AN EXAMINATION OF COGNITIVE AND NONCOGNITIVE FACTORS AND
ACADEMIC SUCCESS IN THE PREENGINEERING CURRICULUM
AT A FOURYEAR SOUTHEASTERN UNIVERSITY
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in the collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
_________________________________
Jennifer Lewis Bell
Certificate of Approval:
_________________________________ _________________________________
Glennelle Halpin, CoChair Gerald Halpin, CoChair
Professor Professor
Educational Foundations, Educational Foundations,
Leadership, and Technology Leadership, and Technology
_________________________________ _________________________________
Jennifer Good George T. Flowers
Coordinator, Assessment and Evaluation Interim Dean
College of Education Graduate School
AN EXAMINATION OF COGNITIVE AND NONCOGNITIVE FACTORS AND
ACADEMIC SUCCESS IN THE PREENGINEERING CURRICULUM
AT A FOURYEAR SOUTHEASTERN UNIVERSITY
Jennifer Lewis Bell
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
August 9, 2008
iii
AN EXAMINATION OF COGNITIVE AND NONCOGNITIVE FACTORS AND
ACADEMIC SUCCESS IN THE PREENGINEERING CURRICULUM
AT A FOURYEAR SOUTHEASTERN UNIVERSITY
Jennifer Lewis Bell
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
__________________________
Signature of Author
_________________________
Date of Graduation
iv
VITA
Jennifer Lewis Bell, daughter of Ted Lewis, Jr. and Judy Lewis Dewell, was born
October 27, 1975, in Thomasville, Georgia. She graduated from the University of West
Georgia with a Bachelor of Science in Education (Special Education: Mental
Retardation) in 1998, Master of Science in Education (Special Education: Interrelated) in
2002, and Education Specialist (Special Education: Curriculum and Instruction) in 2005.
She taught special education in the Troup County (GA) School District for 8 years before
leaving on an educational sabbatical in 2006 to attend Auburn University. In 2002, she
received National Board Certification in Exceptional Needs. While at Auburn University,
Jennifer worked as a graduate research assistant with Drs. Glennelle and Gerald Halpin.
Her responsibilities included managing three grantfunded program evaluation projects,
training data collection personnel, and analyzing program data. In addition, she served as
the Tiger Eyes Director for the Auburn University Marching Band.
v
DISSERTATION ABSTRACT
AN EXAMINATION OF COGNITIVE AND NONCOGNITIVE FACTORS AND
ACADEMIC SUCCESS IN THE PREENGINEERING CURRICULUM
AT A FOURYEAR SOUTHEASTERN UNIVERSITY
Jennifer Lewis Bell
Doctor of Philosophy, August 9, 2008
(Ed.S., University of West Georgia, 2005)
(M.Ed., University of West Georgia, 2002)
(B.S.Ed., University of West Georgia, 1998)
232 typed pages
Directed by Glennelle and Gerald Halpin
A large amount of empirical research has been conducted on academic
achievement with college students. The empirical studies have revealed the significance
of high school preparation, more specifically mathematical preparation, for academic
success in postsecondary institutions; however, limited research exists for predicting
academic success using cognitive and noncognitive factors (i.e., selfconcept, study
habits, and inquisitiveness). The nature of engineering college courses tends to be
quantitatively oriented, and calculus tends to serve as the gateway course for academic
success within these majors. Conversely, noncognitive factors significantly contribute to
college mathematics achievement beyond standardized test scores or high school ranks.
vi
The purpose of this study was to determine if cognitive factors mediate the effect
of noncognitive factors on quantitative grade point average and to determine if these
cognitive and noncognitive factors can predict admission status in engineering
education. With College Freshman Survey results from a sample of 2,276 college
freshman students who intended to major in engineering, the following statistical
analyses were used: exploratory factor analysis, confirmatory factor analysis, structural
equation modeling, and discriminant function analysis.
The structural model analysis revealed that cognitive factors (ACT math scores,
high school math grades, and high school ranks) mediated the effects of noncognitive
factors (lack of confidence in academic ability, mathematical ability, difficulty with
problem solving, and selfappraised abilities) on the quantitative GPA for the pre
engineering curriculum. The results of the discriminant function analysis suggested that
participants who were admitted to engineering and those participants who left
unsuccessful were classified correctly based on the cognitive and noncognitive factors.
The overall percentage of correctly classified cases was 51.6% with this analysis.
Moreover, the model accounted for 29% of the variance in the quantitative GPA.
As a concluding part of this study, a secondary mathematics curriculum was
developed to improve mathematical skills and problemsolving abilities. Within the
Mathematics Curriculum for Advanced Mathematical Proficiency, the mathematical
concepts are taught within realworld contexts. Each unit has an engineering connection
to familiarize the students with the various fields of engineering.
vii
ACKNOWLEDGEMENTS
To my committee members, Dr. Gerald Halpin, Dr. Glennelle Halpin, and Dr.
Jenny Good, each of you gave me, a little girl from rural southeast Georgia, a chance of a
lifetime. Boss Man and Mi Lady, you introduced me to the world of statistics and
research. The experience far exceeds any knowledge that I gained in the classroom
setting. Mrs. Dr. Good, you introduced me to the Auburn Experience and to the Auburn
University Band. Because of you all, I was fortunate to pursue my love for mathematics
and music simultaneously. My last 2 years have been a whirlwind, but it has been worth
it. I also appreciated the insight and assistance of my Outside Reader, Dr. Craig Darch
Also, I would like to thank the faculty and staff of EFLT who tolerated my endless
questions.
To my cohort, Lt. Joe Baker, you supported and guided me through the tedious
process of writing, editing, and revising each chapter. You gave me your time when you
did not have the time for yourself. I am grateful and appreciative for your support.
Lastly, to my daddy, Ted Lewis, Jr., who has never taken the credit for his
contributions to my success, you nurtured my quantitative mind without stifling my
creativity or thirst for logical reasoning. We share more than our birthdays; my
quantitative abilities derive from you.
viii
Style manual or journal used: Publication Manual of the American
Psychological Association, 5
th
Edition.
Computer software used: SPSS 15, AMOS 7.0, Windows XP, and Microsoft
Word 2007
ix
TABLE OF CONTENTS
Page
LIST OF TABLES ................................................................................................ xii
LIST OF FIGURES ............................................................................................. xiv
CHAPTER
I. INTRODUCTION .................................................................................1
Statement of Problem ...........................................................................1
Rationale ...............................................................................................6
Research Questions ..............................................................................9
Definition of Terms ..............................................................................9
Brief Methodology .............................................................................11
Exploratory Factor Analysis ....................................................11
Confirmatory Factor Analysis..................................................11
Structural Equation Model .......................................................12
Discriminant Function Analysis ..............................................12
Limitations of Study ...........................................................................12
Significance of the Study ...................................................................13
Organization of the Study ...................................................................16
II. REVIEW OF LITERATURE ..............................................................17
Predicting Success in Undergraduate Curriculum ..............................17
Cognitive Factors .....................................................................19
NonCognitive Factors .............................................................24
Cognitive and NonCognitive Factors .....................................26
Predicting Success in Engineering Curriculum ..................................32
Cognitive Factors .....................................................................32
NonCognitive Factors .............................................................36
Cognitive and NonCognitive Factors .....................................40
Summary ............................................................................................48
Cognitive Factors .....................................................................48
NonCognitive Factors .............................................................49
III. METHODS ..........................................................................................50
Participants ........................................................................................50
Procedures .........................................................................................52
x
Measures ...........................................................................................52
College Freshman Survey ........................................................52
Institutional Data ......................................................................55
Design and Analysis .........................................................................56
Exploratory Factor Analysis ....................................................56
Confirmatory Factor Analysis..................................................58
Structural Equation Model .......................................................60
Discriminant Function Analysis ..............................................66
Summary ...........................................................................................67
IV. RESULTS ............................................................................................69
Relationship between Cognitive and NonCognitive Factors
and Quantitative GPA ..............................................................69
Effects of Cognitive and NonCognitive Factors
on Engineering Admission Status ............................................76
Summary ...........................................................................................85
V. SUMMARY, CONCLUSIONS, DISCUSSION,
AND RECOMMENDATIONS ...........................................................86
Summary of Methods ........................................................................87
Exploratory Factor Analysis ....................................................87
Confirmatory Factor Analysis..................................................87
Structural Equation Model .......................................................87
Discriminant Function Analysis ..............................................88
Findings of the Study ........................................................................88
Future Research ................................................................................90
VI. CURRICULAR IMPLICATIONS.......................................................92
Context ..............................................................................................94
Historical Development for Secondary
Mathematics Curriculum .........................................................95
Secondary Mathematics Curriculum ......................................105
Needs Assessment ...........................................................................107
Number Sense, Properties, and Operations ............................108
Measurement ..........................................................................109
Geometry and Spatial Sense ..................................................110
Data Analysis, Statistics, and Probability ..............................111
Algebra and Functions ...........................................................113
Mathematical ProblemSolving Ability .................................116
Proposed Strategy ...........................................................................120
Goal #1 ...................................................................................120
Goal #2 ...................................................................................121
Goal #3 ...................................................................................121
Method ...................................................................................121
xi
Outcome Evaluation ...............................................................127
Expected Findings ..................................................................129
REFERENCES ....................................................................................................131
APPENDICES .....................................................................................................142
APPENDIX A. INSTITUTIONAL REVIEW BOARD .........................143
APPENDIX B. LOGIC MODEL ...........................................................145
APPENDIX C. GEOMETRY CURRICULUM UNITS ........................147
APPENDIX D. ALGEBRA II CURRICULUM UNITS ........................159
APPENDIX E. PRECALCULUS/RIGONOMETRY
CURRICULUM UNITS ...............................................170
APPENDIX F. ADVANCED PLACEMENT CALCULUS AB
CURRICULUM UNITS ...............................................185
APPENDIX G. LESSON PLAN DESIGN RATING SYSTEM ...........196
APPENDIX H. SAMPLE MATHEMATICAL PROBLEMSOLVING
EXAMINATION AND SCORING RUBRIC ..............199
APPENDIX I. PROFESSIONAL DEVELOPMENT
EXIT SURVEY ............................................................201
APPENDIX J. WEEKLY INFORMAL OBSERVATION FORM ......206
APPENDIX K. NCTM STANDARDSBASED EXPECTATIONS .....209
xii
LIST OF TABLES
Table Page
1 Frequencies by Academic Year ...................................................................51
2. Scales for the College Freshman Survey ....................................................53
3. Alpha Reliability Coefficients for NonCognitive Factor Scales ................55
4. Alpha Reliability Coefficients for Revised NonCognitive Factor Scales ..58
5. Standardized Beta Weights by Confirmatory Factor Analysis
for the SevenFactor Model .............................................................................59
6. Means and Standard Deviations for Each High School
Mathematics Course .........................................................................................63
7. Intercorrelations for the Cognitive and Revised NonCognitive Factors ....65
8. Means and Standard Deviations for the Cognitive
and Revised NonCognitive Factors ...............................................................65
9. List of Possible Quantitative Courses in PreEngineering Curriculum .......66
10. Frequencies for Admission Status by Academic Year ..............................67
11. Correlation of Predictor Variables with Discriminant Functions
(Structure Matrix) and Standardized Discriminant Functions Coefficients ....78
12. Means and Standard Deviations by Group ...............................................80
13. Post Hoc Test Results: Mean Differences by Group ................................80
14. Classification Analysis for Admission Status ............................................82
15. Crossvalidation: Classification Analysis for Admission Status ...............83
16. Means and Standard Deviations by Group ................................................84
xiii
17. CrossValidation: Means and Standard Deviations by Group ...................84
xiv
LIST OF FIGURES
Figure Page
1. Structural Equation Model ...........................................................................70
2. Structural Equation Model: Restricted Model ............................................71
3. Crossvalidation Structural Equation Model ..............................................74
4. Crossvalidation Structural Equation Model: Restricted Model .................74
1
CHAPTER I
INTRODUCTION
Statement of Problem
According to the National Science Foundation (NSF) (National Science Board,
2006b), in 1983, 11.5% of the US college freshmen declared engineering as their
intended major. This percentage slightly decreased to 9.6% in 2004. In addition to the
trend of decreased interest, the rate of retention in the field of engineering has decreased.
Of the 1983 college graduates, 7.4% of them earned a bachelor?s degree in engineering.
As a comparison, 4.6 % of the college graduates in 2002 completed a program in
engineering. These percentages indicate that a disproportionately high number of students
switch out of engineering majors because they either lose interest in engineering or have
academic difficulties (Wulf & Fisher, 2002). From 1975 to 1999, the number of US
students who completed bachelor?s degrees in the natural science and engineering fields
has dropped from 3rd to 14th compared to 19 other countries (National Science Board,
2006a). The declining interest in engineering fields and increasing attrition rates of pre
engineering majors have led to a serious shortage of engineers (Felder, Forrest, Baker
Ward, Dietz, & Mohr, 1993).
In the middle of the 20th century, President John F. Kennedy inspired a nation of
scientists and engineers to win the space race after the Soviet Union?s launch of Sputnik
in 1957. These motivated individuals are reaching retirement age in the beginning of the
2
21st century, yet the declining interest and increasing attrition rates have reduced the
number of scientists and engineers to replace them. This shortage of prepared scientists
and engineers can be linked to poor preparation in mathematics and science instruction at
the K12 level (National Science Board, 2006a).
As an indicator of academic difficulties and confirmation of the NSF?s
conclusion, the percentage of college freshman engineering majors who reported the need
for remediation in mathematics has increased since 1984, from 11.7% in 1984 to 14.0%
in 2002 (National Science Board, 2006b). Between the years of 1992 and 2000, 20% of
the freshmen who entered a doctoral institution took at least one remedial course in
mathematics (National Science Board, 2004).
The student?s decision to persist or change occurs during the first year of study at
the college level. Often, this decision is based on successful completion of a gateway
course (e.g., calculus) because the culture in these engineering courses tends to be
quantitatively oriented (Gainen & Willemsen, 1995). Moreover, the knowledge gained
from these quantitative courses is essential for the nation to compete successfully in
today?s global society (National Science Board, 2006a).
Possibly one reason college freshmen have problems in mathematics could be the
apparent stagnation in mathematical ability for 12thgrade students. Since 1969, the
National Assessment of Educational Progress (NAEP) has yielded assessments in
reading, mathematics, science, writing, social studies, and the arts among 4th, 8th, and
12thgrade students from public and private schools. An examination of the 1999 NAEP
average mathematics score for the nation revealed that the 9yearold and 13yearold
participants continued to improve their scores each year, but the 17yearold students?
3
scores have remained stagnant since 1973 (Campbell, Hombo, & Mazzeo, 2000).
Furthermore, 97% of the 17yearolds achieved a level of 250 that indicated math
proficiency in the four basic math operations and solving of onestep word problems.
However, only 8% of all 17yearold students scored at the 350 level indicating that they
were capable of understanding and computing multiplestep problems. These data
indicated that nationally 92% of all 17yearold students who took the NAEP test could
not comprehend or solve multiplestep problems.
More promising results of students? ability to complete mathematics problems
were found in a study by Mitchell, Hawkins, Stancavage, and Dossey (1999). Using the
same NAEP mathematics data, these researchers focused on the disaggregated data for
students in 8th and 12th grade who took higher level mathematics courses. These
researchers found that 30% of advanced 12th graders correctly solved problems involving
two or more steps. However, all of the students who comprised the disaggregated data
group indicated on surveys that their mathematics courses included a heavy emphasis on
problemsolving skills. To support the findings of this NAEP study, of the 108,437
students who took the Advanced Placement (AP) calculus AB exam in 1997,
approximately 59% of them earned a passing score of 3 or higher on a 5point scale. Nine
years later, in 2004, the pass percentage was nearly equivalent. These findings along with
others, which have employed the NAEP data, indicate that many students lack
proficiency when presented with mathematics problems that involve higher order
thinking skills (National Science Board, 2006b).
Another indicator of mathematical achievement is the number of advanced
mathematics courses taken at the high school level. Despite increasing percentages of
4
advanced mathematics courses being offered at the high school level (i.e., 26.8% increase
for statistics and probability and 13.4% increase for calculus since 1990), of the 2000
graduating class nationwide, only 5.7% of them completed a statistics and probability
course, and 12.6% completed a calculus course (National Science Board, 2006b).
Courses in mathematics, such as calculus, can open or close the gate for students
interested in mathematical, scientific, or technological careers (Gainen & Willemsen,
1995).
Mathematics requires fundamental knowledge of concepts and procedures;
however, it requires critical and analytical thinking skills. These mathematical problem
solving skills allow the students to apply their fundamental knowledge in various
contextual situations. Students need to practice problemsolving skills in reallife
situations. By practicing these skills, the students can increase their engagement with the
content of mathematics, increase their ability to think critically, and increase their
performance on higher order cognitive questions (Mitchell, Jakwerth, Stancavage et al.,
1999; Wulf & Fisher, 2002). Based on these reasons, there is a need to prepare the
students to solve contextual problems. Thus, they will be prepared for the everchanging
society (National Academy of Engineering, 2005; Litzinger, Wise, & Lee, 2005; Wulf &
Fisher).
Education must academically prepare those potential engineers for the world of
tomorrow (National Academy of Engineering, 2005). Anthony, Hagedoorn, and Motlagh
(2001) suggested problemsolving and application skills would increase the likelihood of
success in engineering (e.g., correlating the calculus and physics content). Unfortunately,
5
traditional classroom instruction provides minimal preparation for inquirybased learning
or critical thinking during performancebased tasks.
The nation must prepare students in K12 education for tomorrow?s demands in
the workforce and society. With continuing advances in technology, students must have a
solid foundation in mathematics to be productive members in their communities
(National Science Board, 2006a). External forces of society, economy, and profession
challenge the stability of the engineering workforce. This instability affects recruitment
of the most talented students into the engineering profession (National Academy of
Engineering, 2005).
The NSF recommends further research regarding teaching and learning
mathematics. For the students, the NSF recommends student exposure to science,
technology, engineering, and mathematics careers through activities (National Science
Board, 2006a). Similarly, Gainen (1995) and Klingbeil, Mercer, Rattan, Raymer, and
Reynolds (2005) recommend early intervention programs in high school and a strong
emphasis on application and appreciation of mathematical inquiry to increase student
success in quantitative courses.
Mathematics ability is the strongest predictor of success in the field of
engineering (LeBold & Ward, 1988). A correlational study conducted by van Alphen and
Katz (2001) with electrical engineering majors supports this notion. The researchers
found that a strong relationship existed between admission to engineering and academic
background. Likewise, Klingbeil et al. (2005) pointed to a lack of high school
quantitative preparation as the most notable factor that influences success in engineering.
6
Without a strong foundation in algebra, the doors are closed for subsequent mathematics
courses (Edge & Friedberg, 1984; Klein, 2003).
Heinze, Gregory, and Rivera (2003) stated that high school mathematics
sequences are very important for students interested in engineering. Similarly, Buechler
(2004) found that grades during the firstsemester calculus course predicted student
performance in the engineering core classes. In addition to cognitive skills, students?
attitudes relating to mathematical achievement can predict success in mathematics
courses. These attitudes include expectation of success, their mathematical ability
compared to their peers, and confidence in their own ability (House, 1995c). Similarly,
success in engineering not only depends on mathematical knowledge and skill, but also it
depends on the attitudes that the students bring with them to college. By measuring these
initial attitudes, academic success can be improved. Students may have been
academically successful in high school, but they may lack the confidence in their
mathematical ability (BesterfieldSacre, Atman, & Shuman, 1997).
Rationale
Several researchers have examined the relationship between cognitive factors
[i.e., high school grade point averages (GPAs), standardized test scores, and high school
ranks] and academic success in postsecondary education (House, 2000). Zhang,
Anderson, Ohland, and Thorndyke (2004) examined 15 years of student data across nine
universities. The purpose of the study was to determine the academic factors for degree
completion in engineering. Using logistic regression, the researchers found that high
school GPAs and Scholastic Aptitude Test (SAT) quantitative scores were positively
correlated with graduation rates across all universities. The coefficient of determination
7
(R
2
= .126) indicated that other preexisting variables would account for the variance.
These results supported the findings of Gilbert (1960) who found that standardized test
scores could not be used as the single predictor of success in engineering. Factors other
than academic aptitude should be examined to predict graduation in engineering. Smith
and Schumacher (2005) suggested the additional research with mathematical ability,
interest, study habits, and academic selfconcept to explain further the variance in college
GPA.
The empirical studies have revealed the significance of high school preparation
for academic success in postsecondary institutions. More specifically to mathematical
preparation, Buechler (2004) examined the mathematical background of electrical
engineering students over a 4year period. Based on pretest results, he found that a
significant number of students did not have the proper mathematical background to be
successful in electrical engineering. Consequently, a large portion of instructional time
during the course was used for remediation. More than 46% of the students had
deficiencies in either algebra or trigonometry. These findings reflected the lack of high
school mathematical preparation for quantitative majors.
The results of the BesterfieldSacre et al. (1997) study suggested that students
with adequate mathematical abilities might not be successful in the field of engineering
because their selfassessed abilities and confidence level were not equivalent to their
actual abilities. For the participants who left engineering in poor standing, they had
confidence in their abilities but were not academically prepared, which may be related to
their inability to manage time and study course material effectively. In a similar study,
Burtner (2004) investigated the relationship between persistence in the engineering
8
curriculum and high school achievement, attitudes, and college academic performance.
He concluded that high school GPA indicated the student?s ability to persevere and study,
which are significant predictors for academic success in college.
Limited research exists for predicting academic success using cognitive and non
cognitive factors (i.e., selfconcept, study habits, and inquisitiveness) (House, 1995b,
2000). A study conducted by LeBold and Ward (1988) investigated the cognitive and
noncognitive variables associated with college and engineering retention using a national
and institutional sample. In addition to academic background factors, the researchers
found selfperceived abilities in math and problem solving were strong predictors of
persistence in engineering. House (1995b) reported similar findings with his study to
predict achievement in an introductory college mathematics course.
In a study by Blumner and Richards (1997), study habits were found to be
associated with academic achievement because participants with high GPAs had
significantly lower levels of distractibility and higher levels of inquisitiveness. The
research of Shaughnessy, Spray, Moore, and Siegel (1995) connected noncognitive
factors with academic success. They investigated personality factors, SAT scores, and
screening test scores to predict the final grade in calculus I for pharmacy majors. While
the screening test for calculus was the most significant predictor, three personality factors
(reasoning, emotional stability, and privateness) were significant contributors to the
model. These results supported the notion that participants who were abstract thinkers
tended to be more successful in calculus I than concrete thinkers.
Numerous research studies found cognitive factors, such as academic background
and standardized test scores, to be significant predictors of academic success in
9
engineering; however, limited research investigated the relationship between non
cognitive factors and academic success in engineering (French, Immekus, & Oakes,
2005; House, 2000; Shuman et al., 2003). The purpose of this study was to determine if
cognitive factors mediate the effect of noncognitive factors on quantitative GPA and to
determine if these cognitive and noncognitive factors can predict admission status in
engineering education.
Research Questions
The study addressed the following research questions:
1. Do cognitive factors (ACT math scores, high school math grades, and high
school ranks) mediate the influence of noncognitive factors on quantitative GPA in the
preengineering curriculum?
2. Can cognitive and noncognitive factors predict engineering admission status?
Definition of Terms
Cognitive factors ? Academic background factors including standardized test
scores, high school ranks, and high school GPAs (House, 2000).
Core curriculum ? The requirements for core curriculum differs by engineering
major. In general, the following quantitative units are required for each major: 3 calculus,
1 linear algebra, 1 differential equations, 2 chemistry, and 2 physics courses. The
humanities requirements are 2 English composition, 2 world literature, and 2 history
courses. Other requirements include an orientation to engineering and an introduction to
computing courses. The remaining courses taken during the first 2 years of study are
majorspecific requirements.
10
Mathematics ? A broad field of study encompassing numbers and operations,
algebra, geometry, measurement, and data analysis (National Council of Teachers of
Mathematics, 2000).
Mathematical problem solving ? The ability to apply and generalize various
mathematical concepts in order to solve contextual and multiplestep problems (National
Council of Teachers of Mathematics, 2000).
Noncognitive factors  Perceived abilities, personal attitudes, and confidence in
mathematical skills (Burtner, 2005).
Quantitative courses ? A group of college courses whose conceptual foundation is
based in mathematics (Gainen & Willemsen, 1995). The following courses from the pre
engineering curriculum are considered quantitative courses in this study:
1. College Algebra (MA1000)
2. PreCalculus Trigonometry (MA1130)
3. PreCalculus Algebra Trigonometry (MA1150)
4. Calculus I (MA1610)
5. Honors Calculus I (MA1617)
6. Calculus II (MA1620)
7. Honors Calculus II (MA1627)
8. Calculus for Engineering and Science I (MA1710)
9. Calculus for Engineering and Science II (MA1720)
10. Calculus III (MA2630)
11. Calculus for Engineering and Science III (MA2730)
12. Survey of Chemistry I (CH1010)
13. Survey of Chemistry II (CH1020)
14. Fundamentals of Chemistry I (CH1030)
15. Fundamentals of Chemistry II (CH1040)
16. General Chemistry I (CH1110)
17. Honors General Chemistry I (CH1117)
18. General Chemistry II (CH1120)
19. Honors General Chemistry II (CH1127)
20. Foundations of Physics (PH1000)
21. General Physics I (PH1500)
22. General Physics II (PH1510)
23. Engineering Physics I (PH1600)
11
24. Honors Physics I (PH1607)
25. Engineering Physics II (PH1610)
26. Honors Physics II (PH1617)
Brief Methodology
The participants in this study were a sample of 2,276 students who entered
Auburn University during the fall semester of 2000 through the fall semester of 2004.
The participants who had an intended engineering major included 1,857 (81.6%) males
and 419 (18.4%) females. Of these cases, the racial classification was 1,885 (82.8%)
White, 258 (11.3%) Black, and 133 (5.8%) students who reported that they belonged to
other racial groups.
Requirements for participation in this study included completion of the College
Freshman Survey (Halpin & Halpin, 1996) and at least two quantitative courses in the
preengineering curriculum at Auburn University. The final letter grades in each
quantitative course were coded using the 4point scale, A = 4, B = 3, C = 2, D = 1, and
F = 0, and were averaged together to create the quantitative GPA.
Exploratory factor analysis. The sample was randomly divided into two databases
of comparable size. With one database, an exploratory factor analysis using principal axis
factoring with an oblimin rotation was conducted using the following scales from the
College Freshman Survey: Math SelfConcept, SelfAppraisal, Perceived Difficulty,
ProblemSolving Ability, Need Help, Academic Difficulty, Study Habits, and Academic
SelfConcept. The purpose of this factor analysis was to discover the factor structure of
the selected items and the correlations between the factors.
Confirmatory factor analysis. With the second database, a confirmatory factor
analysis was conducted to determine how the theoretical structure fits with the data and to
12
crossvalidate the factor structures created with the exploratory factor analysis (Meyers,
Gamst, & Guarino, 2006).
Structural equation model. After the initial frequencies, descriptives, and bivariate
correlations were assessed, a structural equation model was created using AMOS 7.0 to
determine the relationship between cognitive and noncognitive factors and quantitative
GPA in the preengineering curriculum with 60% of the sample. The exogenous variables
were the noncognitive factors that were created and were confirmed with the factor
analyses and the cognitive factors [American College Testing Program (ACT) math
score, mean high school math grades, and high school ranks]. Since these data have not
been previously analyzed with structural equation modeling, the model created with the
60% sample was applied to the 40% crossvalidation sample to confirm the analysis
results.
Discriminant function analysis. A discriminant function analysis was conducted
using the 60% sample to develop a weighed linear combination to predict group
membership (i.e., admitted to engineering, switched to another major at the university, or
left the university unsuccessful). The analysis used admission status as the grouping
variable and the cognitive and noncognitive factors as the independent variables. The
40% holdout sample was used to crossvalidate to results with the analyzed 60% sample.
Limitations of Study
The College Freshman Survey: Engineering Form (Halpin & Halpin, 1996) was
administered during freshman orientation. Due to the time frame for survey
administration, incoming engineering students may have elected to participate in other
activities and did not complete the survey. During the academic years used in this study,
13
approximately 71% of the preengineering freshmen completed the survey. Another
limitation of the study was that the results cannot be generalized due to the use of a single
university data source. In addition, this sample included traditionalaged college students
who were primarily White males. Furthermore, the effects of background variables, such
as socioeconomic status, were not investigated in this study. Lastly, attitudes and self
assessed abilities collected with the College Freshman Survey may change over time and
may be affected when students transition from high school to college.
Significance of the Study
To predict academic success in postsecondary education or within a specific
curriculum, standardized test scores and high school ranks have been found to be
significant predicting variables. The findings of Baron and Norman (1992), Edge and
Friedberg (1984), and Smith and Schumacher (2005) indicated that high school ranks
were the most significant contributor to academic success. Other researchers (e.g., House,
1995b; House, Keely, & Hurst, 1996; Wilhite, Windham, & Munday, 1998) found that
ACT scores were the most significant contributors to academic success and persistence.
In the previous studies, cognitive factors were used to predict academic success,
but the results were used to advise and place students within the college curriculum, to
restructure the college curriculum and instruction, or to retain the engineering majors.
Smith and Schumacher (2005) specifically addressed mathematical preparation (i.e., high
school calculus grades), but the predictive model used college students who were
enrolled in an actuarial program of study.
Pertaining to the field of engineering, Buechler (2004) and Burtner (2004)
investigated college mathematical background with electrical engineering majors. The
14
nature of science, engineering, and mathematics college courses tends to be quantitatively
oriented, and calculus tends to serve as the gateway course for academic success within
these majors (Gainen & Willemsen, 1995). According to House (1995b), noncognitive
factors significantly contribute to college mathematics achievement beyond standardized
test scores or high school ranks. House (1993) examined the effects of noncognitive
factors on academic success after covariating for the effects of the ACT scores and found
that students with lower academic selfconcepts tended to earn lower grades in college
algebra courses.
Numerous studies have examined the relationship between academic success and
academic selfconcept (Burtner, 2004; Burtner, 2005; House, 1995a, 1995b, 1995c,
2000), math abilities (Burtner, 2004; Burtner, 2005; House, 1995a, 1995c; LeBold &
Ward, 1988; Shuman et al., 2003), and problemsolving ability (BesterfieldSacre et al.,
1997; Blumner & Richards, 1997; Brown, 1994; Burtner, 2004; Lackey, Lackey, Grady,
& Davis, 2003; LeBold & Ward; Litzinger et al., 2005; Shaughnessy et al., 1995). Fewer
studies have investigated the influence of selfawareness (BesterfieldSacre et al.; Brown;
Brown & Cross, 1993; Harackiewicz, Barron, Tauer, & Elliot, 2002; House, 1995b;
Shuman et al.) and study habits (BesterfieldSacre et al.; Blumner & Richards, 1997;
Burtner, 2004; Harackiewicz et al.; Nixon & Frost, 1990; Wesley, 1994) on academic
success.
