the open-ended post-test in the methods class
on campus. For both science and mathematics portions of the methods class, participants
completed science and mathematics modules using the learning cycle approach. Phase II of the
study took place in the natural setting of what was being studied: elementary preservice teachers
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in learning environments practicing and learning how to teach during student teaching (Bogdan
& Biklen, 1982; Hatch, 2002; Merriam, 1998). The settings varied based on where the preservice
teachers were placed. The researcher studied the preservice teachers in the natural setting and
made an attempt to cause as little disruption as possible when in the varying learning
environments (Merriam, 1998). Being in the environment allowed the researcher to acquire
information firsthand, learn about the daily routines, and become familiar with the context of the
learning environment for each preservice teacher involved in the study (Bogdan & Biklen, 1982;
Glaser & Strauss, 1967; Hatch, 2002; Marshall & Rossman, 1995; Merriam, 1998). Furthermore,
a test or written survey would not have provided an entire picture of the preservice teachers?
practice.
The researcher served as the primary investigator who attempted to record the
phenomena, person, and/or interactions being studied (Hatch, 2002; Lancy, 1995). The
researcher recorded observations and took notes in the field or field notes (Hatch, 2002).
?Qualitative researchers build toward theory from observations and intuitive understandings
gained in the field? (Merriam, 198, p. 7). The researcher served as an instrument for data
collection since the researcher?s sense-making influenced what the researcher distinguished as
important in the setting (Hatch, 2002; Merriam, 2009). Moreover, the data that was collected was
limited by not only the constraints and limitations of the setting, but by the personal lens of the
researcher collecting the information (Denzin & Lincoln, 2003; Merriam, 1998). In essence, due
to human factors, ?mistakes are made, opportunities are missed, and personal biases interfere?
(Merriam, 1998, p. 20). Thus, in order for the researcher to gain insight into what is being
studied extended time in the field was a necessity (Hatch, 2002).
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Design and Research Methods
Case Study Approach
For this study a case study approach was used (Hatch, 2002; Merriam, 1998; Stake, 2005)
based on a constructivist grounded theory perspective (Charmaz, 2005), and used a grounded
theory approach to analysis (Glaser & Strauss, 1967). A case study is an in-depth study of a real
life situation with specific boundaries (Hatch, 2002; Merriam, 1998; Yin, 2009). A case study is
used when aspects of a phenomenon need to be studied up close (Merriam, 1998; Yin, 2009).
Since case studies involve examinations of real life, then field work is a necessary component
(Merriam, 1998; Yin, 2009). The human factor of the researcher allowed the researcher to be
flexible in the data collection process if unforeseen events occurred (Merriam, 1998). A couple
of time schedule changes and special assemblies altered observation times, shortened the length
of the teaching episode, and altered when the follow-up interview could take place. The
researcher had to be sensitive in the data collection process with note taking, interview style, and
in data analysis (Merriam, 1998). Sensitivity in data collection meant the researcher was careful
to record as much as possible about the teaching episode in order to keep the written observation
as true to the teaching episode. The students in the class were not recorded by name as to protect
their identity. Recognizing the hard work and effort student teachers put into their work, the
researcher approached the follow-up discussions in a diplomatic manner so as not to discourage
the student teacher. The human factor also meant that bias was inherent in the observations and
analyses due to the fact that a human collected, investigated, and made determinations based on a
human?s knowledge (Merriam, 1998).
Data collection of a case study typically involves many different types of information
(Yin, 2009). Multiple sources means that the similarities and differences within the information
83
are identified (Yin, 2009). This process of examining the converging information from the
multiple sources is known as triangulation (Yin, 2009). Specifically, this study used a multiple
case design in which multiple cases were examined (Merriam, 1998; Yin, 2009). Furthermore,
evidence in case studies may take the form of observations, interviews, documents, artifacts, or
records (Merriam, 1998; Yin, 2009).
In order to try to determine preservice teachers? conceptions of pedagogical approaches,
tools for conceptual development, and processes for meaningful learning, case studies were the
best approach. The pre-test on the first day of class established the preservice teachers?
conceptions of tools, processes, and approaches for science and mathematics teaching. The
weekly blogs presented a progression of thinking from science methods to mathematics methods.
The post-test at the end represented their culminating knowledge about tools, processes, and
approaches for science and mathematics teaching. All of these documents for each person
became the evidence for each case. Case studies enabled the researcher to see the progression of
thought with individuals and compare that with the thoughts of other participants. For Phase II of
the study case studies allowed the researcher to observe the preservice teachers in their teaching
environment (Merriam, 1998; Yin, 2009). Furthermore, the researcher was able to conduct
follow-up interviews to gain insight into the preservice teachers? choices and understanding
about practices for teaching mathematics (Merriam, 1998; Yin, 2009).
Participants
Participants of qualitative research are often a purposeful sample rather than a random
sample (Merriam, 1998). This study used a purposeful selection of preservice teachers. Phase I
of the study involved 22 preservice elementary teachers in a 10 week semester course which
required dual enrollment in a science methods class and a mathematics methods class. Preservice
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teachers in the elementary education program at the large southeastern university were organized
into cohorts prior to taking methods courses. Methods courses occurred before internship.
Methods courses focused on practice, theory, and reflection (Hill, 1997). The students in the
science methods class were the same students in the mathematics methods class. The course was
divided into two five-week segments. The first five weeks were devoted to science methods. The
university assigned professor taught the science methods portion of the class. During those five
weeks participants attended class on campus as well as attended lab sites. In the lab sites
participants taught science lessons to students in grades K-5. The second five weeks of the
semester focused on mathematics. The researcher of this study was the instructor of the
mathematics methods class. Preservice teachers worked in a different lab site with K-5 students
during the second half of the semester teaching mathematics.
Preservice teachers attended class on campus as well as worked with students in lab sites.
Although the science lab sites were different from the mathematics lab sites, practice teaching
children provided an opportunity for new teachers to learn how to teach as well as validate new
practices being put into place (Hill, 1997). Due to the nature of the classes as a methods class,
preservice teachers learned pedagogy, observed modeling of science and mathematics, peer-
taught lessons, and then practiced their teaching skills with students in the field. Howitt (2007)
found that preservice teachers identified that science activities that could be used with students in
the field helped build the confidence of the preservice teachers in the teaching of science.
The science methods course and the mathematics methods course were jointly planned by
the researcher, who was the mathematics methods instructor, and the science methods professor
to have commonalities between the designs of the sections. The 5-E learning cycle with inquiry
embedded instruction (Marek & Cavallo, 1997; Withee & Lindell, 2006), national standards for
85
science (NRC, 1996) and national standards for mathematics (NCTM, 2000) were foci of the
classes. The methods classes focused on connecting content and pedagogy (Molina et al., 1997).
Content in the on-campus portion of class focused on inquiry-based materials. For science
methods class, those materials included science kits that were developed from an inquiry
perspective with the learning cycle as the framework. Science lessons in the on-campus portion
of the science class included lessons from Full Option Science System (Foss) (Lawrence Hall of
Science, 2003), Science Technology for Children (STC) (National Science Resources Center,
2003), and Insights (EDC Science Education, 2003). For the mathematics portion of class,
materials consisted of materials from NCTM and the Investigations curriculum (Pearson
Education, 2007). In both classes preservice teachers participated in investigations and
experiments, discussions, and readings (Chick, 2007; Heywood, 2007; Hill, 1997). The learning
cycle was used as a common approach in the science methods and carried into mathematics
methods. Preservice teachers were required to design their lessons for both science and
mathematics field experiences using the learning cycle as a framework.
Since the fragmented knowledge of preservice teachers is well documented (Ball, 1988;
Heywood, 2007; Manouchehri, 1997) along with an instrumental understanding of the subject
(Heywood, 2007; Hill, 1997), experiments, discussions, and readings were selected to challenge
preservice teachers? conceptions of science and mathematics to promote conceptual change
(Green et al., 2008; Heywood, 2007). Howitt (2007) found that modeling by the teacher educator
to be influential in pedagogical content knowledge of the preservice teachers. In both methods
classes, modeling was used to help preservice teachers gain insight into the material for content
knowledge as well as gain insight into the how of teaching the content. Green et al. (2008) found
that focusing on manipulative based tasks that allowed students to explore and address
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misconceptions of mathematics were very beneficial to the preservice teachers. Preservice
teachers, in a study conducted by Hatton (2008), felt that the class investigations aided them in
their growth and development. Manipulatives and tools for conceptual development were
similarly used in both classes during investigations to foster development of content knowledge
and pedagogical content knowledge. In science methods class, one of the modules completed
focused on earth science. Preservice teachers examined rocks and completed a series of inquiries
with the rocks following the learning cycle approach. In the mathematics methods class,
preservice teachers used three-dimensional solids and rice in one module that focused on volume
of three-dimensional figures. Preservice teachers used the rice to determine the volume of figures
and make comparisons between figures. From their findings they determined the relationship
among the volume for three-dimensional figures. Specifically, in both methods classes preservice
teachers engaged in group discussion, produced representations of thinking, conducted
investigations to test phenomena, were held accountable within a peer group, and reflected on
mathematical and scientific justifications (Gresham, 2007; Heywood, 2007; NCTM, 2000).
During the volume module preservice teachers worked in groups, recorded their information in a
journal, and presented the information to the class. Similarly, with the rock module, participants
completed a series of tasks intended to expand their understanding of rocks, as well as provide a
level of comfort with such tasks. Science notebooks were used by the preservice teachers to
record data and their thinking as they completed the modules. Preservice teachers shared their
understandings of the rock inquires within their groups and with the class.
?Mathematics anxiety is a feeling of helplessness, tension, or panic when asked to
perform mathematics operations or problems? (Gresham, 2007, p. 182). Gresham (2007) found
that the use of manipulative with elementary preservice teachers in a methods course aided the
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preservice teachers in reducing mathematics anxiety. With the documentation of anxiety of
mathematics and avoidance of science (Appleton & Kindt, 1999; Howitt, 2007; Sanger, 2006),
group investigations and discussions were also conducted to provide an opportunity for students
to experience mathematics and science at their own pace and possibly with less anxiety since
they did not have to do it individually (Gresham, 2007; Howitt, 2007).
Since the researcher served as one of the methods instructors for Phase I of the study, the
participants were invited to participate at the end of the semester. Twenty-two of the twenty-five
students agreed to participate in the study. A neutral third-party administered the consent forms
and released them to the researcher at the beginning of the following semester after all grades
had been posted. Since the researcher did not serve in any supervisory role for the internship, the
researcher presented the invitation to participate in Phase II of the study which would take place
during their student teaching. Only participants who had consented for Phase I of the study were
invited to participate in Phase II of the study. Five preservice teachers who had participated in
Phase I agreed to participate in Phase II. The researcher sent e-mail requests to obtain permission
to observe interns to superintendents of the school systems. Once the participants agreed to
participate, the researcher also sent e-mails to the principals of the individual schools in which
the participants were completing their student teaching to obtain permission to observe.
Data sources
The study was conducted in two phases. Phase I occurred while the preservice teachers
were in the mathematics and science methods classes. Phase II occurred while the preservice
teachers were in their internship. In qualitative research data gathering instruments vary (Lancy,
1993; Marshall & Rossman, 1995; Yin, 2009). For this study data gathering instruments
included observations, interviews, field notes, and written artifacts such as pre-tests, post-tests,
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blogs, and final reflections (Lancy, 1993; Marshall & Rossman, 1995; Yin, 2009). Merriam
(1998) recommended that researchers chose data collection instruments that are ?sensitive to the
underlying meaning? (p. 1) of the research.
Phase II of the study also used direct observation (Yin, 2009). Direct observation means
the researcher observed the participants in the field; the field being the place of occurrence for
the phenomena (Yin, 2009). Qualitative research relies on observation of behavior since it
indicates underlying values and beliefs (Marshall & Rossman, 1995). Observations included
direct notations of what was observed and heard as well as notes taken after the fact by the
researcher (Lancy, 1993; Marshall & Rossman, 1995). Field interviews followed the field
observations (Yin, 2009).
The study consisted of multiple data sources during both phases (Merriam, 1998; Patton,
2002; Yin, 2009). Phase I data sources consisted of an open-ended pre-test (see Appendix A in
overall appendix section), an open-ended post-test (see Appendix B in overall appendix section),
and weekly blogs (see Appendix C in overall appendix section). Participants completed an open-
ended pre-test at the beginning of the science methods course. On the last day of the semester in
the mathematics methods course they completed the open-ended post-test. The pre-test questions
consisted of knowledge questions intended to elicit information from the participants about their
knowledge of common approaches to teaching mathematics and science (Patton, 2002). The
post-test questions consisted of knowledge questions about similarities in mathematics and
science, questions of common practices to mathematics and science, as well as opinion questions
on how the participants thought about their teaching experiences (Patton, 2002).
After examining preservice teachers? learning of the learning cycle, Lindgren and
Bleicher (2005) found discussion and journal writing were key components of students gaining
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understanding of the learning cycle. In order to document reflection and change of thinking,
computer-mediated communication (CMC) in the form of weekly weblogs (blogs) provided a
venue for participants to describe experiences, past and present, and reflection on science and
mathematics teaching (Heywood, 2007; Howitt, 2007; Lindgren & Bleicher, 2005). Online
postings were required each week (Hatton, 2008). Blogs provided an opportunity for preservice
teachers to reflect upon their experiences in methods class and experiences in the field practice
teaching as a means to help them improve their learning and teaching (Howitt, 2007). Blogs
were visible to students and the instructors of the courses who were asked to reply to other
students? blogs to provide a venue for discussion. Some students are reticent to speak up in class
but may feel more freedom to express themselves through computer mediated communication
(Wiegel & Bell, 1996). The blog offered the students the opportunity to share and discuss their
thoughts and ideas about the learning and teaching of mathematics and science. Blog questions
asked in the science methods portion in the first five weeks were revisited in the mathematics
methods portion the second five weeks.
Phase II of the study consisted of five of the Phase I participants and occurred during
their student teaching experience which occurred the final semester of their senior year. At the
end of the study only two of the five who agreed to participate were included. Two of the
participants who agreed to participate in the study were not viable cases because they were
placed in classrooms in which reform-based methods of instruction were not allowed. The other
participant who agreed to participate in the study was placed in a school that was geographically
too far from the researcher to complete multiple observations. The two participants, that were
included in the study, were placed in schools with teachers who had participated in state science
and math initiatives. The participants had access to materials and tools to develop and teach
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reform-based mathematics lessons. Finally, the cooperating teachers gave them freedom in
designing, planning, and implementing lessons for the two-week full-time teaching period. Both
participants in the study were also included because they presented two different views of
teaching and learning mathematics based on their responses during the methods classes. Data
from Phase II consisted of teaching observations, written as field notes, follow-up interviews,
lesson plans, and an open-ended final summation (Merriam, 1998). Lesson plans were a
weakness of the data collection. The lesson plans, when given to the researcher, were very sparse
and contained the bare minimum such as objective and page numbers.
However, observations and follow-up conversations about the observed teaching lesson
yielded more information about the thoughts of the preservice teachers. Observations consisted
of detailed descriptions of preservice teachers teaching mathematics (Merriam, 1998; Patton,
2002). Observations have an inherent constraint in that the presence of the observer may change
the actions of the one being observed (Patton, 2002). Observations, written as field notes, were
taken as participants were teaching their lessons (Merriam, 1998; Patton, 2002; Yin, 2009). Field
notes contained the actual descriptions of events and what occurred in the teaching episode
(Lancy, 1993; Merriam, 1998; Patton, 2002).
Since observations alone cannot provide enough information about the thinking of the
participants, interviews were also conducted (Merriam, 1998; Patton, 2002; Posner & Gertzog,
1982; Yin, 2009). Interviews were another data gathering instrument used in this research
(Lancy, 1993; Marshall & Rossman, 1995; Merriam, 1998; Yin, 2009). The clinical interview,
which takes its roots from the work of French psychologist Jean Piaget, is a face-to-face
interview with an interviewer and an interviewee (Posner & Gertzog, 1982). Interviews allowed
the participants to express their personal thoughts and feelings on the phenomena being studied
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(Lancy, 1993; Patton, 2002). ?One?s goal in this type of interviewing is to obtain information,
but also to remove any constraints on the interviewee?s responses so that her conceptualization of
the phenomena emerges rather than having her fit her views into the investigator?s framework?
(Lancy, 1993, p. 17). Of the varying types of interviews, Phase II of the study used the general
open-ended question interview (Patton, 2002; Posner & Gertzog, 1982). Interviews were framed
around open-ended questions (See Appendix D in overall appendix) meant to illicit participant?s
thinking about teaching practices for mathematics (Marshall & Rossman, 1995; Merriam, 1998).
Interview questions were framed similarly to questions in Phase I (See Appendix A in overall
appendix) to provide a continuum of thinking from methods course to student teaching
experience. Interviews were flexible to allow for articulation of thoughts by the interviewee
(Patton, 2002; Posner & Gertzog, 1982; Yin, 2009). Additional questions were asked in order to
gain clarification from the participant (Merriam, 1998; Patton, 2002; Posner & Gertzog, 1982).
Finally, a follow-up open-ended assessment (See Appendix E in overall appendix) was
used as another data source. This final reflection consisted of questions similar to questions
posed in Phase I (See Appendix B in overall appendix) of the study. Questions focused on tools
for conceptual development, approaches to teaching mathematics, and processes for meaningful
learning.
Data Management
The information was managed in several ways. Once the investigator had the consent
letters from Phase I she created a code list. Each consenting participant was designated with a
name and number. The first participant on the list was designated as Person 1. The second person
was designated as Person 2 and so on. She then logged into the secure blog site, converted all of
the files for the consenting participants into text files, deleted any names, and replaced the names
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with the code name. Participants were designated as Person 1, Person 2, and Person 3. etc. The
blog for week one was then titled Person 1 Blog 1. The pre-test and post-test had been hand
written by the participants. The researcher typed all of the pre-tests and post-tests for the
consenting participants, deleting any names, and replaced with the name and number. All of
these were placed in a file labeled for each participant. This process continued for all 22
participants.
Since the researcher was not a supervisor during the participants? internship experience,
she passed out the consent forms to the preservice teachers for Phase II of the study. All
information for Phase II was treated the same way as Phase 1. Each document was labeled and
coded as the participant?s same code name from Phase 1. Files were created with all of the
documents for each of the participants for Phase II. The code name list was destroyed as well as
all original hand written documents. All documents remaining were only designated by the code
name.
Data Analysis: Phase I
For Phase I there were twenty-two cases to code. For Phase II there were two cases to
code. First level coding occurred with each blog, pre-test, post-test, interview, and observation
field notes (Merriam, 1998). Merriam (1998) called this initial coding as category construction.
Keeping in mind the research questions the researcher attempted to create categories that would
?reflect the purpose of the research? (Merriam, 1998, p. 183). Pieces of information that were
striking to the researcher were noted in Person 1 pre-test (Merriam, 1998). Categories were
determined based on words or phrases from the participants, conclusions from the researcher, or
connections to existing research (Glaser & Strauss, 1967; Merriam, 1998). For example, when
Person 1 on the pre-test described students in the field placement taking apart machines to
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understand how the machines were designed (Phase I, 2008) the researcher coded based on the
phrasing of the respondent as ?took apart machines for understanding?. Then the first pre-test
was read through again for the case to determine if commonalities existed that could group some
of the items together or if other generalizations became apparent (Merriam, 1998). After
rereading the Person 1 pre-test the researcher made a second code next to the code ?Teacher will
need to know what will happen? as ?teacher control?. Then the first blog for Case 1 was read
through for category construction. New categories were designated based on new information:
?need for fun?, ?benefits?, and ?traditional approach criticism? (Phase I, Person 1-Blog 1, 2008).
Then the pre-test categories and the blog categories were compared to generate any new
categories. After rereading the codes for Person 1 pre-test and Person1-blog 1, there weren?t any
codes the researcher wanted to add or change. The researcher then went on to code Person 1-
blog2. The respondent indicated that the learning cycle were five steps to follow and the
researcher coded this as ?LC as steps?. The researcher then reread for Person 1 the pre-test, blog
1, and blog 2. The researcher saw a thread of the preservice mentioning hands-on. The researcher
made a list of the codes from the first three data sets (Merriam, 1998). From the list of initial
codes the researcher referred to the research questions (Merriam, 1998). This process continued
for all 22 cases.
The research questions dealt with five main areas: tools for conceptual development
(designated with a T in the coding process), processes for meaningful learning (designated with
PFML), pedagogical approaches (designated with PA), and more specifically common
approaches between mathematics and science (designated with CA). Since the learning cycle
was used as a common approach in the science and mathematics methods course, then aspects of
the learning cycle were designated with an LC. Preservice teachers also added many personal
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experiences about learning mathematics and science. A category was designated for personal
experience (PE) in order to differentiate it from their current practice (see Appendix F for coding
guide).
The researcher added the prefixes to the initial codes. For example, on Person 1-pretest,
the researcher noted that the respondent indicated group activities and hands-on activities as
being a common to teaching mathematics and science. The researcher coded as ?CA-group
activities?. For the next data set the researcher used both the category designations and codes.
For codes from the post-test the researcher noted post-test at the end of the code. At the end of
the coding for each person the researcher again read through all of the codes and made a memo
for thoughts, ideas, questions, or potential themes (Charmaz, 2006; Merriam, 1998). This
process constant comparison continued for each set of data for each case (Charmaz, 2006; Glaser
& Strauss, 1967; Merriam, 1998).
A chart (See Appendix G in overall appendix for sample) was made for each participant
that included the T, P, PA, CA, LC, and PE coding of labeling. Codes were placed in the chart
for each person in chronological order with a source code at the end. ?Pre? and ?Post? were
source codes for the pre-test and post-test. The letter B and a number were used for blog source
codes. For example on Person 1 under T, ?Took apart machine in FEP b3, Promoted fun b3?.
Based on the research questions the researcher examined the code list for each person
based on each category. Based on the codes for tools the researcher refined the code list (See
Appendix H in overall appendix for sample). For example, the researcher recorded for Person
1?Moved from seeing tools in science as teacher controlled experiments to students
understanding the phenomena for themselves? (Person 1, Coding level 2 Tools). This continued
for each participant for the tools, processes, pedagogical approaches and commonalities of
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mathematics and science. Once all of the coding had been completed, then patterns were
generated within same cases and across different cases for both phases (Glaser & Strauss, 1967;
Merriam, 1998).
After all of the level two coding was completed the researcher examined data across the
cases to make further generalizations. These generalizations were organized into the themes for
each category (See Appendix I in overall appendix for sample). Themes were organized based on
initial conceptions and end conceptions. For example, three themes stood out for participants?
initial conceptions of the use of tools in science: saw tools in a traditional role, expanded view of
tools, or more progressive view of tools. The researcher examined the development of
participants? conceptions of tools to determine how their thinking about the use of tools in
science had changed over the course of the semester. This organization of themes continued for
processes, pedagogical approaches, and common approaches of mathematics and science.
Traditional ideas and progressive ideas were noted for each area. Traditional concepts were those
that mirrored traditional mathematics and science instruction in which instruction and lesson
design focus on fact learning rather than conceptual learning. This includes the use of tools for
teacher directed lessons that focus in mathematics on getting right answers and in science on
teacher led experiments or following a science textbook.
Based on their responses on the pre-test, participants? conceptions of tools were
categorized into three categories: traditional view of tools, expanded view of tools, or
progressive view of tools. Traditional views of tools followed traditional methods of science
instruction and were indicated by teachers conducting experiments for the students to observe,
textbook learning, or science projects. Responses that had a mixture of students conducting
experiments, worksheets, textbook learning, and students looking at pictures were classified as
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expanded view of tools for science. The progressive category included comments about students
exploring in nature, physical knowledge activities, or students completing hands-on tasks for
conceptual development.
Categories for conceptions of tools in mathematics were categorized the same as for
science: traditional role of tools, expanded view of tools, or progressive view of tool. Initial
responses in which tools were for computation or facts were placed in the traditional view.
Responses in which tools were used for primary concepts such as shapes, money, or counting
were included in the expanded view. Progressive views of tools in mathematics included
comments about tools for hands-on learning or for understanding the concept.
In examining how preservice teachers conceived of processes for learning, their
responses fell into three distinct categories. Processes for learning were categorized as
traditional, expanded, or progressive. Processes in which students focused on students finding
answers were categorized as traditional. Preservice teachers in this category saw experiments as
a means for students to find predetermined outcomes or mathematics lessons that focused on
students getting the right answer. Participants with expanded responses would vaguely talk about
students involved in schemes of thinking but were not able to elaborate. Notions of students
making connections, applications, and reasoning were categorized as progressive.
Themes for conceptions of pedagogical approaches were also categorized as traditional,
expanded, or progressive. Traditional approaches were ones in which the lessons were teacher
directed and instruction focused on fact learning. Expanded approaches represented approaches
that were not completely teacher directed but did not focus on conceptual development.