Beyond the variables used in these studies, the majority of statistical analyses
used with these previous studies were correlational and multiple regression analyses.
These predictive and correlational studies accounted for the variance in the cumulative
GPAs at the college level, which contain humanities and elective courses. Other studies
15
predicted first semester GPAs (BesterfieldSacre et al., 1997; Brown, 1994) or final
grades in a psychology (Harackiewicz et al., 2002; House et al., 1996), science course
sequence (House, 1995a; Sadler & Tai, 2001), and in a calculus sequence (Edge &
Friedberg, 1984; House, 1995b, 1995c; Shaughnessy et al., 1995; Wilhite et al., 1998).
One study (Smith & Schumacher, 2005) used a multiple regression analysis to predict
math GPA with actuarial graduates. A few studies used logistic regression (Besterfield
Sacre et al., 1997; French et al., 2005; House, 1995a; MollerWang & Eide, 1997; Zhang
et al., 2004) and discriminant analyses (Burtner, 2005; Gilbert, 1960) to predict group
membership in mathematics courses or admission status.
Structural equation analyses have not been used to explain the relationship
between cognitive and noncognitive factors and quantitative GPA, more specifically for
engineering majors. By using a structural equation model, the relationship between the
latent variables can be assessed while controlling for measurement error. In addition, the
extent to which the theoretical framework and empirical data are consistent can be
determined. Finally, these multiple regression analyses can determine the extent to which
the measured variables define the respective latent variables (Meyers et al., 2006).
In addition, previous studies have not addressed the need for developing
mathematical problemsolving abilities at the secondary level so the students will be
better prepared for the quantitative courses within the rigorous preengineering
curriculum (Litzinger et al., 2005; Wise, Lee, Litzinger, Marra, & Palmer, 2001; Wulf &
Fisher, 2002). According to the National Academy of Engineering (2005), future
generations need to be educated to be lifelong learners who are critical thinkers and able
to visualize multiple solutions for a given situation. Unfortunately, according to the
16
NAEP data, the majority of 12thgrade students cannot solve multiplestep word
problems despite enrollment in advanced mathematics courses (Campbell et al., 2000).
Considering the limited research that focuses on the predictive relationship
between cognitive and noncognitive factors and academic success in quantitative
courses, the need for further research is apparent. Therefore, this research study builds
upon and expands the findings of French et al. (2005), House (2000), and Shuman et al.
(2003) by investigating the relationship between cognitive and noncognitive factors and
quantitative GPA of incoming freshmen who have indicated they will major in
engineering.
Organization of the Study
Chapter I introduces the study, statement of the problem, research questions, and
definition of terms. Chapter II includes a review of the current literature considering
academic success in undergraduate curriculum and academic success in the engineering
curriculum. Chapter III describes the participants, instrumentation, data collection, and
data analyses. The results of the statistical analyses are presented in Chapter IV. Chapter
V contains summarized findings and future research implications. Chapter VI outlines the
contextual information about secondary mathematics, proposed secondary mathematics
curriculum, and proposed program evaluation plan.
17
CHAPTER II
REVIEW OF LITERATURE
The purpose of this study was to determine if cognitive factors mediate the effect
of noncognitive factors on quantitative GPA and to determine if these cognitive and non
cognitive factors can predict admission status in engineering education. To define and
investigate the issue, the following external forces pertaining to secondary mathematics
and students who enroll as preengineering majors were examined: (a) predicting success
in the undergraduate curriculum and (b) predicting success in the engineering curriculum.
Since there is limited literature that has investigated the relationship between cognitive
and noncognitive factors and success in a preengineering curriculum, research in this
area requires further examination (French et al., 2005; House, 2000; Shuman et al.,
2003).
Predicting Success in the Undergraduate Curriculum
Noble, Roberts, and Sawyer (2006) investigated factors (background
characteristics, high school academic achievement, educational accomplishments, and
selfperceptions) related to performance on the ACT. The ACT composite score, which
ranges from 1 to 36, is an average of the four subject subtests: English, Mathematics,
Reading, and Science. The curriculumbased measure was developed to measure
knowledge acquired in high school and to measure the skills needed for academic success
in postsecondary institutions. The participants for this study were selected by school to
18
maximize within school sample sizes. The sample included high school juniors and
seniors who registered to take the ACT in February 2002 (n = 1,906) or in April 2002 (n
= 1,400). Two weeks after the ACT administration, participants were mailed a survey to
measure noncognitive factors.
Using structural equation model analyses, the researchers found high school
academic achievement (high school mathematics courses, honors or AP courses, and high
school GPA in the four core subjects) had a direct effect on the ACT score (R
2
= .63 for
African American students and R
2
= .64 for Caucasian American students). Family
background (household income, parents? level of education, and number of negative
situations present in the home), educational accomplishments (extracurricular activities
and outofclass accomplishments), and psychosocial factors (selfconcept, positive
attributions, selfefficacy, problemsolving skills, and interpersonal communication
skills) had an indirect effect on the ACT score and a direct effect on high school
academic achievement (R
2
= .42 for African American students and R
2
= .28 for
Caucasian American students)
. The number of high school mathematics courses
completed, number of honors or AP courses completed, and high school GPA had a
moderate relationship with the ACT score for both ethnic groups; correlation coefficients
ranged from .35 to .56 (Noble et al., 2006).
These results suggest that academic achievement in high school, such as rigorous
curriculum and successful grade performance, can increase one?s performance on the
ACT. Furthermore, according to Noble et al. (2006), the development of noncognitive
factors, such as positive coping skills and realistic selfappraisal of abilities, can assist
19
students in overcoming background factors that might otherwise affect their likelihood of
academic success.
Cognitive factors. Standardized tests, such as the ACT and SAT, are examined in
various empirical studies in regard to predicting success in the undergraduate curriculum.
Other cognitive factors related to cognitive factors include high school GPAs and high
school ranks (Edge & Friedberg, 1984; Sadler & Tai, 2001; Smith & Schumacher, 2005;
Wilhite et al., 1998).
A study conducted by Edge and Friedberg (1984) investigated the relationship
between cognitive factors and academic success in the first calculus course in college.
The cognitive variables were ACT scores, high school ranks, high school GPAs, high
school algebra grades, and scores from an algebra pretest. Other predictor variables
include gender, birth order, family size, and high school size. The participants were three
groups of students at Illinois State University. The first group (n = 235) enrolled at the
university during the fall of 1976, the second group (n = 157) enrolled during the fall of
1978, and the third group (n = 397) enrolled during the fall of 1980. The researchers
found a high correlation between the algebra pretest scores and the final grades in the
calculus course.
A stepwise multiple regression analysis was conducted for each group. For the
first group, the algebra pretest scores and high school ranks were the significant
contributors to the model (R
2
= .524). Similar results occurred with the second group. For
the third group, algebra pretest scores, high school GPAs, ACT math scores, and high
school ranks were significant contributors to the model (R
2
= .746). Edge and Friedberg
(1984) hypothesized that high school ranks represent a measure of competitiveness and
20
longterm characteristic of emotional adjustment; therefore, high school ranks can
account for the variance in calculus course grades.
The study by Baron and Norman (1992) confirmed the findings of Edge and
Friedberg (1984). The purpose of this study was to predict cumulative college GPAs
using high school ranks, SAT scores, and average achievement test scores as predicting
variables. The sample included 4,170 freshman students who enrolled at the University of
Pennsylvania during the fall terms of 1983 and 1984. Of these participants, 2,781 were in
the field of arts and sciences, 647 in business, 585 in engineering, and 157 in nursing. A
multiple regression analysis was conducted to predict cumulative college GPAs. High
school ranks was the best single predictor in the model (R
2
= .093). When high school
ranks were combined with College Board Achievement Tests, the model accounted for
13.6% of the observed variance in the college GPA. SAT scores were found to have a
small contribution to the overall prediction model.
A study conducted by Burton (1989) examined 741 students who took the final
exam in a calculus I course. The participants were asked to describe their high school
preparation for calculus on a continuum from no previous experience to a full year in an
AP calculus course. Burton conducted frequency and descriptive counts to analyze his
data. He found 56.5% of the participants who reported minimal or no experience made
Ds and Fs as final course grades. For the students who reported they had had one full year
of calculus experience, 16.6% of the participants made a D or F as a final course grade.
Furthermore, Burton reported that previous high school calculus experience may not be
sufficient for the participant to exempt the college introductory calculus course. These
21
results suggest that success in calculus I was heavily dependent on the participant?s high
school experience.
To explain further the variance within calculus grades, Wilhite et al. (1998)
investigated the effects of high school calculus and academic achievement variables on
the undergraduate achievement in calculus I. The participants were selected as a stratified
random sample from 1,542 calculus I students at the University of Arkansas. Of the 182
selected participants, a stepwise multiple regression was conducted to predict the final
grade in calculus I. The researchers explained 29.9% of the variance in the calculus I
final grade. The most significant predictor was ACT math scores followed by high school
ranks, ages, and high school mathematics GPA. High school calculus background had a
low positive correlation with the dependent variable and was not significant in the
regression model. The researchers suggest that high school calculus be taught to students
who intend to earn college credit concurrently; however, to accomplish this goal, the
mathematics curriculum needs to be restructured to include the skills to master college
level calculus.
Murtaugh, Burns, and Schuster (1999) investigated the cognitive factors
associated with student retention at Oregon State University. The participants in this
study were 8,867 undergraduate students who enrolled during the fall quarters of 1991
through 1995. Predictor variables included demographic variables (gender, ethnicity,
residency, college, and age at first enrollment) and academic characteristics (high school
GPAs, SAT scores, and first quarter college GPAs), and involvement (participation in
educational program and freshman orientation).
22
With the KaplanMeier method, a series of univariate analyses was conducted to
provide a nonparametric estimate of retention probability over time: 1, 2, and 4 years. For
participants with a high school GPA between 3.3 and 4.0, after 1 year, their estimated
retention rate was 86.2%. That percentage decreased to 68.9 after 4 years. On the other
hand, for the participants with a high school GPA between 2.0 and 2.7, their estimated
retention rate was 65.9% after 1 year, and it decreased to 35% after 4 years. First quarter
GPA had similar percentages. The participants with a first quarter GPA between 2.7 and
3.3 had an estimated retention rate of 87.6% after 1 year, and it decreased to 65.9% after
4 years. When the participants had a first quarter GPA less than 2.0, their estimated
retention rate was 57.2%, and it decreased to 33% after 4 years (Murtaugh et al., 1999).
A stepwise regression model was developed using the Cox proportional hazards
regression model. Significant contributing variables were first quarter GPA, freshman
orientation, high school GPA, residency, college, ethnicity, and age. The results of the
univariate and multivariate analyses indicated the strong association between student
retention and high school and first quarter academic performance. The researchers noted
that high school GPA surpassed the SAT score as a predictor of student retention
(Murtaugh et al., 1999).
Sadler and Tai (2001) investigated demographic and high school background
factors to account for the variance in the grades in an introductory college physics
courses. The participants were 1,933 from 18 colleges and universities (9 public state
institutions, 8 private institutions, and 1 military academy). A multiple regression
analysis was conducted to predict the college grade in introductory physics. The resulting
model accounted for 25.8% of the variance in the college grade.
23
Significant predicting variables were type and location of high school, ethnicity,
parent?s level of education, high school GPA, high school courses in calculus and
physics, college year, and professor?s gender being the same as the student?s gender.
Variables, which were not statistically significant and excluded from the model, were
class climate (i.e., instructional practices), problem solving (i.e., quantitative problems
presented during class), school issues (i.e., high school size), student decisions (i.e., high
school courses in chemistry and biology), teacher attributes (i.e., pedagogy), labs and
project work (i.e., student participation in science fairs), and gender. Sadler and Tai
(2001) noted that some participants may tend to avoid more rigorous coursework in high
school in order to avoid a negative impact on their high school GPA, but exposure to high
school calculus can increase the likelihood of success in college physics courses (Sadler
& Tai, 2001).
Smith and Schumacher (2005) examined predicting variables for success in
undergraduate mathematics courses. The participants in this study were 106 actuarial
graduates from Bryant College from 1996 to 2003 and 776 freshman students who
entered during the 2003 academic year. A multiple regression was conducted to predict
the overall math GPAs using the SAT quantitative scores, SAT verbal scores, high school
ranks, and the college?s mathematics placement scores. The model was significant with a
R
2
of .454. All variables were significant contributors to the model; however, high school
ranks were the most significant.
Another purpose of this study was to predict the postcalculus GPA. A multiple
regression analysis was conducted using the same independent variables with the addition
of the average of the two calculus courses. The restricted model, with high school ranks
24
and calculus grades, accounted for 57.9% of the variance after the twocourse sequence
of calculus was completed. These results suggest that college calculus is a significant
predictor for academic success in college. The researchers suggested the additional
research for mathematical ability, interest, study habits, and academic selfconcept to
explain further the variance in college GPA (Smith & Schumacher, 2005).
In summary, researchers found standardized test scores, high school ranks, and
high school GPAs were significant predictors of academic success in the undergraduate
curriculum. These predictors accounted for the variance in retention probability
(Murtaugh et al., 1999), college calculus courses (Burton, 1989; Edge & Friedberg, 1984;
Smith & Schumacher, 2005; Wilhite et al., 1998), introductory college physics courses
(Sadler & Tai, 2001), and cumulate college GPAs (Baron & Norman, 1992).
Noncognitive factors. The vast amount of empirical research that predicts success
in postsecondary institutions involves cognitive factors. A few studies exist that use non
cognitive factors, such as personality traits and attitudes, to predict academic success.
College admission officers want to quantify the individual differences (e.g., study habits
and academic selfconcepts) among undergraduate students and use these differences to
account for the variation of college GPAs (Nixon & Frost, 1990; Wesley, 1994).
Nixon and Frost (1990) developed a 37item inventory to measure study habits
and attitudes about overall academic ability. Using this instrument, the researchers
examined the study habits and attitudes of 128 undergraduate and graduate students to
predict their cumulative college GPAs. The researchers found those participants who
scored lower on the inventory tended to have higher GPAs, but the t tests were not
statistically significant.
25
There was a significant correlation between academic goals and college GPAs (r
= .58; p < .001). Likewise, there was a significant correlation between academic self
concept and college GPAs (r = .56; p < .001). Both of these relationships suggested that
participants who were goaloriented and had high academic selfconcepts tended to have
higher GPAs than those participants with lower selfconcepts and who lacked goal
setting ability. Those participants who reported moderate levels of study time tended to
have higher GPAs compared to those participants who reported high or low levels of
study time. These results suggested that increased study time can lead to higher GPAs,
but a curvilinear relationship existed between grades and study time which means more
study time may indicate poor study skills or lack of ability (Nixon & Frost, 1990).
Shaughnessy et al. (1995) used the 16 Personality Factor Questionnaire to assess
87 pharmacy majors. In addition to the personality factors, the researchers collected SAT
scores and screening test scores for algebra and calculus. A multiple regression analysis
was conducted to predict the final grade in calculus I. The screening test for calculus was
the most significant predictor; however, three personality factors (i.e., reasoning,
emotional stability, and privateness) were significant contributors to the model. The
model accounted for 28% of the variance in calculus I grades. These results suggest
individuals who are abstract thinkers tend to be more successful in calculus I than
concrete thinkers.
Another study conducted by House (1995c) examined 218 freshmen at a large
U.S. university. Prior to their first semester at the university, the participants were given
an attitude questionnaire. The purpose of this study was to determine the predictive
relationship between the participants? attitudes and achievement in their first college
26
calculus course. More specifically, the attitudes that were examined in this study were (a)
overall academic ability, (b) mathematics ability, (c) drive to achieve, and (d) intellectual
ability. Using a multiple regression analysis, he found overall academic ability and
mathematical ability to be the most significant predictors, which accounted for 13.4% of
the variance in the calculus grades.
In summary, researchers (House, 1995c; Nixon & Frost, 1990; Shaughnessy et al.,
1995) found academic selfconcept, academic goals, reasoning, academic ability, and
mathematical ability to be significant contributors for academic success in the
undergraduate curriculum. These predictors accounted for the variance in a college
calculus courses and cumulative college GPAs.
Cognitive and noncognitive factors. As the interest in noncognitive factors has
increased, the interest in the relationship between cognitive and noncognitive factors and
academic success has increased. A number of noncognitive factors have been identified
by empirical findings, such as interests and attitudes. Several of these factors have been
found to have a significant relationship with academic achievement at the postsecondary
level. Further research is needed to investigate the effects of cognitive and noncognitive
factors on academic success in undergraduate curriculum (House, 1995a).
House (1993) examined the relationship between achievementrelated
expectancies, academic selfconcept, and mathematical performance with students who
were admitted to the university through academic services program for students who lack
the collegepreparatory course sequence. The participants were firstgeneration college
students from low socioeconomic backgrounds. Academic expectations and selfconcept
were measured with a survey, which was administered during an orientation session prior
27
to enrollment. The independent variables were gender, ethnic group, achievement
expectations, and academic selfconcept, and the dependent variable was the final grade
in college algebra.
An analysis of covariance (ANCOVA) was conducted with ACT composite
scores as the covariate. The main effect of academic selfconcept was significant, F(1,
178) = 5.65; p < .05, and Duncan?s multiple range procedure revealed students with
lower selfconcepts earned significantly lower grades in the college algebra course. The
main effect for gender was also significant, F(1, 178) = 4.90; p < .05, and followup
procedures revealed that females earned statistically significant higher grades.
Achievement, F(1,178) = 3.79; p < .10, and ethnic group, F(1, 178) = 3.08; p < .10, were
not statistically significant nor were the twoway interaction effects for the variables
(House, 1993).
These results indicated that academic selfconcept was related to mathematical
ability for students who lack college preparation, and, with this sample, the
underprepared students may have graduated from secondary school with an unrealistic
selfconcept of their abilities and an unrealistic understanding of the demands in post
secondary institutions. A limitation of this study may be the use of course grades as the
dependent variable because grades can be affected by effort and persistence with
classroom assignments as well as mathematical ability (House, 1993).
Wesley (1994) examined ability, high school achievement, and procrastinatory
behavior as predictors of cumulative college GPAs with 244 undergraduate students
enrolled in a psychology course. Procrastinatory behavior was measured with the 10item
SelfHandicapping Scale and the 5item Procrastination Assessment Scale. The cognitive
28
factors were achievement (high school grade averages and GPAs) and ability (SAT
scores). A series of stepwise multiple regression analyses were conducted by gender. The
significant contributing variables were high school averages, procrastination, and SAT
scores for the male (R
2
= .50) and female models (R
2
= .29). The results of this study
suggest procrastination accounts for a significant portion of the observed variance in
cumulative GPAs beyond SAT scores and high school grade averages.
House (1995a) investigated the predictive relationship between initial student
attitudes, standardized admission test scores, high school curriculum, and grade
performance in an introductory college chemistry course. The participants included a
sample of 179 freshman students with a mean age of 18.11 years. During an oncampus
orientation session, all participants completed a survey to measure academic abilities and
expectations of academic success. Multiple regression analyses were conducted with the
following predictor variables: (a) overall academic ability; (b) mathematics ability; (c)
drive to achieve; (d) intellectual ability; (e) ACT composite scores; and (f) number of
high school mathematics courses completed. All predictor variables were significantly
correlated with the dependent variable, grade performance in the chemistry course.
The results of the stepwise multiple regression analysis revealed selfratings of
mathematical ability and overall academic ability as statistically significant contributors
to the model (R
2
= .23). A crossvalidation analysis was conducted using cognitive and
noncognitive factors to examine the consistency of ordering the predictor variables. This
analysis revealed mathematical ability and ACT composite scores as the statistically
significant contributors to the model (R
2
= .31) (House, 1995a).
29
A logistic regression analysis was conducted to investigate the relationship
between the cognitive and noncognitive factors and passing the introductory college
chemistry course (final grade of A, B, or C). The only statistically significant contributor
was mathematical ability. The results of this study suggest initial attitudes (e.g., self
ratings of mathematical ability) are better predictors of academic success than ACT
composite scores or number of high school mathematics courses completed (House,
1995a).
House (1995b) further investigated the relationship between cognitive and non
cognitive factors with his study of academic success in introductory college mathematics
courses. This sample included 958 freshman students. As in the previous study, House
administered a survey to the participants during an oncampus orientation session.
Multiple regression analyses were conducted with the following predictor variables: (a)
overall academic ability; (b) mathematical ability; (c) drive to achieve; (d) intellectual
ability; (e) ACT composite scores; and (f) number of high school mathematics courses
completed. Four of the six predictor variables were statistically significant contributors to
the model: mathematical ability, overall academic ability, intellectual ability, and ACT
composite scores (R
2
= .12).
As a followup, a stepwise logistic regression analysis was conducted to
determine the relationship between each cognitive and noncognitive factor, respectively,
and satisfactorily passing the college mathematics course. The significant contributing
predicting variables were mathematical ability, drive to achieve, and ACT composite
scores. These results suggest that selfratings of mathematical ability are the most
significant predictor variable of academic success in mathematics courses. The number of
30
high school mathematics courses completed was not a significant contributor; however,
House (1995b) noted the enrollment in more high school courses may increase the
students? selfconfidence in their mathematical ability.
A study conducted by House et al. (1996) examined the relationship between
initial attitudes, prior achievement, and academic achievement in a general college
psychology course. The participants were 335 students, with 102 of them from a private
urban university and 233 from a rural public university. At an orientation session prior to
fall semester, each participant completed an attitudinal survey. After conducting bivariate
correlations, multiple regression analyses were conducted with the same predictor
variables used in the previous House studies (1995a, 1995b, 1995c).
Mathematical ability and overall academic ability were highly correlated with the
number of high school mathematics courses completed for both universities. Moreover,
all noncognitive factors and the number of high school mathematics courses completed
were highly correlated with the ACT composite scores. In the regression model, ACT
composite scores and the number of high school mathematics courses completed were the
two statistically significant predictor variables. With a stepwise logistic regression
analysis to determine the relationship between the predictor variables and earning a
satisfactory course grade, ACT composite scores were the only statistically significant
contributor for both universities (House et al., 1996).
Harackiewicz et al. (2002) investigated the relationship between cognitive and
noncognitive factors and students? interest and performance in an introductory
psychology course. The participants were 471 freshman and sophomore students who
were enrolled in an introductory psychology course. The predictor variables were
31
ACT/SAT scores, high school ranks, achievement motivation (as measured by the Work
and Family Orientation Questionnaire), and interests and goals in psychology (as
measured by a selfreported questionnaire). The dependent variables were the final course
grade and semester GPA.
A series of multiple regression analyses were conducted using the predictor and
dependent variables. The model predicting the final course grade had a R
2
of .34. The
significant contributors were performanceapproach goals, work avoidance goals,
ACT/SAT scores, high school ranks, instructor, and gender. The results suggest that
participants who displayed work avoidance goals tended to have lower course grades.
Those participants who had higher ACT/SAT scores, performanceapproach goals (i.e.,
competitiveness and workmastery), and high school ranks tended to earn higher course
grades (Harackiewicz et al., 2002).
In summary, several researchers have investigated the relationship between
cognitive and noncognitive factors and the undergraduate curriculum. For example,
Wesley (1994) found cognitive and noncognitive factors accounted for the variance in
cumulative college GPAs. Similarly, Harackiewicz et al. (2002), House (1995a, 1995b),
and House et al. (1996) investigated predictors of academic success in introductory
psychology, chemistry, and college mathematics courses. Their findings revealed
mathematical ability, overall academic ability, and academic selfconcept to be
significant predictors of academic success in the undergraduate curriculum in addition to
standardized test scores and high school GPAs.
32
Predicting Success in Engineering Curriculum
Cognitive factors. Mathematical ability serves as a foundation for the
quantitativeoriented curriculums, and it is considered a critical factor for achieving
success in engineering. In addition, mathematical problemsolving skills are necessary for
solving engineering problems (Heinze et al., 2003). Historically, standardized test scores
and other cognitive factors have been linked by researchers to academic success in
engineering (Lackey et al., 2003).
Gilbert (1960) conducted a study to determine the effectiveness of College
Entrance Examination Board exams (i.e., SAT, Advanced Mathematics test, and Science
test) to predict graduation from the School of Engineering at Princeton University. The
participants in this study were 123 students who entered the university during the fall of
1953. A discriminant analysis technique was conducted to predict group membership.
Gilbert did not find a statistically significant difference between the mean scores for the
attrition and survival groups. The results suggest that standardized test scores cannot be
used as the single predictor of success in engineering. Therefore, factors other than
academic aptitude should be examined to predict graduation in engineering.
MollerWong and Eide (1997) conducted a study to determine why engineering
students stayed and why they chose to leave the major at the College of Engineering at
Iowa State University. A cohort of 1,151 students who entered in the fall of 1990 was
selected as participants for this study. A backward elimination logistic regression analysis
was conducted to predict higher or lower risks of attrition (i.e., graduation from
engineering, changed to another major in good standing, left the university, and still
enrolled) with a possible 100 predicting variables.
33
MollerWong and Eide (1997) found marital status, ethnic classification, ACT
composite scores, number of semesters in English while in high school, and number of
semesters in art while in high school were significant predicting variables for the
participants who left the university. For the participants who successfully graduated from
engineering, the number of transfer credits, residency status, high school ranks, ACT
math scores, number of semesters in physics while in high school, and number of
semesters in social science while in high school were significant predicting variables. The
logistic regression model was able to classify correctly 81% of the successful graduates
and 75% of the participants who left the university.
In a correlational study, van Alphen and Katz (2001) investigated the relationship
between college GPAs, grades in prerequisite classes, scores on prerequisite assessment
quizzes, and math readiness as measured by math placement exams. Another significant
indicator of success for freshmen included high school GPAs. The researchers found that
SAT quantitative scores were not significantly correlated with success in electrical
engineering. The study included 229 participants from California State University who
completed the Electrical Engineering Fundamentals course over a 2year period.
Conversely, Devens and Walker (2001) conducted a study using SAT scores and
baseline math tests to predict success in a first semester engineering course. The
participants in this study were 3,087 freshman engineering students from fall 1997
through spring 1999. The researchers found a correlation of 50 SAT points for every 1.0
increase in the course grade. Furthermore, students with SAT scores greater than 1300
were more likely to pass the course compared to those students with SAT scores less than
1000. The researchers recommended that in future studies SAT quantitative and verbal
34
scores should be examined separately to predict student performance in the first semester
courses.
Heinze et al. (2003) collected 6 years of data at Texas Tech University using over
4,000 students. The researchers analyzed the data to determine student success in
sequential mathematics courses. After initial placement according to previous college
math courses or the results of the math placement exams, the researchers found that the
majority of the students began the sequence at calculus I. The students who began the
sequence at calculus II had an 85% success rate through calculus III; however, those
students who began the sequence at college algebra had a 25% success rate through
calculus III. Their results suggest that the math placement could serve as a predictor of
academic success in engineering. As noted by Heinze et al., the math placement exam in
this study was administered to the students after their admission to the university.
Therefore, the quantitative section of the SAT could be a predictor of success in
engineering because it is an assessment tool for algebra and below.
Burtner (2004) examined the mathematical background of 142 students who
graduate in electrical engineering. Of these 142 participants, 82 completed their calculus
sequence at the university. Frequencies and descriptives were assessed for GPA and
initial math placement. According to their math placement test, 43.9% of the entering
freshmen were calculus ready, and 24.4% of the freshmen began their math sequence in
intermediate algebra or below. Final grade in the first semester of calculus was found to
be an indicator of student performance in the engineering curriculum. Furthermore, for
those students who were poorly prepared for the mathematics curriculum, if they were
35
successfully placed within the sequence and understood the content, they tended to be
successful in the calculus sequence.
Buechler (2004) examined the mathematical background of students who
graduated from University of WisconsinMilwaukee with a degree in electrical
engineering over a 4year period. Based on pretest results, he found that a significant
number of students did not have the proper mathematical background to be successful in
electrical engineering. Therefore, a large portion of the coursework had to be used for
reviewing prerequisite material. Using the transcripts of 142 students, the researcher
conducted analyses using the degree GPA, calculus sequence GPA, and electrical
engineering core course GPA. He found that 43.9% of the graduates were calculus ready
when they entered the university. More than 46% of the students had deficiencies in
either algebra or trigonometry. Students who began their calculus sequence with
trigonometry had the highest GPA in all three categories, which indicated a higher
success rate for those students, but students who received a C or better in calculus I were
more likely to be successful in the electrical engineering core curriculum. Moreover,
Buechler noted indicators for future success were ACT math scores, SAT quantitative
scores, high school GPAs, and high school ranks. A significant number of students had to
retake more than one mathematics course, and these students tended to have lower GPAs.
Zhang et al. (2004) examined 15 years of student data across nine universities.
The purpose of the study was to determine the demographic and academic factors upon
graduation for engineering students. The participants were 47,065 freshmen from eight
schools of engineering. A logistic regression model was conducted using six predicting
variables (i.e., ethnicity, gender, citizenship status, high school GPAs, SAT quantitative
36
scores, and SAT verbal scores) to predict graduation in engineering. The researchers
found that high school GPAs and SAT quantitative scores were positively correlated with
graduation across all universities. The coefficient of determination indicated that the
predicting variables accounted for a significant and meaningful variation in graduation
(R
2
= .126), which indicates there are other preexisting variables that would account for
the variance.
In summary, empirical findings (Buechler, 2004; Burtner, 2004; Devens &
Walker, 2001; Gilbert, 1960; Heinze et al., 2003; MollerWong & Eide, 1997; van
Alphen & Katz, 2001; Zhang et al., 2004) identified cognitive factors as significant
predictors of academic success in engineering. More specifically, standardized test
scores, high school GPAs, and high school coursework were found to be significant
contributors to retention within engineering majors. Future research suggested from these
empirical studies the inclusion of quantitative scores from standardized tests and high
school ranks.
Noncognitive factors. The previous research demonstrated the significant
contribution of cognitive factors to academic success in engineering; however, some
researchers (e.g., Brown & Cross, 1993; Lackey et al., 2003) have found empirical
support for the validity of noncognitive factors and their contribution to academic
success in engineering.
Brown and Cross (1993) compared the personality profile of freshman
engineering students (n = 129) to students who had persisted in the engineering majors (n
= 85) at Old Dominion University. The Adjective Checklist was used to assess
personality of the participants. A series of analysis of variance procedures was conducted
37
to compare the two groups. Statistically significant differences by group were found for
achievement, dominance, endurance, abasement, and nurturing parent scales. The results
of this study suggest personality can play a vital role in retention of engineering students.
Furthermore, if the characteristics of previously retained students differ greatly from the
incoming freshmen, then the existing structure may not be effective with the incoming
freshmen class and should be revised to meet their needs.
Blumner and Richards (1997) wanted to determine if students who earned higher
grades tended to have low levels of distractibility and high levels of inquisitiveness. From
a required introductory course in engineering at the University of Virginia, 27 women
and 42 men were selected as participants for this study. The Inventory of Study Habits
was used to measure study habits. The 18item survey assessed distractibility,
compulsiveness, and inquisitiveness. Other independent variables included SAT verbal
scores, SAT quantitative scores, and firstyear college GPAs. Multiple regression was
conducted to predict GPA. SAT verbal and quantitative scores accounted for 18% of the
variance in the GPA. When the study habits were added to the model, the R
2
increased to
.33.
Based on the median of the GPAs, the continuous variable was dichotomized into
high and low GPAs. Multivariate analyses of variance (MANOVA) were conducted with
high and low GPAs and the inventory scales. The results were significant, which means
that study habits were associated with achievement. Participants with high GPAs had
significantly lower levels of distractibility and higher levels of inquisitiveness. The
researchers suggested the promotion of strategies through the curriculum for students
who reported low levels of inquisitiveness (Blumner & Richards, 1997).
38
Lackey et al. (2003) conducted a predictive study with freshman engineering
students to determine the effectiveness of a critical thinking notebook on predicting
academic success (i.e., GPA after first two college semesters). The participants in this
study were 109 students who were enrolled in an engineering professional practices
course at Mercer University. Based on the admission criterion, the researchers assumed
that all participants were academically prepared in their K12 curriculum for success in
engineering. The predicting variables were SAT combined scores, SAT verbal scores,
SAT quantitative scores, high school GPAs, and the grade for the critical thinking
notebook. With the critical thinking notebook, the participants took notes in class and
reflected upon their learning throughout the semester.
Using a stepwise multiple regression model, the researchers found that SAT
quantitative scores, high school GPAs, and the critical thinking notebook were significant
predicting variables for college GPA. These variables accounted for 60.3% of the
variance in college GPA. The critical thinking notebook accounted for 33.4% of the
variance. These results suggested that the grade for the critical thinking notebook
measures adequate study habits and problemsolving skills. Therefore, variables
representing both ability and interest were the best predictors for college GPA (Lackey et
al., 2003).
Litzinger et al. (2005) examined the trends in selfdirected learning across the
undergraduate years. The participants were randomly selected from all of the engineering
majors at Pennsylvania State University. The researchers used two Likerttype survey
instruments: SelfDirected Learning Readiness Scale and Continuing Learning Inventory.
The results from the analysis of variance revealed a significant difference in selfdirected
39
learning across the academic years in college. A TukeyKramer pairwise comparison
revealed significant differences between the first and second year and the first and fifth
year in college. The researchers hypothesized that the differences could be contributed to
the presence of more openended questions in the classroom. Using a regression analysis
to predict the score from the SelfDirected Learning Readiness Scale, the researchers
found that academic year and GPA were significant predicting variables..
Another purpose of this study was to investigate the effect of problembased
learning on the SelfDirected Learning Readiness Scale. The survey was administered as
a pretest and posttest to the participants in a twocourse sequence in industrial and
manufacturing engineering. Of the 18 participants, nine participants increased their self
directed learning scores. The problembased learning used in the course increased the
mean selfdirected learning score. The results suggested that students needed to be
exposed to multiple problemsolving experiences that occurred within a realworld
context. By modifying the curriculum, the researchers hypothesized that the students
would be able to improve their critical thinking skills (Litzinger et al., 2005).
Burtner (2005) identified noncognitive factors that influence the persistence of
students to remain in the engineering curriculum. To measure the noncognitive factors,
the researcher used the Pittsburgh Freshman Engineering Attitudes Survey. Using
discriminant analysis to predict group membership (i.e., stay, switch, or leave) after 1
year of college, Burtner found job security, engineering characteristics, communication
skills, and basic math, science, and technology skills significantly differentiated the three
groups. The classification percentage was 85.2%. To predict group membership after 3
years of college, another discriminant analysis was conducted using the same four
40
independent variables. Again, the analysis significantly differentiated the groups. In the
sample, 76.5% were correctly classified. These results suggest that mathematics and
science abilities of engineering students may influence their decision to leave
engineering.
In summary, despite the use of cognitive factors as predictors of academic
success, success in the engineering curriculum is dependent upon the attitudes that the
students bring with them to college (Lackey et al., 2003). These students may have been
academically successful in high school, but they may lack the confidence in their
mathematical ability. By measuring these initial attitudes, academic success can be
achieved (BesterfieldSacre et al., 1997).
Cognitive and noncognitive factors. Cognitive factors have the strongest
relationship with academic success in engineering fields (Brown, 1994); however,
success has been linked to noncognitive factors as well (Lackey et al., 2003).
LeBold and Ward (1988) investigated the cognitive and noncognitive variables
associated with college and engineering retention with a national and institutional sample.
The participants began their studies during 1980 and 1981 at 20 different engineering
institutions. The researchers found that the strongest predictors associated with retention
in engineering were math ability, high school math grades, SAT quantitative scores, and
first semester GPAs. Selfperceived abilities in math, science, and problem solving were
strong predictors of engineering persistence.
With the institutional sample at Purdue University, the researchers conducted a
bivariate correlation to determine the relationship between the precollege variables and
engineering retention after six semesters. They found that SAT quantitative scores
41
(r = .22), high school math grades (r = .24), and high school science grades (r = .21) had
moderate relationships with engineering retention. Grades in the following high school
classes had a moderate relationship with engineering retention: trigonometry (r = .25),
calculus (r = .24), chemistry (r = .24), and physics (r = .27). When the bivariate
correlation was conducted with first semester GPAs, a strong relationship existed with all
of the previous precollege variables and high school ranks (r = .40) (LeBold & Ward,
1988).
Felder et al. (1993) examined attitudes and expectations of 124 undergraduate
students enrolled in an introductory chemical engineering course. The collected data were
correlated with the final grade in the engineering course. Statistically significant
differences were found with students who had a high academic selfconcept (p < .001),
students who felt they lacked in academic ability and had poor study habits (p < .01), and
students who reported low levels of distractibility (p < .05). For students who reported
better than average high school preparation, 80% of them passed the course with a C
average or higher (p < .05). Moreover, grades in the firstyear quantitative courses
(calculus, chemistry, and physics) were highly correlated (from .47 to .62) with academic
success in the engineering course. These results suggest that the likelihood of passing the
introductory course to chemical engineering depended upon high quantitative ability and
academic selfconcept and low distractibility.
Brown (1994) investigated cognitive and noncognitive factors to predict the first
semester GPAs for 124 freshman engineering students. The participants were
administered the College Major Interest Indicator (to identify educational interests),
42
Adjective Checklist (to measure personality traits), Ship Destination Test (to measure
symbolic reasoning), and Logical Reasoning Test (to measure verbal reasoning). In
addition to these noncognitive factors, the participants? SAT verbal and quantitative
scores were used as predictor variables.
A multiple regression analysis was conducted to predict the first semester GPA.
When the 37 scales of the Adjective Checklist were the only predictor variable, the model
accounted for 60% of the variance in the first semester GPA. When all cognitive and non
cognitive factors were entered into the model, the model accounted for 66% of the
variance in first semester GPA. The researcher noted that cognitive variables may not be
as important for predicting academic success when the sample has been screened and
admitted to a postsecondary institution on the basis of the cognitive factors, such as high
school transcript and standardized test scores. Thus, noncognitive factors can play a
large role in predicting academic success (Brown, 1994).
BesterfieldSacre et al. (1997) conducted a predictive study for attrition and
performance in a freshman engineering program with 417 college freshmen who entered
the University of Pittsburgh during the fall of 1993 and 1994. The participants were
asked to rate (a) their opinion about the engineering profession and the reasons why they
selected engineering as a major, (b) their confidence in the prerequisite knowledge and
skills and their abilities to be successful in engineering, and (c) their study skills and their
interest to work in groups using the 50item Pittsburgh Freshman Engineering Survey.
The participants rated the attitudinal statements with a 5point Likerttype scale (i.e.,
Strongly Disagree to Strongly Agree) and the confidence statements with a 5point
Likerttype scale (i.e., Not Strongly Confident to Strongly Confident).
43
Using the MannWhitney nonparametric test, the researchers found that the
participants who left engineering in good standing had a significantly lower general
impression of engineering and perception of the engineering profession. Likewise, this
group of participants who left engineering in good standing had a significantly lower
confidence level for the basic engineering knowledge and skills (e.g., problemsolving
and creative thinking skills). They also reported significantly lower levels of enjoyment
in mathematics and science courses. These results suggested that students with adequate
mathematical abilities may not be successful in the field of engineering because their
selfassessed abilities and confidence levels were not equivalent to their actual abilities.
The participants who left engineering in poor standing had the confidence in their
abilities but were not academically prepared. The researchers suggested that this
incompatibility may be related to poor time management skills and ineffective study
habits (BesterfieldSacre et al., 1997).
Using a multiple linear regression (stepwise) analysis with fall quarter GPA as the
dependent variable, the researchers found that 8 out of 20 possible predictor variables
were significant contributors to the regression model, which resulted in a R
2
of .29. Those
significant predicting variables were received at least one scholarship, high school ranks,
SAT quantitative scores, adequate study habits, enjoyment of math/science, and financial
influences for studying engineering (BesterfieldSacre et al., 1997).
The researchers used a stepwise logistic regression analysis to predict freshman
engineering students who would leave in poor standing. They found that SAT
quantitative scores, high school ranks, involvement in the freshman academic support
program, and financial influences for studying engineering were significant contributors
44
to the model; however, the predicting percentage was 15%. Another logistic regression
analysis was conducted to predict the participants who would leave in good standing.
High school ranks, general impression of engineering, enjoyment of math/science, and
family influences to study engineering were significant contributors to the model
(BesterfieldSacre et al., 1997).
A third logistic regression was developed to predict participants who would leave
in good standing, but the researchers only included the attitude and selfassessment
measures. In this model, general impression of engineering, enjoyment of math/science,
confidence in basic engineering knowledge, and confidence in communication skills were
significant predictor variables. The overall predicting percentage for the logistic
regression model was 50%. The researchers found that the participants reported a lack of
confidence in their abilities and a lower level of interest in math and science.
Furthermore, these participants tended to have lower impression of the field of
engineering. BesterfieldSacre et al. (1997) suggested an introduction of more realworld
applications into the classroom and the opportunities for the students to participate in
field experiences so they can understand the job specifics for a variety of engineering
jobs.
House (2000) examined the predictive relationship between achievement in
science, engineering, and mathematics curriculums, academic background, and non
cognitive factors. The participants were 658 freshman students over four consecutive fall
semesters who indicated an initial science, engineering, or mathematics major. The
participants completed the Cooperative Institutional Research Program Annual Freshman
Survey during an oncampus orientation session. Based on the data collected from the
45
survey, seven variables were created: (a) achievement expectancies; (b) academic self
concept; (c) financial goals; (d) social goals; (e) desire for recognition; (f) parental
education; and (g) high school curriculum. ACT composite scores, high school ranks, and
cumulative college GPAs after 1 year were collected.
Two multiple regression analyses were conducted to predict the cumulative
college GPA. The first model was created using only the noncognitive factors. Academic
selfconcept and financial goals were significant contributors to the model (R
2
= .12).
Those participants with higher GPAs reported higher levels of academic selfconcept and
the participants with lower GPAs reported higher levels of financial goals. The second
model was created using the academic background and noncognitive variables. In
addition to academic selfconcept and financial goals, high school ranks and ACT
composite scores were significant contributors to the model (R
2
= .29). House (2000)
suggested further research to assess the joint relationship between academic background
variables and noncognitive factors as predictors of academic success for students who
major in science, engineering, and mathematics.
Burtner (2004) investigated the relationship between persistence in the
engineering curriculum and high school achievement, attitudes, and college academic
performance. The participants were 116 freshman students at Mercer University. Of the
cognitive skills, there were significant group differences for retention status for high
school GPA and college performance during the first year. The Pittsburgh Freshman
Engineering Attitudes Survey was used to assess attitudes. Using ANOVA, the researcher
found that enjoyment of engineering, job security, problemsolving abilities, basic skills,
and study habits were significant. Burtner hypothesized that high school GPA indicated
46
the participant?s ability to persevere and study, which are significant predictors for
academic success in college.
Shuman et al. (2003) used the Cooperative Institutional Research Program and the
Pittsburgh Freshman Attitudes Survey to determine the selfassessed abilities and predict
academic success in the calculus sequences for 394 freshmen who entered the University
of Pittsburgh in 2002. Descriptives and frequencies revealed that 17.2% reported needing
remediation in mathematics. However, after a followup with the tutoring records,
approximately half of the students who indicated that they needed remedial studies did
not attend tutoring sessions. Of the 394 freshmen, 32.7% reported that their mathematics
abilities were among the top 10%. Using a crosstabulation, the researchers found that
87.5% of the participants indicated that they had had high school calculus. Of those
students, their calculus grades were at least one letter grade higher than those participants
who did not report taking high school calculus. Another crosstabulation revealed students
with higher SAT quantitative scores tended to perform better in calculus I.
A Learning Vector Quantization neutral network model was developed to predict
initial math placement in the calculus sequence. The predictor variables were gender,
high school ranks, SAT scores, placement exam results, attitudes toward mathematics,
and background in differential calculus. The model correctly classified 89.5% (Shuman et
al., 2003).
French et al. (2005) examined the cognitive and noncognitive variables that
predict success and persistence in engineering. The participants for this study were two
cohorts of engineering undergraduate students from a large U.S. midwestern university.
The independent variables included gender, participation in orientation class, high school
47
ranks, SAT verbal and quantitative scores, motivation as measured by the Academic
Intrinsic Motivation Scale, and institutional integration as measured by the Institutional
Integration Scale. Three regression models were conducted: hierarchical multiple linear
regression for predicting GPAs and hierarchical logistic regressions to predict university
enrollment and major enrollment with GPA as the control variable. The independent
variables were entered in the following order: (a) background, (b) motivation and
integration, and (c) participation in the orientation class.
In the regression model for predicting GPAs, SAT verbal scores, SAT
quantitative scores, high school ranks, and gender were significant contributing variables.
These variables accounted for 18% of the variance in GPA. For the logistic regression
model for predicting university enrollment, GPA was the only significant contributing
variable. The remaining variables did not significantly contribute to the model beyond
GPA. The correct classification rate was 89%. To predict major enrollment, the
significant contributing variables were GPAs, SAT verbal scores, SAT quantitative
scores, high school ranks, and motivation. The overall classification rate was 65%. The
noncognitive variables and participation in the orientation course did not contribute
additional information beyond the academic variables (French et al., 2005).
Few empirical studies, which use cognitive and noncognitive factors to predict
academic success in engineering, exist. The previous findings (BesterfieldSacre et al.,
1997; Brown, 1994; Burtner, 2003; Felder et al., 1993; French et al., 2005; House, 2000;
LeBold & Ward, 1988; Shuman et al., 2003) revealed that noncognitive factors could
significantly contribute beyond cognitive factors. However, further research is needed to
48
determine the joint relationship between cognitive and noncognitive factors and how that
relationship contributes to academic success in the preengineering curriculum.
Summary
High school background in quantitative content has a direct effect on the student?s
academic success in undergraduate coursework (Gainen, 1995). In addition to cognitive
factors, students? attitudes relating to overall academic achievement can predict success
in quantitative courses. These attitudes include expectation of success, their academic
ability in relation to their peers, and confidence in their abilities (House, 1995c). By
assessing these noncognitive factors, college and university staff could predict academic
success after the initial admission screening of applicants (BesterfieldSacre et al., 1997).
Limited research exists for the effects of cognitive and noncognitive factors on long
term academic measures, such as cumulative college GPAs (House, 1995b; LeBold &
Ward, 1988). Despite the fields of engineering having a quantitative content focus, fewer
researchers have attempted to investigate the joint relationship between cognitive and
noncognitive factors and academic success in quantitative coursework.
The focus of this study was to determine if cognitive factors mediate the effect of
noncognitive factors on quantitative GPA and to determine if these cognitive and non
cognitive factors can predict admission status in engineering education. After reviewing
the literature, the following cognitive and noncognitive factors were selected for the
statistical analysis based on the significant findings of empirical research studies.
Cognitive factors. Three cognitive factors were selected for incorporation in this
study. The dominant predictor of academic success in engineering was standardized test
scores, SAT and ACT, (BesterfieldSacre et al., 1997; Blumner & Richards, 1997;
49
Brown, 1994; Buechler, 2004; Devens & Walker, 2001; French et al., 2005; Heinze et al.,
2003; Harackiewicz et al., 2002; House, 1995a, 1995b, 2000; House et al., 1996; Lackey
et al., 2003; LeBold & Ward, 1988; MollerWong & Eide, 1997; Shuman et al., 2003;
Wesley, 1994; Zhang et al., 2004). High school rank was significant predictor of
academic success in engineering (BesterfieldSacre et al.; Buechler; Edge & Friedberg,
1984; French et al.; Harackiewicz et al.; House, 2000; Lackey et al.; LeBold & Ward;
MollerWong & Eide, 1997; Shuman et al.; Smith & Schumacher, 2005; Wilhite et al.,
1998; Zhang et al.). Lastly, to target quantitative ability specifically, high school math
grades were found to be significant predictors of academic success in engineering (House
et al.; LeBold & Ward; Sadler & Tai, 2001; Smith & Schumacher; Wilhite et al.).
Noncognitive factors. Eight noncognitive variables were chosen to be
investigated based on the findings of the reviewed research: academic selfconcept
(Brown, 1994; Burtner, 2004, 2005; House, 1993, 1995a, 1995b, 1995c, 2000; Nixon &
Frost, 1990), math selfconcept (Burtner, 2004, 2005; House, 1995a, 1995b, 1995c;
LeBold & Ward, 1988; Shuman et al., 2003), problem solving (BesterfieldSacre et al.,
1997; Brown; Burtner, 2004; Blumner & Richards, 1997; Lackey et al., 2003; LeBold &
Ward; Litzinger et al., 2005; Shaughnessy et al., 1995), selfappraisal (BesterfieldSacre
et al.; Brown; Brown & Cross, 1993; Harackiewicz et al., 2002; House, 1993, 1995b),
study habits (BesterfieldSacre et al.; Blumner & Richards; Burtner, 2004; Harackiewicz
et al.; Nixon & Frost; Wesley, 1994), need help (BesterfieldSacre et al.), perceived
difficulty (BesterfieldSacre et al.), and academic difficulty (BesterfieldSacre et al.).
50
CHAPTER III
METHODS
Numerous empirical studies have addressed the relationship between cognitive
skills and academic success in undergraduate and engineering curriculums (Baron &
Norman, 1992; Buechler, 2004; Burtner, 2004; Edge & Friedberg, 1984; House, 1995b;
House et al., 1996; Smith & Schumacher, 2005; Wilhite et al., 1998). Limited research
exists for predicting academic success using cognitive and noncognitive factors (i.e.,
selfconcept, study habits, and inquisitiveness) (House, 1995b, 2000). By increasing
interest and mathematical skills, the percentage of college engineering graduates can
increase (French et al., 2005; Wilhite et al.). Therefore, the purpose of this study was to
determine if cognitive factors mediate the effect of noncognitive factors on quantitative
GPA and to determine if these cognitive and noncognitive factors can predict admission
status in engineering education. This study builds upon the findings of French et al.,
House (2000), and Shuman et al. (2003).
Participants
Participants in this study were from a sample of 3,052 undergraduate students
who entered Auburn University in preengineering from the fall semester of 2000 through
the fall semester of 2004. Table 1 displays the frequencies for each admission year. Of
these cases, 2,276 participants were selected for the study because their survey responses
could be matched with the standardized test scores and college grades provided by the
51
University Planning and Analysis Office. The fall semesters of 2000 through 2004 were
selected because Auburn University switched from the quarter system to the semester
system beginning with the fall term of 2000. This change to the academic calendar
revised the preengineering curriculum when courses were consolidated for the 16week
semester system. The participants who entered engineering after fall 2004 have not had
enough time to complete the preengineering curriculum.
Table 1
Frequencies by Academic Year
Entire sample Sample cases
Year n % n %
2000 609 20.0 451 19.8
2001 608 19.9 454 19.9
2002 626 20.5 439 19.3
2003 641 21.0 494 21.7
2004 568 18.6 438 19.2
Total 3,052 100.0 2,276 100.0
Of the participants with a declared major in engineering, 1,857 (81.6%) were
male, and 419 (18.4%) were female. The racial classification of the group was 1,885
(82.8%) White, 258 (11.3%) Black, and 133 (5.8%) students who reported they belonged
to other racial groups. The majority of the participants (54.6%) reported a master?s
degree as the highest education level they expected to attain. The participants in this
study represented 40 of the 50 U.S. states, and 1,644 (72.2%) reported Alabama as their
home state. When asked to describe the place where they lived before enrolling in
college, 736 (32.3%) participants reported small town, 654 (28.7%) reported suburbia,
52
534 (23.5%) reported large town, 176 (7.7%) reported big city, and 176 (7.7%)
participants reported rural. The range of their high school graduating class size was less
than 50 to more than 500 students with a median of 200.
Procedures
Eight oncampus orientation sessions were held during the summer prior to fall
enrollment for each preengineering cohort in this study. During the first day of the
orientation, the participants completed the College Freshman Survey. The administration
of the survey took approximately 1 hour. If the participants were unable to attend the
sessions, they were invited to attend a makeup session during the first week of fall
semester.
Measures
College Freshman Survey. The College Freshman Survey (Halpin & Halpin,
1996) is a 248item measure designed to assess variables related to success in
engineering. The beginning questions elicited demographic information, standardized test
scores, and high school grades. The remaining questions (200 items) determined the
importance of various subjects, rank of abilities, likelihood of various events, and
agreement with various statements. For the strength statements, the response scale
progressed from a rating of 1, which represented Very Weak, to a rating of 5, which
represented Very Strong. For the likelihood statements, the response scale progressed
from a rating of 1, which represented Very Unlikely, to a rating of 4, which represented
Very Likely. With the remaining noncognitive items, respondents were asked to indicate
the extent of their agreement using a 4point scale, from a rating of 1, which represented
Strongly Disagree, to a rating of 4, which represented Strongly Agree. These items were
53
rationally combined to form 32 scales (Table 2) which measured possible influences for
retention and attrition in engineering education. Based on the review of literature, 49 of
the 248 items on the College Freshman Survey, which comprised eight noncognitive
factor scales, were selected for this study. Items 84, 190, 200, and 214 were reverse
coded before combining with other items to form the scales.
Table 2
Scales for the College Freshman Survey
Scales
Academic Difficulty Like and Excel in English
Academic SelfConcept
Likelihood of Leaving
Auburn University
SelfAppraisal
Academic Success in
Engineering
Likelihood of Leaving
Engineering
Social Skills
Close Friendships Locus of Control Socioeconomic Status
Communication Skills Math SelfConcept Study Habits
Competitiveness Need for Structure Study Hard
Computer Capability Need Help Teacher Importance
Extrinsic Motivation Note Taking Team Membership
Intrinsic Motivation Organizational Skills Test Anxiety
Knowledge and Confidence Perceived Difficulty Work Ethnic
Leadership Problem Solving
54
In terms of noncognitive factors, eight scales were created as follows:
1. Math SelfConcept consists of five items assessing one?s preference for
mathematics, selfassessed ability in mathematics, and grades in mathematics
compared to other subjects.
2. SelfAppraisal consists of six items assessing one?s strengths, weaknesses, and
confidence in his or her ability to handle problems and change.
3. Study Habits consists of seven items assessing one?s ability to stay focused on
tasks, to recognize relevant information, to manage study time, and to selfassess
study effectiveness.
4. Problem Solving consists of five items assessing one?s ability to think analytically
and critically, to recognize and solve problems, and to integrate knowledge and
information.
5. Academic SelfConcept consists of four items assessing one?s ability in
mathematics, academic preparation for college, and motivation to achieve.
6. Perceived Difficulty consists of 11 items assessing one?s loneliness during his or
her freshman year of college, need for assistance from others, anxiety about the
expectations of the curriculum, and inability to control time or events.
7. Need Help consists of six items assessing the likelihood of one needing assistance
with testtaking, basic math, reading, writing, and study skills.
8. Academic Difficulty consists of five items assessing the likelihood of failing a
course, performing less than expected in academics, becoming bored with the
coursework, and becoming stressed with one?s studies.
55
Reliability analyses were conducted to test that the scales provided internally
consistent measurements. A Cronbach?s alpha of .50 or greater was established as the
criterion for reliability following Thorndike (1951). The established criterion for
reliability had not changed since Kelly in 1929 also proposed .50 as the criterion. Hair,
Black, Babin, Anderson, and Tatham (2006) suggested .60 as a criterion for reliability.
The reliability coefficients for the eight noncognitive factor scales were good (ranging
from .63 to .89 with a median of .71). (See Table 3.) The results suggest that the scales
within the survey are internally consistent measures.
Table 3
Alpha Reliability Coefficients for NonCognitive Factor Scales
Scale
Coefficient
Alpha
Math SelfConcept .89
SelfAppraisal .66
Study Habits .72
Problem Solving .79
Academic SelfConcept .69
Perceived Difficulty .70
Need Help .63
Academic Difficulty .63
Institutional data. Data were obtained through the University Planning and
Analysis Office for freshmen who entered the College of Engineering from 2000 through
2004. The collected data included (a) ACT composite, English, math, reading and science
scores; (b) SAT quantitative and verbal scores; (c) grades in preengineering quantitative
courses; and (d) preengineering GPAs.
56
The ACT is a curriculumbased assessment that is composed of four subtests:
English, Mathematics, Reading, and Science. The purpose of this assessment is to
measure the knowledge and skills gained during high school and the expectations for
success in postsecondary institutions. The score on each subtest ranges from 1 to 36.
These subtest scores are averaged to create the ACT composite score, which ranges from
1 to 36. The estimated reliability coefficient for the ACT composite score is .96 (Noble et
al., 2006). For the purposes of this study, the score from the ACT Math Subtest was used.
The SAT is composed of a verbal and quantitative subtest. The purpose of this
assessment is similar to the ACT. The score for each subtest ranges from 200 to 800. The
scores from the two subtests are added together to create the total SAT score. The
estimated reliability for the SAT quantitative subtest is .92 (Ewing, Huff, Andrews, &
King, 2005). For purposes of this study, the quantitative score was used for those
participants who only took the SAT.
Design and Analysis
Exploratory factor analysis. The sample was randomly divided into two databases
of comparable size. With one database, an exploratory factor analysis using principal axis
factoring with an oblimin rotation was conducted using the following scales from the
College Freshman Survey: Math SelfConcept, SelfAppraisal, Perceived Difficulty,
Problem Solving, Need Help, Academic Difficulty, Study Habits, and Academic Self
Concept. The purpose of this factor analysis was to discover the factor structure of the
selected items and the correlations between the factors. This method consisted of a
principal component analysis followed by rotation of factors that had eignvalues greater
57
than 1.00. A criterion of .40 or higher was established for the structure coefficients
(Meyers et al., 2006).
The KaiserMeyerOlkin (KMO) Measure of Sampling is a summary index,
which ranges from 0 to 1, of the interrelatedness among the items and the extent to which
the items will yield an appropriate factor analysis. With the randomly selected sample,
the KMO was .91, which exceeds the suggested minimum value of .70. Bartlett?s Test of
Sphericity (approximate ?
2
= 22,413.24, p < .001) indicates that the intercorrelations
among the items are statistically significant and of a sufficient magnitude to conduct a
factor analysis (Meyers et al., 2006).
With a total of 49 items, 11 factors emerged with initial eigenvalues above 1.0.
The communalities ranged from .34 to .76. After an oblimin rotation was conducted, four
factors were eliminated because they had one to two items meaningfully related to the
factor. With this analysis and modifications, which contained 36 items, 47.57% of the
variance was explained.
Seven noncognitive factors were created as a result of the exploratory factor
analysis. Due to the confidentiality of the secure test items, the specific items with their
respective factor loadings could not be presented. Factor 1, Lack of Confidence in
Academic Abilities, contains four items with the following factor loadings: .51, .64, 
.68, and .71. Factor 2, Mathematical Ability, contains seven items with the following
factor loadings: .62, .79, .79, .80, .84, .86, and .86. Factor 3, SelfAppraised Abilities,
contains four items with the following factor loadings: .55, .57, .67, and .75. Factor 4,
Adaptability, contains five items with the following factor loadings: .43, .45, .66, .68,
and .73. Factor 5, Difficulty with Problem Solving, contains five items with the
58
following factor loadings: .70, .71, .73, .74, and .78. Factor 6, Academic Frustration,
contains four items with the following factor loadings: .44, .65, .67, and .71. Factor 7,
Distractibility, contains seven items with the following factor loadings: .48, .49, .52, .54,
.67, .76, and .77.
The reliability coefficients were conducted for the seven revised noncognitive
factor scales. The results indicated that the coefficients were good (ranging from .59 to
.90 with a median of .70). (See Table 4.)
Table 4
Alpha Reliability Coefficients for Revised NonCognitive Factor Scales
Scale
Coefficient
Alpha
Lack of Confidence in Academic Abilities (LCAA) .70
Mathematical Ability (MA) .90
SelfAppraised Abilities (SAA) .59
Adaptability (A) .62
Difficulty with Problem Solving (DPS) .79
Academic Frustration (AF) .61
Distractibility (D) .74
Confirmatory factor analysis. With the second database, a confirmatory factor
analysis was conducted using AMOS 7.0 to determine how the theoretical structure fits
with the data and to crossvalidate the factor structures created with the exploratory factor
analysis.
According to Meyers et al. (2006), acceptable model fit indexes include the
Comparative Fit Index (CFI) and the Root Mean Square Error of Approximation
(RMSEA). A value of .95 or greater for the CFI is deemed as an acceptable fit. For the
RMSEA, a value of .08 or less indicates good fit. The model had a significant chisquare
59
(?
2
= 6,964.196; p = .00) and a low CFI (.839) but low RMSEA (.05). Since the sample
contained missing data, modification indices could not be utilized. Based on the RMSEA,
the model was deemed to have good fit with the data. Table 5 displays the standardized
beta weights for each item.
Table 5
Standardized Beta Weights by Confirmatory Factor Analysis for the SevenFactor Model
Factor
Item LCAA MA SAA
128 .753
126 .611
127 .601
135 .493
149 .868
177 .806
124 .849
190 .751
137 .742
208 .758
84 .593
154 .584
169 .549
213 .487
142 .458
(Table 5 continues)
60
(Table 5 continued)
Factor
Item A DPS AF D
243 .562
173 .542
219 .529
206 .499
200 .442
111 .715
112 .712
132 .691
116 .689
131 .527
101 .638
100 .629
88 .498
93 .322
201 .759
155 .707
214 .576
170 .515
168 .500
160 .407
143 .394
Structural equation model. Structural equation modeling is an extension of the
general linear model and the multiple regression analysis procedure. An advantage of
structural equation modeling is that it can be used to analyze the relationships between
latent variables and the relationship between the latent variables and multiple measures.