Progressive views were indicated by data that focused on inquiry-based approaches to develop
understanding.
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Knowledge and development of conceptions of common approaches in teaching
mathematics and science varied. Responses were categorized based on recognition of similarities
in approaches for teaching mathematics and science. Quotations were pulled from participants to
reflect the big ideas for each category. Results are explained according to conceptions of tools,
processes for learning, pedagogical approaches, and commonalities in science and mathematics
teaching.
Data Analysis: Phase II
Data sources for Phase II included five mathematics teaching observations for Jane and
four mathematics teaching observations of Kate, follow-up interviews after each observation,
lesson plans, and an open-ended final reflection. The lesson plans served as a means of seeing at
a glance if the observed lesson was what the preservice teacher had planned. The follow-up
discussions were important to gain information on the student teacher?s goals for the lesson and
decisions that went into the structure and implementation of the lesson. The final reflection was
similar to the final reflection from the methods course and provided another opportunity for the
preservice teacher to reflect on their teaching experiences.
The constant comparative method (Glaser & Strauss, 1967) was used to find conceptions
of mathematics teaching that emerged from the varying data sources. For each student teacher,
each interview, observation, lesson plan, final reflection was coded based on the research
questions (Merriam, 1998). Pieces of information that were striking to the researcher with regard
to preservice teachers? thinking about how they linked ideas in practice across disciplines were
noted for each piece of data (Merriam, 1998). Codes were determined based on words or phrases
from the participants, conclusions from the researcher, or connections to existing research
(Glaser & Strauss, 1967; Merriam, 1998). Data about tools, use of tools, plans involving tools
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and so forth were labeled with a T. Data about processes that took place during the lesson were
labeled with a P. Information about pedagogical approaches was labeled with a PA. Data that
referred to the learning cycle was coded as LC. Lesson plans were labeled as LP, observations as
O, follow-up interviews as FI, and final reflection as FR. For example, Jane?s lesson plan for the
first observation stated ?Review of measuring to the nearest inch? (J, LP, 4/14). In the observed
lesson Jane did review how to measure to the nearest inch using items on the overhead projector.
These statements were both coded as ?tools for measurement?. Since Jane led the students step
by step through a series of measurements this was also coded as ?teacher directed?. During the
follow-up discussion when asked ?In what ways did you use tools for conceptual development?
Jane said ?Showing them the different items on the projector? (J, FI, 4/14). This was coded as
?tools for measurement-teacher directed?. Then the first teaching observation and follow-up
discussion was read through again for the case to determine if commonalities existed that could
group some of the items together or if other generalizations became apparent (Merriam, 1998).
Comparing lesson plans to teaching observations to follow-up discussions provided verification
of the data. For example, the researcher could see that in the lesson plan the students would be
measuring, the students somewhat measured in the teaching episode, and in the follow-up
discussion the student teacher confirmed that the lesson did not go as she had planned. This
process of constant comparison continued for each set of data for each case (Charmaz, 2006;
Glaser & Strauss, 1967; Merriam, 1998). Once all of the coding had been completed a table was
constructed for each participant (See Appendix J for sample in overall appendix section). All of
the codes for tools were placed in one section in chronological order. All of the codes for
processes were placed in another section in chronological order. Codes for pedagogical
approaches and the learning cycle were also placed in sections by chronological order. Codes
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were then regrouped to fit like items together (Saldana, 2009). For example, if the researcher
noted that students made towers and then this came up during the follow-up discussion then
those items were placed together. Examining the codes together allowed the researcher to then
narrow the codes to succinctly represent the data. These narrower codes were placed in another
column next to the lengthy code list (Saldana, 2009).
Once the data collected from the student teaching experience had been coded, the
information was compared to the data from the methods course study. Data from the methods
course had been examined in the same way as the student teaching. Therefore, the code list for
the participants was pulled from the methods course data. The coding for the methods course was
also in a similar table for tools, processes, pedagogical approaches, and the learning cycle. Codes
for tools during methods class and tools for student teaching were read through to determine
similarities, differences, or changes. This continued for processes, pedagogical approach, and
learning cycle codes. The researcher tried to determine what changes, if any, occurred from
mathematics-science methods class to student teaching in understanding and development of
approaches in mathematics. The codes were examined in chronological order to determine shifts
or changes in practice. Systematic searches were made to find corroborating or contradictory
evidence. Generalizations then were made within each case across the different periods of data
collection. Finally the two cases were compared.
In student teaching Jane and Kate had different conceptions of tools, processes, and
approaches for teaching mathematics. Quotations were pulled from the data to reflect the big
ideas for each category. Results are explained according to conceptions of tools, processes for
learning, and pedagogical approaches.
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Ethical Considerations
Following the guidelines of the Internal Review Board for the institution participants
were informed of any risks or discomforts in the informed consent letter (American Educational
Research Association (AERA), 2004). Participants were at risk for loss of confidentiality and
coercion. To address the coercion factor at the end of Phase I a third neutral party distributed the
consent forms. The consent forms were placed in a sealed envelope and delivered to the
researcher at the beginning of the next semester. Coercion was not a risk for Phase II since the
researcher was not an intern supervisor and presented the consent forms herself.
In order to ensure confidentiality a code list was developed. All documents for Phases I
and II were given a code name to replace the name of the participant. The code name list was
destroyed as well as all original hand written documents. All documents remaining were only
designated by the code name. Furthermore, the names of schools systems, schools, cooperating
teachers, or students were not recorded in any way as to maintain the confidentiality of the sites
in which the research took place. In the narrative of the cases, pseudonyms were given for each
of the participants.
Participants also were informed that the research may not have any direct benefit for
individual but that the information may help others working with preservice teachers gain a
better understanding of how preservice teachers think about common approaches to mathematics
and science (AERA, 2004). Finally, the participants were able to withdraw from the study at any
time (AERA, 2004). They were informed that if they decided to withdraw that it would not hurt
any relations with the institution or the department.
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Data Triangulation
Triangulation of information was used to ensure reliability of the study (Scott, 2007; Yin,
2009). This process of examining the converging information from the multiple resources is
known as triangulation (Yin, 2009). Specifically this study used a multiple case design in which
multiple cases were examined (Merriam, 1998; Yin, 2009). In data triangulation data sets were
collected from the same participants at different points in the study: a pre-test was given on the
first day of the semester, participants completed weekly blogs, a post-test was given on the last
day of the semester, and notes and observations were made throughout the semester during Phase
I (Denzin, 1970 in Scott, 2007). Multiple data sets mean that redundancy or repeatability of
information may occur (Stake, 2005). Reoccurrences in the data help support the findings (Stake,
2005).
Triangulation of data meant comparing the different data sources, blogs, pre-tests, post-
test, and field notes, to substantiate the findings (Oliver-Hoyo & Allen, 2006). Denzin and
Lincoln (2003, 2005) proposed that triangulation was an ?alternative to validation?. Similarly,
Adami and Kiger (2005) determined that triangulation should be for the purpose of
completeness. Richardson (In Denzin & Lincoln, 2003) likened triangulation to seeing the
different facets of a crystal. ?Triangulation is the display of multiple, refracted realities
simultaneously? (Denzin & Lincoln, 2003, p. 8). In examining the different facets, researchers
look for convergence of information (Oliver-Hoyo & Allen, 2006). Sometimes near
approximations are determined if complete convergence cannot be made (Oliver-Hoyo & Allen,
2006). ?The higher the convergence, the greater the confidence that the measure was capturing
the phenomenon being studied? (Adami & Kiger, 2005, p.20).
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Data for each participant were compared to determine converging ideas (Adami & Kiger,
2005; Denzin & Lincoln, 2003; Oliver-Hoyo & Allen, 2006). Information from the pre-test and
the post-test were compared to see a change in thinking from beginning to end. This information
was compared in relation to blog entries. This analysis was conducted for each person in the
study. Then the information was examined across participants to establish similarities and
differences among the group. This same process occurred for participants in Phase II as well.
After the data from participants for Phase II were coded their responses were compared to Phase
I to determine any changes from methods course through student teaching. For participants in
Phase II this meant a triangulation of information from both sets of data.
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CHAPTER 4: ELEMENTARY PRESERVICE TEACHERS? CONCEPTIONS OF
COMMON APPROACHES TO MATHEMATICS AND SCIENCE DURING METHODS
COURSE
Abstract
This study investigated preservice teachers? thinking about common approaches to
mathematics and science education for elementary children in grades K-6. Specifically, this
study focused on preservice teachers' thinking on the use of tools for conceptual development,
processes for meaningful learning, and common pedagogical approaches for mathematics and
science. The study took place during jointly enrolled science and mathematics methods courses.
The learning cycle was a common approach used in the methods courses and used by elementary
preservice teachers in the field. The nature of the preservice teachers? understandings was
examined through several data sources: open-ended pre and post-course tests, and weekly blogs.
Results indicated varied conceptions of tools, processes, and approaches in science and
mathematics teaching at the beginning of the methods courses. Many of the participants initially
thought of science and mathematics as being approached in different ways, such as science
involves experiments and mathematics focuses on solving problems. All of the participants
expressed broadened ideas about teaching mathematics and science at the end of both methods
courses. At the end of the semester 82 percent of preservice teachers recognized commonalities
in teaching approaches for mathematics and science, including use of inquiry as well as the use
of the learning cycle.
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Introduction
In the United States, as well as other countries, concerns have been raised about the
quality of science and mathematics teaching (National Commission on Excellence in Education,
1983; National Science Board (NSB), 2004). The National Council for Teachers of Mathematics
(NCTM) (2000), the National Research Council (NRC) (1996), and the American Association
for the Advancement of Science (AAAS) (1993) have constructed frameworks for reform to
promote quality science and mathematics programs. Examining the mathematics and science
standards documents reveals similarities in goals for science and mathematics, to move away
from memorization and rote learning and instead to focus on conceptual understanding (Kind,
1999; NCTM, 2000; NRC, 1996; Steen, 1990). However despite reform efforts, the problem
persists due to the fact that preservice teachers are often products of a traditional science and
mathematics classroom that focused on repetition and memorization with little attention to
understanding (Taylor, 2009). Since many of today?s preservice teachers lack a model and
understanding of standards-based reform teaching (Conference Board of the Mathematical
Sciences (CBMS), 2001), approaching teacher development from a perspective of commonalties
in mathematics and science teaching could help elementary preservice teachers better understand
how to teach mathematics and science.
Since technology in the workplace has eliminated positions that require minimal skills,
development of approaches in preservice teacher education programs that promote the
preparation of elementary teachers with 21st century skills, creativity, critical thinking,
communication, and collaboration, is important (Partnership for 21st Century Skills (P21), 2009).
These same skills are foundational principles in the science and mathematics standards (NCTM,
2000; NRC, 1996). The standards documents promote active learning, learning for understanding
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of concepts, use of a variety of assessment techniques, and a move from lower order thinking
skills to higher order thinking skills (NCTM, 2000; NRC, 1996; Tal, Dori, Keiny, & Zoller,
2001). Furthermore, standards in mathematics and science call for similar approaches in teaching
including; use of tools for concept development; use of processes that include problem solving,
reasoning and proof, communication, connections, and representations; and inquiry-based
pedagogical approaches (Goodnough & Nolan, 2008; Justi & van Driel, 2005; NCTM, 2000).
In response to the standards documents efforts have been made in preservice teacher
methods courses to prepare elementary teachers to appropriately teach science and mathematics
in reform-minded ways (Kistler, 1997; McGinnis, Kramer, & Watanabe, 1998; Plonczak, 2010).
Teacher development programs have designed mathematics classes for elementary preservice
teachers that focus on problem solving and reasoning (Kistler, 1997), created courses that
fostered connections between science and mathematics teaching (McGinnis et al., 1998), and
developed courses focused on inquiry-based teaching in both mathematics and science
(Plonczak, 2010). Teacher educators have also worked on the development of methods courses
that integrated mathematics and science and used common instructional strategies (Beeth &
McNeal, 1999; Lonning & DeFranco, 1994, 1997; Lonning, DeFranco, & Weinland, 1998;
Stuessy, 1993; Steussy & Naizer, 1996). However, their concerns were in the development and
integration, or combining, of mathematics and science. They recognized the common approaches
within the methods classes but the conceptions of the commonalities of the preservice teachers
were not the focus of their research.
This research attempted to address how preservice teachers think about the processes that
occur similarly within science and mathematics teaching. This research focused on preservice
teachers? conceptions of teaching science and mathematics, while in methods courses that
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utilized common approaches, and how their conceptions changed during the science and
mathematics methods classes. In order to examine the response preservice teachers had to those
similar processes, the participants? science and mathematics methods courses were approached
from a similar pedagogical perspective utilizing learning cycle approach to teaching.
For this study the learning cycle offered a way of incorporating similar teaching methods
of using tools for conceptual development (Chick, 2007; Fuller, 1996; Hill, 1997), discourse
(Akerson, 2005; Heywood, 2007; Williams & Baxter, 1996), assessing student knowledge
(Manouchehri, 1997; NCTM, 2000; NRC, 1996) inquiry-based teaching (Morrison, 2008;
Manouchehri, 1997; NRC, 1996; Weld & Funk, 2005), and reflection (Bleicher, 2006; Hill,
1997; Manouchehri, 1997). Although the learning cycle has its roots in the science field (Bybee,
1997; Atkin & Karplus, 1962), the tenets align closely with the national mathematics standards.
Preservice teachers were then expected to use the learning cycle approach in their courses and its
laboratory teaching experiences.
The purpose of this study was to investigate how preservice elementary teachers relate,
understand, and use the common or shared teaching practices for effective mathematics and
science learning. The study examined preservice teachers? conceptual understanding of the
learning cycle, an inquiry-embedded teaching approach for mathematics and science, while the
preservice teachers were in their science and mathematics methods courses. Conceptions of
pedagogy, tools for conceptual development, and processes for meaningful learning were
examined for mathematics and science teaching from a qualitative perspective. The research
questions of this study were: (1) What are preservice teachers? conceptions of tools for
conceptual development, processes for meaningful learning, and pedagogical approaches prior to
and after taking the mathematics-science methods courses? (2) What changes, if any, in
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development of knowledge and understanding of the common approaches between mathematics
and science occurred while taking the mathematics-science methods course?
Background
Standards Documents
Like the launching of Sputnik, the release of A Nation at Risk by the National
Commission on Excellence in Education (1983) sparked renewed concerns with the education
system of the United States (Wong, Guthrie, & Harris, 2003). In particular, the resulting efforts
focused on raising expectations with the development of standards throughout the educational
system (Wong et al., 2003). Those standards came about from the efforts of the National
Research Council (NRC) (1989, 1996), National Council of Teachers of Mathematics (NCTM)
(1989, 1991, 1995, 2000, 2006, 2009), and the American Association for the Advancement of
Science (AAAS) (1993). The National Research Council published Adding it up (2001): a report
that described proficiency in mathematics as conceptual understanding, procedural fluency,
strategic competence, adaptive reasoning, and productive disposition. Most recently the
Common Core State Standards Initiative (2010), headed by the National Governors Association
and Council for Chief State School Officers, developed standards for language arts and
mathematics that focused on reasoning, critical thinking, and problem solving. The National
Science Education Standards (NRC, 1996) focus on processes for students to develop an
understanding and reasoning about science in an active learning process.
Standards have been developed for teachers and teacher preparation programs as well.
Interstate New Teacher Assessment and Support Consortium (INTASC) developed general
standards for new teacher candidates (n.d.). The Conference Board of the Mathematical Sciences
(2001), in The Mathematical Education of Teachers, focused their standards and
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recommendations on what teachers and teacher preparation programs needed to have successful
mathematics programs for educators. The National Science Teachers Association (NSTA)
created a similar document, Standards for Science Teacher Preparation (NSTA, 2003). The
resulting documents call for teacher candidates to be able to design, implement, and integrate
teaching that makes content meaningful through inquiry and problem solving processes.
Commonalities in Standards Documents. When examining the elements of science and
mathematics it is clear that similarities and differences exist between the two subjects.
Mathematics is considered to be the language of science and science is dependent upon
mathematics (Shapiro, 1983; Steen, 1990). Although there are differences in the elements that
make up science and mathematics fields, there remain common ways of approaching the subjects
(Steen, 1990). The standards documents promote mathematics and scientific literacy; being able
to ask, find, and determine answers (Kind, 1999; NCTM, 2000; NRC, 1996; Steen, 1990).
Science and mathematics standards call for students to develop deep conceptual understanding
(NCTM, 2000; NRC, 1996; Steen, 1990). In order for students to become scientifically or
mathematically literate and develop conceptual understanding, teachers have to create active
learning environments (Kind, 1999, Steen, 1990). The active learning environment,
recommended by the standards documents, is often likened to a constructivist philosophy of
teaching in which students engage, explore, examine phenomena, and explain their
understanding (Kind, 1999, Steen, 1990).
Just as mathematics and science content are intertwined so are the common approaches to
teaching science and mathematics (NCTM, 2000; NRC, 1996). At the heart of the science and
mathematics standards are common processes for meaningful learning, use of tools for
conceptual development, and pedagogical approaches (NCTM, 2000; NRC, 1996). These
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processes include problem solving, reasoning and proof, communication, connections, and
representations (Goodnough & Nolan, 2008; Justi & van Driel, 2005; NCTM, 2000). In
mathematics, students are asked to solve problems, reason about their answers and demonstrate
proof, represent their work, and communicate their findings (NCTM, 2000). Similarly in science,
students are expected to be able to ask questions and then find the answers to those questions,
validate findings, explain their findings, and be able to evaluate the conclusions of others (NRC,
1996). Furthermore, in mathematics and science students are expected to make connections or
identify relationships between and among concepts and areas of study (NCTM, 2000; NRC,
1996).
Learning Theory and the Learning Cycle
A complex framework of generalizations, ideas, and relationships composes the nature of
mathematics (Molina, Hull, Schielack, & Education, 1997) and the nature of science (NOS)
(Akerson & Donnelly, 2008). In order to develop the skills to make generalizations, understand
ideas, and examine relationships conceptual learning must be developed (Molina et al., 1997).
Underlying assumptions about the nature of learning mathematics and science for conceptual
understanding as articulated in the standards documents are founded on the idea that learning is
an active process (Kind, 1999; NCTM, 2000; NRC, 2000; Tal et al., 2001). On a physiological
level, Zull (2002) found that sensory experiences entered in the sensory cortex of the brain, were
processed and actualized in the integrative cortex, and new ideas were tested in the motor cortex.
From a sociological perspective, adults have observed children tasting, touching, smelling, and
feeling the world around them, incorporating and reflecting on the new ideas, and actively testing
the new ideas (Schmidt, 2008; Zambo & Zambo, 2007). Similarly adults undergo the same
processes when learning new material: consciousness of a new idea, investigation of the new
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idea, and finally use of the new idea (Schmidt, 2008). This process of active learning has been
called the learning cycle.
The learning cycle reflects a natural form of learning that has persisted with child and
adult (Lawson, Abraham, & Renner, 1989; Schmidt, 2008). Schmidt (2008) described the natural
learning in this cycle as consisting of ?awareness, exploration, inquiry and action? (p.12). It was
during the late 1950s and early 1960s, in which educational reforms were in demand, that Atkin
and Karplus (1962) developed their model for teaching and learning science which focused on
the natural active learning of children and later became known as the learning cycle. The
Atkin/Karplus model consisted of three phases: exploration, invention, and discovery.
Furthermore, the phases of the learning cycle in the Atkin/Karplus model align closely with
Piaget?s theory of learning and in particular the processes of assimilation, accommodation, and
organization (Abraham and Renner, 1983; Bybee, 1997; Kratochvil & Crawford, 1971; Renner
& Lawson, 1973). The three phases of the Atkin/Karplus model, exploration, invention, and
discovery, were expanded to five phases (Bybee, 1997). The five phase model still incorporated
the initial phases of the Karplus/Atkin model (Bybee, 1997) and became known by Lawson and
others on the Biological Sciences Curriculum Study (Withee & Lindell, 2006). The five phases
included engagement, exploration, explanation, elaboration, and evaluation (Bybee, 1997).
Although the learning cycle has its roots in the science field (Bybee, 1997; Atkin &
Karplus, 1962), the tenets align closely with national mathematics standards. Marek (2008), in
his explanation of the learning cycle, also provided examples of the learning cycle for science
and mathematics lessons. The mathematics standards promote students working together
(NCTM, 2000), as recommended in the exploration and elaboration phases (Bybee, 1997). Using
concrete tools and solving problems to develop a deep understanding of the concepts is core to
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the mathematics standards (NCTM, 2000) and aligns with the exploration, explanation, and
elaboration phases of the learning cycle (Bybee, 1997). According to the mathematics standards
students generate their own mathematical questions (NCTM, 2000), a part of the engagement
phase (Bybee, 1997), and students generate conclusions (NCTM, 2000), a part of the explanation
phase (Bybee, 1997). The exploration, explanation, and elaboration phase supports a focus on
thinking and reasoning skills that lead to conjectures or arguments about the mathematics being
examined (Bybee, 1997; NCTM, 2000).
In addition, components of inquiry-based teaching are evident in the learning cycle (Gee,
Boberg, & Gabel, 1996; Tracy, 1999; Withee & Lindell, 2006). Inquiry-based teaching involves
the exploration of students around a central idea, formulation of questions, investigations to
answer the questions, and reflection of learned ideas (Morrison, 2008; Tracy, 1999).
Furthermore, the learning cycle approach provides a method for structuring inquiry-based
lessons, and is considered an inquiry embedded approach (Marek, Maier, & McCann, 2008;
Marek & Cavallo, 1997).
Research on the Learning Cycle
Research has been conducted with the learning cycle in the K-12 setting (Boddy, Watson,
& Aubusson, 2003; Cardak, Dikmenli, & Saritas, 2008; Liu, Peng, Wu, & Lin, 2009; Oren &
Tezcan, 2009; Soomor, Qaisrani, Rawat & Mughal, 2010), in higher education (Illinois Central
College, 1979; Walker, McGill, Buikema, & Stevens, 2008), with preservice teachers (Haefner
& Zembal-Saul, 2004; Hampton, Odom, & Settlage, 1995; Hanuscin & Lee, 2008; Morrison,
2008; Park Rogers & Abel, 2008; Settlage, 2000), and with in-service teachers (Gee et al., 1996).
Researchers have concluded that K-12 students using the learning cycle perform better on
content questions than students in a traditional science classroom (Cardak, Dikmenli, & Saritas,
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2008; Ergin, Kanli, & Unsal, 2008; ?ren & Tezcan, 2009; Soomer et al., 2010). Furthermore,
Boddy et al. (2003) found that the use of the 5E learning cycle promoted higher order thinking
skills in primary age students. When working with a primary teacher to help the teacher develop
skills in using inquiry with the 5E model, the researcher found the students were more involved
and more on task than they would have been in a traditional science lesson (Clark, 2003).
Research with elementary pre-service teachers and the learning cycle have indicated
similar results. Hanuscin and Lee (2008) used the learning cycle with their elementary preservice
teachers during the science methods course. They found the learning cycle was an effective
approach with elementary preservice teachers. After using the learning cycle as a model for
teaching science their preservice teachers were able to apply that knowledge to deign tiered
lessons with the learning cycle. Also, direct teaching of the learning cycle approach positively
influenced elementary preservice teacher?s efficacy of teaching science with the learning cycle
during a science methods course (Settlage, 2000). Results from studies in science methods
classes also indicated that elementary preservice teachers embraced the learning cycle due to
dissatisfaction in the way they experienced science in school (Lindgren & Bleicher, 2005). Their
understanding of the learning cycle improved with the continued exposure of the learning cycle
in methods class, but some had a difficult time conceptually changing their ideas about teaching
science (Gee et al., 1996; Lindgren & Bleicher, 2008). Elementary preservice teachers had
greater understanding of some aspects of the learning cycle, but their conceptions of
implementation of the learning cycle did not always follow the way the intended learning cycle
(Hampton et al., 1995; Marek, Laubach, & Pedersen, 2003).
There is a scarcity of research on the learning cycle and mathematics education. The
research that exists on preservice teachers and the learning cycle focuses primarily on science
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(Lindgren & Bleicher, 2005; Marek et al, 2008; Urey & Calik, 2008). However, researchers
teaching preservice teachers have provided examples of the learning cycle and mathematics
(Marek & Cavallo, 1997; Marek et al., 2008; Simon, 1992). Marek (2008) described a science
lesson in which students measured the circumference of objects to make a conclusion about the
relationship of the circumference and diameter of circles. His description illustrates how the
learning cycle could be used in mathematics teaching. Although his example was for a science
lesson, it very easily could have been for a mathematics lesson on understanding area of circles.