61
Unobserved latent variables cannot be measured directly so they are indicated by
observed variables. Also, it is applicable to and appropriate for longitudinal data since
those datasheets generally consist of a large sample size. The goal of structural equation
modeling is to determine if the hypothesized theoretical model, which depicts the causal
pattern of relationships that was determined a priori, is reflected by the empirical data
(Lei & Wu, 2007).
The estimation procedure for this analysis was maximum likelihood. Maximum
likelihood is similar to the generalized least squares method because they both give
proportional weight to the variables based on the interrelationships. As an alternative to
least squares, which compares error or deviations from the mean score, maximum
likelihood estimates assume the data are normally distributed with some unknown mean
and variance. These unknown values are estimated at the highest likelihood of the actual
data and often are required iterative or repetitious solutions. Thus, the procedure
maximizes the probability of sampling the population (Meyers et al., 2006).
Items within each factor were combined to simplify the model (Meyers et al.,
2006). After the initial frequencies, descriptives, and bivariate correlation were assessed,
a structural equation model was created using AMOS 7.0 to determine the relationship
between the cognitive and noncognitive factors and quantitative GPA in the pre
engineering curriculum with 60% of the sample. The noncognitive factors were
exogenous variables for the cognitive factors. The cognitive factors were exogenous
variables for the dependent and endogenous variable, quantitative GPA during the pre
engineering curriculum. A crossvalidation sample, which contained the remaining 40%
62
of the sample cases, was used to confirm the findings of the first structural model
analysis.
Seven revised factor scales served as multiple measures of the latent construct of
noncognitive skills. These factor scales were employed as indictors of a single construct
after a series of exploratory and confirmatory factor analyses. To simplify the model, the
noncognitive scales were grouped into two factors based on their intercorrelations.
Factor 1 contained Lack of Confidence in Academic Abilities; Mathematical Ability;
SelfAppraised Abilities; and Difficulty with Problem Solving. Factor 2 contained
Adaptability, Academic Frustrations, and Distractibility.
The cognitive factors included three measures: ACT math scores, high school
math grades, and high school ranks.
For 292 cases, the participants only took the SAT. A bivariate correlation was
conducted with the SAT quantitative and ACT math scores to determine the extent of
their relationship given this sample. The results yielded by the SAT quantitative and ACT
math scores were highly correlated (r = .79; p < .001). To linear equate the SAT
quantitative and ACT math scores, a multiple regression analysis was conducted
(Peterson, Kolen, & Hoover, 1989). Of the participants who took both the ACT and SAT,
a bivariate correlation was conducted to determine the validity of the analysis between
the adjusted ACT math and the actual ACT math scores (r = .79; p < .001). The adjusted
ACT math scores were used for the participants who only took the SAT.
On the College Freshman Survey, selfreported grades were collected using the
following response scale: 1 = A+, 2 = A, 3 = A to B+, 4 = B to B, 5 = C+ to C, 6 = D+
or less, and 7 = did not take the course. The raw data was recoded to reflect the following
63
scale: 1 = D+ or less, 2 = C+ to C, 3 = did not take the course, 4 = B to B, 5 = A to B+,
6 = A, and 7 = A+. The 7point scale was used to weight empirically the responses in
order to account for the class not being taken in high school and to differentiate between
A+, A, and A. Mean high school math grade was computed using the criterion of self
reported responses in at least three courses. Descriptives for the grades in the high school
mathematics courses (i.e., algebra I, algebra II, geometry, trigonometry, and calculus)
(Shettle et al., 2007) were assessed. The high school math grades for the participants
ranged between an A and B+. Of these participants, 35% did not take a calculus course in
high school. Table 6 displays the means and standard deviations for each high school
mathematics course.
Table 6
Means and Standard Deviations for Each High School Mathematics Course
Scale M SD
Algebra I 5.63 1.39
Geometry 5.57 1.36
Algebra II 5.48 1.39
Trigonometry 5.00 1.60
Calculus 4.46 1.65
According to Edge and Friedberg (1984), high school GPA and high school rank
are highly correlated because they are similar measures of longterm achievement. In this
study, high school rank will be used in place of high school GPA. High school ranks were
collected from the selfreported measure, the College Freshman Survey. The participants
responded using the following scale: 1 = lowest 40%, 2 = middle 20%, 3 = next 10%, 4 =
next 10%, 5 = next 10%, 6 = next 7%, and 7 = highest 3%. Kuncel, Cred?, and Thomas
64
(2005) found with their metaanalysis that selfreported grades had construct validity as a
measure of achievement. Furthermore, selfreported grades and class ranks tend to be
reliable measures of the actual grades and high school ranks when reported by college
students with reasonably high cognitive abilities. The reliabilities are likely to increase by
asking specific grades in specific classes.
To measure discriminate validity, a bivariate correlation was conducted using the
three cognitive and seven revised noncognitive factors. With a Pearson correlation
coefficient less than or equal to .80 as a criterion (Meyers et al. 2006), these results
suggested that the factors have discriminate validity and are not measuring the same
concept. Table 7 displays the intercorrelation matrix for the cognitive and noncognitive
factors. The means and standard deviations for the cognitive and revised noncognitive
factors are presented in Table 8.
The dependent variable of quantitative GPA in the preengineering curriculum
was measured with at least two quantitative courses in the preengineering curriculum at
Auburn University. A quantitative course was defined as a college course whose
conceptual foundation is based in mathematics (Gainen & Willemsen, 1995). Table 9
displays the courses from the preengineering curriculum, which were considered
quantitative courses in this study. The final letter grade in each quantitative course was
coded using the 4point scale (i.e., A = 4, B = 3, C = 2, D = 1, and F = 0) and was
averaged together to create the quantitative GPA. The mean score was 2.33 with a
standard deviation of 1.03. A bivariate correlation was conducted to determine the
relationship between the preengineering quantitative GPA and the total preengineering
GPA. A strong positive relationship existed between the GPAs (r = .87; p < .001).
65
Table 7
Intercorrelations for the Cognitive and Revised NonCognitive Factors
Scale 1 2 3 4 5 6 7 8 9 10
1. Lack of
Confidence in
Academic Abilities
 .37** .41** .25** .47** .32** .42** .19** .29** .30**
2. Mathematical
Ability
 .13** .16** .33** .25** .16** .44** .54** .26**
3. SelfAppraised
Abilities
 .27** .35** .20** .28** .03 .03 .02
4. Adaptability  .29** .38** .32** .05* .07** .02
5. Difficulty with
Problem Solving
 .28** .18** .26** .14** .13**
6. Academic
Frustration
 .41** .08** .06* .02
7. Distractibility  .06** .14** .19**
8. ACT Math
Scores
 .43** .34**
9. High School
Math Grades
 .58*
10. High School
Ranks

Note: * p < .05; ** p < .01.
Table 8
Means and Standard Deviations for the Cognitive and Revised NonCognitive Factors
Scale M SD
Lack of Confidence in Academic Abilities 3.79 0.49
Mathematical Ability 3.37 0.56
SelfAppraised Abilities 3.13 0.34
Adaptability 2.30 0.41
Difficulty with Problem Solving 3.88 0.55
Academic Frustration 2.29 0.46
Distractibility 2.46 0.43
ACT Math Scores 26.58 4.23
High School Math Grades 5.23 1.07
High School Ranks 4.90 1.70
66
Table 9
List of Possible Quantitative Courses in PreEngineering Curriculum
Course Number
College Algebra MA1000
PreCalculus Trigonometry MA1130
PreCalculus Algebra Trigonometry MA1150
Calculus I MA1610
Honors Calculus I MA1617
Calculus II MA1620
Honors Calculus II MA1627
Calculus for Engineering and Science I MA1710
Calculus for Engineering and Science II MA1720
Calculus III MA2630
Calculus for Engineering and Science III MA2730
Survey of Chemistry I CH1010
Survey of Chemistry II CH1020
Fundamentals of Chemistry I CH1030
Fundamentals of Chemistry II CH1040
General Chemistry I CH1110
Honors General Chemistry I CH1117
General Chemistry II CH1120
Honors General Chemistry II CH1127
Foundations of Physics PH1000
General Physics I PH1500
General Physics II PH1510
Engineering Physics I PH1600
Honors Physics I PH1607
Engineering Physics II PH1610
Honors Physics II PH1617
Discriminant function analysis. A discriminant function analysis was conducted
using the 60% sample to develop a weighted linear combination to predict group
membership (i.e., admitted to engineering, switched to another major at the university, or
left the university unsuccessful). The analysis used admission status as the grouping
variable and the cognitive and revised noncognitive factors as the independent variables.
To be admitted to the Samuel Ginn College of Engineering, the participants were
required to have at least a 2.2 GPA. Participants who were identified as switched majors
67
had at least a 2.2 GPA but chose to pursue another major outside the field of engineering
at Auburn University. For those participants who left the university unsuccessful, they
earned a GPA below 2.2 and left the university without earning a degree for unknown
reasons. For the admission year 2003, 11 participants were still enrolled as pre
engineering majors, and 39 participants from the 2004 admission year were still enrolled
as preengineering majors. They were excluded from the sample for the purposes of this
analysis. Table 10 provides the frequencies for admission status by admission year.
Table 10
Frequencies for Admission Status by Academic Year
Admitted to
engineering
Switched
majors
Left unsuccessful Total
Year n % n % n % n %
2000 262 58.1 89 19.7 100 22.2 451 100.0
2001 264 58.1 83 18.3 107 23.6 454 100.0
2002 279 63.5 85 19.4 75 17.1 439 100.0
2003 267 55.3 100 20.7 116 24.0 483 100.0
2004 252 63.2 78 19.5 69 17.3 399 100.0
Total 1,324 59.5 460 20.7 442 19.8 2,226 100.0
Summary
Participants in this study were a sample of undergraduate students who entered
the preengineering curriculum in the fall semester of 2000 through the fall semester of
2004. These participants completed the College Freshman Survey (Halpin & Halpin,
1996). In addition, institutional data were collected for the participants? standardized test
68
scores and preengineering course grades. An exploratory and confirmatory factor
analyses were conducted to create and validate seven revised noncognitive factors. A
structural equation model analysis and discriminant factor analysis were conducted to
answer the two research questions. In these statistical procedures, the seven revised non
cognitive factors, three cognitive factors, and dependent variable, quantitative GPA, were
analyzed. The results are presented in Chapter IV.
69
CHAPTER IV
RESULTS
The research was guided by two research questions: (a) Do cognitive factors
mediate the effect of noncognitive factors on quantitative GPA? and (b) Can cognitive
and noncognitive factors predict engineering admission status?
Relationship between Cognitive and NonCognitive Factors and Quantitative GPA
A structural equation model was created using AMOS 7.0 to determine the
relationship between cognitive and noncognitive factors and quantitative GPA in the
preengineering curriculum with 60% of the sample (n = 1,375). Directional arrows were
used to indicate the hypothesized causal or direct relationship. The curved arrows were
used to indicate unexplained covariance or correlations. The rectangular boxes indicated
the observed variables, and the elliptical shapes indicated the latent, or unobserved,
variables (Meyers et al., 2006).
The initial model was recursive, which means the paths were directional between
the predictor variables and the outcome variable (Hair et al., 2006). The resulting model
containing two latent variables measured by seven noncognitive factor measures and one
latent variable measured by three cognitive factor measures is presented in Figure 1.
With the criteria for acceptable model fit indexes, according to Meyers et al.
(2006), a value of .95 or greater for the GFI and CFI is deemed as an acceptable fit. For
the RMSEA, a value of .08 or less indicates good fit. This initial model had a low GFI
70
(.896) and CFI (.786), high RMSEA (.123), and a significant chisquare (?
2
= 898.894; p
< .001). Thus, the model needs to be modified to reflect the empirical data. This initial
model had a R
2
of .30 for the cognitive factors, which means the two latent variables
measured by the seven noncognitive factors accounted for 30% of the variance in the
latent variable, cognitive factors. The R
2
for quantitative GPA, a singleitem measure,
was .34. These results suggested that the noncognitive and cognitive factors explained
34% of the variance in the dependent variable, quantitative GPA.
Figure 1
Structural Equation Model
F1
F2
.30
Cognitive
Factors
.26
MAe2
.51
.61
LCAAe1 .78
.40
AFe6
.63
.30
Ae5
.55
.71
.32
e11
.22
SAAe3
.47
.36
DPSe4 .60
.39
De7
.62
.33
ACT
Math
e8
.57
.63
HS Math
GPA
e9
.80
.50
HS
Rank
e10
.71 .34
quantGPA
.58
e12
.67
The initial hypothesized model was rejected based on the goodnessoffitness
statistics. With the modification indices, a restricted model, presented in Figure 2, was
created. The model had a higher GFI (.977) and CFI (.961). The RMSEA value was .081,
which improved by .042. The chisquare was significant (?
2
= 131.110; p < .001);
however, the chisquare tends to be overly sensitive with large sample sizes (Meyers et
71
al., 2006). All of these indices indicated a good fit between the restricted model and the
empirical data.
The restricted model had a R
2
of .14 for the cognitive factors and a R
2
of .29 for
the quantitative GPA. The R
2
change
between the initial model and the restricted model for
cognitive factors was .16. The F
change
formula was used to determine if the change in R
2
was statistically significant (Pedhazur, 1997):
F
change
= [(R
2
initial
? R
2
restricted
) / (k
initial
? k
restricted
)]
[(1 R
2
initial
) / (n  k
initial
? 1)],
where R
2
initial
= R
2
for the initial model, R
2
restricted
= R
2
for the restricted model, k
initial
=
number of predictors for the initial model, k
restricted
= number of predictors for the
restricted model, and n = number of participants in the sample.
Figure 2
Structural Equation Model: Restricted Model
F1
.14
Cognitive
Factors
.20
MAe2
.44
.63
LCAAe1 .79
e8
.25
SAAe3
.50
.39
DPSe4 .62
.27
ACT
Math
e5
.52
.67
HS Math
GPA
e6
.82
.52
HS
Rank
e7
.72 .29
quantGPA
e9
.78
.24
.37 .54
.25
.44
.23
.28
The change in R
2
was statistically significant for the latent variable, cognitive
factors, [F(3, 1367) = 937.37; p < .001]. The overall variance accounted for in
72
quantitative GPA using the latent variable measured by the four noncognitive and the
latent variable measured by the three cognitive factors reduced by 5%, .34 with the initial
model and .29 with the restricted model. The change in R
2
was statistically significant
[F(3, 1364) = 310.00; p < .001]; therefore, by restricting the model, the amount of
variance in quantitative GPA which was explained was significantly affected. The
residuals were correlated according to the model specifications and modification indices.
These correlations ranged from .23, weak to moderate relationship, to .78, strong
relationship. Subsequently, the variance explained in the outcome variables, cognitive
factors and quantitative GPA, reduced by correlating these residuals.
The standardized beta weights for the restricted model ranged from .44 to .82 and
were statistically significant at or below the .05 level. A criterion of .40 or higher was
established for the standardized beta weights as a measure of association (Meyers et al.,
2006). These results suggested the observed variables were statistically significant
measures for the latent variables, and these beta weights indicated the model has
construct validity. The standardized beta weight (.37) for the path from the noncognitive
factors to the cognitive factors was statistically significant and considered to have a
moderate relationship. Likewise, the standardized beta weight (.54) for the path from the
cognitive factors to the outcome variable, quantitative GPA, was also statistically
significant and considered as a strong relationship.
According to Hair et al. (2006), a direct effect is defined as the relationship
between two constructs. An indirect effect is defined as the relationship that results from
the sequence of two or more direct effects. Noncognitive variables did not have a direct
effect on the quantitative GPA, but they did have an indirect effect through the cognitive
73
variables. The results suggested that cognitive factors mediate the effects of lack of
confidence in academic abilities, mathematical ability, selfappraised abilities, and
difficulty with problem solving on the quantitative GPA. The interpretation for these
effects followed the criterion established for correlation coefficients where .10 is weak,
.30 is moderate, and .50 is strong according to Cohen (Meyers et al., 2006). The direct
effect for the relationship between cognitive factors and quantitative GPA was strong
(.54), and the direct effect for the relationship between noncognitive and cognitive
factors was moderate (.37). The indirect effects between noncognitive factors and
quantitative GPA were weak to moderate (.20). These effects were interpreted as a
measure of association between the predictor and outcome variables. Due to the
simplified nature of this model, these effects paralleled the standardized beta weights.
As a method of validating the initial structural equation model analysis, a 40%
crossvalidation model (n = 901) was analyzed using the two latent variables, non
cognitive factors 1 and 2, and one latent variable defined as cognitive factors. The results
are presented in Figure 3. With the same criteria for acceptable model fit indexes
according to Meyers et al. (2006), the initial crossvalidation model had a low GFI (.876)
and CFI (.776), high RMSEA (.132), and a significant chisquare (?
2
= 684.698; p <
.001).
The crossvalidation sample was analyzed using the same restricted model from
the 60% sample. Results are presented in Figure 4. The crossvalidation restricted model
had a higher GFI (.974) and CFI (.958). The RMSEA value was .086, which improved by
.046. The chisquare was significant (?
2
= 98.765; p < .001). All of these indices
indicated a good fit between the crossvalidation restricted model and data.
74
Figure 3
Crossvalidation Structural Equation Model
F1
F2
.31
Cognitive
Factors
.32
MAe2
.56
.59
LCAAe1 .77
.46
AFe6
.68
.30
Ae5
.55
.76
.36
e11
.22
SAAe3
.46
.35
DPSe4 .59
.40
De7
.63
.34
ACT
Math
e8
.59
.64
HS Math
GPA
e9
.80
.45
HS
Rank
e10
.67 .42
quantGPA
.65
e12
.72
Figure 4
Crossvalidation Structural Equation Model: Restricted Model
F1
.13
Cognitive
Factors
.25
MAe2
.50
.64
LCAAe1 .80
e8
.25
SAAe3
.50
.37
DPSe4 .61
.28
ACT
Math
e5
.53
.69
HS Math
GPA
e6
.83
.48
HS
Rank
e7
.69 .35
quantGPA
e9
.79
.19
.37 .59
.28
.42
.24
.26
The test of significance for the change in R
2
(Pedhazur, 1997) was used to
compare the change in R
2
from the full model to the restricted model. The full model had
a R
2
of .31 for the cognitive factors, and the restricted model had a R
2
of .13 for the
75
cognitive factors. The R
2
change
between the full model and the restricted model for
cognitive factors was .18. [F(3, 893) = 77.65; p < .001]. The full model had a R
2
of .42
for the quantitative GPA, and the restricted model had a R
2
of .35 for the quantitative
GPA. The overall variance accounted for in quantitative GPA using the latent variable
measured by the four noncognitive and the latent variable measured by the three
cognitive factors reduced by 7%. The change in R
2
was statistically significant [F(3, 890)
= 35.80; p < .001].; therefore, by restricting the model, the amount of variance in
quantitative GPA which was explained was significantly affected.
The standardized beta weights for the restricted model ranged from .50 to .83 and
were statistically significant at or below the .05 level. A criterion of .40 or higher was
established for the standardized beta weights as a measure of association (Meyers et al.,
2006). These results suggested that the observed variables were significant measures for
the latent variables. The standardized beta weight (.37) for the path from the non
cognitive factors to the cognitive factors was statistically significant. Likewise, the
standardized beta weight (.59) for the path from the cognitive factors to the outcome
variable, quantitative GPA, was also statistically significant. These statistically
significant beta weights indicated that the model has construct validity. The results
suggested that cognitive factors mediate the effects of lack of confidence in academic
abilities, mathematical ability, selfappraised abilities, and difficulty with problem
solving on the quantitative GPA. As a mediator variable, cognitive factors serve as a
construct that intervenes between the two other constructs, noncognitive variables and
quantitative GPA.
76
The direct effect for the relationship between cognitive factors and quantitative
GPA was strong (.59), and the direct effect for the relationship between noncognitive
and cognitive factors was moderate (.37). The indirect effects between noncognitive
factors and quantitative GPA were weak to moderate (.22). Again, these results suggested
that cognitive factors mediate the effects of lack of confidence in academic abilities,
mathematical ability, selfappraised abilities, and difficulty with problem solving on the
quantitative GPA. The crossvalidation analysis confirmed the results yielded with the
initial analysis.
Effects of Cognitive and NonCognitive Factors on Engineering Admission Status
A discriminant function analysis was conducted to develop a weighted linear
combination to predict group membership (i.e., admitted to engineering, switched to
another major at the university, or left the university unsuccessful). Admission status was
the grouping variable. The sample for this analysis was the same sample used for the
initial structural equation modeling analysis. This randomly selected sample included
60% of the participants (n = 1,347). Betz (1987) recommended a crossvalidation
procedure when using discriminant function analysis for predictive purposes. Because the
nature of the analysis is to maximize the linear relationship, there is a tendency to
overestimate the accuracy of the classification.
The cognitive and noncognitive factors included in the restricted structural
equation model (lack of confidence in academic abilities, mathematical ability, self
appraised abilities, and difficulty with problem solving, adjusted ACT math scores, high
school math grades, and high school ranks) were used as the independent variables. This
analysis was appropriate to understand the differences between two or more groups and
77
to determine the predictor variables that characterize the group differences. Similar to the
multiple regression analysis procedure, a discriminant function analysis yields a linear
equation with beta weights to maximize the predictability of the group membership
(Betz, 1987).
Results of the discriminant function analysis characterizing each admission status
with the noncognitive and cognitive factors with are presented in Table 11. The analysis
revealed adjusted ACT math scores, high school math grades, high school ranks, lack of
confidence in academic ability, and difficulty with problem solving formed Function 1,
and mathematical ability and selfappraised abilities formed Function 2. Canonical
correlation is equivalent to an eta computed through an analysis of variance procedure
and interpreted using the guidelines for correlation coefficients (Betz, 1987). For
Function 1, the canonical correlation, a measure of association between the discriminant
scores and group membership, was .38, p < .001, meaning the function has a moderate
effect size. For Function 2, the canonical correlation was .09. Function 2 was not
statistically significant, p = .13. Thus, it will not be used for subsequent analysis or
discussion.
78
Table 11
Correlation of Predictor Variables with Discriminant Functions (Structure Matrix) and
Standardized Discriminant Functions Coefficients
Correlation Standardized
Predictor variable Function 1 Function 2 Function 1 Function 2
Adjusted ACT math score .79* .07 .58 .24
High school math grades .76* .06 .41 .44
High school rank .64* .08 .18 .02
Lack of confidence in
academic ability
.47* .18 .34 .09
Mathematical ability .48 .86* .09 1.21
Difficulty with problem
solving
.27* .21 .03 .03
Selfappraised abilities .02 .21* .14 .10
* = largest absolute correlation between variable and discriminant function.
An eigenvalue is the ratio of betweengroups to withingroups sum of square and
is on a continuum from 0 to1 (Betz, 1987; Hair et al., 2006). If the eigenvalue is closer to
1, the function is considered good. The eigenvalue for Function 1 was .17, which means
17% of the variance was accounted for using Function 1.
The group centriod, the mean of the discriminant scores within a group, is
calculated by applying the discriminant beta weights to the group means on each variable
(Betz, 1987). With Function 1, participants who were admitted to the College of
Engineering were likely to have higher cognitive factors, less difficulty with problem
solving, and more confidence in their academic abilities (z = 0.30); however, participants
who were unsuccessful (GPA < 2.2) were likely to have lower cognitive factors, more
difficulty with problem solving, and less confidence in their academic abilities
(z = 0.74). The centriod for the participants who switched majors at the university was
79
0.16, which means the function did not discriminate with these characteristics.
A formula for the optimally weighted combination of predictor variables,
predicted discriminant score (D), was created using the discriminant functions (Betz,
1987):
D = .145(ACT) + .409(MATH) + .111(RANK) + .717(LCAA)
+ .060(DPS) + .167(MA) + .413(SAA) + 7.170,
where ACT = adjusted ACT math scores, MATH = high school math grades, RANK =
high school ranks, LCAA = lack of confidence in academic ability, DPS = difficulty with
problem solving, MA = mathematical ability, and SAA = selfappraised abilities. After
computing this variable, the predicted discriminant scores were converted into z scores to
standardize them. The crossvalidation cases were selected (n = 879) to apply the linear
function developed with the initial discriminant function analysis. An analysis of variance
(ANOVA) was conducted with the crossvalidation sample to determine group difference
between the participants who were admitted to engineering, switched to another major at
the university, and left the university unsuccessfully using the synthetic dependent
variable, the standardized predicted discriminant score.
The assumption of equal variance was not violated according to Levene?s Test of
Equality of Error Variances [F(2, 876) = 0.06; p = .94]. The results revealed a statistically
significant difference among the three groups [F(2, 876) = 76.65; p < .001; ?
2
= .15].
Table 12 displays the means and standard deviations by group. As a followup procedure,
a Bonferroni post hoc analysis was conducted, which revealed statistically significant
differences for all three pairwise comparisons. The greatest mean difference was between
the participants who were admitted to engineering and those participants who left the
80
university academically unsuccessful. Table 13 presents the pairwise comparisons by
group.
Table 12
Means and Standard Deviations by Group
Scale M SD
Admitted to Engineering 0.45 0.91
Switched Majors 0.09 0.89
Left Unsuccessful 0.46 0.89
Table 13
Post Hoc Test Results: Mean Differences by Group
Mean difference
Scale
Admitted to
engineering
Switched majors Left unsuccessful
Admitted to Engineering 
Switched Majors 0.54* 
Left Unsuccessful 0.90* 0.37* 
*Mean difference was statistically significant at the .05 probability level.
The statistical significance of the difference between the canonical correlation and
the eta square from the ANOVA conducted with the crossvalidation sample was
determined using Fisher?s z
r
transformation to convert the correlation coefficients into z
scores. With the transformed correlations, the difference in the two z scores was divided
by the standard error of the difference (Ferguson & Takane, 1989). The algorithm was
utilized to determine if the discriminant function, which was created with the analyzed
sample, was generalizable to the holdout sample:
z = z
r1
? z
r2
_______
?[1/(n
1
? 3) + 1/(n
2
? 3)],
81
where z = difference in z scores, z
r1
= Fisher?s z
r
transformation for the analyzed sample,
z
r2 =
Fisher?s z
r
transformation for the crossvalidation sample, n
1
= number of
participants for the analyzed sample, and n
2
= number of participants for the cross
validation sample. The canonical correlation for the analyzed sample was .38. The square
root of the eta square (.15) from the ANOVA with the crossvalidation sample was .39.
The Fisher?s z
r
transformation for the analyzed sample was .40, and the Fisher?s z
r
transformation for the crossvalidation sample was .41. With the formula, the difference
in z scores was 0.00, which indicated that there was not a statistically significant
difference between the analyzed and crossvalidation samples. Thus, the formula created
with the analyzed sample was generalizable to the holdout sample.
The overall percentage of correctly classified cases was 51.6% (see Table 14).
The participants who left the university unsuccessful were more accurately classified
(60.8%). For the participants who were admitted to the College of Engineering, 56.1% of
them were classified correctly. These results suggested that stronger cognitive and non
cognitive skills could yield higher quantitative GPAs. More specifically, academic
preparation, problem solving, and perception of overall academic abilities could
subsequently increase the quantitative GPAs.
With higher GPAs, it is likely that those participants who are interested in
engineering would have an increased probability of admission to the College of
Engineering. Approximately one third of the participants who switched majors were
classified in each of the categories. The results suggested that cognitive and non
cognitive factors do not discriminate for the participants who are academically successful
82
but choose other majors at the university. Thus, they likely differ with their interest level
for the field of engineering.
Table 14
Classification Analysis for Admission Status
Admitted to
engineering
Switched majors
Left
unsuccessful
Actual group
membership
n n % n % n %
Admitted to
Engineering
809 454 56.1 158 19.5 197 24.4
Switched Majors 270 99 36.7 78 28.9 93 34.4
Left Unsuccessful 268 56 20.9 49 18.3 163 60.8
Note: Overall percentage of correctly classified cases = 51.6%.
With the analyzed sample, a crossvalidation procedure was conducted during the
discriminant function analysis for obtaining a crossvalidated classification table (Table
15). The overall percentage of correctly classified cases (50.6%) was nearly equivalent to
the initial analysis. Similar results were seen with the rate of correctly classified
participants in the admitted to engineering group (56.0%) and those participants who left
unsuccessfully (59.7%). In addition, the participants who switched majors were classified
across the three groups without differentiation.
83
Table 15
Crossvalidation: Classification Analysis for Admission Status
Admitted to
engineering
Switched majors
Left
unsuccessful
Actual group
membership
n n % n % n %
Admitted to
Engineering
809 453 56.0 159 19.7 197 24.4
Switched Majors 270 105 38.9 69 25.6 96 35.6
Left Unsuccessful 268 56 20.9 52 19.4 160 59.7
Note: Overall percentage of correctly classified cases = 50.6%.
Means and standard deviations were conducted for the quantitative GPA, three
measures for the latent variable (cognitive factors), and the four measures for the latent
variable (noncognitive factors) to compare the three groups according to the results of
the discriminant function analysis. The means and standard deviations by group for the
initial analysis sample are presented in Table 16. The quantitative GPA, adjusted ACT
math scores, high school math grades, high school ranks, and mathematical ability were
higher for the participants who were admitted to engineering. The participants who left
the university unsuccessfully reported higher levels of agreement with lack of confidence
in academic ability and difficulty with problem solving. Due to the inverse nature of the
lack of confidence in academic ability and difficulty with problem solving, the lower
numerical values represent higher levels of agreement. The means and standard
deviations were similar across the three groups for selfappraised abilities, which were
confirmed by the low standardized beta weight with the discriminant function analysis.
Similar descriptives are displayed in Table 17 for the crossvalidation sample.
84
Table 16
Means and Standard Deviations by Group
Admitted to
engineering
Switched majors Left unsuccessful
Scale M SD M SD M SD
Quantitative GPA 2.78 0.81 2.13 0.95 1.37 0.83
Adjusted ACT Math
Scores
27.54 4.08 26.01 3.92 24.23 3.86
High School Math
Grades
5.47 0.99 5.10 1.02 4.68 1.01
High School Ranks 5.22 1.57 4.76 1.69 4.12 1.75
Lack of Confidence
in Academic Ability
3.86 0.47 3.73 0.50 3.62 0.50
Difficulty with
Problem Solving
3.91 0.52 3.82 0.56 3.76 0.54
Mathematical
Ability
3.47 0.51 3.26 0.54 3.22 0.59
SelfAppraised
Abilities
3.12 0.33 3.11 0.37 3.13 0.34
Table 17
CrossValidation: Means and Standard Deviations by Group
Admitted to
engineering
Switched majors Left unsuccessful
Scale M SD M SD M SD
Quantitative GPA 2.77 0.87 2.17 0.95 1.24 0.74
Adjusted ACT Math
Scores
27.77 3.92 25.77 3.90 24.39 4.23
High School Math
Grades
5.50 1.01 5.06 1.08 4.64 1.04
High School Ranks 5.26 1.65 4.72 1.65 4.18 1.71
Lack of Confidence
in Academic Ability
3.87 0.49 3.69 0.50 3.70 0.51
Difficulty with
Problem Solving
3.95 0.55 3.79 0.60 3.86 0.54
Mathematical
Ability
3.46 0.53 3.22 0.84 3.17 0.60
SelfAppraised
Abilities
3.15 0.35 3.12 0.36 3.11 0.36
85
Summary
The results from structural equation model analyses supported the theory that
cognitive factors mediate the effects of noncognitive factors on quantitative GPA. With
the measures from the restricted model, a discriminant function analysis yielded
distinguishing characteristics among the three groups: admitted to engineering, switched
to another major, and left the university unsuccessful. The measures for the latent
variable of cognitive factors were higher for the participants who were admitted to
engineering and were lower for the participants who left the university unsuccessful.