Similarly, Marek and Cavallo (1997) described the use of the learning cycle for students to
understand measurement and geometry concepts. Simon (1992) recognized the use of the
learning cycle in mathematics methods courses. He described a mathematics methods class in
which teacher candidates solved a problem situation, discussed solutions, and extended new
ideas into other problem situations.
Methods
Research Design
Twenty-two elementary preservice teachers were invited to share their understandings of
common approaches in science and mathematics teaching while in their methods classes.
Multiple sources of data were collected to provide triangulation of data (Yin, 2009).
Furthermore, evidence in the study took the form of an open-ended pre-test, open-ended post-
test, and weekly weblogs, or blogs (Merriam, 1998; Yin, 2009). This study used a multiple case
design in which multiple cases were examined (Merriam, 1998; Yin, 2009).
Context of the Study
Elementary preservice teachers jointly enrolled in science and mathematics methods
classes were the focus of this study. The principal researcher was the mathematics methods
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instructor who planned and worked with the science methods class professor. As the
mathematics methods instructor, the researcher focused on the development of the preservice
teachers in the mathematics methods course and in the mathematics field placements. The
sample included 22 of the 25 students enrolled in the undergraduate courses.
Participants completed the science portion the first five weeks of the semester and the
mathematics portion the second five weeks of the semester. Since the study took place during the
summer semester, participants attended class or a field experience every day of the week.
Science lessons in the on-campus portion of the science class included lessons from Full Option
Science System (Foss) (Lawrence Hall of Science, 2003), Science Technology for Children
(STC) (National Science Resources Center, 2003), and Insights (EDC Science Education, 2003).
Example activities in the science methods class included a series of inquiries on earth science
concepts in which rocks were explored. For the mathematics portion of class, materials consisted
of materials from NCTM and the Investigations curriculum (Pearson Education, 2007). Example
activities in the mathematics methods class included a series of activities in which three-
dimensional figures were used for a volume unit. In both classes preservice teachers participated
in investigations and experiments, discussions, and readings (Chick, 2007; Heywood, 2007; Hill,
1997). The learning cycle was used as the common pedagogical approach.
Participants who completed elementary science and mathematics activities taught with
the learning cycle approach in the role of a student. After these teaching episodes, students then
took on the role of a teacher in peer teaching and examining the pedagogy of the teaching
episode. Participants completed weekly blogs, as a course requirement, pertaining to their
teaching practice and aspects of the learning cycle each week of the science portion of the
semester. The themes of those blogs were repeated during the mathematics portion of the
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semester (see Appendix C). Preservice teachers also participated in field experiences to give
them an opportunity to practice teach mathematics and science. Although participants were in
differing field placements for the science and mathematics portions of the methods classes, they
were expected to design and teach lessons using the learning cycle. Field placements for science
were at summer academic camps including an outdoor ecology preserve and a local school. Field
placements for mathematics were at two summer programs held by the local Boy?s and Girl?s
Clubs of America.
Data Collection
Multiple methods of data collection took the form of an open-ended pre-test, open-ended
post-test, and weekly weblogs, or blogs (Merriam, 1998; Yin, 2009). The open-ended pre-test
was administered on the first day of class at the beginning of the semester (see Appendix A). A
similar open-ended post-test (see Appendix B) was administered on the last day of the semester.
Questions on the pre-test and post-test were designed to elicit information about participants?
conceptions of tools and processes for learning, pedagogical approaches, and commonalties in
approaches to mathematics and science. Participants completed weekly blogs (see Appendix C)
concerning science and mathematics teaching. Blogs for the first five weeks focused on tools,
processes, and approaches to science teaching. Similar blogs were used the second five weeks
that focused on tools, process, and approaches to mathematics teaching. In addition, the
researcher recorded notes on the actual teaching practices of the participants while in the field.
Data Analysis
The constant comparative method (Glaser & Strauss, 1967) was used to find conceptions
of mathematics and science teaching that emerged from the varying data sources. First level
coding for category construction occurred with each blog, pre-test, and post-test (Merriam,
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1998). Keeping in mind the research questions, the researcher attempted to create categories that
would ?reflect the purpose of the research? (Merriam, 1998, p. 183). Pieces of information that
were striking to the researcher in regard to conceptions of tools, processes, pedagogical
approaches, and similarities in teaching science and mathematics were noted for each piece of
data (Merriam, 1998). Categories were determined based on words or phrases from the
participants, conclusions from the researcher, or connections to existing research (Glaser &
Strauss, 1967; Merriam, 1998). For example, when Person 1 on the pre-test described children in
the field placement taking apart machines to understand how the machines were designed, the
researcher coded based on the phrasing of the respondent as ?took apart machines for
understanding?. Then the first blog for Case 1 was read through for category construction. New
categories were designated based on new information: ?need for fun?, ?benefits?, and
?traditional approach criticism? (Phase I, Person 1-Blog 1, 2008). The researcher made a list of
the codes from the first three data sets (Merriam, 1998).
From the list of initial codes the researcher referred to the research questions and added
prefixes to the codes (Merriam, 1998). For example, on Person 1-pretest, the researcher noted
that the respondent indicated group activities and hands-on activities as being a common to
teaching mathematics and science. The researcher coded as ?CA-group activities?. The research
questions dealt with five main areas: tools for conceptual development, designated with a T in
the coding process; processes for learning, designated with P; pedagogical approaches,
designated with PA; and more specifically common approaches between mathematics and
science, designated with CA; and change in thinking, designated as CIT. Since the learning cycle
was used as a common approach in the science and mathematics methods course then aspects of
the learning cycle were designated with an LC. Preservice teachers also added many personal
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experiences about learning mathematics and science. A category was designated for personal
experience (PE) in order to differentiate it from their current practice.
A chart was made for each participant that included the T, P, PA, CA, CIT, LC, and PE.
Codes were placed in the chart for each person in chronological order with a source code at the
end. ?Pre? and ?Post? were source codes for the pre-test and post-test. The letter B and a number
were used for blog source codes. For example on Person 1 under T, ?Took apart machine in FEP
b3, Promoted fun b3?. Based on the codes for each area the researcher refined the code list.
After all of the level two coding was completed the researcher examined data across the
cases to make further generalizations. These generalizations were organized into the themes for
each category. Three themes emerged for tools and processes: traditional, expanded, and
progressive. Themes for approaches were categorized as traditional, mixed, or hands-on.
Categorical Themes. Based on their responses, participants? conceptions of tools for
math and science were categorized into three categories: traditional view of tools, expanded view
of tools, or progressive view of tools. Traditional views of tools represented following a
traditional textbook or only focused on computation. Responses that had a mixture of students
conducting experiments, worksheets, textbook learning, and students looking at pictures were
classified as expanded view of tools for science. Responses in which tools were used for primary
concepts such as shapes, money, or counting were included in the expanded view. The
progressive category for science and mathematics included comments about students exploring
in nature, physical knowledge activities, or students completing hands-on tasks.
In examining how preservice teachers conceived of processes for learning their responses
fell into three distinct categories. Processes for learning were categorized as traditional,
expanded, or progressive. Processes in which students focused on students finding answers were
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categorized as traditional. Preservice teachers in this category saw experiments as a means for
students to find predetermined outcomes or mathematics lessons that focused on students getting
the right answer. Expanded processes were process indicated beyond just the right answer, but
still limited in scope. Examples of expanded processes indicated responses that indicated
students should see or observe. However, notions of students making connections, applications,
and reasoning were classified as progressive.
There were varying thoughts on pedagogical approaches for science and mathematics.
Pedagogical approaches were categorized as traditional, expanded, or progressive. Traditional
approaches were ones in which the lessons were teacher directed and instruction focused on fact
learning. Expanded approaches in science represented ideas that science should include
experiments, observation, or research. Expanded approaches in mathematics focused on
approaching math with real-world mathematics Progressive views were indicated by data that
focused on inquiry-based approaches to develop understanding.
Quotations were pulled from participants to reflect the big ideas for each theme. Results
are explained according to conceptions of tools, processes for learning, pedagogical approaches,
and commonalities in science and mathematics teaching.
Results
Conceptions of Tools
Science. Participants more readily accepted the use of concrete tools for concept building
in science than for math, as indicated in Table 1. Hands-on learning was a common idea of tools
in science at the beginning of methods class. By the end of methods class seventeen preservice
teachers thought tools in science should be for hands-on learning. A few maintained a limited
view of tools in science for measurement or for right answers.
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Table 1
Thematic Categories Conceptions of Tools
Tools Traditional Expanded Progressive
Science Teacher controlled
experiments;
Science projects;
understand the
science book;
Stories to tell; use
tools for
measurement
Experiences,
worksheets,
activities, and
experiments;
observe nature
Hands-on learning;
Physical knowledge
activities;
Memorable concrete
experiences
Pre (n) 6 3 13
Post (n) 3 2 17
Mathematics To do a problem or
get a right answer,
computation,
for skills practice,
facts, and review
games
Tools as a question
money, tools for
counting, time or
shapes
Games and activities,
to teach the concept,
creative way for
students to
understand; To see
relationships or
understand the lesson
better
Pre (n) 8 8 6
Post (n) 8 5 9
Six participants initially held a traditional view of tools in science. They considered tools to
include teacher-controlled experiments, science projects, the science book, or stories. These
tools were used in a teacher led manner. In such a context, students were being informed of the
science concept rather than learning about it for themselves. By the end of the semester three
participants held a traditional view of tools in science. The preservice teachers continued to see
tools as a means of performing calculations. They did not see tools in a larger context as a means
to foster conceptual understanding:
A concrete tool that would be used in a science lesson would be a thermometer. Children
can be given a problem such as: find the average body temperature or measure this glass
of water to see how hot/cold it is. (Person 16, Post-test)
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Three preservice teachers had an expanded view of tools in science at the beginning of
the methods courses. They recognized tools in science for more than measurement tasks, but they
did not see the full potential of tools. They also were vague in their articulation of the role of
tools. They would say tools were for students to act like a scientist, but they did not provide
further explanation of what was meant by the statement: ?By having students use hands-on tools
in experiments themselves, they are acting like scientists? (Person 9, Post-test). By the end of the
semester only two students held an expanded view because one of them now held a more
progressive view.
Thirteen participants expressed the use of tools for hands-on learning in science.
Respondents with this view believed that, ?When students have a hands-on experience with the
lesson they are able to learn more from the lesson? (Person 2, Pre-test). By the end of the
semester 17 out of the 22 participants believed tools in science were to develop concepts. The
field experiences provided preservice teachers with opportunities to use tools to teach science:
I taught a lesson on how much water is on Earth. They were able to use concrete
materials to pour beans into a gallon container to see how many pints are in a gallon and
so on. They made a measurement book to see it for themselves. (Person 5, Post-test)
Mathematics. Participants more readily accepted tools in science than in math initially.
Traditional notions of tools in mathematics remained an underlying theme for eight of the
participants, as indicated in Table 1. Although preservice teachers talked about using tools for
hands-on or inquiry-based learning in mathematics, eight participants were merely using tools to
reinforce procedures. Their conceptions of tools were limited to computation tasks.
Furthermore, those tasks were often for primary grades for such concepts as addition,
subtraction, and counting money. Using tools to solve rote computation problems focuses the
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mathematics on getting a right answer rather than using tools for conceptual knowledge building.
A typical response was: ?Students understand the material being covered when they can use
manipulatives; using cubes is great when teaching adding, subtracting, and even dividing?
(Person 12, Pre-test). Those with a traditional view of tools in mathematics were focused on
students using tools to determine the right answer. Five of the participants maintained a
traditional view of tools at the end of methods classes. They continued to view tools for teaching
primary concepts:
Using play money because it makes counting money a lot easier when they can actually
see it. Children are going to need to count money in real life situations so it is important
that they know what it looks like. (Person 17, Blog 8).
Participants with an expanded view of tools in math saw tools for more than just
computation but in limited ways. They relegated tools in math to visual representations of
primary concepts, such as time and money. They did not consider that tools could be used to
develop conceptual understanding of concepts. Often their school experiences provided the
framework for tools in mathematics:
One specific memory I have of using concrete tools in math was in my first grade class.
Every week, we would be rewarded with pretend money for good behavior completed
assignments, etc. Each Friday, we were allowed to use our ?money? we accumulated
throughout the week to buy something from the pretend store in the classroom. (Person
18, Blog 9)
Six preservice teachers held a progressive view of tools at the beginning of the methods
classes. They believed that tools could help students understand concepts. These participants
were asked to use tools to teach mathematics in a manner in which most of them did not
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experience as children. Their blogs often referred to memorization along with skill and drill
experiences in school. However, they expressed dissatisfaction in the way they experienced
mathematics in school:
I like the fact that students can be life-long learners when using the learning cycle. If the
learning cycle is applied in math it makes for better instruction and learning. If I would
have had more hands-on activities when I was in school, then math would not have been
so bad. I find the students do learn better with hands-on experiences and learning is fun
and engaging.? (Person 2, Week 6)
When they followed the standards based lesson, they saw the success in learning that
could happen with the use of tools. The following is an example of a successful lesson in which
the preservice teacher was uncomfortable at first using tools to teach a fraction lesson:
I did the lesson with the equivalent fraction strips. They were a little confused at first but
as we continued and played the game, they began to understand. When they placed their
fraction, I had them place it where they thought it would go and tell me what percent it
was. Rather than just having a sheet with equivalent fractions and percents for them to
look at and learn, they actually had a chance to place actual cards in a place where it
should go and strategize with their fractions of how to block others and what fractions
they might have. (Person 4, Blog 10)
The field experiences verified their conceptions that tools in mathematics helped students
understand and deepened their understanding of tools in mathematics. Nine of the 22 participants
at the end of the semester recognized that tools in mathematics could be used to understand a
concept and develop relationships between concepts. They described tools in terms of student
using the tools to help build their knowledge: ?Students can use geoboards to help their
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understanding of polygons. These experiences are very important for students to create their own
knowledge? (Person 7, Post-test).
Conceptions of Processes
Traditional View. At the beginning of the semester a majority of the students had
limited conceptions of processes for learning science and mathematics, as indicated in Table 2.
Table 2
Thematic Categories for Conceptions of Processes
Processes Traditional Expanded Progressive
Science Teacher led lessons
in which students
are passive learners
Articulates processes in a
limited way
Ex: Students observe
experiments.
Views processes in a
larger context to develop
schemes of learning.
Ex: Students conduct
experiments to solve their
own questions
Pre (n) 3 14 5
Post (n) 3 6 13
Mathematics Processes for
answers/ facts
Articulates processes in a
limited way
Ex: Students make
observations
Views processes in a
larger context to develop
schemes of learning.
Ex: Students figuring out
and reasoning about
mathematics
Pre (n) 7 11 4
Post (n) 6 4 12
Three preservice teachers thought of science processes in traditional ways and seven
preservice teachers thought of processes in mathematics in traditional ways at the beginning of
methods class. Traditional conceptions placed the students in passive roles with the teachers
giving all of the information. Those with a traditional view of process articulated processes in
teacher-directed and controlled lessons. The teacher?s role is not one of a guide but rather the
authority in the classroom. A typical statement for traditional processes in science was, ?Teach a
lesson, do the activity, let students explain how it worked and how they go together? (Person 4,
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Pre-test). A typical statement for traditional processes in mathematics was, ?Teach the lesson,
work together as a class and have students work independently? (Person 4, Pre-Test). In these
scenarios the students are following the examples provided by the teacher and are not involved in
problem solving. In both scenarios, students are completing a science or mathematics lesson in
which the method has been given by the teacher. Since the method is known, they are, therefore,
not reasoning, communicating, or making connections about the mathematics or science either.
By the end of the semester those with traditional views had very little change in thinking.
By the end of the semester, three of the participants believed students should be making
connections and seeing real-life applications in science. They expressed in some manner that
students should be ?doing? in science. However, six participants believed procedures and
problems should be the focus for mathematics. They conceived of mathematics as ?an exact
answer? and science as ?changing?. They believed that students finding answers for themselves
indicated good mathematics teaching. Participants with these views often interpreted not giving
the students answers to rote problems as the focus of teaching. They would describe using
aspects of the learning cycle to foster processes in mathematics but then describe a teaching
episode in more traditional ways. Answers or facts were the goals of processes:
Students worked to determine how many steps they would take in one mile and then one
hundred miles. I observed the students as they worked in pairs. I listened carefully to
their conversation to hear student thinking and to determine whether or not their thinking
would lead them to the correct answer. (Person 3, Week 9)
They also considered students solving practice problems to be problem solving. On a lesson in
which students were completing practice problems on making money change, a preservice
teacher reflected, ?I promoted inquiry by making it real life applicable and showing students how
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they could use what they are learning? (Person 3, Week 10). Thirdly, they interpreted
participation in an activity as an example of inquiry-based teaching. A preservice teacher
commented:
In my lesson, the students were actually the concrete materials. I would use them to say 3
out of 6 are wearing jeans so what is the fraction and percent. They enjoyed this, and I
could tell they were learning. We played Guess My Rule, and the children were very
good at this activity. My lesson promoted inquiry because they were able to participate
in the activity and think about what fraction and percent of things the students have in
common. (Person 5, Week 10)
It is clear from these examples that what preservice teachers labeled as inquiry was a very naive
view of inquiry.
Expanded View. Fourteen of the twenty-two participants initially held an expanded view
of processes in science. They considered processes for science were more than understanding a
science textbook. They recognized the importance of students experiencing science for
themselves, but did not articulate processes beyond the level of observation. In a lesson on the
Laws of Motion a preservice teacher responded, ?These children were able to see what the laws
were by creating things that dealt with it and learning first hand? (Person 5, Week 5). Eleven of
the preservice teachers initially held expanded views of processes in mathematics in which
processes were for more than answers, but still very limited in nature. By the end of the semester
six participants had expanded views of processes in science and four participants held expanded
view of processes in mathematics. Eight of the participants with an expanded view of science
and seven of the participants with an expanded view in mathematics shifted to a progressive
126
view by the end of the semester. The field experiences provided a framework for them to help
students develop problems solving, reasoning, and communication in science and mathematics.
Progressive View. Initially five preservice teachers for science and four preservice
teachers for mathematics held progressive views of processes. Preservice teachers who started
out with expanded views in math developed more cohesive understanding of processes in science
and mathematics teaching. They believed that students should reason for themselves about
concepts and relate that knowledge to other areas. The field experiences provided a means for the
preservice teachers to put the learning cycle into practice. By the end of the semester 13 of the 22
participants for science and 12 of the participants for math held progressive views of processes.
Data sources indicated they were able to articulate processes in relation to lessons they had
taught:
So, I had the students go around and tell me why they thought there might be more or less
raisins in each box and the students suggested that because the raisins were different sizes
there might be more big ones that took up more space and made less as many raisins in
the box or there might have been more smaller raisins making there more raisins in the
box because of the room. I was really proud of the students? observations and predictions.
I provided the students with an opportunity to build knowledge through real-life and
hands-on activity. (Person 19, Week 10)
They conceived of processes as students reasoning, making connections, finding relationships,
and justifying conclusions. They often articulated this in terms of students reasoning, students
discussing their observations or findings with each other, or experiencing the concept: ?I thought
it was so interesting to see what these kids could design and everything they used to create their
designs, and the reasoning they had behind their creations? (Person 6, Week 5).
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Conceptions of Pedagogical Approaches
As with tools and process, participants held varying views about approaches to teaching
science and mathematics, as indicated in Table 3.
Table 3
Thematic Categories for Conceptions of Pedagogical Approaches
Approaches Traditional Expanded Progressive
Science Teacher the lesson,
do the activity, let
students explain,
teacher-led
demonstrations in
which students are
passive learners,
teacher shows
models
Experiments, centers,
nature walks
Hands-on Activities to
construct knowledge,
experience science for
themselves
Pre (n) 7 6 9
Post (n) 4 7 11
Mathematics Teach the lesson,
work together, then
students work
individually, work
by skill level
Math games, centers,
real-life (i.e. play store)
Hands-on to build
understanding, to see the
math unfolding
Pre (n) 8 9 5
Post (n) 6 5 11
Traditional View. Initially, seven participants held traditional views about approaches
for teaching science and eight participants held traditional views about approaches for teaching
mathematics. Preservice teachers with this view believed that science and mathematics should be
approached with the teacher demonstrating and the students completing a verification activity.
With this approach the students are being told the information rather than experiencing it for
themselves. Four of the seven for science and six of the eight for mathematics maintained their
view of traditional approaches by the end of the semester. They tried to justify traditional
approaches in science and mathematics with the learning cycle and hands-on approaches:
128
I also believe that more traditional approaches can still be combined with the learning
cycle. Reading in the textbook, completing worksheets and defining terms can be helpful
for students as long as they are exposed to other approaches as well. (Person 7, Week1)
They liked the idea of a learning cycle and hands-on but still tried to negotiate how to
incorporate hands-on into more traditional science and mathematics instruction. Some
participants believed in a hands-on approach to teaching science and mathematics but under
certain conditions. One participant believed in teaching with hands-on but only when the
students had earned it through good behavior:
I understand these kinds of activities must be implemented with a certain structure,
perhaps spread out occasionally over a period of time. Kids view these types of activities
as a treat, so promising them to get to participate in them would be something they could
work towards. (Person 18, Week 5)
Similarly, a participant believed that the teacher must establish her authority and be in
control of the classroom in order to use hands-on teaching: ?The kids have a lot of fun and don?t
see the lessons as lessons but as fun activities. This laid back time is great, but when there is a
lack of discipline and consequences, it can become an issue? (Person 3, Week 5). Her need for
control over the classroom and students influenced her approach to teaching mathematics and
science. Three of the participants indicated that hands-on approaches in science were for special
occasions or when it was convenient for the teacher to implement:
I do think there is usually some way to incorporate hands-on activities with each
scientific concept you teach. The students will enjoy it more, and will be more likely to
retain the information they learn. That said, I know that there are other times that call for
more traditional instruction, but as long as the teacher keeps it interesting and
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incorporates hands-on activities where it is possible, the students will have an enjoyable,
meaningful experience. (Person 18, Week 4)
For three of the participants with traditional views in science and two with traditional views in
math, the field experience and methods class impacted their conceptions of how to approach
science and mathematics. They moved to progressive or expanded views by the end of the
semester.
Expanded View. Six participants held expanded views for teaching science and nine
participants held an expanded view for teaching mathematics. They believed in science that
students should be involved in a combination of experiments, centers, observation, and research:
?Having learning centers for children to learn or review different science centers? (Person 10,
Pre-Test). In mathematics, they believed that the approach should focus on real-life mathematics
and centers: ?Real-life projects have value in everyday life like counting money. Role play like
grocery store? (Person 14, Pre-test). Their description included more than the teacher delivering
the knowledge, but not on the level of approaching science and mathematics for conceptual
understanding. By the end of the semester there were seven participants who held expanded
views in science, one more than the beginning of the semester. This can be explained in three
preservice teachers with traditional views moving into the expanded view and three preservice
teachers in the expanded view moving into the progressive view. Some preservice teachers with
an expanded view also shifted into the progressive view by the end of the semester. For
mathematics, four of the preservice teachers moved into the progressive view by the end of the
semester, which left five in the expanded category.
Progressive View. Nine participants in science and five in mathematics believed science
and mathematics should be taught with a hands-on approach. At the beginning of the semester
130
they talked in general terms about hands-on or constructivist learning. They talked about students
completing physical knowledge activities. They described students working in nature to learn
about plants and insects. They described mathematics using learning centers and hands-on
activities to develop understanding. Their teaching experiences in the field reinforced their
conceptions and expanded their notions of teaching science and mathematics:
At the Forest Ecology Preserve I taught a lesson about the basic needs for survival
including food, water, shelter, and space. During the lesson I elaborated on shelter as the
basic need and had the students build their own shelter. I followed the 5E?s teaching
model. I also provided the students with a very hands-on approach to learning by having
them create their own shelter. By having them create their own shelter, they were able to
make personal connections to what they were learning. (Person 19, Post-test)
Eleven participants believed that science and mathematics should be taught with a hands-
on approach by the end of the semester. They felt they benefited from the learning cycle
approach to hands-on lessons as well as the students:
The lesson I did converting the gallon of beans into cups, pints, and quarts helped me
learn so much better. I was able to actually see a gallon be converted into cups first
hands. I hate to admit it, but before I only remembered the formulas used and did not
exactly remember how many cups were in a quart and so on. This played a role in my
learning and helped me remember why it works out as it foes. Using this concrete
experience not only added to my learning, but it did to the children I taught. (Person 5,
Week 9)
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Common Approaches between Mathematics and Science
Initially most of the preservice teachers thought of mathematics and science as being
approached in different ways, as indicated in Table 4.
Table 4
Thematic Categories for Approaches Between Mathematics and Science
No similarities in
approaches
Partial connections Recognizes similar
approaches for both
mathematics and science
n 13 2 7
Pre Sees as two different
subjects with two
different ways of
teaching
For example: science
uses experiments and
solve problems in
mathematics
Sees connection of
mathematics and science, but
sees teaching them differently.