The interesting finding of the discriminant function analysis was that the
participants who left the university unsuccessful reported lower levels of agreement with
lack of confidence in academic ability and difficulty with problem solving. Another
finding was the similar descriptives for the participants who were admitted to engineering
and for those participants who switched to another major at the university. One
explanation for this finding could be interest in the field of engineering. The classification
table from discriminant function analysis supported this possible explanation because
those participants who switched majors were grouped similarly across the three groups.
To increase the percentage of students who were admitted to the College of Engineering,
cognitive factors, noncognitive factors, and interest in the field of engineering need to be
addressed before entering postsecondary institutions.
86
CHAPTER V
SUMMARY, CONCLUSIONS, DISCUSSIONS, AND RECOMMENDATIONS
A large amount of empirical research has been conducted on academic
achievement with college students. The empirical studies have revealed the significance
of high school preparation, more specifically mathematical preparation, for academic
success in postsecondary institutions; however, limited research exists for predicting
academic success using cognitive and noncognitive factors (i.e., selfconcept, study
habits, and inquisitiveness) (French et al., 2005; House, 2000; Shuman et al., 2003). The
nature of science, engineering, and mathematics college courses tends to be quantitatively
oriented, and calculus tends to serve as the gateway course for academic success within
these majors (Gainen, 1995). Conversely, noncognitive factors significantly contribute to
college mathematics achievement beyond standardized test scores or high school ranks.
The purpose of this study was to determine if cognitive factors mediate the effect
of noncognitive factors on quantitative GPA and to determine if these cognitive and non
cognitive factors can predict admission status in engineering education. The study
addressed the following research questions: (a) Do cognitive factors (ACT math scores,
high school math grades, and high school ranks) mediate the influence of noncognitive
factors on quantitative GPA in the preengineering curriculum? and (b) Can cognitive and
noncognitive factors predict engineering admission status?
87
Summary of Methods
The participants in this study were a sample of 2,276 students who entered
Auburn University during the fall semester of 2000 through the fall semester of 2004.
Requirements for participation in this study included completion of the College Freshman
Survey (Halpin & Halpin, 1996) and at least two quantitative courses in the pre
engineering curriculum at Auburn University.
Exploratory factor analysis. The sample was randomly divided into two databases
of comparable size. With one database, an exploratory factor analysis using principal axis
factoring with an oblimin rotation was conducted using the following scales from the
College Freshman Survey: Math SelfConcept, SelfAppraisal, Perceived Difficulty,
ProblemSolving Ability, Need Help, Academic Difficulty, Study Habits, and Academic
SelfConcept. The purpose of this factor analysis was to discover the factor structure of
the selected items and the correlations between the factors.
Confirmatory factor analysis. With the second database, a confirmatory factor
analysis was conducted to determine how the theoretical structure fits with the data and to
crossvalidate the factor structures created with the exploratory factor analysis (Meyers et
al., 2006).
Structural equation model. After the initial frequencies, descriptives, and bivariate
correlations were assessed, a structural equation model was created using AMOS 7.0 to
determine the relationship between cognitive and noncognitive factors and quantitative
GPA in the preengineering curriculum with 60% of the sample. The exogenous variables
were the noncognitive factors that were created and were confirmed with the factor
analyses and the cognitive factors (ACT math scores, mean high school math grades, and
88
high school ranks). Since these data have not been previously analyzed with structural
equation modeling, the model created with the 60% sample was applied to the 40% cross
validation sample to confirm the analysis results.
Discriminant function analysis. A discriminant function analysis was conducted
using the 60% sample to develop a weighed linear combination to predict group
membership (i.e., admitted to engineering, switched to another major at the university, or
left the university unsuccessful). The analysis used admission status as the grouping
variable and the cognitive and noncognitive factors as the independent variables. The
40% holdout sample was used to crossvalidate to results with the analyzed 60% sample.
Findings of the Study
The goal of this study was to identify the cognitive and noncognitive factors that
significantly contributed to academic success in quantitative courses within the pre
engineering curriculum. With these factors, this quantitative study sought to differentiate
those participants who were admitted to engineering from those participants who either
switched to another major at the university or left the university unsuccessful.
The significance contribution of ACT math scores from the structural equation
model analysis supports the findings of BesterfieldSacre et al. (1997), Blumner and
Richards (1997), Brown (1994), Buechler (2004), Devens and Walker (2001), French et
al. (2005), Heinze et al. (2003), Harackiewicz et al. (2002), House (1995a, 1995b, 2000),
House et al. (1996), Lackey et al. (2003), LeBold and Ward (1988), MollerWong and
Eide (1997), Shuman et al. (2003), Wesley (1994), and Zhang et al. (2004). In addition to
ACT math scores, the cognitive factors of high school ranks and high school math grades
support the significant findings of BesterfieldSacre et al., Buechler, Edge and Friedberg
89
(1984), French et al., Harackiewicz et al., House (2000), House et al., Lackey et al.,
LeBold and Ward, MollerWong and Eide (1997), Sadler and Tai (2001), Shuman et al.,
Smith and Schumacher (2005), Wilhite et al. (1998), and Zhang et al.
The four noncognitive variables (lack of confidence in academic ability,
mathematical ability, difficulty with problem solving, and selfappraised abilities) from
the restricted structural equation model confirm the findings of BesterfieldSacre et al.
(1997), Blumner and Richards (1997), Brown (1994), Brown and Cross (1993), Burtner
(2004, 2005), Harackiewicz et al. (2002), House (1993, 1995a, 1995b, 1995c, 2000),
Lackey et al. (2003), LeBold and Ward (1988), Litzinger et al. (2005), Nixon and Frost
(1990), Shaughnessy et al. (1995) and Shuman et al. (2003).
The structural model analysis revealed that cognitive factors, (ACT math scores,
high school math grades, and high school ranks) mediate the effects of noncognitive
factors on the quantitative GPA for the preengineering curriculum. Mathematical skills
were proven to be the gatekeeper with this sample. In addition, the noncognitive factors
that were found to have a relationship with cognitive factors and were also found to have
a significant contribution to the academic success of these participants.
The results from the study conducted by Gilbert (1960), who examined College
Board examination scores to predict group membership in engineering retention,
suggested that standardized test scores could not be used as the single predictor of
success in engineering. In the study conducted by Burtner (2005), noncognitive factors
were used to predict group membership in engineering admission. These results
suggested that perceived abilities in mathematics and science could influence students?
decision to leave engineering. By using the combination of the cognitive and non
90
cognitive factors, the discriminant function analysis from the present study correctly
classified over 50% of the participants who were admitted to engineering and those
participants who left unsuccessful. For the participants who switched to another major,
the classification was less accurate and the discriminant function was unable to determine
distinguishing factors, which may suggest a lack of interest in the field of engineering
that contributes to their decision to switch majors.
As a result of this study, the College of Engineering and K12 educational
systems will become more aware of the effects of cognitive and noncognitive factors on
academic success in preengineering. Furthermore, this study has addressed the need for
developing mathematical skills and problemsolving abilities at the secondary level so the
students will be better prepared for the quantitative courses within the preengineering
curriculum.
Future Research
In the present study, there was an inability to determine the distinguishing
characteristics between the participants who were admitted to engineering and those
participants who switched majors at the university. Both of these groups had similar
cognitive and noncognitive skills. Future research is needed to discriminant between the
two groups. This research could use a focus group of students who switched majors at the
university to determine possible barriers and reasons for their leaving the field of
engineering. Based on the topics discussed during the focus group, a survey could be
developed to administer to a random sample of students who have switched to other
majors at the university since 2000 when the university changed to the semester system.
91
Another future research topic could be the application of the structural equation
model and discriminant function analysis to other southeastern universities with colleges
of engineering. This research could determine if the effects of cognitive and non
cognitive factors on the quantitative GPA are unique to this sample or generalizable to
other preengineering students. Provided the sample of females was sufficient, this
research could determine the moderation effect of gender with the cognitive and non
cognitive facts and quantitative GPA.
92
CHAPTER VI
CURRICULAR IMPLICATIONS
The purpose of this study was to determine if cognitive factors mediate the effect
of noncognitive factors on quantitative GPA and to determine if these cognitive and non
cognitive factors can predict admission status in engineering education. The structural
model analysis revealed that cognitive factors (ACT math scores, high school math
grades, and high school ranks) mediated the effects of noncognitive factors (lack of
confidence in academic ability, mathematical ability, difficulty with problem solving, and
selfappraised abilities) on the quantitative GPA for the preengineering curriculum.
Furthermore, the noncognitive skills were found to have a direct relationship with
cognitive factors and were also found to have a significant contribution toward the
academic success of these participants.
The results of the discriminant function analysis suggested that participants who
were admitted to engineering and those participants who left unsuccessful were classified
correctly based on the cognitive and noncognitive factors. For the participants who
switched to another major at the university, the classification was less accurate, and the
discriminant function was unable to determine distinguishing factors. These results
suggest that a lack of interest in the field of engineering may contribute to their decision
to switch majors.
93
With both statistical analyses, structural equation model and discriminant function
analysis, all three cognitive measures, lack of confidence in academic ability, and
difficulty with problem solving were significant contributors. These results suggest that
the need to develop secondary curriculum to increase mathematical proficiency, to
increase the selfappraised noncognitive factors, such as lack of confidence in academic
ability and difficulty with problem solving, and to increase the exposure to engineering
fields.
Therefore, as a concluding part of this study, a secondary mathematics curriculum
was developed to improve mathematical skills and problemsolving abilities. The
Mathematics Curriculum for Advanced Mathematical Proficiency (Appendices C, D, E,
and F) has the central content of the NCTM Standards (2000) and College Board (2007)
recommendations for AP calculus. Within the curriculum, the mathematical concepts are
taught within realworld contexts. Each unit has an engineering connection to familiarize
the students with the various fields of engineering. The curriculum was devised by
consulting with professionals from the fields of naval science, business, construction, and
engineering. By integrating various professional fields into the context of each unit, the
student will see how mathematical fundamentals integrate throughout all subjects and
professions. In addition to key concepts and learning objectives, each curriculum unit
contains a summative evaluation. This evaluation serves as a measure of student progress
with a culminating performance assessment where the students use their conceptual and
procedural knowledge of mathematics to solve an existing problem.
The Mathematics Curriculum for Advanced Mathematical Proficiency will
address the following longterm outcomes (see Appendix B for the logic model): (a) To
94
increase the mathematical proficiency of secondary students, (b) To increase the
mathematical problemsolving ability of secondary students, and (c) To increase the
interest in engineering fields. Implementation will occur over a 4year period beginning
with the first mathematics course, geometry, and phasing in the curriculum with each
sequential mathematics course: algebra II, precalculus/trigonometry, and AP calculus.
Context
According to Saylor, Alexander, and Lewis (1981), the common formula for
developing a subjectspecific curriculum is the use of expert judgment, use of interest and
sequence criterion, and use of planning to implement appropriate instruction. Curriculum
should be viewed as a plan that encompasses the subject matter, outlines longterm goals
and shortterm objectives, and provides learning opportunities. Instruction facilitated by
the teacher implements the curricular plan. Effective plans assign appropriate weights to
the basis of curriculum: society, learners, and knowledge. The values of a society shape
the aims of education, but the curriculum planning should be guided by the interests and
needs of the learner. Within the curriculum plan, knowledge should be organized to
ensure generalizability in future situations. In addition, consideration must be given to the
external forces: legal requirements, research, and professional knowledge.
In order to meet the criteria for curriculum development, as prescribed, the five
steps for sitespecific curriculum development are (a) identify needs, (b) survey available
curricular materials, (c) adapt materials to meet the identified needs, (d) develop site
specific materials, and (e) implement new curriculum. When identifying the needs, the
data should be collected from the following entities: students, society, knowledge,
learning processes, goals, external policies, and resources. When surveying available
95
materials, Saylor et al. (1981) suggested an examination of current textbooks,
instructional packages, and local and national curriculum guides. If appropriate materials
were available, then one could adapt them to meet the identified needs. If the available
materials were not appropriate, then one could develop sitespecific materials. After
adapting or developing instructional materials, implementation of the new curriculum
could occur.
Mathematics curricula are organized in a spiral sequence where concepts and
procedures taught in one class are built upon in later classes. Sequential arrangement in a
curriculum fosters continuity and integration of learning for a student. The fundamental
goal of formal schooling is to develop lifelong learning because 12 years of formal
education cannot provide students with all of the skills, knowledge, and attitudes that
they will need in the future; however, education can provide a student with the ability to
become a lifelong learner for an everchanging society (Saylor et al., 1981).
Historical Development for Secondary Mathematics Curriculum
When schools developed, the primary goals of the institution were to teach
reading and writing skills (Jones & Coxford, 1970). Beginning with the Puritans in
Massachusetts, the purpose of education was to convey Christian morality (Litz, 1975).
Society believed that by teaching individuals to read and write they would obey the laws
of God. With the requirements of the Massachusetts Law of 1642 and Old Deluder Satan
Law in 1647, ministers and parents were required to educate the town?s youth about the
principles of religion and the laws of the colony. The Old Deluder Satan Law required
communities of 50 or more families to teach reading and writing. If 100 or more families
lived in the town, the law required a Latin Grammar School and hired teacher. The
96
curriculum used at these Latin Grammar Schools was a highly classical one dominated by
Latin and Greek. As grammar schools were established throughout the northern states,
the focus of the instruction was memorization and recitation of religious catechisms from
the New England Primer and passages from the Bible. These catechisms derived from the
beliefs of the Protestant religion (Henson, 2001; Kaestle, 1983; Spring, 2005).
Nineteenth century. The main application of arithmetic was business, and
computation skills dominated the content of instruction. During colonial times, the rule
method was used for instruction. The students were given ciphering books with a
systematic procedure, referred to as a rule, and practice problems to solve (Bidwell &
Clason, 1970; Jones & Coxford, 1970). Similar to the rote memorization required for
reading, the teachers would repetitiously ask a series of questions, such as ?ten and one
are how many?,? or give the student a term and expect a definition in return (Jones &
Coxford).
During the antebellum period, the purpose of education changed from moral
education to citizenship and industry. For instance, children?s stories portrayed
completing chores without talking. The teachers used moral persuasion to develop the
individual character of the students. The antidotes of Benjamin Franklin?s Poor Richard?s
Almanac were used in this capacity. Noah Webster?s textbooks integrated the themes of
republicanism, Protestantism, and capitalism into the classroom. The philosophical views
of John Locke promoted these educational themes. According to Locke, the child?s mind
is a blank slate, which means he/she was capable of great moral and intellectual
development. This view changed the educational methods from rote memorization to
internalizing and understanding the content (Kaestle, 1983). This change in educational
97
philosophy affected the high school mathematics curriculum. When the college
requirements increased, algebra and geometry were moved from the college curriculum
to the high school, which required students to internalize and apply the concepts instead
of memorize and recite them (Jones & Coxford, 1970).
Education is a function of the state according to the Bill of Rights of the U.S.
Constitution. State legislatures have the authority to prescribe the curriculum for public
schools. At the local level, all statemandated courses must be offered, but local schools
boards have great latitude in supplementing the curriculum. In 1874, the Michigan
Supreme Court ruled in Stuart v. S. D. No. 1 of Village of Kalamazoo that local school
boards had the responsibility to maintain high schools; however, with this decision, local
schools were given implied powers. These implied powers can be used to make specific
curricular elements and methods for curriculum implementation (Lunenberg & Ornstein,
2004). Often, curriculum implementation by local school boards differ across a given
state because there are differing educational philosophies between school boards, district
personnel, and school administrators and different student and teacher populations within
a given district. These implemented curricular policies affected the students? experience
in mathematics education and thus affected their mathematical performance (Hawkins,
Stancavage, & Dossey, 1998).
In 1893, the Committee of Ten rejected the idea for differentiating the curriculum
for elementary and secondary schools. The Commission on Reorganization of Secondary
Education would reverse this decision in 1918 with their report on Cardinal Principles of
Secondary Education. Different abilities among the students were recognized; therefore, a
differentiated curriculum was recommended (Saylor et al., 1981). Substantial changes
98
would occur for secondary mathematics curriculum and instruction as the 20th century
began.
Twentieth century. In 1920, the National Council of Teachers of Mathematics
(NCTM) was founded to keep the interests of mathematics at the top of the education
agenda. The first president of NCTM felt that mathematics curriculum should derive
from teachers and not educational reformers. With the establishment of NCTM, the
struggle between instructional methods and mathematical content was brought to the
forefront. In the classroom, when the focus is on content, the pedagogy that focuses on
studentcentered instruction tends to be restricted. Conversely, if the methods are the
focus, the amount of content tends to be limited (Klein, 2003).
In 1925, William Heard Kilpatrick (as cited in Klein, 2003), in his book,
Foundations of Method, rejected the idea of contentdriven instruction. Mathematics
should be taught through independent discovery based on the students? needs and
interests. Despite his preference for studentcentered pedagogy, Kilpatrick felt instruction
in algebra and geometry should be discontinued at the high school level. In his report,
The Problem of Mathematics in Secondary Education, he stated that the content in
secondary mathematic should have proven value. Furthermore, he recommended that the
traditional high school curriculum involving algebra and geometry was appropriate for a
select few individuals. Instruction in basic skills and conceptual knowledge should be
differentiated for different students (Commission on the Reorganization of Secondary
Education, 1920; Klein). Kilpatrick referred to specialized and elective mathematics
courses for 10th, 11th, and 12thgrade students who plan to pursue careers in
mathematics and sciences. For the other students, their mathematics curriculum should be
99
geared toward vocational training. At the junior high level, the curriculum should teach
procedural knowledge and basic arithmetic (Commission on the Reorganization of
Secondary Education).
In response to Kilpatrick, the Mathematical Association of America (MAA)
published The Reorganization of Mathematics for Secondary Education in 1923. This
comprehensive report included an extensive survey of secondary curricula, teacher
preparation, psychology of learning mathematics, and mathematical applications. MAA
recommended curriculum that incorporated algebra for everyone (Klein, 2003).
According to the National Committee on Mathematical Requirements (1923), since the
majority of students end their formal education years in the ninth grade, the mathematics
curriculum should include basic arithmetic with fundamentals of algebra, geometry, and
trigonometry. Thus, every student can benefit from understanding the language of algebra
and the ability to set up and solve equations based on realworld situations. For those
students who remained in high school and had intrinsic mathematical interests, the
National Committee recommended elementary calculus for the 12th year in school. The
content for this calculus course would be focused on the rate of change and not the
superficial.
Changes in the nation?s society influenced the shift in content. Up to the 20th
century, youth learned their vocational skills from their parents or other adults in the
community via apprenticeships (Saylor et al., 1981). As the national economic situation
worsened during the Great Depression, the idea emerged to use education as a resource
for reconstructing society. Lack of employment caused more students to stay in school;
100
however, mathematics was unpopular in secondary curriculum (Bidwell & Clason, 1970).
It became an elective subject, and enrollment decreased (Jones & Coxford, 1970).
George Counts promoted this social reconstruction philosophy at the 1932 annual
meeting of the Progressive Education Association. At this meeting, he attacked
capitalism and business control of education in his speech, ?Dare Progressive Education
Be Progressive?? He suggested that teachers take the lead in this progressive movement
and reconstruct society; however, this movement deemphasized the need for rigorous
curriculum content in mathematics and sciences (Spring, 2005).
Mathematics gained attention during the 1940s. As young men volunteered for
military service, these men needed remediation in basic arithmetic. Officers in the Army
and Navy complained about the mathematical deficiencies of new recruits. Based on
these complaints, the Life Adjustment Movement emerged. The secondary mathematics
curriculum focused on consumer mathematics (i.e., insurance, taxation, and home
budgeting). These functional mathematics skills replaced the emphasis on algebra,
geometry, and trigonometry. The declining interest in higher level mathematics was
evident by enrollment in higher level mathematics courses at the secondary level which
had reduced by half since 1910, 56.9% (Klein, 2003).
By the end of the decade, there were advances in technology (i.e., radar,
navigation, and atomic energy). These technological advances emphasized the
importance of mathematics in the modern society and defeated the social reform
movement (Bidwell & Clason, 1970; Klein, 2003). Following World War II, the demand
for collegetrained personnel grew, which affected college preparatory programs in high
school. There were pressures to teach analytic geometry, calculus, and statistics (Jones &
101
Coxford, 1970). In reality, the mathematics curriculum was unable to keep up with the
demand (Bidwell & Cason).
During the Cold War Era, beginning in the 1950s, education was the country?s
weak link for defense against communism according to Spring (2005). These tensions
resulted in a greater emphasis on science and mathematics. The ideology was that by
emphasizing science and mathematics the United States could stay technologically ahead
of the Soviet Union. In 1957, Russia launched the first satellite, Sputnik I, into space. In
response to Sputnik I, Congress passed the National Defense Education Act (NDEA).
Title III of the NDEA appropriated nearly $300 million dollars for hiring faculty and
purchasing materials for science, mathematics, and foreign language. President
Eisenhower wanted to increase the quality of mathematics and science education in order
to be competitive in the space and arms race with the Soviet Union (Spring).
In 1960s, ?new mathematics? developed as a result of the NDEA. Instead of
students learning rote mathematics skills (e.g., add, subtract, multiply, and divide), the
students were taught mathematical concepts, such as set theory and functions (Spring,
2005). According to Saylor et al. (1981), these programs were developed by scholars who
were content specific. Little emphasis was placed on the needs or interests of the
students. Curricula were developed to provide logical explanations for procedural
mathematics. For the first time in mathematics education, mathematicians were actively
involved in K12 curriculum development. One of the contributions of the new
mathematics movement was the introduction of the calculus course at the high school
level (Klein, 2003).
102
In 1958, the American Mathematical Society established the Science
Mathematics Study Group (SMSG). The group was led by Edward G. Begle from Yale
University. During the new mathematics movement, this group created junior and senior
high school curricula. These curricula were formal and did not emphasize basic rote skills
or application skills. As the curriculum content focus shifted to the abstract, students,
parents, and teachers had difficulties with the rigors of the curriculum (Klein, 2003).
Under the Cooperative Research Act, the federal government established the first
of several Research and Development Center in 1963. The purpose of these centers was
to conduct specialized education research using new programs and instructional strategies
(Saylor et al., 1981). President Lyndon Johnson signed the Elementary and Secondary
Education Act of 1965 (ESEA). With the passage of ESEA, the federal government used
its funds to control educational policies (Spring, 2005). In addition, after the passage of
ESEA, there were 30 Research and Development Centers and educational laboratories
established by the federal government by the end of 1966. Unfortunately, by the 1970s,
the number of these centers and laboratories had declined to 17. These trends marked the
backtobasics (i.e., reading, writing, and arithmetic) movement of the 1970s (Klein,
2003; Saylor et al.).
According to Mann (1976), the mere exposure to calculus did not replace a strong
foundation in algebraic fundamentals. Beginning in the mid1970s, the majority of the
states had developed minimum competency exams. These exams assessed basic skills,
and half of the states required the students to pass them to earn their high school diploma
(Klein, 2003).
103
The NCTM released An Agenda for Action in 1980. This report, which developed
later into the national standards, prescribed a new focal point for mathematics education.
NCTM recommended problem solving as the focus in mathematics; therefore, skill
mastery should not interfere with acquisition of problemsolving skills. Likewise, the
report suggested increased calculator use and decreased paperpencil computations. Most
controversial, the NCTM recommended that the role of calculus in the high school
curriculum be reevaluated because the new integrated mathematics curriculum did not
warrant a specific course in calculus. The students received portions of algebra,
geometry, and trigonometry through their integrated textbooks; however, the components
were not systematic (Klein, 2003).
As the industrial competition with Japan and West Germany increased, Nation at
Risk, released by Secretary of Education Terrel Bell and the National Commission on
Excellence in Education, in 1983 blamed public schools for the America?s failure to
compete. The Commission wanted to assess the quality of teaching and learning in the
nation?s schools, student achievement, and education programs. In addition, the
Commission compared the nation?s schools to those schools in other advanced nations.
This report called for increasing the academic rigor, reforming the curriculum, and
improving the quality of the teachers in the school (Henson, 2001; National Commission
on Excellence in Education, 1983; Spring, 2005).
The findings of the report stated that there was a rising level of mediocrity, which
threatened the future of the nation. Based on their collected data, student achievement
(e.g., SAT and NAEP scores) was stagnant since the launch of Sputnik I. Furthermore,
the report noted an increasing need for remedial mathematics at postsecondary, business,
104
and military institutions (National Commission on Excellence in Education, 1983).
According to Spring (2005), Americans wanted the education system to solve the
economic and social problems in society and subsequently close the gap between
education and industry.
To address this desire for educational reform, the National Commission on
Excellence in Education (1983) wanted to increase the requirements for high school
graduation, especially for science and mathematics. Likewise, they demanded a more
rigorous curriculum. Science and mathematics curriculum development at the secondary
level was targeted to improve performance on standardized achievement tests (e.g., SAT)
(Henson, 2001). The National Commission recommended the following implementations,
specifically for mathematics curricula: (a) understand geometric and algebraic concepts;
(b) understand elementary probability and statistics; (c) apply mathematics to reallife
situations; and (d) estimate, measure, and test the accuracy of their calculations (National
Commission on Excellence in Education, 1983).
This thought continued when President George Bush presented Goals 2000. This
plan suggested creating model schools, instituting national standards, establishing a
national testing program, and providing incentives for parental choice (Spring, 2005). To
facilitate these goals, the Education and Human Resources Division of the National
Science Foundation (NSF) developed a series of grants to promote fundamental changes
in science and mathematics education. Beginning in 1991, these statewide initiative
grants encouraged alignment with the NCTM standards. NSF felt mathematical ideas
should be taught using meaningful and realworld situations. The instruction should
105
incorporate inquirybased learning and problemsolving skills so the students will be
effectively and actively engaged students (Klein, 2003).
In 2001, President George W. Bush sponsored the No Child Left Behind (NCLB)
Act, which Congress passed in 2002. It mandated that all children had to reach specific
levels of academic proficiency in specific subject areas including mathematics by the
20132014 academic year. In addition, this law mandated that all children be assessed
based on statewide standards in reading, mathematics, and science that would help
determine whether a state met its adequate yearly progress goals (NCLB, 2001). This
high stakes testing increased the need for mathematical achievement.
The preceding historical developments demonstrated how secondary mathematics
curriculum has emerged since the dame kitchen schools. With the primary external force
of NCLB, many forget that curriculum development is an evolving process based on a
variety of entities (Saylor et al., 1981).
Secondary Mathematics Curriculum
Analyses of the secondary mathematics curricula and textbooks indicated a large
scope of content with little indepth coverage. The recent reform efforts of the NCTM
recommended that mathematics curricula cover fewer topics but in greater depth. In
addition, they suggested the use of inquirybased methods and the emphasis on
conceptual understanding of the mathematics content. NCLB, passed in 2001, required all
states to establish mathematics standards. Resulting from the passage of NCLB, many
states have revised their standards and curriculum frameworks and developed a
systematic evaluation and modification systems (National Science Board, 2004).
106
Beginning in 1999, American Association for the Advancement of Science
(AAAS) Project 2061 designated teams of mathematics professors and K12 educators to
evaluate textbooks based on the NCTM standards. Of the 12 mathematics textbooks, only
four were rated as excellent. In general, the textbooks were deemed inadequate because
they tended to teach basic number and geometry skills. These textbook failed to teach
higher order thinking skills or to develop the students? reasoning abilities (National
Science Board, 2004).
Science curricula and collaborations for K12 education have integrated science
and mathematics using engineering applications. The majority of these curricula were
implemented in science classrooms (e.g., Kimmel & Rockland, 2002; Rigby & Harrell,
2005; Schwartz, Regan, & Marshall, 1997). Of the mathematics curricula that incorporate
science and technology, the greater part of them was geared toward middle school
populations. Furthermore, a minimal number of high school mathematics curricula
addressed problem solving and NCTM standards (Klein, 2003). These curricula
integrated the content for the algebracalculus sequence. This integration can prove to be
problematic for students and teachers who feel the pressure to pass the high stakes
graduation assessments and endofcourse tests (National Science Board, 2004). The
pressures of high stakes testing (NCLB, 2001) and standards reform movement in
mathematics (NCTM, 2000) demand increased proficiency in mathematics for adequate
yearly progress.
According to Shettle et al. (2007), if students take algebra I in ninth grade, then
they were more likely to end with algebra II as their highest level of mathematics;
however, if the students took geometry in the ninthgrade year, then they were more
107
likely to take calculus before graduating from high school. Thus, a mathematics
curriculum at the high school level should be restructured to include the skills to master
collegelevel calculus.
Needs Assessment
A major source of student achievement and performance is the National
Assessment of Educational Progress (NAEP). Established in 1969, the NAEP measures
the nation?s educational progress by routinely administering subjectarea assessments to a
representative sample of 4th, 8th, and 12thgrade students. With scientifically selected
samples, reading, mathematics, social studies, and science are assessed by performance
tasks and multiplechoice questions (Saylor et al., 1981).
The mathematics section of the NAEP contains a combination of multiplechoice
(54%), short constructedresponse (40%), and extended constructedresponse (6%) items.
The constructedresponse questions assess the students? ability to reason and
communicate mathematical knowledge. The NAEP allows the use of calculators, rulers,
protractors, and manipulatives (e.g., geometric shapes, threedimensional models, and
spinners). No one student completes the entire assessment. A portion of the assessment is
administered to each student. The data are combined across students to provide estimates
of mathematical achievement for 4th, 8th, and 12thgrade students. The estimates are
disaggregated by gender and ethnic group (Mitchell, Hawkins, Jakwerth, Stancavage, &
Dossey, 1999).