Science strategies can be
used in mathematics.
Both require critical
thinking, physical
knowledge activities
Both hands-on activities
and games to make
interesting and fun.
n 2 2 18
Post Sees as two different
subjects with two
different ways of
teaching
Commented that hands-on
could be used for both but
described mathematics for
finding the right answers.
Learning cycle for both;
both involve inquiry,
learning experiences,
using lab activities,
let students figure things
out.
None of the students mentioned the learning cycle on their pre-test. Thirteen preservice
teachers initially indicated that there were no commonalties between teaching science and
mathematics. They indicated that science was for teaching plants and animals and mathematics
was for teaching numbers. ?Math deals with numbers, geometry, fractions, etc. Science deals
with animals, plants, biology, etc? (Person 22, Pre-test). Only two preservice teachers saw
mathematics and science as being approached differently at the end of the semester.
Two participants indicated partial connections initially between teaching science and
mathematics. They saw that they had connections but should be approached differently. ?Math
132
and science both use critical thinking. Math and science are different in that science has more
experimentation and math has more number problem solving? (Person 11, Pre-test). Two
preservice teachers commented at the end of the semester that both science and mathematics is
hands-on, but they described mathematics as finding the right answer.
Seven participants indicated that there were commonalties between science and
mathematics teaching at the beginning of the semester. They indicated that science and
mathematics both used hands-on learning: ?Math is like teaching science because there are
different types of math and science. Math and science also provide hands-on activities for
learning. Science and math are similar in that we use each subject everyday? (Person 8, Pre-test).
Eighteen participants indicated at the end of methods class that the learning cycle approach
should be used for both science and mathematics teaching.
Students developed understanding of the learning cycle through methods classes and field
placement teaching. Concepts of the learning cycle varied from seeing the learning cycle as steps
to recognizing the learning cycle as a means for fostering developmental understanding. Some
grappled with the learning cycle approach in comparison to the manner in which they were
taught mathematics and science. With the learning cycle approach students are engaged and
participate in an exploratory activity to bridge prior knowledge with new knowledge.
Explanations often follow this exploratory stage. However, exploration followed by discussion is
at odds with cookbook science lessons of traditional classrooms. Due to prior science teaching
experiences, a preservice teacher believed that explanations were important before a hands-on
activity:
When they see things happen they tend to learn more. I think that hands-on is always a
good way to go, but I also think there?s a time for explaining most likely before the
133
hands-on activity. Students need instruction and explaining or they will just look at
hands-on activities as play time. (Person 17, Week 4)
The learning cycle lessons in methods class provided an example for them to reflect on in
their blog posts. Preservice teachers discussed the idea of explanation and hands-on and when
explanation should occur in the lesson. Preservice teachers had to rethink the learning cycle
approach in relation to their experiences. Preservice teachers thought:
The important characteristic of the learning cycle is that it involves concept introduction
following exploration. This surprised me when I first discovered this, because I originally
assumed teachers explained the concepts to students, and then allowed the students to see
examples of the concepts through activities. (Person 18, Week 1)
Preservice teachers also described advantages to the learning cycle in their blogs. They
liked the idea of students being actively engaged in exploration. They felt that this active
engagement helped to build students? background knowledge. They also felt that exploration set
the stage for learning.
The learning cycle, I feel, is a method of teaching that needs to be incorporated into
mathematics. This is true because math, much like science, is easier to retain knowledge
when there are activities that allow students to create his or her own knowledge.? (Person
6, Week 6)
Eighteen out of 22 participants at the end of the methods classes did recognize the
learning cycle could be used for both science and mathematics. They recognized the important
role of concrete experiences in science and mathematics lessons. A typical statement was, ?I now
see how closely teaching mathematics and science can be related. I noticed that they both have
an importance of using hands-on or concrete experiences to help the understanding of the lesson?
134
(Person 7, Post-test). They recognized the importance for students to figure concepts out.
Furthermore, they recognized the learning cycle approach as a common approach to teaching
science and mathematics. Some of them referred to their blogs as the source that helped them
realize the learning cycle could be used for both subjects: ?I was able to see similarities in math
and science because some of the posts were the same such as dealing with the 5Es, concrete
materials, hands-on activities and assessment in both math and science? (Person 16 Post-test).
Discussion
There were several positive outcomes from this study. First of all, preservice teacher
seemed to benefit from the design of the science and mathematics classes. In their blogs they
mentioned new ideas about tools, processes, and approaches in science and mathematics that
they learned from the class investigations. Once they began working with children in the field,
they recognized how well students responded to use of tools, hands-on processes, and the
learning cycle format of lessons. They would comment that teaching the lesson with the hands-
on on approach helped them personally have a better understanding of the concept. Secondly,
they felt the learning cycle was appropriate for science and mathematics. Using the learning
cycle in both methods classes provided a consistent approach for the preservice teachers. Bt the
end of the semester 18 out of 22 preservice teachers recognized the learning cycle as a common
approach to teaching science and mathematics. Teaching preservice teachers a common approach
enabled them to focus on tools, processes, and implementation of the approach. Finally, many of
the preservice teachers deepened their ideas about the use of tools, processes, and approaches in
teaching science and mathematics. Eight preservice teachers shifted towards expanded or
progressive views with the use of tools and processes in science. Only three participants shifted
in their conceptions of tools in mathematics, but eight participants shifted in their views of
135
processes for mathematics teaching. Five participants shifted in their views of how to approach
science. Six preservice teachers shifted in their conceptions of how to approach mathematics.
The class investigations, peer-teaching, and field experiences opened them the idea of teaching
for conceptual understanding. Moreover, Hill (1997) found that when preservice teachers used
concrete experiences in courses and with students in field placements, it set the stage for
preservice teachers to achieve conceptual change due to bolstering self-confidence in teaching,
gaining a sense of accomplishment, and deepening of mathematical understanding.
This research attempted to find out if elementary preservice teachers in a dually
combined methods course using the same approach would recognize commonalities of those
approaches. Preservice teachers consider science a subject in which students are supposed to be
?doing? something (Gee et al., 1996). Research tells us that students should be ?doing? in
mathematics as well (NCTM, 2000). The ?doing? of mathematics is not computation problems to
solve or solving word problems that follow a pattern to practice a problem solving strategy
(NCTM, 2000). In the case of the preservice teachers in this study they recognized science was
for ?doing?. They wrote about science in terms on hands-on or experiments. We do not know
what they mean by experiments, but their intent is clear that students are involved with concrete
materials. For mathematics, it was different. Even though methods class focused on using tools
and developing processes within the learning cycle approach, a few of the preservice teachers
continued to try to conform to their primarily traditional experiences. Furthermore, those with
traditional views of tools rigidly held onto those views. Their view of tools for answers is similar
to an instrumental view. An instrumental view is a view of mathematics as a set of memorized
formulas and rules (Skemp, 1976). Preservice teachers? instrumental view of mathematics
perhaps creates interference in the way they understand the learning cycle, and commitment they
136
have for it as a teaching method. Gee et al. (1996) were also disappointed to find that preservice
teachers were not fully committed to the learning cycle as an approach for science. They
believed that is was difficult for the preservice teachers to accept a different way to approach
teaching science. Lindgren and Bleicher (2008) drew similar conclusions when some of the
strongest science students were reluctant about the learning cycle as an approach for teaching
science. However, students, who had negative science experiences or were dissatisfied with the
way they were taught, embraced the learning cycle approach. For those in this study, with a
traditional view of tools in science, it seemed they could shift to an expanded or progressive
view because it still fit with the ?doing? notion of science.
Processes similarly mirrored the use of tools. For those with traditional views of tools,
they also commonly held traditional views of processes. They wanted students finding the right
answer. Designing lessons requires teachers to understand the ideas and related concepts (Hill et
al., 2008). The challenge lies in implementing lessons for students to use processes to see those
ideas and relationships for themselves (Hill et al., 2008). For new teachers who are learning to
teach, this is a challenging prospect. Many of the preservice teachers were able to articulate
teaching in terms of processes for understanding by the end of the semester. Processes for
reasoning and justification developed as preservice teachers worked with children in the field.
The learning cycle approach requires teachers to have strong pedagogy in science and
mathematics teaching. This approach is different from what preservice teachers experienced in
school. Therefore, they have to be willing to accept the approach used in methods class. The
process of altering beliefs is known as conceptual change (Posner, Strike, Hewson, & Gertzog,
1982). Reflection, in the form of weekly blogs, was used as a tool in the conceptual change
process by providing time for the preservice teachers to examine conflict in ideas and examine
137
new ideas (Heywood, 2007). Preservice teachers did overall recognize common approaches in
mathematics and science teaching by the end of the methods classes. Besides the learning cycle
approach, they also included the use of reasoning, concrete experiences, students making
connections, and assessment for commonalities in teaching mathematics and science. It is hoped
that this experience with common approaches helped set the stage for the preservice teachers in
their science and mathematics teaching.
Impact of Beliefs and Prior Experiences
Teachers make decisions about teaching practices based on underlying beliefs and past
experiences (Kelly, 2000; Manouchehri, 1997). Although a few students said that hands-on,
inquiry-based teaching, or the learning cycle were ways to approach teaching science and math,
their blogs reflected contrasting beliefs about how science and mathematics should be
approached. The participants shared their school experiences in the blogs. Some blogs would
refer to hands-on learning. Then within the same blog or the next blog they would describe a
teaching episode which was more traditionally oriented and focused on right answers rather than
conceptual understanding. Those few, even with learning cycle lessons, would focus on whether
or not students came up with right answers. Their beliefs impacted their decisions in how they
implemented the lessons (Ball, 1991; Chick, 2007; Davis & Petish, 2005). The participants?
school experiences interfered with their teaching practice and made it difficult for a few of them
to be completely committed to the use of the learning cycle for mathematics teaching (Gee et al.,
1996).
Research based materials were used for the lessons. However, curricula materials alone
do not ensure a reform based practices (Battista, 1999; Huang, 2000). Battista (1999) found that
even with reform based materials that teachers could potentially distort the ideas in the materials
138
if the teacher had misconceptions. The variety of responses from students when given the same
curriculum ??provides evidence of the interplay between curriculum as designed and
curriculum as ?wrapped? around the ongoing action and interaction of students and teacher?
(Cannizzaro & Menghini, p. 376, 2006). For preservice teachers in a methods course they are
beginning to understand the ?ongoing action? of teaching. Preservice teachers were using an
approach which was unfamiliar to them and expected to use it at the same time in their field
placements. Curricular materials were also unfamiliar in nature and goal. For elementary
preservice teachers, who learn to teach multiple subjects, this could be a daunting task. Using the
learning cycle approach in both science and mathematics can then provide a consistency and
familiarity for preservice teachers to focus on their practice.
Issues with Inquiry
Inquiry-based teaching involves the exploration of students around a central idea,
formulation of questions, investigations to answer the questions, and reflection of learned ideas
(Morrison, 2008; Tracy, 1999). Some of the preservice teachers believed they were teaching
inquiry-based lessons. Their descriptions of how they promoted inquiry were quite revealing.
They held a na?ve view of an inquiry-based approach. Gyllenpalm, Wickman, and Holmgren
(2010) similarly found that teachers were ?conflating methods of teaching with methods of
inquiry [sic]? (p. 1151). They did not have a clear understanding of inquiry and the processes
students should undergo in an inquiry-based lesson. Their reflections demonstrated that they did
not have a deep understanding of what inquiry or problem solving means. Therefore, it can be
inferred that they did not have a deep understanding of the learning cycle as an inquiry-based
teaching approach. It was clear form their responses that they wanted to teaching inquiry-
embedded lessons. For elementary preservice teachers, conceptions of inquiry may need to be
139
explored more in methods classes to help them develop a clear understanding of how to fully
implement inquiry-embedded approaches, such as the learning cycle.
Implications
Focusing on common approaches in mathematics and science helped many of the
preservice teachers in this study to recognize the importance of an active learning environment.
Evidence of preservice teacher development from the data indicated that a majority of the
preservice teachers, who entered the methods class with expanded of progressive views of the
use of tools, processes, and pedagogical approaches, deepened their conceptions throughout the
semester. However, there were a few who entered the methods class with na?ve or traditional
views of teaching science and mathematics and had difficulty accepting new methods of
teaching. Hill (1997) similarly found that elementary preservice teachers initially held an
instrumental view of mathematics: mathematics as a set of memorized formulas and rules. This
indicates a need for support for those students who initially hold na?ve or traditional views of
what it means to teach science and mathematics. Tasks in the methods class and careful
placement with strong reform-minded mentors in the field may help them in their teacher
development.
This study did use common approaches to mathematics and science through the learning
cycle as a means of preparing future elementary teachers to teach science and mathematics. A
majority of the preservice teachers recognized common approaches to teaching science and
mathematics through the use of the learning cycle. Some focused more on certain aspects of the
learning cycle than other aspects. The exploration and engagement aspects were appealing to
preservice teachers. They wanted students to be engaged in learning. The exploration aspect of
the learning cycle also fit in with their idea of hands-on learning. This partial understanding of
140
the learning cycle can be expected with novice teachers learning how to teach. As preservice
teachers enter the field it is important to provide coaching and support of reform-based
approaches.
Participants? ideas about common approaches to mathematics and science changed for
most of the participants. Learners moved from seeing science and mathematics as completely
separate to being able to provide concrete examples from field experiences that articulated
common approaches of science and mathematics in the elementary classroom. Findings indicated
that purposeful planning and design of science and mathematics methods courses can yield
changes in development of thinking about the teaching of mathematics and science. Additional
studies are needed to examine if and how preservice teachers articulate common approaches to
mathematics and science beyond the methods classes.
141
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Appendix A
Pre-Test
1. How is teaching math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/experiences play a role in a math lesson? Give examples.
4. Name and describe a few teaching strategies or approaches used to teach science for
meaningful understanding.
5. Name and describe a few teaching strategies or approaches used to teach math for
meaningful experiences.
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Appendix B
Post-Test
1. How is math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/ experiences play a role in a math lesson? Give examples.
4. A. Name and describe a few teaching strategies pr approaches used to teach science for
meaningful understanding.
B. Name and describe a few teaching strategies or approaches to teach math for
meaningful understanding.
5. A. Think back to a science lesson that you taught this summer and briefly describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
6. A. Think back to a math lesson that you taught this summer and briefly describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
7. How has the blog helped you in your development of?
A. Ideas about teaching science?
B. Ideas about teaching math?
C. Similarities between the teaching of math and science?
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Appendix C
Weekly Blogs
Week 1: Why is following a Learning Cycle so important in teaching science? Won?t
more traditional approaches such as giving information first to students, such as in reading the
textbook, completing worksheets, and writing notes/definitions work just as well? Why not? I am
not convinced!
Week 2: We have learned about inquiry and the associated process skills for teaching
science through ?doing science?. We have learned that the Learning Cycle for planning and
teaching a series of lessons is ?best practice? to maximize student engagement and understanding
of the science we are teaching - often called a ?hands-on, minds-on? approach. So, how does the
Science-Technology-Environment-Society piece fit into all of this? What really is it anyway?
How does it work, and is it important in my science teaching? Please explain and help clear up
my confusion.
Week 3: So, this week you had the chance to finally practice teach about either ecological
or technological ideas to kids, and followed some portion of the Learning Cycle to do it!
(whether an ?exploration? activity to first develop students? common understandings OR an
?elaboration? activity to get them to apply their previous learning to a new situation or use). Also,
assessment was on everyone?s mind. So, how did you assess your students? attitudes,
understanding, or performance in your lesson this week? Do you feel your assessment strongly
aligned with your learning objective(s)? Was it authentic enough? Why is assessment so
important anyway? Share your thoughts about your thinking and how you are feeling about
assessment.
Week 4: This week we have been doing many hands-on activities in our FOSS Earth
Materials kit curriculum. All of the hands-on activities have been pretty fun, or at least
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interesting. Most of us really believe that ?hands-on? is the best way to go in teaching science,
but is there more to it? What do you think? Are all hands-on activities equal? Is hands-on best no
matter what you do, when you do it, or how you do it? Explain to me your thinking now about
?hands-on? activities in science to best help student learning. I know that you can help me
understand this approach better and are pretty knowledgeable about how to do it best.
Week 5: Kids and stuff everywhere! Inventing and building and Newton?s Laws of
Motion can certainly seem to be unruly in the classroom, but is this O.K.? Taking kids outdoors
to learn about science in nature also has its own planning and managing hurdles, but is it worth
it? Even in doing the state of Alabama?s science teaching in the classroom (AMSTI) with kits,
there is a level of uncertainty and messiness with kids and materials ?in motion?, but it seems to
work. How are you now feeling about these issues? Where do you begin personally in your
future classroom? What is your current thinking and your plan?
Week 6: Think about the Learning Cycle. Explain how the Learning Cycle pertains to the
teaching and learning of mathematics. Support with examples. Then respond to two other
people's responses.
Week 7: So far in class we have discussed inquiry in mathematics, assessing
mathematical understanding, developing number sense, and participating in tasks to develop our
own mathematical knowledge. Think about all we have talked about, experienced for ourselves,
and experienced with students. Explain which part of the Learning Cycle you find to be the most
important in developing a true understanding of mathematics and why. Support with examples of
your own experiences or mathematics field experiences.
Week 8: In class we have been learning about how to assess and different types of
assessment. Think about one of your math teaching experiences this semester. How did you
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determine student understanding of the topic? Be specific. Support with examples. Based on
your assessment what judgments and decisions will you have to make about teaching/learning?
Would you teach the lesson differently if you taught it again? Be specific. Support with
examples.
Week 9: We have used concrete materials in class and with students in lab. I want you to
think about the role that concrete experience plays in learning. Think of an instance in which
concrete experiences played a role in your own learning of mathematics. Describe that learning
experience. Describe how you have used concrete experience in one teaching lesson this term.
Week 10: Think about the two consecutive lessons you taught this week. What growth
did you see in your students' understanding of the topic? What role did concrete experience play
in your lessons? How did you promote inquiry?
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CHAPTER 5: ELEMENTARY STUDENT TEACHERS?
CONCEPTIONS AND USE OF TOOLS, PROCESSES, AND APPROACHES FOR
MATHEMATICS TEACHING
Abstract
This study focused on two elementary student teachers? thinking and practice in the use
of tools for conceptual development, processes for meaningful learning, and pedagogical
approach for teaching mathematics. The study took place during their student teaching
experience and is a continuation from a study that took place during the participants? science and
mathematics methods class. The nature of the students? understandings was examined through
several data sources: observations, open-ended questionnaire, final reflection, and researcher
field notes of practice. Data sources from the methods class were compared with data from the
student teaching. The case studies for ?Jane? and ?Kate? are presented. Jane and Kate are
pseudonyms. Jane and Kate participated in a jointly enrolled science and mathematics methods
class that used the learning cycle approach as a method for teaching science and mathematics.
The cases of Jane and Kate were selected because of their different development in the methods
courses. Kate thought of approaching mathematics using real-world mathematics. Her
conceptions of tools for math were manipulatives for teaching real-world concepts, such as time
and money. Processes were controlled in a teacher led lesson. Her approach for math, initially,
was the teacher explained and students worked problems. Jane thought of mathematics, initially,
as being a subject that should have a hands-on approach. She wanted students to see for
themselves and figure out the mathematics. However, by the end of the methods course they both
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conceived of science and mathematics as being taught with the learning cycle approach but in
different ways. Jane thought the learning cycle would help make mathematics meaningful and
make students actively involved in learning mathematics. Kate considered the exploration and
evaluation phases of the learning cycle to be important in teaching mathematics. She believed
that the exploration phase helped students construct knowledge. She believed evaluation was
important in order for the teacher to see the students? levels of understanding. Implications for
preservice teachers and teacher educators are provided.
Introduction
Preservice teachers are often products of a traditional classroom that focuses on repetition
and memorization with little attention to understanding (Taylor, 2009). Elementary preservice
teachers enter the undergraduate program with beliefs about mathematics that have been shaped
by those experiences (Manouchehri, 1997). Examining the mathematics standards documents
reveals goals that are at odds with the traditional approach to teaching mathematics: to move
away from memorization and rote learning and instead to focus on conceptual understanding
(Kind, 1999; National Council of teachers of mathematics (NCTM), 2000; Steen, 1990). The
standards documents promote mathematics and scientific literacy; being able to ask, find, and
determine answers (Kind, 1999; NCTM, 2000; National Research Council (NRC), 1996; Steen,
1990). Since many of today?s preservice teachers lack a model of standards-based reform
teaching from school experiences (Conference Board of Mathematical Sciences (CBMS), 2001),
the use of the same reform-based approach to teaching in mathematics and science, like the
learning cycle, could help in their understanding and teaching. In addition, the use of inquiry
processes embedded within the learning cycle for both mathematics and science is another area
of commonality that requires further study. A lack of research exists on the development of
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elementary preservice teachers and their thinking about use of inquiry as processes in
mathematics as they progress through their teacher preparation program.
Common inquiry processes for science and mathematics are founded in the natural
curiosity of a child (Neal, 1962). This natural curiosity causes children to pose questions about
phenomena around them and seek out the answers to those questions (Neal, 1962). Inquiry is
often historically associated with science (Neal, 1962). Yet, inquiry is inherent in mathematics as
well. Children seek to understand numbers and patterns in the world around them. Young
children explore science and mathematics concepts as a means of understanding (Hoffer, 1993;
Speilman & Lloyd, 2004; Zull, 2002). The physical experience enables them to develop abstract
thinking and make generalizations (Renner & Lawson, 1973). Tools, therefore, become a means
for children to explore and develop understandings. Tools in science are often thought of as
materials and equipment used to answer and pose questions (Neal, 1962). Mathematics uses tools
as well. Tools in mathematics for young children start off as objects to count, classify, or
describe (NCTM, 2000). As children answer questions about science or mathematics concepts
through exploration, they are involved in thinking processes. These processes help develop
sense-making, reasoning and proof, and communication (NCTM, 2000).
However, the description of mathematics being taught from an inquiry oriented
perspective, although a natural form of learning for children, is often not the approach taken by
elementary mathematics teachers (Taylor, 2009). Mathematics is often reduced to a series of
procedural problems or word problems to practice a procedure (Hill, 1997). Furthermore, tools
may become a source of procedural focus as well, rather than a means of understanding
mathematical relationships. The CBMS (2001) in their standards document determined that
teacher preparation programs had the daunting task of preparing teachers with primarily
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traditional classroom experiences to teach in alternative ways. Using tools to foster processes and
inquiry is, therefore, a challenging task for future educators (CBMS, 2001). They are not used to
experiencing mathematics through inquiry and may associate mathematics with formulas and
word problems (Hill, 1997).
Similarly, Interstate New Teacher Assessment and Support Consortium (INTASC)
developed general standards for new teacher candidates (n.d.). The INTASC principles focus on
teachers developing meaningful lessons using ?tools of inquiry? and designing instruction to
develop students? problem solving and critical thinking skills; teaching approaches also
contradictive to traditional methods of teaching. Teacher education programs have to establish a
process that allows preservice teachers opportunities to develop those recommended teaching
skills (Manouchehri, 1997; Noori, 1994; Speilman & Lloyd, 2004) and understand the complex
task of teaching (Grossman et al., 2009). Since new teachers may not understand the complexity
of teaching, teacher preparation institutions face the challenge of helping novice teachers see the
array of components that comprise teaching (Bolton, 1997; Grossman et al., 2009). Education
programs have to take into account students prior experiences and provide experiences that allow
the preservice teachers to shift or alter their belief system to envelop the ideologies of the
program (Ball, 1988; Heywood, 2007; Kelly, 2000). Approaching science and mathematics from
a common pedagogical approach has been used as a means of helping preservice teachers deepen
their understandings and make connections between the teaching of the two subjects (McGinnis
& Parker, 1999).
Issue as Conceived in Study
Improvement of science, technology, engineering, and mathematics (STEM) education is
of great importance today (Atkinson & Mayo, 2010). As the language of science (Shapiro, 1983;
162
Steen, 1990), a strong mathematical background is necessary for students to succeed in STEM
fields. In order for students to be able to take the high level mathematics courses in high school
to be eligible for STEM fields, they need a strong foundation in mathematics in the elementary
grades. Teacher preparation programs are tasked in the preparation of elementary teachers to
teach mathematics to build a strong foundation in their students (Manouchehri, 1997; Noori,
1994; Speilman & Lloyd, 2004). Teacher preparation programs implement content courses to
help build content knowledge of teachers, design methods courses to teach the pedagogy of the
subject, and place preservice teachers in a full-time teaching experience as the culmination of
their program (Bales & Mueller, 2008). However, research on elementary preservice teachers
and mathematics often focuses on their development within content courses (Alsup, 2003;
Emenaker, 1996; Even et al., 1996; Gresham, 2007; McLeod & Huinker, 2007), methods courses
(Lonning & DeFranco, 1994; Quinn, 1997; Stuessy, 1993), or student teaching (Clark, 2005;
Johnson, 1980; Lubinski, Otto, Rich, & Jaberg, 1995; Philippou, 2003; van Es, 2009).