NAEP reports provide composite mathematics scales on a 3point scale (i.e.,
basic, proficient, and advanced). Basic level is defined as partial mastery of prerequisite
knowledge and skills expected for the grade level. Proficient level is defined as solid
108
academic performance for grade level, which includes the ability to apply conceptual
knowledge to realworld situations and to use analytical skills within the mathematics
content. At the advanced level, the mathematical performance is considered superior. The
NAEP mathematics assessment is divided into five content strands: (a) number sense,
properties, and operations; (b) measurement; (c) geometry and spatial sense; (d) data
analysis, statistics, and probability; and (e) algebra and functions (Mitchell, Hawkins,
Jakwerth et al., 1999).
Number Sense, Properties, and Operations
The content strand focuses on mathematical operations using whole numbers,
fractions, decimals, integers, and rational numbers. In addition, students are expected to
apply their understanding to realworld situations. For the Grade 12 level, the content
strand includes real and complex numbers. When asked to evaluate expressions for
odd/even, 92% of the 12thgrade students answered at least one entry correctly. For
students who had taken algebra II, 94% of them answered at least one entry correctly.
Likewise, 91% of the students who had taken calculus as their highest level of
mathematics answered at least one entry correctly. When asked to solve a rate versus
time problem, which required application skills, 49% of the students responded correctly.
For students who took algebra II as their highest level of mathematics, 48% of them
responded correctly, but 65% of the students who took calculus as their highest level
answered correctly. When the 12thgrade students were asked to compute a percent
increase, which required problemsolving skills, 56% of the 12thgrade students
answered incorrectly (Mitchell, Hawkins, Jakwerth et al., 1999).
109
At the Grade 12 level, 33% of the students who took calculus I as their highest
level of mathematics responded incorrectly as opposed to 57% of the students who took
algebra II as the highest level. Based on these results, the majority of the students were
successful when solving explicit computational problems; however, the 12thgrade
students tended to experience difficulties when they were asked to respond to problems
involving multiple steps, application of conceptual knowledge, and problemsolving
skills. For number sense, in the 1996 NAEP, 12thgrade students who were enrolled in
calculus had an average score of 338 from a range of 0 to 500, and the average score for
those 12thgrade students who ended their mathematics sequence with algebra II was 304
(Mitchell, Hawkins, Jakwerth et al., 1999).
Measurement
The content strand involves (a) units of measurement; (b) measurement
instruments; (c) perimeter, area, and volume; and (d) estimation of measurements. The
questions involve complex application of volume and surface area. In addition, there are
questions involving proportions with maps and scale drawings. When 12thgrade students
were asked to recognize the best unit of measurement, 87% of the students responded
correctly. There was not a statistical difference among the mathematics courses. In
contrast, the students were asked to use a protractor to draw a 235? arc on a circle, 25%
of the students responded correctly with a maximum 5? margin of error. When examining
the highest level of mathematics, 24% of those students who took algebra II responded
correctly, but 56% of those students who took calculus I responded correctly. There was a
32% difference between the two mathematics courses. The differences in mathematical
abilities were evident when the 12thgrade students were asked to find the volume of a
110
cylinder. The 12thgrade students who took calculus I answered with 56% accuracy as
opposed to 31% of the students who took algebra II as their highest level of mathematics
(Mitchell, Hawkins, Jakwerth et al., 1999).
Parallel results were seen when the students were asked to find the perimeter of a
quadrilateral, to use a ruler to find the circumference of a circle, and to compare the area
of two shapes. Moreover, students who took more advanced mathematics courses tended
to perform better compared to those students in less advanced mathematics (i.e., calculus
I versus algebra II). In the measurement strand, for students enrolled in calculus, the
average score was 337 from a range of 0 to 500; however, the students who were enrolled
in algebra II had an average score of 305 on the 1996 NAEP (Mitchell, Hawkins,
Jakwerth et al., 1999).
Geometry and Spatial Sense
This content strand concentrates on the conceptual understandings of geometry
figures and their properties. At Grade 12 level, the strand involves transformational
geometry and application of proportional thinking and geometric formulas. When 12th
grade students were asked to use similar triangles, 37% of the students answered
correctly. Sixtytwo percent of the students who took calculus responded accurately, and
only 32% of the students who took algebra II responded accurately. Furthermore, when
the 12thgrade students were asked to draw a parallelogram with perpendicular diagonals,
18% of them performed the task correctly. For the students who took calculus, 56% of
them responded correctly; however, only 18% of the students who took algebra II
responded correctly. Similar accuracy levels were seen when the 12thgrade students
were asked to use a protractor to draw perpendicular lines and measure angles and to
111
describe the geometric process for finding the center of a disk (Mitchell, Hawkins,
Jakwerth et al., 1999).
When the 12thgrade students were asked to assemble pieces to form a shape that
was not a square, 58% of the 12thgrade students performed the task correctly. Students
who took calculus responded with 65% accuracy compared to 59% accuracy for the
students who took algebra II. Most of the questions in this strand required the students to
draw or explain a response and to apply their knowledge of geometric properties. These
types of questions were difficult for the 12thgrade students. Students in the more
advanced mathematics tend to perform better than those students in the less advanced
mathematics courses. There was a 35point difference in average geometry strand scores
for those students who took calculus and those students who took algebra II on the 1996
NAEP (Mitchell, Hawkins, Jakwerth et al., 1999).
Data Analysis, Statistics, and Probability
The content strand focuses on the ability to collect, organize, read, represent, and
interpret data. For the Grade 12 level, the questions require the use of statistical
techniques and the application of probability to dependent and independent events. When
12thgrade students were asked to use data from a chart to make a decision and explain
their rationale, 67% of the 12thgrade students answered correctly. There was not a
substantial difference between students in more advanced mathematics courses and those
students in the less advanced mathematics courses (Mitchell, Hawkins, Jakwerth et al.,
1999).
Likewise, when 12thgrade students were asked to use data in a table to compute
hourly wages and determine when the wage rate changed, only 13% of the 12thgrade
112
students responded correctly. Twentyfive percent of the students who took calculus
responded correctly compared to 15% of the students who took algebra II as their highest
level of mathematics course. Comparable results were seen when 12thgrade students
were asked to compare means and medians. Only 4% of the responses were rated at least
satisfactory on the 4point scale (i.e., extended, satisfactory, partial, and minimal).
Sixteen percent of the responses by students who took calculus were rated at least
satisfactory as opposed to 2% of the responses by students who stopped their
mathematics coursework with algebra II. When the students were asked to compare
probabilities, 28% of the 12thgrade students answered correctly with at least a partial
explanation. There was a 30% difference between students who took more advanced
mathematics course and those students who took less advanced mathematics, 58% and
28%, respectively (Mitchell, Hawkins, Jakwerth et al., 1999).
Consistent with the results with the other content strands, the students tended to
be successful with the explicit questions pertaining to graphs, charts, and tables. When
the students were asked to compute and reason calculations based on displayed data, the
students? performance indicated those types of questions were more difficult. The more
mathematics courses a student took the better their mathematical ability; however,
problem solving and multiple steps requiring answers from previous steps were
challenging for the advanced students. In 1996, students who took calculus had an
average score of 333, from a range of 0 to 500, on the data analysis and probability
strand. The average score for those students who ended their mathematics courses with
algebra II was 307 (Mitchell, Hawkins, Jakwerth et al., 1999).
113
Algebra and Functions
The content strand focuses on algebraic notation and the ability to solve
equations. At Grade 12 level, the strand involves the use of functions to represent and
describe relationships. For the 12thgrade students, when asked to identify a graph of a
specific function, 20% of the students answered correctly. When examined by the highest
course taken in the algebracalculus sequence, 17% of the students, who responded that
algebra II was their highest mathematics course completed, answered the question
correctly, and 55% of the students, who responded that calculus was their highest level of
mathematics, answered the question correctly. Similar results were seen with solving a
pair of equations and using trigonometric identities (Mitchell, Hawkins, Jakwerth et al.,
1999).
On the other hand, when 12thgrade students were given an extended constructed
response question to assess the problemsolving ability of the students, the results were
not statistically significant. When asked to describe the pattern of squares in a particular
figure, only 4% of all 12thgrade students earned at least satisfactory on a 5point scale
(i.e., extended, satisfactory, partial, minimal, and incorrect). For students who had taken
either precalculus or calculus, the percentage who earned at least satisfactory was 18%
and 10%, respectively. Four percent of the students who scored at least satisfactory
responded that algebra II was their highest level of mathematics. This question lends
support to the idea that all students are challenged when asked to apply their problem
solving skills and communicate their ideas in writing. The difference between the average
strand scores of those students who were enrolled in calculus and those students who took
algebra II was 41 points (Mitchell, Hawkins, Jakwerth et al., 1999).
114
According to Grigg, Donahue, and Dion (2007), 61% of high school seniors
performed at or above the basic level on the 2005 NAEP, and only 23% of them
performed at or above the proficient level. As concluded by the 1996 NAEP data,
students who took higher level mathematics courses were more likely to perform better
on the NAEP (i.e., 182 for calculus and 143 for algebra II from a range of 0 to 300).
Students were able to identify a solution or compute values when all information was
given in the mathematics problem. The performance of these students indicated the
students have the ability to make direct applications of concepts or procedures (e.g., using
the Pythagorean Theorem to determine the length of a hypotenuse), but they are unable to
implicitly apply their knowledge with multiple steps.
Successful mathematics achievement requires the completion of more advanced
mathematics courses. Results from the NAEP data revealed that significant differences
existed between students who took calculus I and those students with algebra II as their
highest level of mathematics. Students who took one more high school mathematics
course increased their average by 18 points, approximately 6%, on the 1996 NAEP. For
those students with two additional mathematics courses, the average increase was 33
points (approximately 10%) (Mitchell, Hawkins, Jakwerth et al., 1999).
This trend has continued according to Perie, Moran, Lutkus, and Tirre (2005) and
Shettle et al. (2007), and the average point difference has increased. The average
mathematics score for those student who took calculus was 336 from a range of 0 to 500,
and, for those students with algebra II as their highest level of mathematics courses
completed, the average score was 310 (Perie et al.). This 26point gap widened further in
2005. The average score for calculus students was 192 from a range of 0 to 300 and for
115
algebra II students was 142. Despite the scale?s maximum range changing from 500 to
300, a 50point difference existed for students with 2 more years of advanced
mathematics preparation (Shettle et al.).
In 2005, 89% of the students who performed at the advance level had calculus as
their highest course completed, and less than 1% of the students who reported algebra II
as their highest mathematics course performed at the advance level (Shettle et al., 2007).
According to Mitchell, Hawkins, Jakwerth et al. (1999), lower performing students may
select or be assigned to less advanced mathematics courses, and the mathematics course
may not be providing them with challenging opportunities. Therefore, the likelihood of
their ending their mathematics courses earlier in their high school career increases. By
taking algebra I during the eighthgrade year, the likelihood increases for students who
take more mathematics courses and more advanced coursework. Despite the percentage
of students taking algebra I during their eighthgrade year increasing (Mitchell, Hawkins,
Jakwerth et al., 1999), the percentage of 12thgrade students who end their mathematics
courses with algebra II has remained unchanged. In 2004, 53% of the 12thgrade students
reported that algebra II was their highest level of mathematics courses (Perie et al., 2005).
This unchanged percentage reflects curriculum concerns in advanced mathematics
courses. At the high school level, the majority of the schools require 3 years of
mathematics, referred to as midlevel curriculum. Since 1990, the average number of
required mathematics courses has increased from 3.2 to 3.8. Only 10% of the high school
graduates in 2005 participated in a rigorous curriculum level where 4 years of
mathematics was required. The traditional sequence of mathematics courses at the high
school level is algebra I, geometry, algebra II, precalculus, and calculus. Based on this
116
sequence, for a student who took algebra I in eighth grade and continued to take courses
until his or her 12thgrade year, the highest level of mathematics would be calculus.
Thus, it is likely that a graduating senior would not be involved in formal mathematics
courses for at least one year (Shettle et al., 2007). The lack of formal exposure affects
students? overall mathematical ability (Mitchell, Hawkins, Jakwerth et al., 1999).
Mathematical ProblemSolving Ability
According to Mayer (2003), mathematical problem solving includes three
components: cognitive, process, and directed. Therefore, problems, which contain
mathematical content that involves problem solving, require the individual to use his or
her cognitive abilities, to apply his or her mental computations to a mental representation,
and to direct the activities to achieve an outcome. In mathematical problem solving, there
are two types of problems: routine and nonroutine. Routine problems are exercises
where the individual immediately sees the solution procedure (e.g., 2 X 2). On the other
hand, with nonroutine problems, the solution procedures are not obvious to the
individual.
Mathematical problem solving is composed of four interdependent cognitive
processes: translating, integrating, planning, and executing. When an individual reads a
mathematical word problem, he or she translates or paraphrases the sentences into a
mental representation. This process employs semantic and linguistic knowledge. The
process of integrating involves schematic or conceptual knowledge because the
individual builds a mental model of the word problem. During the planning process, the
individual devises a plan for how to solve the word problem, which utilizes strategic
knowledge. After representing the word problem, building a model, and devising a plan,
117
in the executing process, the individual executes out the strategic plan using his or her
procedural knowledge (Mayer, 2003).
Each of these cognitive processes requires the application of prior knowledge and
experience with mathematical concepts and procedures. Often, the lack of mathematical
problemsolving ability causes the individual to select the numbers from the word
problem and conduct rote procedures based on key word phrases. With instruction,
individuals can recognize structural similarities among word problems and solve them
effectively. Mayer (2003) noted that computational fluency aids mathematical problem
solving. If the individual can complete computations automatically, his or her cognitive
resources can be devoted to the four cognitive processes of mathematical problem
solving.
Mitchell, Hawkins, Stancavage et al. (1999) conducted the study of mathematics
incontext, referred to as the theme study. The purpose of the theme study was to assess
the students? problemsolving abilities within reallife contexts and to determine how the
students make connections across content areas of mathematics. The participants were
4th, 8th, and 12thgrade students who took the 1996 NAEP mathematics assessment.
For Grade 12, there were 3,860 participants who represented 196 schools. Each
participant was given an assessment booklet with a theme block, which contained grade
appropriate reallife scenarios, and a gradeappropriate block of mathematics questions.
This instrument was administered separately from the main 1996 mathematics
assessment.
The participants were given 30 minutes to complete the theme block and 15
minutes to complete the block of mathematics questions. The 12thgrade participants
118
were given seven questions about buying a car (two multiplechoice and five constructed
response items). With these participants, at least 90% of the items were attempted;
however, the researchers found that the majority of the students tended to have
difficulties with complex multiplestep problems (Mitchell, Hawkins, Stancavage et al.,
1999).
Fortyone percent of the participants responded correctly to the Buying A Car
Theme block. When the participants were asked to find the amount of the down payment,
82% of them answered correctly. Similar results were seen with finding the total amount
paid for the car (83%) and finding the difference between total amount paid and price
(80%). All three of these questions derived from the number sense, properties, and
operations content strand and required procedural knowledge (Mitchell, Hawkins,
Stancavage et al., 1999).
When participants were asked to find the amount to be financed, which required
multiple steps, only 34% of the participants provided a complete and correct response.
When asked to use a formula to find the total cost of the car, 23% of the participants
scored at least satisfactory on a 5point scale (i.e., extended, satisfactory, partial,
minimal, and incorrect). Similar percentages were seen with finding the amount saved if
leased (27%) and comparing the price of leasing and the price of buying (16%). When
questions required conceptual or procedural knowledge with minimum steps and
straightforward calculations, the high percentage of participants responded correctly, but,
when the question required multiple steps and/or problemsolving ability, the percentage
correct dramatically reduced. The researchers noted that the participants appeared to have
difficulties when they were required to isolate the information needed to solve the
119
problem. Extraneous information seemed to complicate the solving of the word problems
(Mitchell, Hawkins, Stancavage et al., 1999).
Another study conducted by Mitchell, Hawkins, Stancavage et al. (1999) was the
study of students taking advanced courses in mathematics, referred to as the advanced
study. The purpose of the advanced study was to assess the level of mathematical
proficiency, primarily in the content strand of algebra and functions, of the students who
were taking or had taken advanced mathematics courses (e.g., precalculus or calculus for
Grade 12). The participants were 2,965 12thgrade students which represented 207
schools. The format of the instrument was similar to the Theme Study with a block of
mathematics questions and a theme block. The 12thgrade assessment contained 7
multiplechoice, 10 short constructedresponses, and 5 extended constructedresponse
items. For Grade 12, 30% of the items were answered correctly.
When asked to use a linear function, 20% of the participants provided a correct
response with a complete explanation of their process. When asked to compare the
volumes of three pyramids, only 10% of the participants? responses were rated at least
satisfactory on a 5point scale (i.e., extended, satisfactory, partial, minimal, and
incorrect). Mitchell, Hawkins, Stancavage et al. (1999) hypothesized that performance on
items which contained content that the participants had not studied since 9th or 10th
grade years tended to be reduced because the participants lacked the continuous exposure
to the mathematical content. Therefore, in addition to the inability to solve mathematical
problems with two or more successive steps, the participants appeared to be unable to
maintain and generalize mathematical concepts.
120
According to the NAEP data, in 1999, only 8% of all 17yearold students across
the nation had the ability to solve multiplestep word problems using reasoning skills
(Campbell et al., 2000). In 2004, 7% of the 12thgrade students performed at the level
which requires students to apply reasoning skills to multiplestep problems. This
percentage has not changed since 1973 despite significant increases with the percentages
for 4th and 8thgrade students (Perie et al., 2005).
According to Perie et al. (2005), 8thgrade students have significantly increased
their performance on the NAEP mathematics assessment, which ranged from 0 to 500,
from 1973 (266) to 2004 (281); however, 12thgrade performance has remained stagnant
from 1973 (304) to 2004 (307). After further examination, 12thgrade students who fell
in the 50th percentile and lower significantly increased their mathematics performance,
but the students in the 75th and 90th percentiles did not significantly improve their
performance (from 325 to 330 and from 345 to 345, respectively).
Proposed Strategy
Goal #1
To increase the mathematical proficiency of secondary students.
Objectives (Outcome)
1. To increase mathematical proficiency levels across implementation years and
across mathematics courses based on benchmark examinations.
2. To increase AP calculus scores across implementation years.
3. To increase graduation exit examination mathematics subtest scores across
implementation years.
121
Goal #2
To increase the mathematical problemsolving ability of secondary students.
Objectives (Outcome)
1. To increase mathematical problemsolving abilities across implementation years
and across grade levels.
Goal #3
To increase the interest in engineering fields.
Objectives (Outcome)
1. To increase the number of students who intend to major in engineering fields as
they enter postsecondary institutions.
2. To increase the number of students who are admitted to a school of engineering.
3. To increase the number of students who graduate with a bachelor?s degree in
engineering.
Method
Participants
The selected participants will be all high school students over a 4year period
within the school district selected as the implementation site. The school district, with a
total enrollment of 12,000, includes three high schools (Grades 9 through 12) with an
approximate enrollment of 3,490. The number of students increases an average of 2%
each academic year. The gender classification is 48% male and 52% female. The racial
makeup of the district is 54% White, 41% Black, and 5% who classify themselves as
belonging to other racial groups. Eight percent of the students receive special education
services. Fiftynine percent of the students are eligible for free or reduced meals.
122
Intervention Activities
Description: Curriculum. The geometry curriculum consists of six units: (a) land
and water navigation, (b) horticulture/landscape design, (c) bridge building, (d) adaptive
devices, (e) simple and complex machines, and (f) friction. The navigation unit covers the
geometric concepts related to triangles and parallel lines. The horticulture unit covers the
properties and theorems associated with circles. In the bridge building unit, the content
includes threedimensional shapes. The adaptive devices unit covers symmetry and
transformations. For the simple and complex machines unit, the content includes Euclid?s
axioms and postulates. The friction unit focuses on the geometric concept of surface area.
The algebra II curriculum consists of five units: (a) thermodynamics, (b) viral
diseases, (c) HVAC systems, (d) cellular respiration, and (e) pipeline design. The
thermodynamics unit covers addition of functions, inequalities, and transformation of
functions. The viral diseases unit covers linear functions, systems of equations, and tree
diagrams. The HVAC systems unit includes area and volume. For the cellular respiration
unit, the content includes additive growth, multiplicative growth, and exponential
equations. The pipeline design unit focuses on the geometric concepts of slope and rate of
change.
The precalculus/trigonometry curriculum consists of seven units: (a) business
plan, (b) electrical circuits, (c) wave motion, (d) aeronautical navigation, (e) optics, (f)
introduction to statistics, and (g) dynamic systems. The business plan unit covers
logarithms, bases, and logarithmic functions. The electrical circuits unit covers the
properties and applications of polynomials. In the wave motion unit, the content includes
the trigonometric functions. The aeronautical navigation unit covers coordinate systems
123
and vectors. The optics unit focuses on analytic geometry. In the introduction to statistics
unit, the content includes the binomial theorem. The dynamic systems unit covers change
with discrete dynamical systems, including constant, linear, and proportional change.
The AP calculus curriculum consists of five units: (a) water supply, (b) market
growth, (c) amusement park design, (d) rocket design, and (e) loglinear analysis. The
water supply unit covers local linearity. The market growth unit covers functions and
limits. The amusement park design unit includes the derivative and applications of
differentiation. For the rocket design unit, the content includes the integral and
applications of integration. The loglinear analysis unit focuses on transcendental
functions. (See Appendix C for specific geometry curriculum unit outlines, Appendix D
for algebra II curriculum unit outlines, Appendix E for precalculus/trigonometry
curriculum unit outlines, and Appendix F for AP calculus curriculum unit outlines.)
Description: Workshops. During the year prior to implementation, the evaluator
and teachers will use the curriculum units to develop instructional lessons and incorporate
applicable lessons from their previous lesson materials. At each professional
development workshop where lesson plans were developed, a lesson plan design rating
system will be conducted (Appendix G). This rating system was adapted for this
application using the Inside the Classroom: Observation and Analytic Protocol (Horizon
Research, 2000). A team of three teachers who were not involved in the development of
the lesson plan will review the lesson?s design and content independently. Based on their
evaluations and recommendations, the lesson plan will be revised or submitted to the
curriculum unit.
124
At subsequent professional development workshops, the evaluator will work with
the high school teachers to develop two benchmark examinations, midterm and final
examinations, for the geometry, algebra II, precalculus/trigonometry, and AP calculus
courses. The examinations will contain items that will be representative of the
expectation instructional content for that time period. As a summative evaluation, a
benchmark examination, which will be a multiplechoice format, will be given every 18
weeks to assess mathematical proficiency based on course content and performance
standards.
In addition to the lesson plans and benchmark examinations, the high school
teacher staff will create the mathematical problemsolving examination, which will be
administered at the end of each course. The items for the mathematical problemsolving
examination will be written, peer reviewed, fieldtested, and data reviewed prior to
placement on the final form. The examination will consist of four tasks (one each from
statistics and probability; algebraic relationships; measurement; and geometry).
High school mathematics teachers will score the examinations after attending 2
days of training. At the training workshops, the evaluators will work on the four sample
tasks at their grade level. When a consensus is reached among the scoring panel, these
criteria responses will be used to illustrate the scoring guide and the variety of possible
solutions for each task during training and scoring. After further training with the criteria
papers, each rater will qualify to score the examinations by accurately scoring a packet of
examinations. (See Appendix H for the scoring rubric.)
At the conclusion of each professional development workshop, all participants
will complete an exit survey to determine the effectiveness of the session and determine
125
future professional development needs. (See Appendix I.) The exit survey was developed
using a variety of preexisting instruments. Questions regarding instructional and student
assessment methods were devised from the National Survey of Science and Mathematics
Education (Westat, 2000). The areas of future professional development needs were
based on the Local Systemic Change: Principal Questionnaire (Horizon Research, 2006).
The items, which relate the importance for the skill to student success in mathematics,
were collected from the Mathematics Teacher Questionnaire: Main Survey (TIMSS
Study Center, 1998).
Procedure: Implementation. The geometry curriculum will be implemented
during Year 1 at each high school in the school district. The teachers will be provided
with curriculum materials and instruction during August at teacher preservice meetings.
One hundred eighty lessons from the Mathematics Curriculum for Advanced
Mathematical Proficiency will be taught in 55minute sessions from August to May.
When students are absent, they will receive makeup lessons. Each teacher will document
that the lesson was taught in his or her daily lesson plan book. These daily lesson plans
will be turned into the school administrative team for review.
Beginning in Year 2, the geometry curriculum will continue with the next
freshman class, and the algebra II curriculum will be implemented. In Year 3, the
geometry and algebra II curricula will continue, and the precalculus/trigonometry
curriculum will be implemented. During Year 4, utilization of the geometry, algebra II,
and precalculus/trigonometry curricula will continue, and the AP calculus curriculum
will be implemented.
126
Procedure: Data collection. For each implementation year, the school
administrative staff will gather the graduation exit examination mathematics subtest and
AP calculus examination scores. In addition, the guidance office staff will collect the
number of students who intended to major in engineering, the number of students who
were admitted to a school of engineering, and the number of students who earned a
bachelor?s degree in an engineering field by contacting the former students.
Process Evaluation
Reach. Reach is the extent to which the targeted populations received the
scheduled intervention dosages. The students? participation in the curriculum activities
will be assessed using the teachers? daily attendance record.
Dosage. The geometry curriculum will consist of 180 lessons at 55 minutes each.
The algebra II curriculum will consist of 180 lessons at 55 minutes each. The pre
calculus/trigonometry curriculum will consist of 180 lessons at 55 minutes each. The AP
calculus curriculum will consist of 180 lessons at 55 minutes each. The lessons will be
taught once a day for 36 weeks during the scheduled mathematics class time block.
Fidelity. With the weekly informal observation forms, school personnel will
monitor the implementation process in the classroom. (See Appendix J.) One of the
following people will conduct these observations: school principal, assistant principal,
curriculum director, or assistant curriculum director. A professional development
workshop will be conducted to train the observers with the weekly informal observation
form and behavioral checklist. Sample videos of classroom instruction will be utilized
during the training workshop. After direct instruction and guided practice, independent
practice will occur until the interrater reliability among the observers is consistent.
127
Formative evaluation. A formative evaluation will be conducted to assess the
attitudes and instructional methods of the teachers throughout the implementation
process. A demographic survey will collect information regarding education level,
certification areas, and years of experience in public education. Qualitative interviews
with the implementing teachers and other faculty members will ascertain their
perceptions and gather feedback for program improvements. The series of interviews will
be conducted during preplanning, midterm, end of the course, and postplanning. Since
adults are more likely to reject the new knowledge that contradicts their beliefs, the
information gathered during these interviews will evaluate existing knowledge, beliefs,
and motivations and will determine the extent to which the implementing teachers have
ownership in the curriculum implementation process (Klingner, Ahwee, Pilonieta, &
Menendez, 2003).
Outcome Evaluation
Participants
Comparison. During the academic year prior to implementation, the students who
are enrolled in geometry, which will be primarily ninth and tenthgrade students, will be
assessed using the two benchmark examinations and the mathematical problemsolving
examination. These grade ahead comparisons will serve as baseline data for which to
compare the mathematical proficiency and mathematical problemsolving ability of the
implementation group. In addition, baseline information will be collected from the
school?s administrative staff regarding the scores from AP calculus examinations and the
scores from the graduation exit examination mathematics subtest. Baseline information
will be collected regarding the number of students during Year 0 who plan to major in
128
engineering and the number of previous students who earned a bachelor?s degree in an
engineering field.
Intervention. Beginning with the first year of implementation, the students who
are enrolled in geometry will be assessed using the two benchmark examinations and the
mathematical problemsolving examination. In the second year of implementation, the
students who are enrolled in algebra II will be assessed using the benchmark and
mathematical problemsolving examinations. During the third year, the students who are
enrolled in precalculus/trigonometry will complete the prescribed assessments and the
graduation exit examination mathematics subtest. Lastly, in the fourth year of
implementation, the students who are enrolled in AP calculus will complete the
benchmark examinations, the mathematical problemsolving examination, and the AP
calculus examination.
Design
Objective 1.1. With the scores from the midterm and final benchmark
examinations, a 4 X 2 analysis of variance (ANOVA) will be conducted to determine if
mathematical proficiency levels changed across implementation years and across
mathematics courses. An intervention sample of students who begin the geometry
calculus sequence in Year 1 will be tracked through Year 4 to assess mathematical
proficiency with the comparison group. In addition, an intervention sample of students
who begin the geometrycalculus sequence in Year 2, in Year 3, and in Year 4 will be
compared with the comparison sample of students. With a profile analysis, the repeated
measure analysis will determine group differences and longitudinal trends between the
intervention and comparison groups.
129
Objectives 1.2 and 1.3. To analyze the longterm outcomes for the Mathematics
Curriculum for Advanced Mathematical Proficiency, with the scores from the AP
calculus examinations and the scores from the graduation exit examination mathematics
subtests, longitudinal trends will be graphed using the percentage of passing scores and
the average score with both examinations across the implementation years.
Objective 2.1. After the initial descriptives are assessed, a repeated measure
ANOVA with one withinsubject factor (time) and two betweensubject factors (group
and grade level) will be conducted to determine if mathematical problemsolving abilities
have changed across implementation years and across grade level and group. An
intervention sample of students who begin the geometrycalculus sequence in Year 1, in
Year 2, Year 3, and Year 4 will be compared to a comparison sample of students.
Objectives 3.1, 3.2, and 3.3. A frequency count of the number of students who
intend to major in engineering at high school graduation, the number of students who
were admitted to a school of engineering, and the number of students who earn a
bachelor?s degree in an engineering field will be assessed. Based on these frequency
counts, a chisquare nonparametric analysis will be conducted to determine if the
observed numbers differ from the expected numbers across implementation years.
Expected Findings
The expected findings will include increased mathematical proficiency and
increased mathematical problemsolving ability as the curriculum was implemented. The
evaluator will expect to see increased scores on the graduation exit examination
mathematics subtest and on the AP calculus examination. In addition, the evaluator will
expect the numbers of students who pursue engineering majors, numbers of students who
130
are admitted to engineering, and numbers of students who graduate with a bachelor?s
degree in an engineering field to increase.
The findings of the program evaluation plan will be reported to the school faculty
each semester as a formative report and each preservice faculty meeting during the
implementation period as a summative report. Once a semester, the evaluator will meet
with the superintendent individually and with the local school board during a caucus
meeting to discuss the results. Afterwards, an annual summative report will be presented
at a public school board meeting.
131
REFERENCES
Anthony, J. M., Hagedoorn, A. H., & Motlagh, B. S. (2001). Innovative approaches for
teaching calculus to engineering students. Paper presented at the ASEE Annual
Conference, Albuquerque, NM.
Baron, J., & Norman, M. F. (1992). SATs, achievement tests, and highschool class rank
as predictors of college performance. Educational and Psychological
Measurement, 52, 10471055.
BesterfieldSacre, M., Atman, C. J., & Shuman, L. J. (1997). Characteristics of freshman
engineering students: Models for determining student attrition in engineering.
Journal of Engineering Education, 86, 139148.