A few researchers have studied elementary preservice teachers beyond just the content
course or methods course (Castro, 2006; Lubinski et al., 1995; Wilcox, Schram, Lappan, &
Lanier, 1991). Castro (2006) studied the way elementary preservice teachers conceived of
curricular materials for mathematics teaching in a mathematics content course and subsequent
mathematics methods course. Preservice teachers conceived of tools as learning tools or as
instructional aids for the teacher. Lubinski et al. (1995) studied 6 elementary preservice teachers
during mathematics methods class and student teaching. The authors determined that they could
not determine the effect of the methods course on mathematics teaching during internship. Other
factors that influenced the preservice teachers during internship included the cooperating teacher,
classroom environment, personal maturity, and depth of mathematical understanding. Wilcox et
163
al. (1992) concluded, after observing beginning teachers through their preparation process, that
teacher conceptions of teaching and learning influence instructional decisions.
Preservice teachers in this study were in science and mathematics methods classes which
were taught with the learning cycle, an inquiry-embedded approach, see Chapter 4. Two
semesters after the science and mathematics methods courses, preservice teachers entered their
student teaching experience. Student teaching is 15 weeks of being in a classroom full-time.
During the student teaching, the classroom teacher gradually releases all of the responsibilities of
the classroom over to the student teacher. In order to ascertain how preservice teachers carried
over practice of tools for conceptual development, processes for learning, and the learning cycle
approach in mathematics into their student teaching, preservice teachers needed to be studied as
they moved through the teacher preparation process.
Research Questions
Since integration of theory and practice continue to be important in teacher preparation
(Fennell, 1993), research needs to be conducted as preservice teachers transition from teaching in
a methods course through the student teaching experience (Jong & Brinkman, 1999). A lack of
research exists on the development of elementary preservice teachers and their thinking about
common pedagogical approaches as they progress through their teacher preparation program.
Manouchehri (1997) called for researchers to conduct long term studies on change of preservice
teachers and how this change is exhibited in teaching practices. Methods of inquiry in teaching
science and mathematics share common processes for meaningful learning and use of tools for
conceptual development (NCTM, 2000; NRC, 1996). This study intended to examine preservice
teachers? conceptions of mathematics teaching in relation to common processes, tools, and
pedagogical approaches while completing the student teaching experience. The question of this
164
study was: How did preservice teachers put into practice in student teaching their thinking from
the methods courses on tools for conceptual development, processes for meaningful learning, and
pedagogical approaches to mathematics teaching?
Literature Review
Common Processes
Processes for meaningful learning are described in the Principles and Standards for
School Mathematics (NCTM, 2000) and the National Science Education Standards (National
Research Council (NRC), 1996). Process skills are also considered thinking skills (Sambs, 1991).
These processes include problem solving, reasoning and proof, communication, connections, and
representations (Goodnough & Nolan, 2008; Justi & van Driel, 2005; NCTM, 2000). Processes
must be facilitated by the teacher in order for students to ?make meaning of their world by
logically linking pieces of their knowledge, communication and experiences? (Jaeger &
Lauritzen, 1992, p.1). In mathematics students are asked to solve problems, reason about their
answers and demonstrate proof, represent their work, and communicate their findings (NCTM,
2000). Similarly in science, students are expected to be able to ask questions and then find the
answers to those questions, validate findings, explain their findings, and be able to evaluate the
conclusions of others (NRC, 1996). Furthermore, in mathematics and science, students are
expected to make connections or identify relationships between and among concepts and areas of
study (NCTM, 2000).
Problem Solving. Problem solving is one of the processes common to science and
mathematics teaching. ?Problem solving means engaging in a task for which the solution is not
known in advance? (NCTM, 2000, p. 52). For science, the problem solving process often occurs
during scientific inquiry in which students are finding out the unknown (NRC, 1996). Whether
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called problem solving, as in mathematics, or a part of inquiry, as with science, students have to
learn how to gather information, record collected data, and offer answers and explanations of
those answers (NCTM, 2000; NRC, 1996). Since not all problems are simple, problem solving
causes the learner to wrestle with alternative solutions and take risks in thinking (Manouchehri,
1997, NCTM, 2000). Thinking about alternatives in solutions allows the learner to expand
problem solving skills and gain insight into the content that can be reapplied in later situations
(Manouchehri, 1997). Swartz (1982) expounded on the importance of problem solving using a
variety of strategies to gain understanding. The problem solving process, therefore, promotes the
development of thinking skills (NCTM, 2000).
Problem solving often begins, for both science and mathematics, in problem posing
(NCTM, 2000, NRC, 1996). Children have a natural curiosity about the world in which they live
(NCTM, 2000). This natural curiosity leads them to question and explore the world around them.
Students build knowledge through the problem solving process (Goodnough & Nolan, 2008;
NCTM, 2000). Problem solving provides a means for teachers to create a mathematics and
science classroom that fosters teaching through conceptual learning rather than rote
memorization (Molina, Hull, Schielack, & Education, 1997; NCTM, 2000).
Reasoning and Proof. Reasoning and proof help students develop logical thinking that
helps one decide if an answer makes sense. Students develop guesses or conjectures about
concepts, experiments, or observations (NCTM, 2000; NRC, 1996). In mathematics, students
note patterns. They use reasoning and proof processes to determine if the ?patterns are accidental
or if they occur for a reason? (NCTM, 2000, p. 56). Reasoning and proof includes the ability of
students to use counterexamples to disprove a conjecture (Chick, 2007; NCTM, 2000). Similarly
in science, students use reasoning and proof processes in experiments and tests to determine if
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the results are consistent or conditions that make the results different (NRC, 1996). Reasoning
and proof are especially critical in science due to the nature of science requiring evidence of
conclusions based on experiments and observations (NRC, 1996).
Communication. As students are involved in problem solving and reasoning and proof
of a situation, communication becomes another common process that emerges. Communication
involves all aspects of communication including talking, writing, and listening about the concept
being learned (NCTM, 2000; NRC, 1996). Communication allows students to refine their
thinking and cement ideas (Goodnough & Nolan, 2008; NCTM, 2000; Zull, 2002). Classroom
discourse becomes a critical arena for students to share, question, and revisit ideas (NCTM,
2000). Students benefit from learning about the way other students in the class think about and
solve problems (NCTM, 2000). Communication also aids students in the developing of more
formal mathematical language (NCTM, 2000). Since science is based on experiments and
observations, communication in science is important for students to express their understanding,
make predictions, and develop conclusions of experiments and observations (Goodnough &
Nolan, 2008; NRC, 1996). In mathematics communication becomes important as students
present their proofs for other students in the classroom to understand the reasoning behind the
proof (NCTM, 2000).
Connections. Connections allow students to have a deeper understanding of a concept
(NCTM, 2000; Zull, 2002). Connections can be made within topics, to other subjects, or to
experiences (NCTM, 2000). Furthermore, connections help students to link concepts rather than
learn isolated facts (NCTM, 2000). In mathematics, students may complete a series of inquires
investigating the relationship of the volume of three-dimensional figures with the same height
and base. Their investigation will aid them to make connections between the two volumes to
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draw conclusions about the volume of cones and cylinders. In science, students may take their
knowledge of what makes a simple circuit to make connections to series or parallel circuits.
Representations. As students solve problems in mathematics and science they rely on
representations to express ideas (Davis & Petish, 2005; Justi & van Driel, 2005; NCTM, 2000).
Representations are not only what occurs on paper, but what also occurs in the mind (NCTM,
2000; Zull, 2002). Symbols, diagrams, graphs, and images are examples of representations
(NCTM, 2000; Posner, Strike, Hewson, & Gertzog, 1982). In science as students test a simple
circuit consisting of a battery, a light bulb, and connecting wire, students represent on paper their
various tests. Similarly, in mathematics, as students determine combinations of outfits they
represent the combinations through colored squares or drawings labeled by their own invention.
Concrete tools allow students to begin to develop more complex forms of representations
(NCTM, 2000). However, there are situations in which representations may serve as an obstacle
if the learner does not know how to interpret the representation (Heywood, 2007). For example,
if students in a classroom did not understand how base 10 blocks represented the number system,
this would be an obstacle in using the representations. Communication among students about
their representations can serve as providing cohesiveness with the mathematical thinking and the
representation of that thinking. Furthermore, representations allow students to relate concepts to
the real world and are important in communication, reasoning and proof, and problem solving
(NCTM, 2000).
Common use of Tools for Conceptual Development
Experiences impact the way students learn and understand (Hoffer, 1993; Speilman &
Lloyd, 2004; Zull, 2002). Concrete tools offer opportunities for direct experiences and added
opportunities for learning (Hoffer, 1993; Marek & Cavallo, 1997; Zull, 2002) as well as
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supporting real world mathematical and science situations (Jurdak & Shahin, 2001; NRC, 1996).
The National Research Council (1996) recommended that students must have tools in order to be
able to directly investigate scientific phenomena. Tools for developing concepts are often in the
form of models (Justi & Gilbert, 2000). Learning aids in mathematics are known as
manipulatives, and student use should precede symbolic notation (Sriraman & Lesh, 2007).
Mathematics and science relies on representations to express ideas (Davis & Petish, 2005; Justi
& van Driel, 2005; NCTM, 2000; NRC, 1996). Concrete tools allow students to begin to develop
more complex forms of representations which allow students to relate concepts to the real world
and are important in the processes of communication, reasoning and proof, and problem solving
(NCTM, 2000).
Merely giving students concrete materials does not ensure that connections will be made
or deep levels of understanding will be attained (Justi & Gilbert, 2000). When teachers plan
lessons, choices are made as to the representations or models to be used in the lesson (Chick,
2007; Goodnough & Nolan, 2008). Teachers determine questions, explanations, and tasks
involving the tool while keeping in mind the learning objective for the lesson (Chick, 2007;
Goodnogh & Nolan, 2008). Teachers must provide a means for students to make connections
from the experiences with concrete tools (Bleicher, 2006).
Tools become a helpful resource in science and mathematics for students to be able to
reason and provide proof of their solution (NCTM, 2000; NRC, 1996). It is recommended that
students use concrete materials as a means of investigating conjectures held about concepts
(NCTM, 2000; NRC, 1996). Tools for mathematical calculations are often used in science as a
means of providing evidence for the conjecture: measurement devices for time, length, capacity,
temperature, and weight. Tools in mathematics for reasoning and proof may involve
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measurement devices but they might also be strings or numbers, calculations, or diagrams. For
example, children in a science classroom may use a magnifying glass to observe organisms or
objects to answer a question. Their observations then lead them to look for patterns or to list
common features. This is very similar to students in a mathematics class examining and sorting
pattern blocks to determine which ones fill a space. In both scenarios students are actively
involved in problem solving with tools as a means of providing reasoning and proof to their
conclusions.
Common Pedagogical Approaches
Focusing on processes and tools for conceptual development makes the learner an active
part of the learning process. Educators have approached teaching based on the idea of active
learning through the process of exploration, investigation, and articulation and have called this
the learning cycle approach or learning cycle. The learning cycle offers a way of incorporating
similar teaching methods of using tools for conceptual development (Chick, 2007; Fuller, 1996;
Hill, 1997), discourse (Akerson, 2005; Heywood, 2007; William & Baxter, 1996), assessing
student knowledge (Manouchehri, 1997; NCTM, 2000; NRC, 1996) inquiry-based teaching
(Morrison, 2008; Manouchehri, 1997; NRC, 1996; Weld & Funk, 2005), and reflection
(Bleicher, 2006; Hill, 1997; Manouchehri, 1997). Components of inquiry are evident in the
learning cycle approach (Gee, Boberg, & Gabel, 1996; Tracy, 1999; Withee & Lindell, 2006).
Inquiry-based teaching involves the exploration of students around a central idea, formulation of
questions, investigations to answer the questions, and reflection of learned ideas (Morrison,
2008; Tracy, 1999). Furthermore, the learning cycle approach provides a framework for
structuring inquiry-based lessons (Marek, Maier, & McCann, 2008; Marek & Cavallo, 1997). As
a framework, the learning cycle provides a means for inquiry and its processes, problem solving,
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reasoning and proof, communications, and connections, to be applied. Within the learning cycle
framework, tools serve as a means for conceptual development across all processes.
Although the learning cycle has its roots in the science field (Bybee, 1997; Atkin &
Karlpus, 1962), the tenets align closely with the national mathematics standards. Similarly,
Marek and Cavallo (1997) explained that the learning cycle could be used in mathematics for
problem solving, and provided examples of learning cycle lessons to teach measurement and
geometry concepts. In the engagement phase teachers have students focus on an event or
problem (Bybee, 1997). This may be accomplished through a question, a situation or problem
(Bybee, 1997; Marek et al., 2008). Teachers may even present a discrepant event as a means of
engaging students (Marek et al., 2008; Tracy, 2003). Students then move into the exploration
phase. Teachers have the responsibility of providing the materials, observing and ensuring
students are conducting the experiment correctly, and interacting with students while students are
collecting data (Marek, 2008). Physical experiences in the exploration phase are necessary
(Bybee, 1997) and allow the learner to move beyond initial observations to generalizations
(Renner & Lawson, 1973). In the explain phase, students are explaining what they discovered in
the explore phase (Withee & Lindell, 2006). Once the students have an explanation for their
experiences in the explore phase, the learning moves to the elaborate phase (Bybee, 1997). In
this phase students are applying or extending what they learned to a new situation (Atkin &
Karplus, 1962; Marek, 2008). Evaluation is the last phase but does not have to occur last
because it can occur throughout the lesson (Bybee, 1997; Marek, 2008; NCTM, 2000).
Student Teaching and Mathematics
The student teaching experience is typically a fifteen week full-time teaching experience
and is intended as a time to help new teachers solidify their practice. Novice teachers are
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expected to take what they learned about teaching from methods courses and put it into practice
during the full-time teaching experience (Grossman et al., 2009). Preservice teachers learn the
day-to day routines of being a teacher during the student teaching experience. This experience is
affected by the culture and norms of the school in which it takes place (Cuenca, 2011). The
cooperating teacher serves as the guide to help the student teacher with the norms of teaching
(Feiman-Nemser & Buchmann, 1986). The context of the school and the norms of the
classroom, established by the cooperating teacher, influence the work of the beginning teacher
(Cuenca, 2011; Feiman-Nemser & Buchmann, 1986).
Inherently the culture of the school may clash with the philosophy of the student teacher
and the preparation institution (Crawford, 2007). Some cooperating teachers are open to the
student teacher teaching reform-based lessons (Philippou, Charalambos, & Leonidas, 2003). In
other cases the traditional norms of the classroom inform the student teacher of the manner in
which to teach mathematics (Lubinski et al., 1995). In the case of mathematics teaching, barriers
may also arise in the use of tools for exploration (van Es & Conroy, 2009). Furthermore,
Crawford (2007) noticed that skepticism of reform practices developed with student teachers
during student teaching due to the culture clash of the assigned classroom and the university
philosophy of teaching.
Preservice teachers, however, improve their mathematics teaching during internship in
several ways. Clark (2005) observed preservice teachers modeling mathematical concepts
effectively. Teacher candidates also developed skills in the selection and design of mathematics
lessons (Johnson, 1980). Their teaching experiences helped them realize that sometimes they
would have to digress from the lesson plan in order to meet students? mathematical needs (Clark,
2005; Lubinski et al., 1995). Likewise, student teaching allowed preservice teachers to teach
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hands-on lessons and design lessons that helped students make real world connections (Clark,
2005; Lubinski et al., 2005). Teacher candidates who did use more hands-on lessons and helped
students make real-world connections also recognized students had a deeper understanding of the
concepts (Clark, 2005).
Methodology
Case Selection
The cases were two elementary preservice teachers who had already participated in a
study on conceptions of common approaches to teaching mathematics and science, and who were
completing their student teaching. These two cases were selected because they were placed in
schools and with teachers who had participated in a state-wide mathematics initiative. The
student teachers had access to reform curricula and materials. The cooperating teachers gave
them the freedom to design and implement their lessons as long as they met the given teaching
objectives. Also, based on their methods course data, they represented two different ways of
approaching mathematics. The result is the following case of ?Jane? and ?Kate?. Jane and Kate
are pseudonyms.
Jane and Kate were in jointly enrolled science and mathematics methods courses two
semesters before their student teaching. Teacher candidates completed science methods the first
part of the semester and mathematics methods the second part of the semester. The purpose of
the methods courses was to prepare the teacher candidates in methods and practice of teaching
reform-minded science and mathematics. The learning cycle, with inquiry embedded, was the
focus of both courses. It was used as a common approach to teaching both methods courses.
Furthermore, Jane and Kate spent time in field placements during the science and mathematics
methods class using the learning cycle as a framework. During the first five weeks of the
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semester they practiced teaching science in field placements using tools and fostering processes
with the learning cycle approach. They continued to use tools and foster processes with the
learning cycle approach when they moved into the mathematics portion of the methods course.
At the beginning of the science and mathematics methods course, Jane and Kate appeared to
have quite different conceptions of science and mathematics and how the two subjects should be
taught. Jane conceived of tools, processes, and approaches for students to see and understand the
mathematics. Kate conceived of tools and processes for real-life mathematics with a teacher
directed approach. However, by the end of the methods courses they both described mathematics
as being taught with the learning cycle approach but in different ways.
Context of the Study
Jane and Kate were both placed at the same school to complete their student teaching.
The school is a rural school that is 80% Caucasian with about 50% of the students on free or
reduced lunch. The teachers in the school had participated in ongoing state sponsored
professional development for teaching reform-based mathematics and science. Since the teachers
had participated in professional development institutes, Investigations in Number, Data, and
Space (Pearson, 2007) kits, materials, and curricula were available for the teachers to use in
teaching mathematics. The student teachers were placed under the supervision of a group of
teachers within the school. The cluster teachers were responsible for the supervision,
observation, and general guidance of the student teachers. The student teachers were placed with
a single cooperating to learn the day-to-day operations of a classroom.
Jane was placed in a third grade self-contained classroom. She was responsible for
teaching all subjects. She had eighteen students in her class. Kate was placed in a fourth grade
classroom. The students in her class switched with two other fourth grade teachers. She was
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responsible for three mathematics classes. The students in the mathematics classes were assigned
to a class based on their mathematics ability level. Students were considered to be low, medium,
or high ability levels.
The study took place in the assigned classrooms of the student teachers. Being in the
classroom environment allowed the researcher to acquire information firsthand, learn about the
daily routines, and become well familiarized with the context of the learning environment for
each preservice teacher involved in the study (Bogdan & Biklen, 1982; Glaser & Strauss, 1967;
Hatch, 2002; Marshall & Rossman, 1995; Merriam, 1998). The researcher served as the primary
investigator who attempted to record the phenomena, person, and/or interactions being studied
and did not serve in any supervisory role (Hatch, 2002; Lancy, 1995). The researcher served as
an instrument for data collection since the researcher?s sense-making influenced what the
researcher distinguished as important in the setting (Hatch, 2002; Merriam, 1998). The human
factor also meant that bias was inherent in the observations and analyses due to the fact that a
human collected, investigated, and made determinations based on a human?s knowledge
(Merriam, 1998).
For this study a case study approach was used (Hatch, 2002; Merriam, 1998; Stake, 2005)
with a grounded theory approach to analysis (Charmaz, 2005). More specifically these were
narrative case studies (Connelly & Clandinin, 1986). Connelly and Clandinin (1986) believed
that the story of the teacher in context serves as a means of understanding. They posit that
teachers themselves may gain new knowledge but that this knowledge may not be actually
reflected in their practice. Furthermore, they suggest that the only way to understand a teacher?s
knowledge is to experience the knowledge in context. They believe that teacher knowledge is
composed of experiences from personal and social contexts (Connelly & Clandinin, 1986).
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The two weeks of full-time teaching required of student teachers was selected as the time
to observe and collect data. In both cases the cooperating teachers provided the student teachers
with a list of objectives to meet during the time frame. The student teachers were responsible for
planning lessons, implementing lessons, and the daily routines of an elementary classroom. The
student teachers had access to the curricular materials for that grade. Jane was observed teaching
six times and Kate was observed teaching five times during that time period. Lesson plans were
provided at the beginning of the week.
Data Collection
Data sources included six teaching observations for Jane and five teaching observations
of Kate, follow-up interviews after each observation (see Appendix A), lesson plans, and an
open-ended final reflection (see Appendix B). The lesson plans served as a means of seeing at a
glance if the observed lesson was what the preservice teacher had planned. The follow-up
discussions were important to gain information on the student teacher?s goals for the lesson and
decisions that went into the structure and implementation of the lesson. The final reflection was
similar to the final reflection from the methods course and provided another data source for the
preservice teacher to reflect on their teaching experiences.
The researcher recorded observations of mathematics lessons taught by the student
teachers in the cases presented (Hatch, 2002). Notes that were taken included the arrangement of
the rooms, the types of materials used in the lesson, questions the teacher asked, how much time
students spent on different parts of a lesson, questions and answers students presented, and
overall impressions of the lessons. After observing and taking notes during the lessons, the
researcher would discuss the lesson with the student teacher. Discussions followed each
observed teaching episode. Discussions included not only questions pertaining to the study but
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the researcher referred back to information from the methods courses. Discussions began with a
general question about how did they think the lesson went. This opening question was used as a
means to have the student teacher begin to reflect on the teaching episode. Questions in the
discussion that followed referred back to the common teaching approaches used in the science
and mathematics methods classes (see Appendix A). Those questions inquired about the learning
cycle as a common pedagogical approach, the role of inquiry as a means of fostering processes,
and how they used tools to develop the concept. The researcher would pose the question, record
the response, and read it back to the student teacher. Additional comments that were made or
revisions from the student teacher were recorded.
As their mathematics methods instructor, the researcher was able to discuss and recall
elements of the methods courses as a means of helping the preservice teachers connect methods
course with student teaching. After the general discussion questions (see Appendix A) the
researcher would then begin to probe the preservice teachers to reflect on the observed lesson
and ascertain the development they were trying to achieve as the week progressed. Based on
their responses more questions were asked to gauge what they remembered from methods class.
This allowed the researcher to mention materials or lessons that the student teachers may have
forgotten about since a semester had lapsed between methods courses and student teaching. It
also provided the student teachers with time to ask questions about tools, teaching approaches,
student management, or organization of lessons. The researcher was intrinsically involved in
helping the preservice teachers make connections to methods class through the debriefings that
occurred after the teaching episodes. The researcher served as part of the teacher development
process by focusing preservice teachers on ideas from the methods course including the learning
cycle as an inquiry embedded approach, use of tools, and development of processes.
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Data Analysis
Triangulation of data was used to ensure trustworthiness of the study (Yin, 2009).
Triangulation relied on checking the information throughout various data sources. The constant
comparative method (Glaser & Strauss, 1967) was used to find conceptions of mathematics and
science teaching that emerged from the varying data sources. For each student teacher each
interview, observation, lesson plan, final summation and memo was coded based on the research
questions (Merriam, 1998). Information that was striking to the researcher in regard to preservice
teachers? thinking about how they linked ideas in practice across disciplines was noted for each
data source (Merriam, 1998). Codes were determined based on words or phrases from the
participants, conclusions from the researcher, or connections to existing research (Glaser &
Strauss, 1967; Merriam, 1998). Data about tools, use of tools, plans involving tools and so forth
were labeled with the code of T. Data about processes that took place during the lesson were
labeled with a P. Information about pedagogical approaches was labeled with a PA. Data that
referred to the learning cycle was coded as LC. Lesson plans were labeled as LP, observations as
O, follow-up interviews as FI, and final reflection as FR. For example, Jane?s lesson plan for the
first observation stated ?Review of measuring to the nearest inch? (J, LP, 4/14). In the observed
lesson Jane did review how to measure to the nearest inch using items on the overhead projector.
These statements were both coded as ?tools for measurement?. Since Jane led the students step
by step through a series of measurements this was also coded as ?teacher directed?. During the
follow-up discussion when asked ?In what ways did you use tools for conceptual development?
Jane said ?Showing them the different items on the projector? (J, FI, 4/14). This was coded as
?tools for measurement-teacher directed?. Then the first teaching observation and follow-up
discussion was read through again for the case to determine if commonalities existed that could
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group some of the items together or if other generalizations became apparent (Merriam, 1998).
Comparing lesson plans to teaching observations to follow-up discussions provided verification
of the data. For example, the researcher could see that in the lesson plan the students would be
measuring, the students somewhat measured in the teaching episode, and in the follow-up
discussion the student teacher confirmed that the lesson did not go as she had planned. This
process of constant comparison continued for each set of data for each case (Charmaz, 2006;
Glaser & Strauss, 1967; Merriam, 1998). Once all of the coding had been completed a table was
constructed for each participant. All of the codes for tools, processes, and approaches were
placed in separate sections with the information in each section placed in chronological order.