Betz, N. E. (1987). Use of discriminant analysis in counseling psychology research.
Journal of Counseling Psychology, 34, 393403.
Bidwell, J. K., & Clason, R. G. (Eds.). (1970). Readings in the history of mathematics
education. Washington, DC: National Council of Teachers of Mathematics.
Blumner, H. N., & Richards, H. C. (1997). Study habits and academic achievement of
engineering students. Journal of Engineering Education, 86, 125132.
Brown, N. W. (1994). Cognitive, interest, and personality variables predicting first
semester GPA. Psychological Reports, 74, 605606.
Brown, N. W., & Cross, Jr., E. J. (1993). Retention in engineering and personality.
Educational and Psychological Measurement, 53, 661671.
132
Buechler, D. N. (2004). Mathematical background versus success in electrical
engineering. Paper presented at the ASEE Annual Conference, Salt Lake City,
UT.
Burtner, J. (2004). Criticaltoquality factors associated with engineering student
persistence: The influence of freshman attitudes. Paper presented at the Frontiers
In Engineering Conference, Savannah, GA.
Burtner, J. (2005). The use of discriminant analysis to investigate the influence of non
cognitive factors on engineering school persistence. Journal of Engineering
Education, 94, 335338.
Burton, M. B. (1989). The effect of prior calculus experience on ?introductory? college
calculus. The American Mathematical Monthly, 96, 350354.
Campbell, C. M., Hombo, C. M., & Mazzeo, J. (2000). NAEP 1999 trends in academic
progress: Three decades of student performance. Washington, DC: U.S.
Department of Education, Office of Education Research and Improvement,
National Center for Education Statistics. Retrieved January 20, 2005 from
http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2000469.
College Board. (2007). Calculus: Calculus AB calculus BC course descriptions. New
York: Advanced Placement Program National Office. Retrieved May 14, 2007
from http://www.collegeboard.com/prod_downloads/ap/students/calculus/apcd
calc0708.pdf.
Commission on the Reorganization of Secondary Education. (1920). The problem of
mathematics in secondary education. Washington DC: Department of the Interior,
Bureau of Education. In J. K. Bidwell, & R. G. Clason (Eds.) (1970), Readings in
133
the history of mathematics education. Washington, DC: National Council of
Teachers of Mathematics.
Devens, P. E., & Walker, T. D. (2001). Freshman engineering student success indicators.
Paper presented at the ASEE Annual Conference, Albuquerque, NM.
Edge, O. P., & Friedberg, S. H. (1984). Factors affecting achievement in the first course
in calculus. Journal of Experimental Education, 52, 136140.
Felder, R. M., Forrest, K. D., BakerWard, L., Dietz, E. J., & Mohr, P. H. (1993). A
longitudinal study of engineering student performance and retention: I. Success
and failure in the introductory course. Journal of Engineering Education, 82, 15
21.
Ferguson, G. A., & Takane, Y. (1989). Statistical analysis in psychology and education
(6th ed.). New York: McGrawHill Publishing Company.
French, B. F., Immekus, J. C., & Oakes, W. C. (2005). An examination of indicators of
engineering students? success and persistence. Journal of Engineering Education,
94, 419425.
Gainen, J. (1995). Barrier to success in quantitative gatekeeper course. In J. Gainen, & E.
W. Willemsen (Eds.), Fostering student success in quantitative gateway course.
New Directions for Teaching and Learning, 61, 13.
Gainen, J., & Willemsen, E. W. (Eds.). (1995). Fostering student success in quantitative
gateway course. New Directions for Teaching and Learning, 61, 13.
Gilbert, A. C. F. (1960). Predicting graduation from an engineering school. Journal of
Psychological Studies, 11, 229231.
134
Grigg, W., Donahue, P., & Dion, G. (2007). The nation?s report card: 12thgrade
reading and mathematics 2005. Washington, DC: U.S. Department of Education,
National Center for Education Statistics (NCES 2007468).
Hair, Jr., J. F., Black, W. C., Babin, B. J., Anderson, R. E., & Tatham, R. L. (2006).
Multivariate data analysis (6th ed.). Upper Saddle River, NJ: Pearson Education,
Inc.
Halpin, G., & Halpin, G. (1996). College freshman survey: Engineering Form. Auburn
University, AL: Auburn University.
Harackiewicz, J. M., Barron, K. E., Tauer, J. M., & Elliott, A. J. (2002). Predicting
success in college: A longitudinal study of achievement goals and ability
measures as predictors of interest and performance from freshman year through
graduation. Journal of Educational Psychology, 94, 562575.
Hawkins, E. F., Stancavage, F. B., & Dossey, J. A. (1998). School policies and practices
affecting instruction in mathematics: Findings from the national assessment of
educational progress. Washington, DC: U.S. Department of Education, Office of
Educational Research and Improvement (NCES 1998495).
Heinze, L. R., Gregory, J. M., & Rivera, J. (2003). Math readiness: The implications for
engineering majors. Paper presented at the Frontiers In Engineering Conference,
Boulder, CO.
Henson, K. T. (2001). Curriculum planning: Integrating multiculturalism,
constructivism, and education reform (2nd ed.). Long Grove, IL: Waveland Press.
Horizon Research, Inc. (2000). Inside the classroom: Observation and analytic protocol.
Chapel Hill, NC: Horizon Research, Inc.
135
Horizon Research, Inc. (2006). Local systemic change: Principal questionnaire. Chapel
Hill, NC: Horizon Research, Inc.
House, J. D. (1993). Achievementrelated expectancies, academic selfconcept, and
mathematics performance of academically underprepared adolescent students. The
Journal of Genetic Psychology, 154, 6171.
House, J. D. (1995a). Noncognitive predictors of achievement in introductory college
chemistry. Research in Higher Education, 36, 473490.
House, J. D. (1995b). Noncognitive predictors of achievement in introductory college
mathematics. Journal of College Student Development, 36, 171181.
House, J. D. (1995c). The predictive relationship between academic selfconcept,
achievement expectancies, and grade performance in college calculus:
Replications and refinements. The Journal of Social Psychology, 135, 111112.
House, J. D. (2000). Academic background and selfbeliefs as predictors of student grade
performance in science, engineering, and mathematics. International Journal of
Instructional Media, 27, 207220.
House, J. D., Keely, E. L., & Hurst, R. S. (1996). Relationship between learner attitudes,
prior achievement, and performance in a general education course: A
multiinstitutional study. International Journal of Instructional Medial, 23, 257
271.
Jones, P. S., & Coxford, Jr., A. F. (1970). Part one: Mathematics in the evolving schools.
In National Council of Teachers of Mathematics, A history of mathematics
education in the United States and Canada: Thirtysecond yearbook. Washington
DC: National Council of Teachers of Mathematics.
136
Kaestle, C. E. (1983). Pillars of the republic: Common schools and American society,
17801860. New York: Hill and Wang.
Kimmel, H., & Rockland, R. (2002). Incorporation of preengineering lessons into
secondary science classrooms. Paper presented at the Frontiers In Engineering
Conference, Boston, MA.
Klein, D. (2003). A brief history of American K12 mathematics education in the 20th
century. In J. M. Royer (Ed.), Mathematical cognition (pp. 175259). Greenwich,
CT: Information Age Publishing.
Klingbeil, N. W., Mercer, R.E., Rattan, K. S., Raymer, M. L., & Reynolds, D. B. (2005).
The WSU model for engineering mathematics education. Paper presented at the
ASEE Annual Conference, Portland, OR.
Klingner, J. K., Ahwee, S., Pilonieta, P., & Menendez, R. (2003). Barriers and facilitators
in scaling up researchbased practices. Exceptional Children, 69, 411429.
Kuncel, N. R., Cred?, M., & Thomas, L. L. (2005). The validity of selfreported grade
point averages, class ranks, and test scores: A metaanalysis and review of the
literature. Review of Educational Research, 75, 6382.
Lackey, L. W., Lackey, W. J., Grady, H. M., & Davis, M. T. (2003). Efficacy of using a
single, nontechnical variable to predict the academic success of freshmen
engineering students. Journal of Engineering Education, 92, 4148.
LeBold, W. K., & Ward, S. K. (1988). Engineering retention: National and institutional
perspectives. Paper presented at the ASEE Annual Conference, Portland, OR.
137
Lei, P., & Wu, Q. (2007). An NCME instructional module on introduction to structural
equation modeling: Issues and practical considerations. Educational
Measurement: Issues and Practice, 26, 3343.
Litz, C. E. (1975). Horace Mann and the sectarian controversy. Education, 95, 280286.
Litzinger, T. A., Wise, J. C., & Lee, S. H. (2005). Selfdirected learning readiness among
engineering undergraduate students. Journal of Engineering Education, 94, 215
221.
Lunenberg, F. C., & Ornstein, A. C. (2004). Educational administration: Concepts and
practices (4th ed.). Belmont, CA: Wadsworth/Thomson Learning.
Mann, W. R. (1976). Some disquieting effects of calculus in high school. The High
School Journal, 59, 237239.
Mayer, R. E. (2003).Mathematical problem solving. In J. M. Royer (Ed.), Mathematical
cognition (pp. 6992). Greenwich, CT: Information Age Publishing.
Meyers, L. S., Gamst, G., & Guarino, A. J. (2006). Applied multivariate research: Design
and interpretation. Thousand Oaks, CA: Sage Publications.
Mitchell, J. H., Hawkins, E. F., Jakwerth, P. M., Stancavage, F. B., & Dossey, J. A.
(1999). Student work and teacher practices in mathematics. Washington, DC:
U.S. Department of Education, Office of Educational Research and Improvement
(NCES 1999453).
Mitchell, J. H., Hawkins, E. F., Stancavage, F. B., & Dossey, J. A. (1999). Estimation
skills, mathematicsincontext, and advanced skills in mathematics: Results from
three studies of the national assessment of educational progress 1996
138
mathematics assessment. Washington, DC: U.S. Department of Education, Office
of Educational Research and Improvement (NCES 2000451).
MollerWong, C., & Eide, A. (1997). An engineering student retention study. Journal of
Engineering Education, 86, 715.
Murtaugh, P. A., Burns, L. D., & Schuster, J. (1999). Predicting the retention of
university students. Research in Higher Education, 40, 355371.
National Academy of Engineering. (2005). Educating the engineer of 2020: Adapting
engineering education to the new century. Washington DC: National Academies
Press.
National Commission on Excellence in Education. (1983) A nation at risk. Washington,
DC: U.S. Department of Education. Retrieved April 26, 2007 from
http://www.ed.gov/pubs/NatAtRisk/intro.html.
National Committee on Mathematical Requirements. (1923). The reorganization of
mathematics in secondary education. Mathematical Association of America. In J.
K. Bidwell, & R. G. Clason (Eds.) (1970), Readings in the history of mathematics
education. Washington, DC: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Science Board. (2004). Science and engineering education indicators 2004.
Arlington, VA: National Science Foundation.
National Science Board. (2006a). America?s pressing challenge ? Building a stronger
foundation: A companion to science and engineering indicators 2006. Arlington,
VA: National Science Foundation.
139
National Science Board. (2006b). Science and engineering education indicators 2006.
Arlington, VA: National Science Foundation.
Nixon, C. T., & Frost, A. G. (1990). The study habits and attitudes inventory and its
implications for students? success. Psychological Reports, 66, 10751085.
No Child Left Behind Act, 20 U.S.C. ? 6301 et seq. (2001). Retrieved January 20, 2005
from http://www.ed.gov/policy/elsec/leg/esea02/index.html.
Noble, J. P., Roberts, W. L., & Sawyer, R. L. (2006). Student achievement, behavior,
perceptions, and other factors affecting ACT scores. Iowa City, IA: ACT, Inc.
(ACT Research Report Series 20061).
Pedhazur, E. J. (1997). Multiple regression in behavioral research: Explanation and
prediction (3rd ed.). Samford, CT: Thomson Learning, Inc.
Perie, M., Moran, R., Lutkus, A. D., & Tirre, W. (2005). NAEP 2004 trends in academic
progress: Three decades of student performance in reading and mathematics.
Washington, DC: U.S. Department of Education, Institute of Education Science,
National Center for Education Statistics (NCES 2005464).
Peterson, N. S., Kolen, M. J., & Hoover, H. D. (1989). Scaling, norming, and equating. In
R. L. Linn (Ed.), Educational measurement (3rd ed.) (pp. 221262). New York:
Macmillan Publishing Company.
Rigby, K., & Harrell, D. (2005). Issues in developing a high school preengineering
program. Paper presented at the Frontiers In Engineering Conference,
Indianapolis, IN.
Sadler, P. M., & Tai, R. H. (2001). Success in introductory college physics: The role of
high school preparation. Science Education, 85, 111136.
140
Saylor, J. G., Alexander, W. M., & Lewis, A. J. (1981). Curriculum planning for better
teaching and learning (4th ed.). New York: Holt, Rinehart, and Winston.
Schwartz, S., Regan, T., & Marshall, D. (1997). A preengineering high school course in
engineering design. Paper presented at the Frontiers in Engineering Conference,
Pittsburgh, PA.
Shaughnessy, M. F., Spray, K., Moore, J., & Siegel, C. (1995). Prediction of success in
college calculus: Personality, scholastic aptitude test, and screening scores.
Psychological Reports, 77, 13601362.
Shettle, C., Roey, S., Mordca, J., Perkins, R., Nord, C., Teodorovic, J., Brown, J., Lyons,
M., Averett, C., & Kastberg, D. (2007). The nation?s report card: America?s high
school graduates. Washington, DC: U.S. Department of Education, National
Center for Education Statistics (NCES 2007467).
Shuman, L., BesterfieldSacre, M., Budny, D., Larpkiattaworn, S., Muogboh, O.,
Provezis, S., & Wolfe, H. (2003). What do we know about our entering students
and how does it impact upon performance? Paper presented at the ASEE Annual
Conference, Nashville, TN.
Smith, R. M., & Schumacher, P. A. (2005). Predicting success for actuarial students in
undergraduate mathematics courses. College Student Journal, 39, 165177.
Spring, J. (2005). The American school: 16422004. New York: McGrawHill.
Thorndike, R. L. (1951). Reliability. In E. F. Linquist (Ed.), Educational measurement.
Washington, DC: American Council on Education.
TIMSS Study Center. (1998). Mathematics teacher questionnaire: Main survey. Chestnut
Hill, MA: Boston College.
141
van Alphen, D. K., & Katz, S. (2001). A study of predictive factors for success in
electrical engineering. Paper presented at the ASEE Annual Conference,
Albuquerque, NM.
Wesley, J. C. (1994). Effects of ability, high school, achievement, and procrastinatory
behavior on college performance. Educational and Psychological Measurement,
54, 404408.
Westat. (2000). National Survey of Science and Mathematics Education. Rockville, MD:
Westat.
Wilhite, P., Windham, B., & Munday, R. (1998). Predictive effects of high school
calculus and other variables on achievement in a firstsemester college calculus
course. College Student Journal, 32, 610615.
Wise, J., Lee, S. H., Litzinger, T. A., Marra, R. M., & Palmer, B. (2001). Measuring
cognitive growth in engineering undergraduates: A longitudinal study. Paper
presented at the ASEE Annual Conference, Albuquerque, NM.
Wulf, W. A., & Fisher, G. M. C. (2002). A makeover for engineering education. Issues in
Science and Technology, 18(3). Retrieved May 7, 2007 from
http://www.issues.org/18.3/p_wulf.html.
Zhang, G., Anderson, T. J., Ohland, M. W., & Thorndyke, B. R. (2004). Identifying
factors influencing engineering student graduation: A longitudinal and cross
institutional study. Journal of Engineering Education, 93, 313320.
142
APPENDICES
143
APPENDIX A
INSTITUTIONAL REVIEW BOARD
144
145
APPENDIX B
LOGIC MODEL
146
Logic Model
147
APPENDIX C
GEOMETRY CURRICULUM UNITS
148
Unit: Land and Water Navigation Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IC1; IC2; IIA1; IIB1; IIB4; IIIA1; IIIA3; IIIB1; IIIB2; IIID1; IIID4; IIID5; VB3; VB5;
VC1; VC2; VC3.
Engineering Connection:
? General Engineering
? Naval Architecture & Marine Engineering
Mathematical Concepts:
? Coordinate planes
? Total, average, and maximum distances in one and two dimensions
? Isosceles and equilateral triangles
? 30?  60? right triangle relationships
? Pythagorean theorem
? Parallel lines, transversals, and corresponding angles
? Scaling
Learning Opportunities:
During this unit, all students will:
? Apply the concepts of coordinates and cardinal directions to navigation.
? Apply the concept of dead reckoning to navigation.
? Contrast relative and absolute locations.
? Explain how to use a map and compass to triangulate two known locations in
order to find an unknown location.
149
? Identify the similarities and differences between navigation on water and land.
? Read, explain, and use compass directions to determine a location on land maps
and nautical charts.
? Use velocity and time to determine location.
? Utilize the formula for distance (distance equals speed multiplied by time).
Key Terms:
? Dead reckoning
? GPS
? Latitude
? Longitude
? Line of position
? Triangulation
? Vectors
Equipment:
? Compass
? GPS
? Land maps
? Nautical charts
? Nautical slide ruler
? Scientific calculator
Performance Assessment:
Develop a grid chart. Chart a course. At 10minute intervals, provide cardinal
directions, latitudinal and longitudinal directions, speed, and distance. Based on your
plotted course, use vectors to determine total distance traveled.
150
Unit: Horticulture: Landscape Design Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IB1; IC2; IIA1; IIB2; IIB4; IIIA1; IIIA3; IIIB1; IIIB2; IIID1; IIID4; IIID5; VB1; VB3;
VB4; VC3.
Engineering Connection:
? Agricultural Engineering
? Environmental Engineering
? Geological Engineering
Mathematical Concepts:
Circles
? Area
? Circumference
? Chords
Learning Opportunities:
During this unit, all students will:
? Describe the specific relationships and properties of arcs and angles that are part
of a circle.
? Detail the rationale for plant selections and other landscaping decisions.
? Explain that lines and secants, which are part of a circle, have relationships and
properties and how to use these relationships and properties to determine the
length of line segments and congruent segments.
151
Key Terms:
? Center
? Chord
? Circle
? Circumference
? Congruence
? Diameter
? Major arcs
? Minor arcs
? Radius
? Secant
? Tangent
Equipment:
? Scientific calculator
Performance Assessment:
Given a parcel of land with an area of 4,800 square feet, design a landscape.
Include rationale for selection of shapes, turfs, plants, and other structures. Projected
budget and scale drawing should also be included.
152
Unit: Bridge Building Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IA1; IB1; IC2; IIA1; IIB1; IIB2; IIB4; IIIA1; IIIB1; IIIB2; IIID1; IIID2; IIID4; IIID5;
IVB4; VA3; VA6; VB3; VB5; VC3; VD1.
Engineering Connection:
Civil Engineering
Mathematical Concepts:
Threedimensional shapes
? Spheres
? Cylinders
? Cones
? Cubes
? Pyramids
Learning Opportunities:
During this unit, all students will:
? Apply the concept of structural design and testing.
? Determine efficiency rating and critical load.
? Explain the impact of construction materials on critical load.
Key Terms:
? Critical load
? Edge
? Efficiency rating
? Face
? Vertex
153
Equipment:
? Scientific calculator
Performance Assessment:
Design a twodimensional sketch for a bridge. Denote the intended purpose,
construction materials, and dimensions. Construct a threedimensional model to scale
from the sketch. Determine efficiency rating and critical load for your structure.
154
Unit: Adaptive Devices Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IA1; IB1; IC2; IIA1; IIB1; IIB2; IIB4; IIIA1; IIIB1; IIIB2; IIIC1; IIIC2; IIID1; IIID2;
IIID4; IIID5; IVB4; VA6; VB3; VB5; VC3; VD1.
Engineering Connection:
? Bioengineering
? Industrial Engineering
? Manufacturing Engineering
? Mechanical Engineering
Mathematical Concepts:
? Symmetry
? Transformations
Learning Opportunities:
During this unit, all students will:
? Describe the symmetry of geometric figures with respect to point, line, and
plane.
? Explain the difference between adaptive and assistive devices.
Key Terms:
? Dilation
? Enlargement
? Isometry
? Projection
? Reduction
? Reflection
? Rotation
? Translation
155
Equipment:
? Scientific calculator
Performance Assessment:
Explain a situation that could benefit from an adaptive or assistive device.
Develop a device or improve an existing device for this situation. Include a drawing,
model, materials list, and estimated costs of production.
156
Unit: Simple and Complex Machines Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IA1; IB1; IC2; IIA1; IIB1; IIB4; IIIA1; IIIB1; IIIB2; IIID1; IIID2; IIID3; IIID4; IIID5;
IVB4; VA3; VA6; VB3; VB5; VC3; VD1.
Engineering Connection:
? Industrial Engineering
? Manufacturing Engineering
? Mechanical Engineering
Mathematical Concepts:
? Euclid?s axioms
? Euclid?s postulates
Learning Opportunities:
During this unit, all students will:
? Identify the types of simple machines and their uses.
? Explain how force and distance impact the amount of work.
? Explain how simple machines reduce the amount of work.
? Compare and contrast simple and complex machines.
Key Terms:
? Distance
? Force
? Inclined plane (wedge, screw)
? Lever (first class, second class,
third class)
? Line
? Plane
? Point
157
? Pulley
? Surface
? Wheel and axle
? Work
Equipment:
? Scientific calculator
Performance Assessment:
Describe a problem that could benefit from a simple machine. Design and
construct a simple or complex machine to resolve the problem. Include the calculations,
diagrams, and rationale.
158
Unit: Friction Subject: Geometry Grade Level: 9
NCTM Standardsbased Expectations:
IB1; IC2; IIB2; IIB4; IIIA1; IIID4; IIID5; VA6; VB3; VB5; VC3.
Engineering Connection:
? Architectural Engineering
? Construction Engineering
? Civil Engineering
Mathematical Concepts:
? Surface area
Learning Opportunities:
During this unit, all students will:
? Identify occurrences of friction in real world settings.
? Explain the difference between static and kinetic friction.
? Describe how weight affects friction.
? Identify methods for reduction in friction.
Key Terms:
? Coefficient of friction
? Friction (static and kinetic)
Equipment:
? Scientific calculator
Performance Assessment:
Develop a product that could be affected by friction. Provide a scale diagram to
illustrate your product?s modifications to decrease/increase friction.
159
APPENDIX D
ALGEBRA II CURRICULUM UNITS
160
Unit: Thermodynamics Subject: Algebra II Grade Level: 10
NCTM Standardsbased Expectations:
IB1; IC2; VA1; VA2; VA4; VA6; VB1; VB2; VB3; VB5; VC2; VC3.
Engineering Connection:
? Chemical Engineering
? Electrical Engineering
Mathematical Concepts:
? Addition of functions
? Absoluterate equations
? Inequalities
? Transformation of functions
Learning Opportunities:
During this unit, all students will:
? Describe the types of functions: linear, absolute value, greatest integer, inverse
variation, quadratic, cubic, exponential, logarithmic, and trigonometric.
? Compare and contrast even and odd functions.
? Explain the similarities between solving equalities and inequalities.
? Describe and illustrate thermodynamic states and processes.
? Explain the first and second law of thermodynamics.
161
Key Terms:
? Property of addition
? Property of subtraction
? Property of multiplication
? Property of division.
Equipment:
? Scientific/graphing calculator
Performance Assessment:
Design three cylindrical metal conductors. Include scale drawing, dimensions,
crosssectional area, and materials of construction. The cylinder will be arranged from
left to right between a HTR and LTR. Calculate the temperatures of the two interfaces
and the rate at which heat is conducted into the LTR.
162
Unit: Viral Diseases Subject: Algebra II Grade Level: 10
NCTM Standardsbased Expectations:
IA1; IB1; IC2; IIB2; IIIC2; IVA2; IVA4; IVB1; IVB2; IVB3; IVB4; IVB5; IVD1;
IVD2; IVD3; IVD4; IVD5; VA3; VA4; VA6; VB2; VB3; VB5; VC3; VD1.
Engineering Connection:
Biomedical Engineering
Mathematical Concepts:
? Linear functions and models
? Systems of equations
? Probability distributions
? Tree diagrams
Learning Opportunities:
During this unit, all students will:
? Describe a viral disease and how it replicates itself.
? Explain how the immune system responds to viral diseases.
? Illustrate the four methods of solving systems of equations: graphing, substitute,
additionsubtraction, and multiplication with additionsubtraction.
? Illustrate the four ways of graphing linear equations: plotting points, using
intercepts, and using slopeintercept formula.
? Use the standard form of a linear equation.
163
Key Terms:
? Epidemiology
? Host
? Virus
Equipment:
? Scientific/graphing calculator
Performance Assessment:
Track a virus (actual or hypothetical). Develop a plan for decreasing the spread
of the viral disease. Include target population and detailed action plan.
164
Unit: HVAC Systems Subject: Algebra II Grade Level: 10
NCTM Standardsbased Expectations:
IC2; IIA1; IIB1; IIB2; IIB4; IIIA1; IIIC1; IIIC2; IIID1; IIID4; IIID5; VA1; VA2; VA4;
VA6; VB3; VB5; VC3.
Engineering Connection:
? Architectural Engineering
? Civil Engineering
? Construction Engineering
? Mechanical Engineering
Mathematical Concepts:
? Voronoi diagram
? Area
? Volume
? Weighted averages
? Algorithm
? Polygons
? Heron?s formula
? Pick?s formula
? Reflection
? Iteration
165
Learning Opportunities:
During this unit, all students will:
? Describe the process for determining volume and area of geometric figures.
? Explain the interrelationship between heating, ventilating, and airconditioning.
Key Terms:
? Air quality
? Ductwork
? Heat transfer
? Room air distribution
? Thermodynamics
Equipment:
? Scientific/graphing calculator
Performance Assessment:
Design a structure. Include a drawing to scale with a heating and air
conditioning diagram based on the square footage.
166
Unit: Cellular Respiration Subject: Algebra II Grade Level: 10
NCTM Standardsbased Expectations:
IA1; IA2; IC2; IIB1; IIB4; IVA1; IVA2; IVA3; IVA4; IVA5; IVB1; IVB2; VA1; VA3;
VA5; VA6; VB1; VB3; VB4; VB5; VC2; VC3; VD1.
Engineering Connection:
? Biological Engineering
? Biomedical Engineering
? Chemical Engineering
? Environmental Engineering
Mathematical Concepts:
? Additive growth
? Multiplicative growth
? Linear equations
? Exponential equations
? Recursive equations
? Rate of growth
? Timeseries graphs
? Web diagrams
? Properties of exponents
? Quadratic functions
167
Learning Opportunities:
During this unit, all students will:
? Explain the effect of environmental factors on cellular respiration.
? Explain the methods for solving a quadratic function: factoring, using the
squareroot property, completing the square, and using the quadratic formula.
? Graph a parabola when given a quadratic function.
? Identify the vertex, axis of symmetry, xintercept, and yintercept when given a
quadratic function.
Key Terms:
? Cellular respiration
? Environmental factors
? Fermentation
? Population growth
Equipment:
? Scientific/graphing calculator
Performance Assessment:
Design an experiment to test cellular respiration based on realworld application.
Decide whether to control for environmental factors during the experiment. Use the
scientific method to conduct the experiment. Include observational data, data analysis,
and conclusions.
168
Unit: Pipeline Design Subject: Algebra II Grade Level: 10
NCTM Standardsbased Expectations:
IC2; IIA1; IIB1; IIIC1; IIIC2; IVA3; IVA4; IVB1; IVB2; IVB3; IVB4; IVB5; VA3;
VA6; VB1; VB3; VB4; VB5; VC2; VC3; VD1.
Engineering Connection:
? Civil Engineering
? Industrial Engineering
? Manufacturing Engineering
Mathematical Concepts:
? Rate of change
? Slope
? Piecewise equations
? Linear transformations
? Symmetric difference quotient
? Parametric equations
Learning Opportunities:
During this unit, all students will:
? Describe the environmental influences on the motion of an object.
? Determine slope when given two sets of coordinates.
? Explain that parallel lines have the same slope and perpendicular lines have
negative reciprocal slopes.
169
Key Terms:
? Force
? Motion
? Position
Equipment:
? Scientific/graphing calculator
Performance Assessment:
Given a specific terrain, design a pipeline to transport a golf ball. Include a scale
drawing, calculations for intended transported material, estimated construction costs, and
threedimensional model. Develop an evaluation plan to test and assess your pipeline.
170
APPENDIX E
PRECALCULUS/TRIGONOMETRY UNITS
171
Unit: Business Plan Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IA1; IB1; IC2; IIB1; IIIC2; IIID4; IIID5; IVB4; IVB5; IVD1; IVD2; VA3; VA4; VA5;
VA6; VB3; VB5; VC3; VD1.
Engineering Connection:
Engineering Management
Mathematical Concepts:
Exponential and logarithmic functions
? Exponential functions
? Base e
? Logarithmic scale
? Graphs of logarithmic functions
? Logarithms and bases
? Changing bases
? Logarithmic functions
? Composition and inverse functions
Learning Opportunities:
During this unit, all students will:
? Describe a market analysis using logarithmic functions.
Key Terms:
? Mission
? Products
? Services
? Strategy
172
Equipment:
? Graphing calculator
Performance Assessment:
You have decided to open your own business. The bank will finance you
business, but they want the following items before approving the loan:
1. Business plan, including name of the proposed business, location, product or
service, and target customers)
2. Marketing plan, including an advertisement with the business?s logo and
slogan
3. Budget, including the initial loan amount and expenses
4. Projected growth, including graphics, variables, and calculations
173
Unit: Electrical Circuits Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IA2; IB1; IC2; IIA1; IIID5; VA5; VA6; VB3; VB5; VC3.
Engineering Connection:
Electrical Engineering
Mathematical Concepts:
Polynomials
? Oneterm polynomials
? Polynomials with more than one term
? Cubic equations
? Complex numbers
? Fundamental theorem of algebra
? Rational functions
? Power series
Learning Opportunities:
During this unit, all students will:
? Distinguish between series and parallel circuits and their effect on current flow.
? Explain the properties of complex numbers: closure, commutative, associative,
identity, distributive, additive, inverse, and multiplicative inverse.
174
Key Terms:
? Active clamping circuit
? Active clipping circuit
? Current flow
? Parallel circuit
? Passive clamping circuit
? Passive clipping circuit
? Resistance
? Series circuit
Equipment:
? Graphing calculator
Performance Assessment:
Develop a series or parallel circuit. Based on your design, graph the electrical
input and output. Include a scale drawing of the circuit and computations for your
graphed electrical input and output.
175
Unit: Wave Motion Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IIIA1; IIIA2; IIIA3; IIIA4; IIIB1; IIIB2; IIID1; IIID4; IIID5; VA2; VA3; VA4; VA5;
VA6; VB3; VB4; VB5; VC3.