Codes were then regrouped to fit like items together (Saldana, 2009). Examining the codes
together allowed the researcher to then narrow the codes to succinctly represent the data. These
narrower codes were placed in another column next to the lengthy code list (Saldana, 2009).
Once the data collected from the student teaching experience had been coded the
information was compared to the data from the methods course study. Data from the methods
course had been examined in the same way as during the student teaching. Therefore, the code
lists for the participants were pulled from the methods course data. The coding for the methods
course was also in a similar table for tools, processes, pedagogical approaches, and the learning
cycle. Codes for tools during methods class and tools for student teaching were read through to
determine similarities, differences, or changes. This continued for processes, pedagogical
approach, and learning cycle codes. The researcher tried to determine what changes, if any,
occurred from mathematics-science methods class to student teaching in understanding and
development of approaches in mathematics. The codes were examined in chronological order to
determine shifts or changes in practice. Systematic searches were made to find corroborating or
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contradictory evidence. Generalizations were then made within each case across the different
periods of data collection. Finally the two cases were compared.
After reviewing the data sets using grounded theory (Charmaz, 2006; Glaser & Strauss,
1967), a number of themes emerged. The remaining sections of this paper present an analysis
and interpretation of Jane and Kate?s development of approaches to mathematics in student
teaching that first emerged in the methods courses.
Jane and Kate: Methods Class
Jane and Kate had varying conceptions in their methods course of tools for conceptual
development, processes for meaningful learning, and pedagogical approaches. These conceptions
changed somewhat from when they initially entered the methods course to when they completed
it. Table 5 and Table 6 provide a summary of those areas for each case. Initially, Jane conceived
of teaching both in a way that would allow students to ?see? and ?do?. In contrast, Kate
conceived of hands-on teaching in mathematics for real-life concepts such as time and money.
Jane focused more on the engage and explore phases since this helped students to ?see? and ?do?
the mathematics. Kate focused on the engage phase as a means of connecting with students?
prior knowledge. At the end of the methods course it appeared as if the learning cycle approach,
particularly the explore and engagement aspects, had affirmed Jane?s ideas of how mathematics
should be taught. It also seemed as if Kate embraced the learning cycle as means for students to
engage and explore concepts but under teacher direction. Observing them during their student
teaching provided information to see if they had maintained or altered their conceptions of
teaching mathematics using tools, processes, and the learning cycle approach.
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Table 5
Jane-Methods Class
Initial Conceptions End Conceptions
Tools Hands-on;
?See? concepts
Students to conduct experiments;
Learn for themselves
Processes Reasoning Peer discussion and questioning
techniques to foster reasoning
Pedagogical
approaches
Experiments; Games and
activities
Inquiry-based for both subjects
Approach to
teaching science and
mathematics
2 different approaches
Science as experiments-
mathematics as games and
manipulatives
LC and tools for concept
development for both subjects.
Notebooks for science-games and
activities for mathematics
Table 6
Kate-Methods Class
Initial Conceptions End Conceptions
Tools In science-to enhance
concepts in science book
In mathematics- tools for
real life (i.e. time and
money)
Teacher controlled use of tools for
both subjects
Processes Teacher directed
experiences
Limited processes (reasoning,
connections, multiple
representations) due to need to
control the lesson
Pedagogical
approaches
Teacher explains-student
see examples
Hands-on in science for reward or
treat
Hands-on in mathematics for real
life (i.e. money, time)
Approach to
teaching science and
mathematics
2 different approaches
Science use hands-on
mathematics use real life
experiences
Learning cycle and inquiry in both
Science use hands-on
Mathematics use everyday concrete
objects
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Results
Jane: Student Teaching
At the beginning of her two weeks of teaching, Jane was overwhelmed with the
responsibilities of designing, planning, and implementing every subject. The first observed
mathematics lesson was poorly designed and implemented. Initially her approach to teaching
mathematics was very inefficient. Her first lesson focused on tools in a procedural way. Students
measured two items in a fifty minute time span. Her struggle with discipline along with poor
lesson planning interfered with her teaching of the lesson. She seemed to take a step backwards
from the way she conceptualized tools in the methods class. In methods class she conceptualized
tools as a way for students to learn abstract concepts through concrete experiences. Her
mathematics teaching in the internship may have been patterned after the way her cooperating
teacher taught mathematics or she may have fallen back on the way she experienced
mathematics.
The second lesson also focused on measurement. She demonstrated how to measure and
distributed buckets. The buckets had materials for the students to measure items to the nearest ?
inch, ? inch, and inch. Tools were once again used for procedures. After observing the second
lesson it was clear that Jane was making up her own mathematics lessons. They were clearly
missing the components of a quality lesson. These missing components led to of-task behavior.
This second lesson was just as poorly done as the first lesson: ?It is torture for me to sit and
watch such off-task behavior. The lack of management is interfering with the mathematics and
the poorly designed lesson is encouraging additional discipline problems? (Researcher journal,
4/15).
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When asked what specific teaching practices she wanted to improve she said, ?Having
the materials prepared and ready. Making sure they know the difference between ? and ?.
Making sure they weren?t fighting.? (J, FI, 4/15). The areas she mentioned she wanted to
improve were not in her lesson the day before. Although she was trying to improve her practice,
she clearly needed more guidance in thinking through all parts of a lesson: how would the
concept be presented? How would students be engaged? How would students be accountable for
their learning? After the second lesson observation those were the types of questions I asked her
to think about in the next mathematics lesson she would be teaching. I told her that until she
became a strong experienced teacher that she needed to use research-based curriculum and think
of the elements of the learning cycle in her lesson development. We discussed resources she had
that she could use to teach a successful mathematics lesson.
Her teaching style was often very teacher directed at the beginning of class. Then she
would have students complete a task to explore or extend the concept further. Throughout her
teaching she used questioning as a means for students to reason about the concept she was
teaching. She did this in her field placement teaching during her methods class as well. She
wanted students to understand for themselves and be able to explain their reasoning. She would
often follow with, ?How did you know?? When asked in the follow-up interview ?What role did
the learning cycle play in your lesson development and implementation? She responded,
?Engage part. I wanted to get them going. Evaluate. I wanted to see how they did.? (J, FI, 4/14).
She did try to engage her students, but they were not actively involved in any new learning.
Furthermore, her ideas about the aspects of the learning cycle were very limited. Engagement in
the learning cycle means students are focused on a problem or event. She interpreted
engagement as the students participating in the assigned task, which was not a problem or event
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in the aspect of inquiry-based teaching. She similarly focused on the engagement and explore
phases of the learning cycle during methods class. In her blogs in methods class she often
referred to the exploration aspect of her lessons. She did use questioning techniques to have
students think about concepts. ?Stack your books in the middle of the table. What did you figure
out about the books? What is something you concluded? What is something you noticed?? (J, O,
4/14).
The third lesson showed dramatic improvement from the first lesson, and seemed to be
the turning point for Jane. In the third lesson she had the students vote on their favorite sports
team and pet when they arrived at school. During the mathematics lesson she had them tally the
information and relate different ways the information could be displayed. She used the NCTM
website, Illuminations, to illustrate how graphs look different based on the information they
represent. She then led them back to the data they collected earlier that morning. Based on the
data, she had them describe what a graph of the data would look like. She gave them tiles to
make a graph on their desk. In the lesson she used a variety of tools to connect different
representations of data. She used drawings, the interactive website, the students, and tiles to help
them ?see? the mathematics. Her questions demonstrated she wanted students to reason and
make connections with the representations. This was a similar pattern for her teaching in
methods class. She used visual representations and questions to help students understand. When
asked about the role of the learning cycle in the lesson Jane responded,
Explore-Thinking about the other day. Thinking about the 5E?s. Engage-Asking
questions who remembered about bar graphs. Explore-Using the computer to see how it
looks. Elaboration-working with the tiles. Sketching it out. How they were doing it on
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their own. Evaluation-Watching them fill in the graph. Used the Handful lesson from
Investigations. (J, FI, 4/16)
The behavior management issues were noticeably less in the third observation as well. The
students were engaged, she moved the lesson along at a good pace, and she connected the data to
the students? interests.
By the end of the two weeks teaching she attempted to develop well thought out lessons.
During the times that she was observed teaching mathematics, her skills in selecting tools and
management of the tools improved. Moreover, by the end of the two weeks she was using a
variety of tools within a lesson to model concepts for students as well as tools for students to use
themselves to understand concepts. For example, she used fraction magnets on the board, animal
pieces on the document camera, and the students had their own fraction circle sets to understand
fractional parts. She thought of more than just the engage phase. When she began to develop
lessons, thinking about all phases of the learning cycle, they were more successful. In the data
lesson she intentionally thought about what students would be doing for each phase of the
learning cycle. However, she interpreted the learning cycle as a set of steps to follow rather than
thinking about it as a cyclical learning process. With additional coaching, she may have
developed a stronger understanding of teaching math with the learning cycle.
Because students were engaged and on-task, the quality of the lessons the second week
was much better than the quality of the lessons the first week:
I am amazed at Jane?s teaching today. By no means was it a perfect lesson. She still had a
couple of behavior issues to deal with, but wow was it different from the first two lessons
I observed. We talked after the last observation about different fractional representations
to help students understand fractional concepts. She did make the fraction circles so the
185
students could each have their own sets. She needed to move the lesson along a little bit
faster. Students were getting bored. But all-in-all the quality of the lesson was so much
better. (Research journal, 4/23)
During the first week when she seemed to be floundering she fell back on a more traditional
approach to mathematics. During the second week when she planned better lessons and used
standards based resources from National Council of Teachers of Mathematics and the
Investigations (Person Education, 2007) series her approach was more hands-on in nature. Her
approach the second week influenced the way she used tools as well. Her use of tools the second
week was for concept exploration and understanding not just for procedures. In her final
reflection she noted,
I taught a fraction review lesson with fraction circles. These were the concrete tools I
used for each student. I engaged them by showing them a program on Illuminations site
with the fraction circle. I asked them what they thought the circle would look like with
certain fractions. I then had them explore with the fractional pieces that were in the bags
to see what fractional parts they had. I had them apply what they knew about fractions to
find equivalents with their pieces (J, FR).
She also spent less time focusing on discipline the second week because of all of the combined
factors: well thought out lesson plans, student engagement, and student involvement throughout
the lesson.
Jane appeared to take a step backwards initially in mathematics teaching from where she
ended in methods class. With some coaching, she was able to develop better lessons, although
not true inquiry-embedded lessons. Her use of tools shifted from tools for procedures to tools for
reasoning. She maintained throughout methods and student teaching the practice students
186
explaining how they understand. From the first day of methods class she indicated that she
wanted students to be able to ?see? the mathematics. With coaching, she was able to develop
lessons that fit her need for students to ?see?. Her approach maintained a traditional approach in
which the teacher demonstrates, students complete practice, and the teacher reviews. The
practice element for the students often involved hands-on materials.
Kate: Student Teaching
Kate?s first observed lesson was also procedure focused and did not involve inquiry
within the learning cycle. In the first lesson Kate directed students as they played various games
to promote speed of multiplication facts. She kept students on task throughout the various fact
games they played. She summarized the lesson by reminding students of tricks or devices to
help them remember certain ones. ?Remember 5,6,7,8? [writes on the board 56=7x8] Who
remembers the 9 trick? [Shows the 9?s with fingers]? (K, O, 4/20). Although the students were
well managed, I was concerned about the quality of task she had designed for her teaching time.
I just observed a lesson in which students practiced multiplication facts for 45 minutes.
Students are not involved in any new learning. This is for speed of facts. That does have
its place but within a context. I know she had the freedom to design this lesson. I am
concerned that she resorted to this type of lesson because it is comfortable for her to
implement (Researcher journal, 4/20).
I did not see where the students were thinking about mathematics in deep and meaningful
ways. Processes in her initial mathematics lesson were for facts. I asked her in the follow-up
interview what the goal of the lesson was and she said, ?for students to get back into the swing of
things? (K, FI, 4/20). After the multiplication fact lesson we discussed what she had planned for
the week. She said she was going to move to division. I asked her how she was going to teach
187
this. She said she was just going to show them how. I asked her if she thought about using an
Investigation (Pearson, Education, 2007) to teach division. I asked her to think on her
mathematics methods classes and ways to make lessons more meaningful for students. We
discussed representations that would help students see the relationship between multiplication
and division. She said she would look into the tower investigation for the next lesson. The tower
lessons involve students working in pairs to record multiples of a given number on a strip of
calculator tape. The tower is then used over several days as students complete tasks and explore
the relationships of the numbers in the tower.
For the second observation I observed her teaching the multiple towers lesson. In that
lesson students were relating multiplication and division, finding patterns, and reasoning about
numbers. The nature of the number tower lesson focused the students towards reasoning and
communication of ideas. Kate modeled how to record the number tower:
I am already at my hip. What number do you think I will be at my shoulder? Let?s make
some predictions [writes on board as student call out]. Who has a prediction of what my
number is going to be when I get to the top? Who is noticing some patterns as we go up??
(K, O, 4/21).
I could tell Kate was nervous as she taught the lesson. She stammered and often referred
to the teacher guide in following the lesson. Her students answered questions about the example
tower and then made their tower with their partner. At the end of the lesson we discussed how
the lesson went. She told me she thought it went ?pretty good for teaching it for the first time?
(K,FI, 4/21).
Her second mathematics lesson in comparison to her first was of much greater quality.
Students used towers as tools to bridge the concepts of multiplication and division. Students used
188
processes that focused on concept building. The mathematics lessons that followed used
research based lessons and the use of tools were for different purposes. Students were using
number towers, calculators, and problems to understand division concepts. In mathematics she
changed her pedagogical approach from a fact based teacher directed one to a more hands-on
student exploring approach but with a great amount of teacher direction.
On the third day she continued with the tower lesson. She had students complete a
number walk around the room. The students had to find the 10th, 15th, and 20th numbers in the
towers. They also had to record patterns they noticed in a tower. In her mathematics methods
class she taught a lesson in which students had to figure out a pattern. She ended up giving
students the pattern instead of giving them time to figure it out. Her number tower lessons were
very similar. If students didn?t figure something out, she would tell them. She did use the tower
as a tool to develop multiplication and division, but she interfered in students figuring the math
out on their own. Her focus on processes was for students to find the answers as to which
numbers were the 10th, 15th, and 20th. She did not recognize the lesson in the larger scope of
multiplication and division. She did think about the learning cycle in her lesson development,
?They were engaged in the beginning. They had to find a pattern. Pretty much it was explore the
whole time with pairs discussing? (K, FI, 4/22).
In her methods class, she often referred to the engage and explore portions of the learning
cycle. She liked the engage part because she felt like it helped get students interested in the
lesson. She thought the explore portion was most important because ?students really construct
their own knowledge? (K, Week 7 Methods). With the number towers students were given plenty
of time to explore the different towers. However, she did not expand the number towers to the
depths of multiplication and division it is intended to develop.
189
At the end of the week, she reverted back to her original plans and had students
completing procedural review problems. Students who finished early were given problems to
solve with the towers. Kate did not go over those answers or discuss strategies students used to
solve the tower problems. Instead she focused on the answers to the procedure review problems.
Lack of teaching experience or a lack of confidence in continuing the use of the investigation
lessons may have been contributing factors. Yet, in the follow-up discussion when asked about
the learning cycle in the lesson she replied, ?Explore-I feel they had to look at the division
problems and look hard and ask, ?What tools do I have? How can I figure it out based on what
tools I know?? (K, O, 4/23). Although some students may have explored solving multiplication
and division problems with the towers, she left out the evaluation phase in this process. She did
not find out how students solved the problems or how they used the tools in the process.
Kate had several interferences in her teaching. Her mathematics lessons that she
designed originally were based on sources that were not research based. When her lessons were
focused on procedures, the tools and processes were also for answers. When she followed the
investigation type lessons, students had more time to explore concepts and tools were used as the
key part of the lesson. She did expand the lessons into multiplication and division but not to the
intended depths of the lesson. She did not fully trust the reform-based lessons. She placed her
confidence in the lessons she created. The lessons she created were very low-level in thinking,
focused on procedures, and reverted back to a traditional approach to teaching mathematics. The
learning cycle was not an approach she used in her lessons. In our discussions after her lessons
when asked about the learning cycle, she would weakly try to relate the lesson to aspects of the
learning cycle? ?Engage-The students played a game last week and worked problems. The kids
were not where we wanted them to be [with their multiplication facts]? (K, FI, 4-20). In methods
190
class she was able to successfully plan and implement mathematics lessons using the learning
cycle approach. In one of her blogs she responded, ?I have always thought of the learning cycle
as a useful tool for teachers to go by when planning their lessons? (K, Week 6). However, she
may have believed the learning cycle useful in planning but she didn?t put that into her own
practice. Similar to Jane, Kate held na?ve views of the learning cycle as an approach to teaching.
Discussion
The cases of Jane and Kate present two beginning teachers in their development process.
Kate appeared to be a somewhat stronger student in methods class than Jane. They both
presented the idea of using the learning cycle in their mathematics teaching at the end of the
methods course. Following them through internship was quite revealing. An outsider may have
perceived Kate as a stronger intern due to the well behaved students in the class. Upon closer
examination however, the teacher-directed style and low-level questioning revealed other aspects
of her teaching that needed improvement. Looking back at her data from the methods class
revealed her beliefs about teaching mathematics and science in limited ways and under
controlled conditions. Richards, Levin, and Hammer (2011) posited that student teachers weren?t
able to maintain reform based practices because they did not alter their belief systems during the
preparation program or they have teaching qualities that make it difficult for them to support that
teaching approach. Her actions may be attributed to her deeply held beliefs about the teaching
and learning of mathematics. Jane, from the outside, may have been perceived as a weaker
teacher due to her struggles in classroom management. However, Jane was quite tenacious in
improving her practice and having students learn in meaningful ways. Examining her pre-test
from the first day of class reveals her concern that student be able to ?see? the mathematics for
themselves.
191
It appeared at the end of methods class that both Jane and Kate had embraced aspects of
the learning cycle in their mathematics teaching. The responses on the post-test could have been
due to inundation with the learning cycle approach throughout the summer semester. They had a
semester in-between mathematics methods and student teaching in which the learning cycle was
not the focus. It became apparent after observing and talking with them in student teaching, that
they maintained their deeply held beliefs about teaching mathematics from methods class
through internship. They both seemed to try to fit aspects of the learning cycle approach within
their belief systems. Jane wanted hands-on to help students ?see? the concepts and became
attached to the engage and explore phases and a means of helping students ?see? the concepts.
Kate wanted mathematics to focus on real-life mathematics. She liked the engagement aspect of
the learning cycle because it helped put students on track for the lesson that she would direct the
rest of the way. She believed other aspects of the learning cycle should be conducted through
teacher guidance and control.
During methods class they both completed required teaching assignments using the
learning cycle. However, during their two weeks of full-time teaching they designed the lessons
based in their level of comfort in how to teach mathematics. Initially both started out with very
traditional types of lessons. It both seemed as if they had taken a step backwards from the
teaching they exhibited in methods class. This is interesting because in methods class they
wanted students to see or connect with real-world mathematics but this was not evident based on
the first lesson observation. When they were responsible for designing, planning, and
implementing multiple subjects they both reverted to a traditional style of teaching.
A noted similarity for both Jane and Kate was that they designed lessons for their two
week lesson plans from ?other? sources. For the case of Jane, since she did not have strong
192
classroom management skills, the poor lessons caused additional off-task behavior. Novice
teachers are not curriculum designers or developers. They have not developed the expertise for
their subject, grade, or needs of the students to be expected to invent their own curriculum. In the
methods class, all of the material that was used had been tested in classrooms and was research-
based. The lessons were easy for them to use and promoted conceptual development of
mathematics. The student teachers did not turn to these materials for their two-weeks of full-
time teaching. The sources for their lessons were off of the top of their head or worksheets they
found in various places. These ?other? sources resulted in poorly designed lessons in which
students were not being challenged to learn new materials in meaningful ways.
In both cases after discussing the mathematics lesson, both preservice teachers changed
their original lessons to investigation type lessons. The researcher supported the student teachers
to alter their lessons to include investigative type lessons. Although Kate did use some of the
Investigations (Pearson Education, 2007) in her mathematics lessons, she did not follow them the
way they were intended. She taught them in a teacher controlled manner that did not allow the
students to fully explore the mathematics. The fact that Kate was willing to teach investigative
lessons that she was unfamiliar with gives hope for Kate?s future mathematics teaching. Jane?s
teaching improved with coaching and guidance, and she seemed to find more of her comfort
zone once she started using NCTM materials and Investigations (Pearson Education, 2007). This
fulfilled her desire for students to be able to ?see? the mathematics. She was always careful to
pose questions that would make the students think. Yet, Jane also maintained a mathematics
approach in which the teacher explains, student completes a task, and the teacher summarizes.
In both cases the learning cycle and inquiry initially seemed to be a forgotten approach
to teaching mathematics. A lack of connection often exists between the experiences of the
193
methods course and actual teaching (Manouchehri, 1999). The learning cycle was not used by
the cooperating teachers and therefore not enforced as an approach with the student teachers.
Putman (2009) likewise noted that student teachers shifted towards the practices of their
cooperating teachers. Hargreaves (1984) also noticed new teachers copying the style of the
veteran teachers. Jane and Kate were not in classrooms in which the learning cycle was used as a
framework for inquiry embedded teaching, despite the fact that the teachers had participated in
ongoing mathematics professional development using reform curricula and science professional
development based on learning cycles. Leonard, Boakes, and Moore (2009) also concluded that
appropriate learning environments were necessary to maintaining inquiry-based practices.
Implications
The nature of the dual science and mathematics methods course provided a unique
methods course experience. The purpose of that common approach to the methods class was to
help preservice teachers see that the learning cycle as an inquiry-embedded approach for science
also applied to mathematics. For Jane, it appeared as if the methods course affirmed her need for
students to experience mathematics. Kate began methods course with a very na?ve view of
mathematics as real-world applications. She appeared to have made changes in her thinking
about how to approach the teaching of mathematics. Kate successfully taught lessons using the
learning cycle for mathematics during methods class. She was thought by the researcher to be a
strong mathematics teacher. She talked about the importance of students? discovering concepts
on their own, the use on concrete experiences to develop understanding, and the importance of
the learning cycle and inquiry. The researcher expected Kate to be a strong mathematics teacher
in student teaching. Perhaps in the case of Kate, it really demonstrated that she is a strong
student. She followed the directions and assignments required for the methods class, including
194
writing a strong blog. Without the structure of the learning cycle imposed on her mathematics
teaching, she designed lessons that were very low-level in nature and represented her comfort
level in teaching.
Although both Jane and Kate initially taught mathematics lessons that were very shallow
they did alter their lesson plans to include investigative type lessons. Through discussion about
the lesson and methods class, both implemented changes in their mathematics lessons. The
researcher?s direct involvement with Jane and Kate impacted their decisions on their
mathematics teaching. Their pedagogy was not one of true inquiry within the learning cycle.
However, it progressed to using tools for meaning and sense-making, processes for reasoning
and communications, and lessons that were more hands-on in nature for the students.
The researcher served as a coach to help Jane and Kate improve their practice. Teaching
inquiry-based lessons can be a challenging and daunting task for novice teachers. Although the
learning cycle provides a framework for inquiry, beginning teachers need help in using it for
mathematics. Perhaps teacher preparation institutions and school partners should think about
establishing coaches within student teaching to provide additional support for student teachers.
This research indicates the roles of supervisors and other support personnel can help influence
and remind new teachers about reform-based teaching practices. Having the preservice teachers
put reform-based approaches in practice not only helps to build their confidence but also their
abilities to teach reform-based lessons. Teaching reform-based practices takes more thought and
complexity of teaching. Mathematics coaches can help student teachers in their development
and implementation of reform-based lessons. Practice teaching using reform-based curricula can
provide the teaching practice, build pedagogical repertoire, and serve as a foundation for future
teaching.
195
Additionally, both Jane and Kate needed more direct advice on planning and teaching
inquiry-based mathematics lessons. They both reverted to a traditional style of mathematics
teaching when it came time for them to design their full time teaching. Imposing parameters for
preservice teachers similar to those of the methods course, including using reform-based
curriculum and more developed lesson plans, may provide additional scaffolding as preservice
teachers develop their craft. Imposing similar parameters assumes the cooperating teacher is of
like mind. Therefore, teacher preparation institutions need to continue to develop relationships
with school partners to help establish those similar parameters.
Developing school partnerships that helped in-service and preservice teachers foster a
practice of inquiry-embedded practices within the learning cycle would be an ideal situation.
Although preservice teachers were expected to use the learning cycle approach in their
laboratory teaching experiences during their methods courses, this did not carry over into their
internship. The observation-conferencing technique made a difference in the mathematics
teaching of both student teachers. The quality of the lessons improved due to the focus of tools
for conceptual understanding, use of processes for reasoning, and the learning cycle approach.
Placing mathematics coaches with student teachers would provide a continuum of training to
help student teachers develop meaningful lessons using ?tools of inquiry? and designed
instruction to develop students problem solving and thinking skills (INTASC, n.d.).
196
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Appendix A
Follow-up Discussion Questions
Pedagogical practices
1. In both science and math methods classes we focused on the Learning Cycle. What role
did the Learning Cycle play in your lesson development and implementation?
2. What specific teaching practices were you working on improving through the teaching of
this lesson?
Processes for meaningful learning
3. What role did inquiry play in the lesson? In what ways do you feel you promoted
inquiry?
4. What connections between mathematics and science did you want students to make?
How did you design the lesson to promote students making those connections between
mathematics and science?
Tools for conceptual development
5. In what ways did you use tools for conceptual development?
6. Would you use the tools differently next time you teach the same lesson? If yes, explain
how you would use the tools differently. If no, explain why you would keep the use of
the tools the same.
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Appendix B
Final Reflection
1. How is math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/ experiences play a role in a math lesson? Give examples.
4. Name and describe a few teaching strategies or approaches used to teach
science for meaningful understanding.
5. Name and describe a few teaching strategies or approaches to teach math for
meaningful understanding.
6. A. Think back to a math lesson that you taught during student teaching and briefly
describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
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CHAPTER 6: CONCLUSIONS
Summary
In conducting science and mathematics methods courses using common tools, processes,
and approaches, preservice teachers? conceptions changed from the beginning of the methods
course to the end of the methods course. Thirteen out of the twenty-two participants considered
tools in science for hands-on learning initially. In contrast, only 6 of the participants initially
recognized tools for hands-on learning in mathematics. By the end of the course, 17 of the
preservice teachers recognized tools in science for building an understanding of concepts and 9
preservice teachers considered tools in mathematics for conceptual development. Participants
articulated processes primarily in limited ways at the beginning of the semester. The design of
the methods course and the field experiences bolstered their repertoire of processes for student
engagement. Participants with more traditional views of science and mathematics teaching, in
which the teacher delivered the knowledge, seemed to have a more difficult time accepting the
learning cycle approach. Four preservice teachers in science and six participants in mathematics
maintained a traditional approach for teaching science and mathematics. The participants with
expanded views and progressive views seemed to accept the learning cycle approach in
mathematics and science more easily; thought of tools in terms of conceptual development; and
spoke of teaching in a manner that promoted reasoning, communication, and explanation by the
end of the semester. By the end of the semester, 18 of the 22 participants recognized the learning
cycle as a common approach to teaching science and mathematics. Table 7 provides detail of
preservice teachers? conceptions of tools, processes, and approaches.
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Table 7
Themes: Combined Chart
Traditional Expanded Progressive
Tools
Science
Pre (n)
6
3
13
Post (n) 3 2 17
Mathematics
Pre (n)
8
8
6
Post (n)
8 5 9
Processes
Science
Pre (n)
3
14
5
Post (n) 3 6 13
Mathematics
Pre (n)
7
11
4
Post (n)
6 4 12
Approaches
Science
Pre (n)
7
6
9
Post (n) 4 7 11
Mathematics
Pre (n)
8
9
5
Post (n)
6 5 11
Approaches
between
mathematics
and science
No similarities in
approaches
Partial connections Recognizes similar
approaches for both
mathematics and science
Pre (n) 13 2 7
Post (n) 2 2 18
Two of the participants were followed into student teaching. At the end of methods these
two seemed to have embraced the learning cycle as an approach for mathematics. However, their
conceptions did not appear to be long-lasting. The lessons that they designed for their student
teaching did not use the learning cycle approach. This could be attributed to the fact that they
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were in classrooms in which reform-based teaching was not the approach regularly used for
mathematics teaching. After coaching by the researcher, they both altered their teaching to use
reform-based mathematics lessons that followed a learning cycle. For Jane, this fit with the way
she wanted to teach mathematics. For Kate, she was very uncomfortable and reverted to a more
traditional style of teaching at the end of the observation time.
Tools for Conceptual Development
Methods Course Cases
Preservice teachers more readily accepted the use of tools for conceptual development in
science than in mathematics teaching. Thirteen out of the twenty-two students thought that
science should be taught with tools to help students see and understand the concept (see Table 7).
They talked about tools in terms of hands-on learning, physical knowledge activities, and
memorable concrete experiences. A typical response for initial thoughts of tools in science for
understanding was, ?Hands-on activities allow students to learn through experiences as well as
trial and error. The concrete experiences are most of the time more memorable than lecture or
book work so students often remember the information learned through experience more readily?
(Person 3, Pre-test). By the end of the methods class, the few preservice teachers with a
traditional view of the use of tools in science more easily accepted a hands-on approach to
science teaching than mathematics teaching. They were also able to express the use of tools in
science in terms of processes: ?In science, concrete experiences help students observe and come
to realizations. They allow students to make discoveries and construct knowledge? (Person 3,
Post-test). By the end of the semester they also articulated tools in terms of concrete experience
within their teaching practice:
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I taught Calcite Quest. The students had to search for evidence of calcite in various rocks.
The student had concrete experiences in discovering what the evidence of calcite is and
the searching for it. This was effective because students understood first-hand and were
able to build on that due to comprehension. (Person 12, Post-test Methods Class)
Most of the preservice teachers initially thought of tools in science as a means for hands-on
learning. Their work with students in the field verified the use of tools as a means to help
students observe, connect with prior knowledge, and make realizations about science concepts.
Those with a traditional view of tools in mathematics were reticent to alter their
conceptions of tools for mathematics teaching. Their personal school experiences as students
were deeply embedded in their notions of teaching mathematics and likely affected their view of
tools in mathematics. ?To be honest, I never really had concrete mathematics experiences in
school. My teachers always showed us methods to work problems on the board and that was it. It
was all just memorization? (Person 9, Blog Methods Class). Some had very limited experiences
in school with concrete materials for mathematics: ?I only recall one experience in mathematics
in school that I used concrete materials. It was in sixth grade and we were talking about
geometry, shapes, and angles and we used geoboards as a concrete material? (Person 8, Blog
Methods Class). Their mathematics lessons, although from hands-on curriculum, would often
end up being taught in very teacher directed ways with the right answer being the sole purpose
for the students in the lesson. However, almost all of the preservice teachers, who initially had a
more expanded view and progressive view of tools for mathematics, were able to expand their
ideas of tools in mathematics for the purpose of conceptual development. They already held
ideas that mathematics should be interesting and creative for the students: ?In a math lesson
students can use the concrete tools or manipulatives to actually see the math lesson unfolding.
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The students will have the numbers as a concrete thing? (Person 20, Pre-test). The use of tools
for concept development rather than for right answers fit in with their existing notions.
Student Teaching Cases
Jane began methods class with a view of tools in science for hands-on learning and tools
in mathematics for students to ?see? and ?do? the mathematics. The methods class reinforced
her ideas of how tools should be used for science and mathematics teaching: ?When discussing
topics that can be experimented, the children need to actually see it and get involved instead of
just listening about it? (Jane, Pre-test Methods Class). Tools for conceptual development fit with
her notions of science as a hands-on subject. Tools in mathematics also offered a means for
students to understand the mathematics. Furthermore, her lessons in the field demonstrated
effective use of tools for conceptual development.
Kate entered methods class with very different ideas from Jane in the use of tools in
mathematics and science. She saw the use of tools in science as students seeing concepts from
the textbook: ?Concrete tools and experiences allow the students to see the concepts that they
read about using textbooks applied in a tangible way? (Kate, Pre-test Methods Class). Tools in
mathematics were thought of in limiting terms of real-world mathematics such as time and
money. ?When you relate mathematics to real-life situations, like counting money or telling
time, they are more likely to become engaged? (Kate, Pre-test Methods Class). She seemed to try
to justify traditional modes of science teaching throughout her science methods:
There are times that call for more traditional instruction, but as long as the teacher keeps
it interesting and incorporates hands-on activities where it is possible, the students will
have an enjoyable, meaningful, learning experience. (Kate, Blog Methods Class)
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However, in her mathematics teaching she described the importance of using tools in
mathematics:
Many students do not gain understanding by simply watching the teacher do a problem in
the board. Most students learn more easily and retain more information if they discover
the concepts on their own, and often times the best way to get students to explore and
discover new ideas is through working with concrete materials. (Kate, Blog Methods
Class)
Although, Kate?s mathematics lessons were very controlled and she sometimes would tell
students answers instead of waiting for them to figure it out, she did use a variety of tools for
students to understand the mathematics concept. During methods class she used tools to teach
patterns, geometry, data, and money:
The first day I taught the investigation lessons in which the students stood as long as they
could on one foot with their eyes closed. Then they placed their times on the line plot and
compared the group data. The second lesson I taught dealt with comparing the ages of
students and adults they knew which built on simply observing individual data with
group data because the students had to observe the several relationships between the data.
Concrete experiences were crucial in each of the activities. The students actually stood on
one foot with their eyes closed in the first activity, and placed their times on an enlarged
plot on the wall. In the second activity, the students used cash register paper strips to
compare the different sizes between their ?life strips? and the adult ?life strips?. This
made the math terms such as ?times as much as? make more sense. (Kate, Blog Methods
Class)
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Based on how Jane and Kate ended their methods class, it was expected that they would
be using tools in their mathematics teaching during the student teaching experience. However,
the first lessons that were observed were very traditional in nature with the teacher explaining
and the students working problems. Tools were used for procedures rather than conceptual
understanding. In Jane?s first and second lessons, students were practicing using a ruler to
measure objects. In Kate?s first lesson, students completed a series of games to build
automaticity of facts. Through the conversations after the lessons, the researcher was able to
recall how and why tools were used for concept development in mathematics. Jane altered her
teaching to include more of a variety of tools to help students understand concepts. Although her
initial lessons were procedure focused, with a little coaching she was able to develop and
implement lessons that used tools for students to ?see? and ?do? the mathematics. In one lesson
she used student data, an interactive website, and tiles to help students understand
representations of data. In another lesson she used fraction magnets on the board, farm animal
toys on the document camera, and fraction circle sets with the students as a means of fostering
understandings of fractional concepts. She embraced the use of tools for conceptual development
due to the dissatisfaction of her own school experiences in mathematics:
I agree with letting students figure things out on their own. When a teacher tried to drill
things in my head, it always made me feel like I had no freedom to do what I wanted to
do. It put more pressure on me because I felt like I had to do it just like everyone else. I
honestly think I would have done a lot better in math if I were given this choice in school.
Math is so flexible and there is no reason that you should have to do something one way.
(Jane, Blog Methods Class)
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Kate also changed her lesson plans to teach more hands-on lessons. However, Kate?s use
of tools was still within a controlled teacher context. She also would give students answers rather
than giving them time to figure information out based on the observations and mathematical
relationships the students had learned from the tools.
Processes for Meaningful Learning
Methods Course Cases
Three preservice teachers thought of science processes in traditional ways and seven
preservice teachers thought of processes in mathematics in traditional ways at the beginning of
methods class (see Table 7). Traditional conceptions placed the students in passive roles with the
teachers giving all of the information. A typical statement for traditional processes in science
was, ?Teach a lesson, do the activity, let students explain how it worked and how they go
together? (Person 4, Pre-test). A typical statement for traditional processes in mathematics was,
?Teach the lesson, work together as a class and have students work independently? (Person 4,
Pre-Test). In these scenarios the students are following the examples provided by the teacher.
The students are not involved in problem solving. ?Problem solving means engaging in a task for
which the solution method is not known in advance? (NCTM, 2000, p. 52). In both scenarios,
students are completing a science or mathematics lesson in which the method has been given by
the teacher. Since the method is known, they are, therefore, not reasoning, communicating, or
making connections about the mathematics or science. There were 3 more preservice teachers
who thought of mathematics processes in traditional ways than those who thought of science in
traditional ways by the end of the semester.
Most of the preservice teachers articulated processes for science in expanded ways at the
beginning of the methods classes. Fourteen preservice teachers held expanded views of processes
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in science and 11 preservice teachers held expanded views of processes in mathematics at the
beginning of the methods classes. Expanded conceptions were indicated as ideas of process that
were more than just students serving in a passive role but not as deep as concept development.
The most change occurred with those who had expanded conceptions of processes in
mathematics. They deepened their understanding to think about processes in mathematics for
students to develop reasoning skills. In their mathematics teaching tools became a means to help
students with processes:
I did an activity called Shifty Shapes which allowed the students to explore shape
composition. The students used pattern block cut-outs to create solutions to completely
cover the hexagons. The students understood the small shapes could be assembled to
make the larger shape. The students were able to solve the problems with different
shapes. They had tor make sure their solutions would completely cover the hexagon.
(Person 21, Blog Methods Class)
Conceptions in which processes in science and mathematics focused on problem solving,
reasoning, communications, making connections, or developing representations were
progressive. On the pre-test they referred to students finding meaning through hands-on
teaching: ?Experiment and hands-on inquiry allow students to use trial and error to find meaning
and understanding? (Person 3, Pre-test). Thirteen of the preservice teachers with expanded views
at the end of methods class recognized processes for students to reason and figure out concepts
for themselves. In one lesson students had to figure out how to communicate on Mars. In the
lesson the teacher promoted problem solving, communications, and reasoning:
The lesson I taught was on Friday, which was the day that they got to make
communication devices. I had to explain to them that there is no air in Mars and sounds
219
need a way to travel. I also explained to them that the inconvenience with radio waves in
space and all of the problems associated with trying to get the internet in space. They got
to brainstorm and come up with a way to communicate while on Mars. I could see that
they learned a lot through these activities because of the questions that were asked while
they worked. (Person 17, Blog Methods Class)
They articulated processes in their lessons as students making connections, using observations to
draw conclusions, and justify their reasoning. By the end of methods classes, 13 of the
participants in science and 12 of the 22 preservice teachers in mathematics conceived of
processes in progressive ways.
Student Teaching Cases
Jane began methods class with conceptions of students actively involved and figuring out
the mathematics and science: For science she believed that: ?The children need to actually see
and get involved instead of just listening about it? (Jane, Pre-test Methods Class). She felt
similarly in mathematics: ?Again the students need to ?see? and ?do? to learn in math. It helps
them get a better understanding of how math really works instead of just plugging numbers into
formulas? (Jane, Pre-test Methods Class). Methods class deepened her conceptions to include not
only reasoning for students but the use of communication between students to develop
understanding. She used questioning techniques to help students to think about the mathematics:
I taught a measuring capacity lesson to K-2 students. They began by learning how to
count, record, and measure how many spoonfuls of rice o took to fill up a small party
cup. The first time they all got different numbers. I asked them to look at the numbers
and tell me what they noticed about them. They couldn?t understand why everyone?s was
so different because they all had the same cup and spoon. Then I began to ask them how
220
they measured. For example, how big a spoonful they used or what they called a full cup.
This began the inquiry stage. They began thinking about why it was so different. So we
established a certain way everyone was going to measure. This time they all got around
25 and 30. They were so amazed at how it worked. (Jane, Blog Methods Class)
Kate considered processes initially as students serving in passive roles while the teacher
imparted the knowledge. In science she initially believed that students used processes to
understand the textbook content: ?Experiences allow students to see the concepts that they read
about in the science book? (Kate, Pre-test Methods Class). By the end of the semester she
believed that students should be allowed freedom to explore concepts in science under teacher
direction: ? Some useful techniques for science include encouraging students to make predictions
based on prior knowledge, and then allowing them free exploration so they can discover
concepts on their own (w/ teacher scaffolding) [sic]? (Kate, Post-test Methods Class). She
expanded her notions for students to reason and make connections. In a lesson that she taught on
patterns, students used pattern blocks to continue the iteration of the pattern. With every iteration
the pattern would grow and the students recorded how many blocks were in the iteration. She
had the students continue the pattern and record. The students were supposed to figure out the
100th pattern based on what they noticed. When they didn?t figure it out right away, she told
them the answer. Her need for continuous control limited the reasoning and connection-making
for the students in the lessons she taught during methods class.
At the beginning of their two weeks of full-time student teaching, both Jane and Kate
focused on processes for right answers. Jane?s lesson focused on using a ruler and Kate?s lesson
focused on basic multiplication facts. They seemed to have forgotten the methods class
instruction of fostering processes for science and mathematics to develop conceptual
221
understanding. When they altered their mathematics lessons to include more investigation type
mathematics lessons, students were required to reason, communicate, and explain their
understandings due to the nature of hands-on mathematics lessons. In the data lesson, that Jane
taught, her students used various representations to understand graphs: ?So our bar would
represent our categories. Let?s try this again. What would our bar graph look like from this
morning? So you have seen the circle graph. Now let?s look at the bar graph. Does anyone have a
prediction?? (Jane, O, 4/16). In her fraction lessons, students used pattern blocks, fractions
circles, and fraction magnets on the board to ?see? and ?do? the mathematics. After one of the
fraction lessons in which students used pattern blocks to make cookies she felt she promoted
processes, ?when they were putting together the hexagon. They had to figure it out, to put it
together themselves? (Jane, FI, 4/22).
In Kate?s teaching, processes even within investigation lessons were limited due to her
controlled style of teaching. She changed her original plans to teach the tower investigation.
Even though she followed the frame of the lesson, she would not give students time to figure out
the mathematics. In the first example, she showed the tower to the students and asked them to
make predictions about what number will end up at the top of the tower. Then she tells the
student the pattern instead of letting them figure it out:
The tens place does seem to go up by 30 and 20 ignoring the ones place. Look at the
numbers you predicted. Most of you picked numbers that made sense. They all ended in 5
and 0. If we look at the bottom it is like 1x 25, 2x 25. (Kate, O, 4/21)
In the same lesson she does a few multiplication and division problems using the tower. The
students are supposed to use the tower to solve the problems. If the lesson is followed, students
understand the inverse relationship between multiplication and division. They then develop ways
222
to reason and figure out the answers using number sense. However, in Kate?s lesson she tells
them how to think and move on the tower:
That?s what we mean when we divide. It doesn?t always come out evenly and we have a
remainder. I am going to try to trip you up. 420 divided by 25. [Kate counts up to 400].
425 is too many so we will stop here. [Kate draws a line at 400]. How much over 420 is
425? (Kate, O, 4/21)
Pedagogical Approaches
Methods Course Cases
Interpretations of Hands-on Approach. Pedagogical approaches were categorized as
traditional view, expanded view, or progressive view (see Table 7). Participants with a traditional
view followed the notion that the teacher explains and the students complete verification
activities in science and mathematics. Initially seven participants in science and eight
participants for mathematics held traditional views. Throughout their blogs, they focused on
teachers explaining and students giving right answers:
It is very important that the hands-on activity is clearly explained and applied? It is best
to first have students learn information and then assess or reinforce with a hands-on
activity?Flash cards h elp students in remembering key facts to build upon?I listened
carefully to their conversation to hear student thinking to determine whether or not their
thinking would lead them to a correct answer. (Person 3, Blog Methods Class).
Four participants in science and six participants in mathematics who conceived of teaching
mathematics and science with a traditional approach initially had little change by the end of the
methods class.
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Six of the participants in science and 9 of the participants in mathematics initially held an
expanded view of teaching science and mathematics. They described science and mathematics as
more than the teacher delivering the knowledge, but not on the level of approaching science and
mathematics for conceptual understanding: ?Having learning centers for children to learn or
review different science centers? (Person 10, Pre-Test Methods Class). In mathematics, they
believed that the approach should focus on real-life mathematics and centers: ?Real-life projects
have value in everyday life like counting money. Role play like grocery store? (Person 14, Pre-
test Methods Class).
By the end of the semester seven people were categorized with an expanded view in
science and five in mathematics. The additional person in the expanded category for science can
be explained by some of the participants in the expanded category moving to the progressive
category and some of the ones with a traditional view moving to the expanded view.
Nine of the participants were categorized as progressive for science and five were
categorized as progressive for mathematics. Those with a progressive view initially expressed
that science and mathematics should be taught with a hands-on approach:
[For science teaching] Use learning centers, group activities, and hands-on activities for
example when you are teaching about insects and plants have the students go outside and
see these insects and plants hands-on. [For mathematics teaching] Group work, hands-on
approaches and everyday uses. For example when teaching division, bring in a cookie
and ask students how will everyone get a piece, and allow the students to think about and
use their knowledge on how to divide the cookie. (Person1, Pre-test Methods Class)
For those with expanded or progressive views at the beginning of methods courses, their beliefs
seemed to fit reform-based practices on use of hands-on for science and mathematics teaching.
224
By the end they often referred to approaching both science and math with the learning cycle
approach:
Teaching math is like teaching science in several ways. Each subject needs a ?hands-on?
approach to learning the material through experience. Teachers in both subjects need to
provide students with concrete materials to explore how things work. The 5E?s is a great
model to use for approaching science teaching. It provides instruction at all levels, all the
way from engagement and evaluation. The 5E?s can be used in math just as in science.
(Person 19, Post-test Methods Class)
Interpretations of the Learning Cycle Approach. Although 18 of the preservice
teachers did indicate that the learning cycle approach was a good approach for science and
mathematics at the end of the semester, they were unfamiliar with the learning cycle initially. In
the learning cycle student exploration precedes explanation and definitions of terms. With the
learning cycle approach exploration provides the base for developing an understanding of
concepts. This is different from the inform-verify-practice model of teaching. Preservice teachers
interpreted the use of the learning cycle in different ways throughout the semester. In the blogs,
the preservice teachers described their understanding of the learning cycle. They felt that the
learning cycle helped students relate to what they were learning through the use of first-hand
experiences:
During the learning cycle students also learn to become problem solvers which help them
to become more autonomous as adults. Another benefit of the learning cycle is that it
allows students to relate new ideas about science to their experiences associated with it. I
believe that through the learning cycle stages students learn more and are better able to
225
construct and relate knowledge from their firsthand experiences compared to just copying
definition and writing notes. (Person 19, Blog Methods Class)
The few who held to more traditional modes of teaching were conflicted with the idea of the
students exploring before explanation. ?I think the traditional approach needs to be kept in the
curriculum, along with the learning cycle approach? (Person 2, Blog Methods Class). Those who
held to more traditional roles of mathematics teaching saw the learning cycle as steps to follow
in mathematics. Their steps often mirrored the teacher explains and the students solve problems.
Solving problems is not the same as problem solving. In problem solving the method of finding
the solution is not known initially. Those who saw the learning cycle as steps also believed that
students working who were working in groups were also exploring. This is not exploration if
students are solving routine problems: ?I believed playing a short clip or film would be good to
show the students during the engage part of the lesson. Then in the explore part we should allow
students to work problems to solve as a group? (Person 10, Blog Methods Class). A majority
believed the learning cycle in mathematics created an environment that allowed students time to
develop understandings about mathematics:
If you don?t get one concept, there is no way that you are going to understand and be able
to comprehend things on a higher level. Thus is true for a lot of students. Teachers get
wrapped up and in a hurry and don?t really allow time for the students to think or fully
understand something before moving on to something totally different! The LC is a good
way, I think to keep teachers in line and on the right track for student learning, especially
in math. (Person 22 Blog Methods Class)
By the end of the methods class, 18 of the beginning teachers thought that the learning cycle
was a common approach for both science and mathematics teaching. A typical response was
226
?teaching math and teaching science both relate to the learning cycle, 5E?s and very important
assessment strategies for the classroom? (Person 20, Post-test Methods Class).
Student teaching Cases
Jane entered methods class with the notion that science involved hands-on experiments
and that mathematics should be taught with games and hands-on activities. Her ideas of how
science and mathematics should be approached were consistent with her notions of tools for
students to ?see? and ?do?. In order for students to ?see? and ?do?, the use of hands-on teaching
provided a means for students to reason and understand the concept. Methods class reinforced
her conceptions of tools, processes and ultimately approaches to teaching science and
mathematics. By the end of the methods class she thought that mathematics was like science in
that, ?they both involve inquiry, 5E learning cycle, and hands-on activities? (Jane, Post-test).
When asked on the post-test to describe a math lesson she had taught and explain how it met
effective approaches, she responded:
I taught a lesson on measurement and capacity with K-2 grades. I taught them how to
measure with non-standard approaches. I had them guess how many spoonfuls of rice it
would take to fill their cup. This involved inquiry. Then I had them explore and fill it
once. We then discussed and made second guesses (more inquiry). We then discussed
measurement concepts. They then did experiment 2 using what we discussed. (Jane, Post-
test Methods Class)
Kate began methods class with mixed views on her pre-test. For tools in science she said
tools were to enhance concepts in the science book. Then when asked about approaches, she said
hands-on was the approach for science. Her response about tools actually reflected her true ideas
about science teaching. Her science blogs discussed hands-on but in terms of a treat for students
227
or something a teacher does if she has time. Her pedagogical approach fit with her notions of
tools in science as a means of enhancing the concepts in the science book as well as processes for
answers. She held very traditional ideas about science teaching. She grappled with accepting a
more hands-on science approach throughout the science methods class: ?I also know that there
are other times that call for more traditional instruction, but as long as the teacher keeps it
interesting and incorporates hands-on where it is possible, the students will have an enjoyable,
meaningful learning experience? (Kate, Blog Methods Class). By the end of the semester, she
conceived of teaching science with hands-on lessons as a reward or treat for the students for
good behavior. Her approach for teaching science also limited processes for students. Teaching
from the science book limited her students to processes articulated for textbook answers.