Engineering Connection:
? Aerospace Engineering
? Ocean Engineering
Mathematical Concepts:
Trigonometric functions
? Periodic functions
? Periods and amplitude
? Sine function
? Cosine function
? Tangent function
? Radians and degrees
? Inverse trigonometric functions
Learning Opportunities:
During this unit, all students will:
? Distinguish between the three trigonometric functions and their inverses.
? Display a sine, cosine, and tangent function graph.
176
Key Terms:
? Amplitude
? Crest
? Period
? Trough
? Wavelength
Equipment:
? Computer/internet access
? Graphing calculator
Performance Assessment:
Design a musical instrument. Graph the sound waves from the instrument using
FlexiMusic software (www.fleximusic.com). Using the graph, provide calculations and
diagrams of the sound waves.
177
Unit: Aeronautical Navigation Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IA3; IB2; IC1; IIA1; IIIA1; IIIA2; IIIA3; IIIA4; IIIB1; IIIB2; IIID2; IIID4; IIID5;
VA6; VB3; VB4; VB5; VC3.
Engineering Connection:
Aerospace Engineering
Mathematical Concepts:
Coordinate systems and vectors
? Polar coordinates
? De Moivre?s theorem
? Vectors
? Vector and parametric equations in two dimensions
? Vector equations in three dimensions
? Matrices
Learning Opportunities:
During this unit, all students will:
? Be able to convert rectangular coordinates to polar coordinates.
? Identify the similarities and differences between navigation on air, water, and on
land.
? Read, explain, and use compass directions to determine a location on
aeronautical charts.
178
Key Terms:
? GPS
? Latitude
? Longitude
? Line of position
? Triangulation
? Vectors
Equipment:
? Graphing calculator
Performance Assessment:
Design a transportation vehicle. Include scale drawing, estimated construction
costs, computations, and actual prototype of the vehicle. The aerodynamics will be
assessed using a wind tunnel.
179
Unit: Optics Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IB1; IC2; IIIA1; IIIA4; IIIB1; IIIB2; IIIC1; IIIC2; IIID1; IIID4; IIID5; VA6; VB3;
VB5; VC3.
Engineering Connection:
? Aerospace Engineering
? Mechanical Engineering
Mathematical Concepts:
Analytic geometry
? Circles
? Parabolas
? Ellipse
? Hyperbola
? Reflection property
Learning Opportunities:
During this unit, all students will:
? Illustrate Fermat?s principle (law of reflection).
? Distinguish between plane and spherical mirrors.
? Discriminate the terms of reflection and refraction.
? Describe how lenses and magnifying instruments transform light rays.
180
Key Terms:
? Concave
? Convex
? Diffuse surface
? Geometric optics
? Light rays
? Mirror
? Optical axis
? Specular surface
Equipment:
? Graphing calculator
Performance Assessment:
Design a spherical mirror (concave or convex). Use the law of reflection to
calculate the geometric relationships with the optical axis and spherical mirror. Include
a scale drawing with thickness and computations.
181
Unit: Introduction to Statistics Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IA1; IB1; IB3; IC1; IC2; IIIB1; IIIB2; IIID1; IIID4; IIID5; IVA1; IVA2; IVA3; IVA4;
IVA5; IVB1; IVB2; IVB3; IVB4; IVB5; IVC1; IVC2; IVC3; IVC4; IVD1; IVD2;
IVD3; IVD4; VA1; VA2; VA3; VA6; VB1; VB2; VB3; VB4; VB5; VC2; VC3; VD1.
Engineering Connection:
General Engineering
Mathematical Concepts:
Counting and the binomial theorem
? Basic multiplication principle
? Addition principle
? Binomial theorem
Learning Opportunities:
During this unit, all students will:
? Explain the process of hypothesis testing
? Describe the strengths and limitations of different research designs
? Identify applications of a wide variety of statistical procedures
? Make accurate interpretations of statistical findings
Key Terms:
? Analysis of variance
? Frequency distribution
? General linear model
? Measures of central tendency
? Measures of variability
? Parameter
182
? Probability
? Regression
? Residuals
? Statistics
? Sampling
? Trend analysis
Equipment:
? Graphing calculator
Performance Assessment:
Create a dataset with at least three variables. Based on the measurement of the
data, conduct a statistical analysis to determine statistical significance. The analysis
results should include the formulas, calculations, and interpretations.
183
Unit: Dynamic Systems Subject: PreCalculus/Trigonometry Grade Level: 11
NCTM Standardsbased Expectations:
IB1; IB3; IC1; IC2; IIB4; IVB4; IVB5; VA3; VB1; VB2; VB3; VB4; VB5; VC1; VC2;
VC3; VD1.
Engineering Connection:
Civil Engineering
Mathematical Concepts:
Change with discrete dynamical systems
? Difference equations
? Constant change
? Linear change
? Proportional change
? Equilibrium values
Learning Opportunities:
During this unit, all students will:
? Distinguish between classic scientific approach and dynamic change approach.
Key Terms:
? Patterns of change
Equipment:
? Graphing calculator
184
Performance Assessment:
Based on a specific area of terrain, design a highway system. Include a
description of the terrain including altitude and dimensions, topographical model, scale
drawing, and estimated construction costs.
185
APPENDIX F
ADVANCED PLACEMENT CALCULUS AB CURRICULUM UNITS
186
Unit: Water Supply Subject: AP Calculus AB Grade Level: 12
NCTM Standardsbased Expectations:
IB1; IB2; IB3; IIIA1; IIIA2; IIIA3; IIIA4; IIIB1; IIIB2; IIID4; IIID5; VA1; VA2; VA3;
VA4; VA5; VB1; VB2; VB3; VB4; VB5; VC1; VC2; VC3; VD1.
Engineering Connection:
? Biosystems Engineering
? Civil Engineering
? Electrical Engineering
? Environmental Engineering
Mathematical Concepts:
Local linearity
? Graphs of basic families of functions
? Graphs of linear equations
? Graphs of algebraic simplifications
Learning Opportunities:
During this unit, all students will:
? Describe the basic families of functions.
? Determine whether an equation is linear or nonlinear.
187
Key Terms:
? Function
? Linear equation
? Slope
? yintercept
Equipment:
? Graphing calculator
Performance Assessment:
Consider a hypothetical city or town. Describe the population demographics and
the municipality?s location (i.e., geographic size and proximity to resources). Develop a
water supply system for the hypothetical municipality. Include a scale drawing from the
top and side views and estimated construction costs.
188
Unit: Market Growth Subject: AP Calculus AB Grade Level: 12
NCTM Standardsbased Expectations
IA2; IA4; IB1; IB2; IB3; IIIA1; IIIA2; IIIA3; IIIA4; IIIB1; IIIB2; IIID4; IIID5; IVB1;
IVB5; VA1; VA2; VA3; VA4; VA5; VB1; VB2; VB3; VB4; VB5; VC1; VC2; VC3;
VD1.
Engineering Connection:
Chemical Engineering
Mathematical Concepts:
Functions and Limits
? Function notation, domain, and range
? Odd and even functions definitions and graphical properties
? Limit notation, including right and lefthand limits
? Asymptotic behavior of rational and exponential functions using limit notation
? Estimating limits from graphs and from tables of values
? Calculating limits using algebra
? Presentation of a definition of a limit only to show students how a formal
definition addresses the idea of ?closeness? and how it excludes concern for
what occurs when x = a
? Continuity and graphical properties of continuous functions, including
Intermediate Value Theorem and Extreme Value Theorem.
? Investigating functions that are not continuous at x = a
189
Learning Opportunities:
During this unit, all students will:
? Utilize the Gompetz equation to fit data to growth curves.
Key Terms:
? Adaptation
? Asymptotes
? Continuity
? Deceleration
? Discontinuity
? Domain
? Fastgrowth
? Limits
? Saturation
Equipment:
? Graphing calculator
Performance Assessment:
Develop a new product or technological innovation to be introduced in the
market. Create data to illustrate its growth pattern. Use a simple exponential model of
the form 1e
x
and the Gompetz equation using the created data. Compare and contrast
the two methods for graphing growth curves.
190
Unit: Amusement Park Design Subject: AP Calculus AB Grade Level: 12
NCTM Standardsbased Expectations:
IA2, IA4, IB1, IB2, IB3, IIIA1, IIIA2, IIIA3, IIIB1, IIIB2, IIIB3, IIID1, IIID2, IIID3,
IIID4, IIID5, IVB1, IVB2, IVB4, IVB5, VA1, VA2, VA3, VA4, VA5, VA6, VB1,
VB2, VB3, VB4, VB5, VC1, VC2, VC3, VD1.
Engineering Connection:
? Aerospace Engineering
? Mechanical Engineering
Mathematical Concepts:
The derivative and applications of differentiation
? Definition of the derivative
? Instantaneous rate of change as the limiting value of average rate of change
? Investigating functions differentiable at x = a as well as those not differentiable
at x = a algebraically and graphically
? The relationship between the graphs of f and f?
? Relative (local) extrema and the first derivative test
? Absolute extrema in the context of applied problems
? Derivatives of algebraic functions: power, sum, constant multiple, and product
rules
? Derivatives of circular functions
? Composite functions
? Chain rule
191
? Implicit differentiation
? Related rates
? Second derivative: concavity and points of inflection
? Differential equations
? Relationship between differentiability and continuity
? Mean value theorem
Learning Opportunities:
During this unit, all students will:
? Use first and second derivatives to determine motion.
? Compare and contrast different guided motions along spatial curves.
? Graph a derivative function.
Key Terms:
? Derivatives
? Extrema
Equipment:
? Graphing calculator
Performance Assessment:
Design an amusement park ride. Include a scale drawing, estimated construction
costs (labor and material), and threedimensional model.
192
Unit: Rocket Design Subject: AP Calculus AB Grade Level: 12
NCTM Standardsbased Expectations:
IA2, IA4, IB1, IB2, IB3, IIIA1, IIIA2, IIIA3, IIIB1, IIIB2, IIIB3, IIID1, IIID2, IIID3,
IIID4, IIID5, IVB1, IVB2, IVB4, IVB5, VA1, VA2, VA3, VA4, VA5, VA6, VB1,
VB2, VB3, VB4, VB5, VC1, VC2, VC3, VD1.
Engineering Connection:
? Aerospace Engineering
? Industrial Engineering
Mathematical Concepts:
The integral and applications of integration
? Riemann sums (left hand, right hand, and midpoint sums) to approximate the
area of a region bounded by continuous functions
? Evaluating limits of Riemann sums over equal subdivisions to determine area of
regions bounded by polynomial functions on the interval [0, b]
? Definite integral as a limit of Riemann sums
? The fundamental theorem of calculus
? Indefinite integrals: antiderivatives of known functions and using simple
substitutions
? Integration by parts
? Numerical approximations to definite integrals using tables and graphs:
Riemann sums and trapezoidal rule
193
? Using definite integrals whose integrands are velocity functions to show that
accumulating rates of change in distance yields net distance traveled
? Rectilinear motion
? Volumes of known cross sections as limits of Riemann sums, including sums of
discs, washers, cylindrical shells, and other crosssectional slices.
? Average value of a function
? Variable separate differential equations involving simple polynomial and
trigonometric functions
Learning Opportunities:
During this unit, all students will:
? Apply the average velocity method.
? Apply the trapezoidal rule.
? Apply polynomial interpolation.
Key Terms:
? Integral
? Velocity
Equipment:
? Graphing calculator
Performance Assessment:
Design a rocket. Include scale drawing, estimated construction costs, materials
lists, and final product. Conduct 10 time trials and record the data. Determine the
distance, ,s covered by the rocket using the velocity data provided.
194
Unit: Loglinear Analysis Subject: AP Calculus AB Grade Level: 12
NCTM Standardsbased Expectations:
IA2, IA4, IB1, IB2, IB3, IIIA1, IIIA2, IIIA3, IIIA4, IIIB1, IIIB2, IIIB3, IIID1, IIID2,
IIID3, IIID4, IIID5, IVB1, IVB2, IVB4, IVB5, VA1, VA2, VA3, VA4, VA5, VA6,
VB1, VB2, VB3, VB4, VB5, VC1, VC2, VC3, VD1.
Engineering Connection:
General Engineering
Mathematical Concepts:
Transcendental functions
? Definition of natural logarithm
? Properties of natural logarithms
? Logarithmic differentiation
? Inverse functions and their derivatives
? Exponential functions as inverses of logarithmic functions
? Definition of e
? Differentiation and integration involving e
u
, a
u
, and log
a
u
? Exponential growth and decay problems
? Inverse trigonometric functions and their derivatives
195
Learning Opportunities:
During this unit, all students will:
? State the properties of natural logarithms.
? Compare and contrast exponential functions and inverse trigonometric
functions.
Key Terms:
? Categorical data
? e
? Natural logarithm
? Parameters
Equipment:
? Graphing calculator
? Statistical Package for Social Sciences (SPSS) software
Performance Assessment:
Design a longitudinal study. Include participants, data collection procedures,
timeline, measures, and research question(s). Using SPSS software, enter mock data to
illustrate your study. Conduct a loglinear analysis to answer your research questions.
196
APPENDIX G
LESSON PLAN DESIGN RATING SYSTEM
197
Lesson Plan Design Rating System
Title of Lesson Plan: _________________________________________________
Subject: _____________________________ Grade Level: _________________
Directions: Using the following rating scale, rate each of these design elements as to
the extent you observed them in this mathematics lesson.
Design 1 2 3 4
Not at all Minimal Adequate Great
Extent
a. The design of the lesson
incorporated tasks, roles,
and interactions consistent
with investigative
mathematics/science.
null null null null
b. The instructional strategies
and activities used in this
lesson reflected attention
to students? prior
knowledge.
null null null null
c. The instructional strategies
and activities used in this
lesson reflected attention
to students? learning styles.
null null null null
d. The resources available in
this lesson contributed to
its purpose.
null null null null
e. The design of the lesson
encouraged a collaborative
approach to learning
among the students.
null null null null
198
Directions: Using the following rating scale, rate each of these content elements as to
the extent you observed them in this mathematics lesson.
Content 1 2 3 4
Not at all Minimal Adequate Great
Extent
a. The mathematics content
was appropriate for the
developmental level of the
intended students.
null null null null
b. Mathematics was
portrayed as a dynamic
body of knowledge.
null null null null
c. Appropriate connections
were made to other areas
of mathematics.
null null null null
d. Appropriate connections
were made to other
disciplines.
null null null null
e. The content had realworld
applications.
null null null null
Directions: Based on your review of this lesson, rate the design of this lesson.
1 2 3 4 5
The design did
not reflect best
practices in
investigative
mathematics.
The design
reflected best
practices in
investigative
mathematics.
null null null null null
Directions: Please respond to the following prompts in the space provided.
1. Give two suggestions for improving this lesson.
________________________________________________________________
________________________________________________________________
2. Provide any additional comments about this lesson.
________________________________________________________________
________________________________________________________________
199
APPENDIX H
SAMPLE MATHEMATICAL PROBLEMSOLVING EXAMINATION
AND SCORING RUBRIC
200
Sample Mathematical ProblemSolving Examination
Directions: Read each of the following prompts. Use your conceptual and procedural
knowledge of mathematics to complete each task. Show all of your work for each step.
1. Your grandmother?s birthday is next month. She would love to have a birdhouse
as a birthday present.
Tasks:
1. Design a birdhouse using a scale drawing.
2. Develop a materials list to build your design.
3. Create a budget for all needed materials.
Sample Mathematical ProblemSolving Examination Scoring Rubric
Domain 1 2 3 4 5
Concepts
Inappropriate
application of
mathematical
concepts was
used.
Minimal
evidence of
concept
application was
shown.
Part of the
response
showed
application of a
mathematical
concept.
Solid
application of
one
mathematical
concept was
shown.
Integration of
two or more
mathematical
concepts was
used.
Procedures
Procedure was
unclear or
ineffective.
Lacked use of
appropriate
procedures.
Procedure was
under
developed.
Procedure was
evident in
partial.
Rote skills or
partial use of
procedures.
Procedure was
clear and
evidence
showed
execution.
Procedure was
complex,
systematic, and
supported.
Full use of
appropriate
procedures.
Accuracy
Answer was
correct.
Response
contained
errors/
mistakes.
Answer was
incorrect.
Communication Ineffective Minimal
Partial
development
Full
development
but contained
unclear
explanations.
Full
development
and clear
explanations
that used
pictures,
symbols,
and/or
vocabulary.
201
APPENDIX I
PROFESSIONAL DEVELOPMENT EXIT SURVEY
202
Professional Development Exit Survey
Subject(s) that you are teaching: _______________________________________
Date: ______________ Location: _________________________
Session: __________________________________________________
Directions: Please rate the listed items using the rating scale.
1 2 3 4
Unsatisfactory Fair Satisfactory Excellent
a. Format of the session.
null null null null
b. Content provided during
the session.
null null null null
c. Usefulness/Application
of the activities to your
classroom instruction.
null null null null
d. Quality of lunch
provided at the session.
null null null null
e. Location (facilities) of
the session.
null null null null
f. Overall professional
development experience.
null null null null
Directions: Please respond to the following prompts in the space provided.
1. List two outstanding components of this session.
________________________________________________________________
________________________________________________________________
2. Give two suggestions for improving this session.
________________________________________________________________
________________________________________________________________
3. Provide any additional comments about this session.
________________________________________________________________
________________________________________________________________
203
Directions: Rate your level of need for the following professional development topics
from 1, which indicates the most need, to 7, which indicates the least need.
______ Deepening my own content knowledge of mathematics.
______ Deepening my own content knowledge of performance standards.
______ Learning how to assess student learning.
______ Learning how to teach students with diverse learning needs.
______ Learning how to use inquiry/investigationoriented teaching strategies.
______ Learning how to use technology in instruction.
______ Understanding studentthinking abilities.
Directions: Using the following rating scale, rate each of these skills regarding how
important you feel they are for a student to be successful in mathematics.
1 2 3 4
Not
Important
Somewhat
Important
Important
Very
Important
a. Remembering
formulas and
procedures.
null null null null
b. Thinking in a
sequential and
procedural manner.
null null null null
c. Understanding
mathematical
concepts, principles,
and strategies.
null null null null
d. Being able to think
creatively.
null null null null
e. Understanding how
mathematics is used
in the real world.
null null null null
f. Being able to
provide reasons to
support their
solutions.
null null null null
Directions: Use the following rating scale to indicate the extent to which you are
familiar with the National Council of Teachers of Mathematics Standards.
1 2 3 4
Not at All Minimal Adequate Great Extent
null null null null
204
Directions: Read each of the following prompts and place a check mark beside the
appropriate items to indicate your response. These prompts require more than one
response; therefore, check all that apply.
1. During the last week, which of the following instructional activities have occurred with
your students?
Demonstration and guided practice sessions
null
Listened and took notes during lecture presentation by teacher.
null
Watched a demonstration.
null
Worked in groups.
null
Took field trips.
null
Used mathematical problemsolving skills.
Independent practice
null
Answered textbook or worksheet questions.
null
Made formal presentations to the rest of the class.
null
Read from a textbook in class.
null
Read other (nontextbook) subjectrelated materials in class.
null
Recorded, represented, and/or analyzed data.
null
Wrote reflections (e.g., in a journal).
Lab activities
null
Completed handson/laboratory activities or investigations.
null
Designed or implemented independent investigation.
null
Followed specific instructions in an activity or investigation.
null
Participated in fieldwork.
null
Prepared written lab reports.
null
Worked on extended investigations or longterm projects (i.e., a week or more in
duration).
Assistive Technology
null
Used computers as a tool (e.g., spreadsheets, data analysis).
null
Used calculators or computers for learning or practicing skills.
null
Used calculators or computers to develop conceptual understanding.
null
Watched audiovisual presentations (e.g., videotapes, CDROMs, television, or
films).
205
2. During the last week, which of the following student assessment activities have occurred
in your classroom?
Summative
null
Gave predominantly shortanswer tests (e.g., multiplechoice, true/false, fill in the
blank).
null
Gave tests requiring openended responses (e.g., descriptions, explanations).
null
Graded student work on openended tasks using defined criteria (e.g., a scoring
rubric).
null
Reviewed student homework.
null
Reviewed student portfolios.
Formative
null
Asked students questions during large group discussions.
null
Conducted a preassessment to determine what students already know.
null
Had students present their work to the class.
null
Had students assess each other (peer evaluation).
null
Observed students and asked questions as they worked individually.
null
Observed students and asked questions as they worked in small groups.
null
Reviewed student notebooks/journals.
null
Used assessments embedded in class activities (e.g., informal assessments).
Thank you for your feedback.
Your opinion is taken into consideration
when planning future professional development sessions.
206
APPENDIX J
WEEKLY INFORMAL OBSERVATION FORM
207
Weekly Observational/Implementation Monitoring Checklist
Subject: ____________________________ Teacher: ________________________
Date: _________ Time: Start _____ End ______ Location: _________________
Lesson Topic/Objective(s): ______________________________________________
Directions: Based on the time spent in the classroom and using the following
continuum, describe the focus of this lesson:
1 2 3 4 5
Algorithms,
Facts, and
Vocabulary
100%
75/25
50/50
25/75
Mathematics/
Science
Concepts
100%
null null null null null
Directions: Using the following rating scale, rate each of these teacher/lesson
indicators as to the extent you observed them during the mathematics lesson.
1 2 3 4
Not at all Minimal Adequate Great
Extent
a. The instructional strategies
were consistent with
investigative
mathematics/science.
null null null null
b. The teacher appeared
confident in his/her ability to
teach mathematics.
null null null null
c. The pace of the lesson was
appropriate for the
developmental level of the
students.
null null null null
d. The teacher?s questioning
strategies were likely to
enhance the students?
problemsolving ability.
null null null null
e. Elements of abstract
mathematics were included
when appropriate.
null null null null
f. Appropriate connections
were made with realworld
contexts.
null null null null
208
Directions: Using the following rating scale, rate each of these student indicators as
to the extent you observed them during the mathematics lesson.
1 2 3 4
Not at all Minimal Adequate Great
Extent
a. Students were actively
engaged with important
ideas, which were relevant to
the focus of the lesson.
null null null null
b. Interactions between students
reflected cooperative learning
relationships.
null null null null
c. The climate of the lesson
encouraged students to
generate ideas, hypothesis,
and questions.
null null null null
Directions: For each of the following behaviors, place a check mark beside the item if the
behavior was included in the observed lesson.
Demonstration and guided practice sessions
null Listened and took notes during lecture presentation by teacher.
null Made formal presentations to the rest of the class.
null Recorded, represented, and/or analyzed data.
null Used manipulatives.
null Watched a demonstration.
null Wrote reflections (e.g., in a journal).
Lab activities
null Completed handson/laboratory activities or investigations.
null Designed or implemented independent investigation.
null Followed specific instructions in an activity or investigation.
null Prepared written lab reports.
Assistive Technology
null Used computers as a tool (e.g., spreadsheets, data analysis).
null Used calculators or computers for learning or practicing skills.
null Used calculators or computers to develop conceptual understanding.
null Watched audiovisual presentations (e.g., videotapes, CDROMs, television, or films).
209
APPENDIX K
NCTM STANDARDSBASED EXPECTATIONS
210
NCTM Expectations
(NCTM, 2000)
I. Number and Operations
A. Understand numbers, ways of representing numbers, relationships among
numbers, and number systems.
In Grades 9 ? 12, all students should:
? Develop a deeper understanding of very large and very small numbers and of
various representations of them (IA1);
? Compare and contrast the properties of numbers and number systems,
including the rational and real numbers, and understand complex numbers as
solutions to quadratic equations that do not have real solutions (IA2);
? Understand vectors and matrices as systems that have some of the properties
of the realnumber system (IA3);
? Use numbertheory arguments to justify relationships involving whole
numbers (IA4).
B. Understand meanings of operations and how they relate to one another.
In Grades 9 ? 12, all students should:
? Judge the effects of such operations as multiplication, division, and
computing powers and roots on the magnitudes of quantities (IB1);
? Develop an understanding of properties of, and representations for, the
addition and multiplication of vectors and matrices (IB2);
? Develop an understanding of permutations and combinations as counting
techniques (IB3).
211
C. Compute fluently and make reasonable estimates.
In Grades 9 ? 12, all students should:
? Develop fluency in operations with real numbers, vectors, and matrices,
using mental computations or paperandpencil computations for simple
cases and technology for morecomplicated cases (IC1);
? Judge the reasonableness of numerical computations and their results (IC2).
II. Measurement
A. Understand measurable attributes of objects and the units, systems, and processes
of measurement.
In Grades 9 ? 12, all students should:
? Make decisions about units and scales that are appropriate for problem
situations involving measurement (IIA1).
B. Apply appropriate techniques, tools, and formulas to determine measurements.
In Grades 9 ? 12, all students should:
? Analyze precision, accuracy, and approximate error in measurement
situations (IIB1);
? Understand and use formulas for the area, surface area, and volume of
geometric figures, including cones, spheres, and cylinders (IIB2);
? Apply informal concepts of successive approximation, upper and lower
bounds, and limit in measurement situations (IIB3);
? Use unit analysis to check measurement computations (IIB4).
212
III. Geometry
A. Analyze characteristics and properties of two and threedimensional geometric
shapes and develop mathematical arguments about geometric relationships.
In Grades 9 ? 12, all students should:
? Analyze properties and determine attributes of two and threedimensional
objects (IIIA1).
? Explore relationships (including congruence and similarity) among classes
of two and threedimensional geometric objects, make and test conjectures
and them, and solve problems involving them (IIIA2);
? Establish the validity of geometric conjectures using deduction, prove
theorems, and critique arguments made by others (IIIA3);
? Use trigonometric relationships to determine lengths and angle measures
(IIIA4).
B. Specify locations and describe spatial relationships using coordinate geometry
and other representational systems.
In Grades 9 ? 12, all students should:
? Use Cartesian coordinates and other coordinate systems, such as
navigational, polar, or spherical systems, to analyze geometric situations
(IIIB1);
? Investigate conjectures and solve problems involving two and three
dimensional objects represented with Cartesian coordinates (IIIB2).
213
C. Apply transformations and use symmetry to analyze mathematical situations.
In Grades 9 ? 12, all students should:
? Understand and represent translations, reflections, rotations, and dilations of
objects in the plane by using sketches, coordinates, vectors, function
notation, and matrices (IIIC1);
? Use various representations to help understand the effects of simple
transformations and their compositions (IIIC2).
D. Use visualizations, spatial reasoning, and geometric modeling to solve problems.
In Grades 9 ? 12, all students should:
? Draw and construct representations of two and three dimensional
geometric objects using a variety of tools (IIID1);
? Visualize threedimensional objects from different perspectives and analyze
their cross sections (IIID2);
? Use vertexedge graphs to model and solve problems (IIID3);
? Use geometric models to gain insights into, and answer questions in, others
of mathematics (IIID4);
? Use geometric ideas to solve problems in, and gain insights into, other
disciplines and other areas of interest such as art and architecture (IIID5).
214
IV. Data Analysis and Probability
A. Formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them.
In Grades 9 ? 12, all students should:
? Understand the difference among various kinds of studies and which types
of inferences can legitimately be drawn from each (IVA1);
? Know the characteristics of welldesigned studies, including the role of
randomization in surveys and experiments (IVA2);
? Understand the meaning of measurement data and categorical data, of
univariate and bivariate data, and of the term variable (IVA3);
? Understand histograms, parallel box plots, and scatterplots and use them to
display data (IVA4);
? Compute basic statistics and understand the distinctions between a statistic
and a parameter (IVA5).
B. Select and use appropriate statistical methods to analyze data.
In Grades 9 ? 12, all students should:
? For univariate measurement data, be able to display the distribution, describe
its shape, and select and calculate summary statistics (IVB1);
? For bivariate measurement data, be able to display a scatterplot, describe its
shape, and determine regression coefficients, regression equations, and
correlation coefficients using technological tools (IVB2);
215
? Display and discuss bivariate data where at least one variable is categorical
(IVB3);
? Recognize how linear transformations of univariate data affect shape, center,
and spread (IVB4);
? Identify trends in bivariate data and find functions that model the data or
transform the data so that they can be modeled (IVB5).
C. Develop and evaluate inferences and predictions that are based on data.
In Grades 9 ? 12, all students should:
? Use simulations to explore the variability of sample statistics from a known
population and to construct sampling distributions (IVC1);
? Understand how sample statistics reflect the values of population parameters
and use sampling distributions as the basis for informal inference (IVC2);
? Evaluate published reports that are based on data by examining the design of
the study, the appropriateness of the data analysis, and the validity of
conclusions (IVC3);
? Understand how basic statistical techniques are used to monitor process
characteristics in the workplace (IVC4).
D. Understand and apply basic concepts of probability.
In Grades 9 ? 12, all students should:
? Understand the concepts of sample space and probability distribution and
construct sample spaces and distributions in simple cases (IVD1);
? Use simulations to construct empirical probability distributions (IVD2);
216
? Compute and interpret the expected value of random variables in simple
cases (IVD3);
? Understand the concepts of conditional probability and independent events
(IVD4);
? Understand how to compute the probability of a compound event (IVD5).
V. Algebra
A. Understand patterns, relations, and functions.
In Grades 9 ? 12, all students should:
? Generalize patterns using explicitly defined and recursively defined
functions (VA1).
? Understand relations and functions and select, convert flexibly among, and
use various representations for them (VA2).
? Analyze functions of one variable by investigating rates of change,
intercepts, zeros, asymptotes, and local and global behavior (VA3);
? Understand and perform transformations such as arithmetically combining,
composing, and inverting commonly used functions, using technology to
perform such as operations on morecomplicated symbolic expressions
(VA4);
? Understand and compare the properties of classes of functions, including
exponential, polynomial, rational, logarithmic, and periodic functions
(VA5);
? Interpret representations of functions of two variables (VA6).
217
B. Represent and analyze mathematical situations and structures using algebraic
symbols.
In Grades 9 ? 12, all students should:
? Understand the meaning of equivalent forms of expressions, equations,
inequalities, and relations (VB1);
? Write equivalent forms of equations, inequalities, and systems of equations
and solve them with fluency ? mentally or with paper and pencil in simple
cases and using technology in all cases (VB2);
? Use symbolic algebra to represent and explain mathematical relationships
(VB3);
? Use a variety of symbolic representations, including recursive and
parametric equations, for functions and relations (VB4);
? Judge the meaning, utility, and reasonableness of the results of symbol
manipulations, including those carried out by technology (VB5).
C. Use mathematical models to represent and understand quantitative relationships.
In Grades 9 ? 12, all students should:
? Identify essential quantitative relationships in a situation and determine the
class or classes of functions that might model the relationships (VC1);
? Use symbol expressions, including iterative and recursive forms, to represent
relationships arising from various contexts (VC2);
? Draw reasonable conclusions about a situation being modeled (VC3).
218
D. Analyze change in various contexts.
In Grades 9 ? 12, all students should:
? Approximate and interpret rates of change from graphical and numerical
data (VD1).