She viewed approaching mathematics in terms of relating mathematics to real-life. Her
view of ?the teacher explains and students see examples? also fit with her ideas of mathematics
teaching. In math, her students were still limited to using tools for procedural ways and focusing
on right answers in mathematics. She expanded her approach for mathematics to include hands-
on but only in limited terms of real-world mathematics. When asked on the Post-test to describe
a math lesson she taught that met effective approaches, she wrote, ?I taught the Shapes on the
Bus activity, which had students relate their knowledge about shapes to everyday objects. The
students created their own school buses? (Kate, Post-test Methods Class).
By the end of the methods class, both Jane and Kate believed that the learning cycle was
an approach for teaching science and mathematics that also involved inquiry-embedded teaching.
Although Kate said that the learning cycle was an approach for science and mathematics, this
was not how she would approach science and mathematics. In her first science blog she seemed
to grapple with the notion that explanation followed hands-on learning. By the end of her science
228
methods class she described hands-on in science as something students could work towards. Her
blogs revealed that she maintained a more traditional approach for teaching science with hands-
on as reward. She did see mathematics as being taught with hands-on approaches for students to
learn and understand real-world mathematics. The explore aspect of the learning cycle fit with
Kate?s ideas for approaching mathematics. Jane, on the other hand, seemed committed to the
learning cycle approach for both science and mathematics. This approach fit in with her notions
of students being able to ?see? and ?do? to develop their learning.
Based on the data from the end of methods class it was expected that Jane would
incorporate more of the learning cycle approach in mathematics than Kate. However, their first
lesson observation indicated that neither one of them had considered the learning cycle approach
in their mathematics lessons. After debriefings with Jane in which the learning cycle as an
approach for mathematics was the topic, she made changes in her lessons to incorporate more of
the learning cycle in her mathematics teaching. Jane did include exploration and engagement
with various materials. Kate also altered her lesson plans to include more hands-on mathematics
teaching. Her lessons had limited engagement, explorations, and elaborations. Her need to
control the lesson, along with the lack of think time for the students to think about the concepts,
limited student learning.
Although Jane and Kate articulated hands-on for mathematics teaching at the end of
methods class, this did not carry over into their student teaching experience. They both
approached mathematics with a traditional style of teaching. They may have been mirroring how
the cooperating teacher approached mathematics teaching. They did alter their lessons upon the
prompting of the researcher to include more hands-on learning. The first two lessons observed
for Jane were for measurement practice using a ruler. The first follow-up conversations focused
229
on management issues she was having with students. Part of her management problems were due
to student boredom. The rest of the follow-up conversation focused on improving the quality of
the mathematics and using resource based materials. The third lesson seemed to be the turning
point for Jane in the implementation of the lesson that matched how she wanted to approach
mathematics. For Jane, this inclusion of hands-on fit her ideas of students being able to ?see? the
concept.
Kate?s initial lesson was on multiplication facts. The follow-up discussion with Kate
revealed that she was moving into division. I discussed the number towers and how they were
designed to foster students? understanding of the relationship between multiplication and
division:
Kate was teaching an Investigation lesson for the first time. It was obvious in her
mannerisms that she was timid about this lesson. It is not a traditional approach to
division where students learn the division algorithm. This way of approaching division is
probably foreign to the way she learned multiplication and division. (Researcher Journal,
4/21)
Kate incorporated more hands-on in some of her lessons but appeared to be very
uncomfortable in teaching those lessons. By the end of her two-weeks she reverted back to a
traditional style of mathematics teaching and appeared to be more at ease. At the end of the week
she had students complete a series of various procedural problems. The ones who finished early
were allowed to get up and solve problems with the number towers. However, in the follow-up
discussion this is how she described her approach for the lesson, ?Explore. I feel they had to look
at the division problem and look hard and ask ?What tools do I have? How can I figure it out
based on what I know?? (Kate, 4/23). Although she talked about students exploring in the lesson
230
she never asked the students about their findings with the towers. Her description of her
approach and what actually occurred were glaringly inconsistent. When using reform-based
materials, it is important to provide avenues for student to build bridges and make connections.
Teachers have to probe and find out about student thinking. Perhaps Kate will develop this in
her teaching practice as she gains experience.
Final Note
Understanding preservice teachers? conceptions of tools, processes, and approaches is
important in teacher education. Preservice teachers? conceptions of the use of tools often
represented their views of processes and approaches. Some would mention hands-on as the
approach they would use but their description of tools indicated a teacher-directed style of
teaching. With the learning cycle approach, tools serve as a means for connecting to prior
knowledge and promoting understanding of new knowledge. Preservice teachers would use the
phrase ?hands-on? to describe their teaching, but this does not mean they viewed hands-on as
intended by the learning cycle approach. Additional research needs to be conducted to determine
what preservice teachers mean by ?hands-on? for science and mathematics teaching.
Since tools are thought of as a necessary component of science teaching, the question
then becomes, ?How can teacher educators help preservice teachers see tools in mathematics as
being necessary for understanding as well?? Planning and developing mathematics courses
throughout the preparation program that use an inquiry-embedded approach would be one means
of establishing tools in mathematics for inquiry-embedded teaching. This would require
cooperation on the part of the teacher educators and mathematicians at institutions.
Another way to foster development of preservice teachers thinking and practice would be
through the use of blogs. In this research the preservice teachers wrote their blogs and responded
231
to blogs by other preservice teachers. If the methods professors had also participated in the blogs,
asking questions, providing responses, and becoming part of the conversation, this may have
added another dimension to the development process. Although time on campus served as an
opportunity to teach concepts and help clarify misconceptions, the professors? involvement in the
blogs may also have provided a means to teach and clarify conceptions. In some of the blogs
conceptions held by preservice teachers were unclear. In the response blogs, the professors also
could have asked the preservice teachers to explain and clarify their thinking.
The intersection of technology, pedagogy, and content knowledge through the blogs
provided a means for preservice teachers to express their understandings or misunderstandings.
The blogs were an assignment intended to help the instructors gain insight into preservice teacher
thinking. The blogs could have been used more directly in the methods classes as a model for
reflective thinking. Ideas expressed in the blogs could have been used for further discussion in
class. This might have been helpful for those who had notions of inquiry that weren?t true
inquiry. Group discussion could have served as another means for clarification. Discussion of the
blog itself could have helped the preservice teachers think about technology use for instruction as
well. Furthermore, it would have been ideal if the student teachers continued to blog about math
during their student teaching and responded to each other. This might have provided additional
insight into their development as mathematics teachers.
Following the two preservice teachers beyond the methods course into student teaching
revealed that the nature of their understandings and conceptions of teaching mathematics from
the start of methods class had changed little by the time they entered the student teaching
experience. The change that appeared to have occurred at the end of methods class could be for
several reasons. First of all, the students were immersed in the learning cycle as an approach to
232
teaching science and mathematics during the methods class. Furthermore, they may have liked
the idea of using the learning cycle in mathematics. However, when it came time for them to
plan and implement mathematics lessons in student teaching, they reverted back to a style of
mathematics teaching that they had experienced in school and that their cooperating teacher
supported.
True change in thinking that impacts practice occurs over time. The time it takes to
complete a methods course, and with limited practice, is not enough time for people to change
years of experience as students in traditional classrooms. Those who shifted their thinking from
the beginning of methods class to the end, shifted in ways that fit with their expectations and
notions for what teaching mathematics should be like. The few students who held traditional
conceptions of tools, processes, and approaches at the onset of methods course had very little
change by the end. This indicates that it may be even more difficult to alter the conceptions of
those with deeply held traditional perspectives. Kate leaned more toward the traditional
approaches than Jane initially. Even in her student teaching, although she tried to teach lessons
with a different approach, her comfort was in a traditional approach. Kate may be viewed as a
good teacher because she had excellent classroom management. However, her need to direct and
inform the students of how to think and solve problems interfered in students figuring it out for
themselves. However, Jane really wanted to teach in a manner that made the concepts easier for
students to understand. Her vision she expressed at the beginning of methods class stayed with
her into her student teaching. For the case of Jane, she improved in her teaching after some
guidance in teaching hands-on lessons.
In order for a person to achieve conceptual change they must become dissatisfied with
their existing notions (Posner et al., 1982). Kate demonstrated how well she was able to perform
233
the assignments required during her methods courses. This gave the impression at the end of the
semester that she had embraced the learning cycle. However, Kate was not dissatisfied with the
way she learned in school. The teacher-directed approach worked for her. Kate appeared to place
her trust in a teacher-directed approach for teaching mathematics in her student teaching. Jane,
however, indicated strong feelings for the manner in which she was taught mathematics. She was
dissatisfied with a teacher-directed approach. It was very important to Jane that students learn for
themselves through experience.
This research also indicated a need to rethink qualities that make methods students appear
to be strong teachers. Jane completed her assignments during methods classes but was not
considered as strong of a student as Kate. This indicated that those who are perceived as being
strong methods students may not actually be strong methods students in teaching through
reform-based approaches. They may be good at school. Following them through student teaching
then becomes an important part of teacher education to provide continual support for the novice
teachers.
The teaching approaches of the methods class needs to follow preservice teachers as they
enter student teaching. A working relationship with cooperating schools and cooperating teachers
would help both parties in the development of the beginning teachers. The learning cycle from
the outside appears to be easy to use in teaching. However, perhaps it is more difficult to
implement for new teachers. In the case of Jane and Kate, they reverted to more traditional styles
of teaching upon taking on the responsibility of full-time teaching. Placing preservice teachers
in classrooms that use the learning cycle may yield different outcomes.
Finally, the researcher?s role as a coach in teaching mathematics was effective in the both
cases. The follow-up conversations provided a time to recall and discuss methods of mathematics
234
teaching. Sometimes discussion focused on resources and lessons that would be applicable to
the objectives they had to teach. Other conversations focused on management of an investigative
type lesson. Both student teachers changed and altered their lesson plans to teach investigative
type lessons. Jane bloomed as a math teacher when she made this change. Kate tried teaching the
lessons but was very uncomfortable and would not give the students freedom in thinking. The
fact that she was willing to try something outside of the norm for her student teaching experience
indicates she is willing to try new methods of teaching. Perhaps as she gains experience she will
become more flexible in her teaching style. Further research needs to be conducted as elementary
teachers move through the teacher preparation process that uses the learning cycle approach into
full-time teaching.
235
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Overall Appendix
269
Appendix A
Pre-Test
1. How is teaching math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/experiences play a role in a math lesson? Give examples.
4. Name and describe a few teaching strategies or approaches used to teach science for
meaningful understanding.
5. Name and describe a few teaching strategies or approaches used to teach math for
meaningful experiences.
270
Appendix B
Post-Test
1. How is math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/ experiences play a role in a math lesson? Give examples.
4. A. Name and describe a few teaching strategies pr approaches used to teach science for
meaningful understanding.
B. Name and describe a few teaching strategies or approaches to teach math for
meaningful understanding.
5. A. Think back to a science lesson that you taught this summer and briefly describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
6. A. Think back to a math lesson that you taught this summer and briefly describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
7. How has the blog helped you in your development of?
A. Ideas about teaching science?
B. Ideas about teaching math?
C. Similarities between the teaching of math and science?
271
Appendix C
Weekly Blogs
272
Weekly Blogs
Week 1: Why is following a Learning Cycle so important in teaching science? Won?t
more traditional approaches such as giving information first to students, such as in reading the
textbook, completing worksheets, and writing notes/definitions work just as well? Why not? I am
not convinced!
Week 2: We have learned about inquiry and the associated process skills for teaching
science through ?doing science?. We have learned that the Learning Cycle for planning and
teaching a series of lessons is ?best practice? to maximize student engagement and understanding
of the science we are teaching - often called a ?hands-on, minds-on? approach. So, how does the
Science-Technology-Environment-Society piece fit into all of this? What really is it anyway?
How does it work, and is it important in my science teaching? Please explain and help clear up
my confusion.
Week 3: So, this week you had the chance to finally practice teach about either ecological
or technological ideas to kids, and followed some portion of the Learning Cycle to do it!
(whether an ?exploration? activity to first develop students? common understandings OR an
?elaboration? activity to get them to apply their previous learning to a new situation or use). Also,
assessment was on everyone?s mind. So, how did you assess your students? attitudes,
understanding, or performance in your lesson this week? Do you feel your assessment strongly
aligned with your learning objective(s)? Was it authentic enough? Why is assessment so
important anyway? Share your thoughts about your thinking and how you are feeling about
assessment.
Week 4: This week we have been doing many hands-on activities in our FOSS Earth
Materials kit curriculum. All of the hands-on activities have been pretty fun, or at least
273
interesting. Most of us really believe that ?hands-on? is the best way to go in teaching science,
but is there more to it? What do you think? Are all hands-on activities equal? Is hands-on best no
matter what you do, when you do it, or how you do it? Explain to me your thinking now about
?hands-on? activities in science to best help student learning. I know that you can help me
understand this approach better and are pretty knowledgeable about how to do it best.
Week 5: Kids and stuff everywhere! Inventing and building and Newton?s Laws of
Motion can certainly seem to be unruly in the classroom, but is this O.K.? Taking kids outdoors
to learn about science in nature also has its own planning and managing hurdles, but is it worth
it? Even in doing the state of Alabama?s science teaching in the classroom (AMSTI) with kits,
there is a level of uncertainty and messiness with kids and materials ?in motion?, but it seems to
work. How are you now feeling about these issues? Where do you begin personally in your
future classroom? What is your current thinking and your plan?
Week 6: Think about the Learning Cycle. Explain how the Learning Cycle pertains to the
teaching and learning of mathematics. Support with examples. Then respond to two other
people's responses.
Week 7: So far in class we have discussed inquiry in mathematics, assessing
mathematical understanding, developing number sense, and participating in tasks to develop our
own mathematical knowledge. Think about all we have talked about, experienced for ourselves,
and experienced with students. Explain which part of the Learning Cycle you find to be the most
important in developing a true understanding of mathematics and why. Support with examples of
your own experiences or mathematics field experiences.
Week 8: In class we have been learning about how to assess and different types of
assessment. Think about one of your math teaching experiences this semester. How did you
274
determine student understanding of the topic? Be specific. Support with examples. Based on
your assessment what judgments and decisions will you have to make about teaching/learning?
Would you teach the lesson differently if you taught it again? Be specific. Support with
examples.
Week 9: We have used concrete materials in class and with students in lab. I want you to
think about the role that concrete experience plays in learning. Think of an instance in which
concrete experiences played a role in your own learning of mathematics. Describe that learning
experience. Describe how you have used concrete experience in one teaching lesson this term.
Week 10: Think about the two consecutive lessons you taught this week. What growth
did you see in your students' understanding of the topic? What role did concrete experience play
in your lessons? How did you promote inquiry?
275
Appendix D
Follow-up Discussion Questions
Pedagogical practices
1. In both science and math methods classes we focused on the Learning Cycle. What role
did the Learning Cycle play in your lesson development and implementation?
2. What specific teaching practices were you working on improving through the teaching of
this lesson?
Processes for meaningful learning
3. What role did inquiry play in the lesson? In what ways do you feel you promoted
inquiry?
4. What connections between mathematics and science did you want students to make?
How did you design the lesson to promote students making those connections between
mathematics and science?
Tools for conceptual development
5. In what ways did you use tools for conceptual development?
6. Would you use the tools differently next time you teach the same lesson? If yes, explain
how you would use the tools differently. If no, explain why you would keep the use of
the tools the same.
276
Appendix E
Final Reflection
1. How is math like teaching science? What is similar and what is different?
2. How would concrete tools/experiences play a role in a science lesson? Give examples.
3. How would concrete tools/ experiences play a role in a math lesson? Give examples.
4. Name and describe a few teaching strategies or approaches used to teach
science for meaningful understanding.
5. Name and describe a few teaching strategies or approaches to teach math for
meaningful understanding.
6. A. Think back to a math lesson that you taught during student teaching and briefly
describe it.
B. How do you think it met effective practice approaches to teaching? Explain.
277
Appendix F
Coding Guide
278
Coding Guide
Codes developed as the data was analyzed. Below are the major codes that developed.
Codes that did not answer the research questions were not included.
Coding Family: Tools
Code Description Example
Procedures Quotations in which tools are
for procedures
?Solve problems to get
answers?
Science book Quotations in which tools are
to enhance science textbook
?Tools allow students to see
concepts read about in science
book?
Real-World Quotations in which tools are
for real-world math
?Math is taught using tools
such as time, money, cooking?
Hands-on Quotations in which tools are
for hands-on learning
?Hands-on-sees things first
hand, understand why things
happen in science?
Within Teaching Approach Quotations in which tools are
key in teaching approach
?use tools for students to see
problem, students develop
their own methods , students
show how they got an answer
Coding Family: Processes
Code Description Example
Answers Quotations in which processes
are for answers
?Work together on a math
problem to figure answer?
Communication Quotations in which
processes to foster
communication
?Students discuss answer with
peers?
Reasoning Quotations in which reasoning
was a focus
?Students explained why
shapes were polygons?
Multiple processes Quotations in which multiple
processes work together
Explore-students work
through problems and come
up with solutions, have group
discussion
Tools Build Quotations in which tools
were used to build processes
?Used tools for students to
reason about measurement?
279
Coding Family: Pedagogical Approaches
Code Description Example
Games Quotations in which the
approach for math are games
?Math games allow for fun
experience with math
concepts?
Hands-on-structured Quotations in which the
approach is hands-on but
structured
?Hands-on has to be structured
and well planned?
Answers Quotations in which the
approach is to elicit right
answers
?Asked questions for students
to answer problems?
Reasoning then formula Quotations in which the focus
of teaching reasoning before
giving formula
?Give experiences that give
skill reason rather than giving
a formula?
Common Approach Quotations in which sees
common approach for science
and math
?Science and math both use
group activities, hands on, and
learning cycle?
Different Approaches Quotations in which sees math
and science as being taught in
different ways
?Hands-on for science, Work
problems in math?
Coding Family: Learning Cycle
Code Description Example
Steps Quotations in which the
learning cycle is described in
prescribed steps
?follow each step?
Partial Quotations in which only
recognizes certain aspects of
the learning cycle
?Engaged ?talking about
something they are interested
in?
Complete Quotations in which fully sees
all parts of the learning cycle
?Important to consider all
parts of the learning cycle?
280
Appendix G
Sample Coding Level 1 Phase I
281
Sample Coding Level I Phase I
T Sci-Hands-on allows for trial and error (pre)
Sci-Remember info through experience (pre)
See math applied (pre)
Math-Learn skills-Pre
Math-More engaging Pre
Ex: money, time , cooking Pre
Stu made art from recyclable materials (b3)
Allows stu to interact with learning (b4)
Hands-on must be clearly explained and applied
Concrete experiences with money ?making change (b9)
Sci-
concrete experiences-help stu observe and come to realizations (post)
Stu make discoveries, construct knowledge
Challenge stu thinking
Math- Post
Find meaning and reasoning
Application for math skills
(post)-saw importance of concrete experiences for both subjects
PA Experiments and hands-on for sci (pre)
Sci-info for key facts
Math-games, fun experiences (pre)
Flash cards ?remember key facts
Checked stu progress with assessment (b3)
Hands-on has to be structured and well planned (b4)
Concerned with est authority and management (b5) (b8)
Asked questions for stu to -Concerned with stu getting the right answer
(b8)
Sci ?Post
Using 5 E?s of LC
Stu experiment-discover truths rather than lecture
Stu observed in lesson
Math (post)
Give experiences that give skill reason rather than giving a formula
Stu folded and cut shapes, discussed
P Realistic application b1
Stu see purpose of sci and applications-connections b2
Use natural curiosity to solve problems b6
Explore phase stu expand their knowledge-reasoing-b7
Discussion to see if stu get right answer b8
Felt like she needed to guide ps more b8
Math-money-real life application b10
CA Can approach both with natural curiosity (b6)
Both requires hands-on interactive learning (post)
282
LC Levels the playing field (b1)
Students given reason why
Gives stu realistic application
Helps stu see many purposes of scie (b2)
Stu more interested in learning
Can be applied to math (b6)-engage, activate PK
Confidence booster
Use engage time to level the playing field (b6)
Without enough exploration-stu will have difficulty grasping concepts
(b7)
If not properly engaged stu will get lost in the lesson
Have to activate PK for stu to understand concepts
283
Appendix H
Sample Coding Level 2 Phase I
CT Moved from seeing tools in science as teacher controlled experiments for students
to see to students understanding the phenomena for themselves
Moved from seeing money as being a concrete tool to the use tools for students to
see problem, students developing their own methods, and students showing how
they got an answer.
PA Initially see science as teacher controlled experiments. She also considers
traditional methods to ok with the LC at the beginning of methods class. After
working in FEP believes science can be other than hand-on with interactive
worksheets and journals. By the end realized the lesson she taught was effective
because of her use of the LC.
Wants math to relate to the students lives. Initially she sees math as being group
work and hands-on. Her FEP experiences show her that students think differently
about math and still come up with the answers. Her conceptions of inquiry in math
are not real inquiry though. Proud of correct answers from students. Wants students
to be able to show her how they came up with their answer. Fun
P Science journals
Encouraging discourse
Asking challenging questions
LC Traditional approach criticism
Wants to explain topics better
Realized students could develop their own methods that would work
Thinks the LC is effective because it helps the students learn the material in depth.
284
Appendix I
Sample Coding Level 3 Phase I
PA Likes the idea of inquiry and hands-on, still tries to figure in traditional, wants
stu to get answers by themselves (1, 7, 16, 17, 20)
Sci-holding onto traditional, math-hands-on (2)
Concerned with facts for math and sci, hands-on can follow-(3)
Started with traditional views about sci and math. FEP influenced and changed
ideas to more hands-on for sci and math (4, 6,10)
Initially believed hands-on for sci and math. FEP reinforced (5, 8, 9, 11, 13, 14,
15, 21, 22)
Initially believed hands-on for science and a more traditional approach for math
?in the end thought what she was teaching in math was ?inquiry? but still more
along traditional (12)
Sees the importance of hands-on but in the context of fun or a reward when
students have been good and earned it (18)
Science went from students to hands-on to students understanding from hands-
on as well as making connections and relating science to real life situations. For
math went from idea of students working in groups by skill level to students
using concrete experiences to understand (19)
285
Appendix J
Sample Coding Phase II Jane
286
Sample Coding Phase II Jane
Initial codes Secondary codes
T Demonstrates how to measure to nearest ? and ?
inch 4-14, O
Showing them how to measure different items on
projector
Students given rulers to measure books, noses 4-
14, O
Wanted them to be able to move around 4-14 FI
Would have had more items 4-14, FI
Wanted them to go around the room and measure-
didn?t get to that 4-14-FI
Labels number line with fractional increments 4-
15 O
Students measuring items
Show circle graph from Illuminations 4-16 O
Has students use tiles to make a bar graph of data
collected 4-16 O and 4-16 FI
Used sticky notes earlier in the day to collect data
4-16 FI
Using the computer to make to circle graphs 4-16
FI
Labels coordinate grid on board 4-21 O
Uses coordinate grid on doc cam with boat cutout
4-21 O same as LP
Uses large floor grid 4-21 O
Students move on coordinates on floor grid 4-21 O
Fraction circle magnets 4-22 O
Wanted students to have their own but that wasn?t
possible 4-22 FI
Used to engage and think about fractional parts
Used whole for them to show me what it means
Uses animals to show how many out of
Uses pattern blocks for students to make cookies
Fraction circles students put together 4-23 O
Students make equivalent fractions with pieces 4-
23 O
Worried about use of fraction circles 4-23 FI
Ended up being more teacher directed 4-23 O
Would add a recording chart so students could
refer to what color means what fraction 4-23 FI
Instead of standard algorithms students should be
able to solve different ways FR
Tools for measurement-
procedural focus
Didn?t use tools often as
wanted
Tools for measurement-
procedural focus
Tools for representations of
information
Tools for active
involvement, engagement
Tools for representations of
information
Tools to reason and make
connections between
representations
Tools for active
involvement, engagement
Tools for engagement
Tools for representations
Tools to make connections
to concept
Tools for student to
demonstrate knowledge
Trying to promote inquiry
with students finding equal
pieces