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PME-NA 2011 Proceedings 787

PRECALCULUS CONCEPT ASSESSMENT: A PREDICTOR OF AP CALCULUS

AB AND BC SCORES

Rusen Meylani

Arizona State University

rusen.meylani@asu.edu

Dawn Teuscher

Brigham Young University

dawn.teuscher@asu.edu

This study establishes a theoretical framework for predicting the American College Testing

(ACT) Mathematics sub-score and AP Calculus AB and BC scores from the Precalculus

Concept Assessment (PCA) exam results and suggests a total of 16 different regression based

models to actually perform the prediction. The strong positive correlation between the actual

and predicted values confirm that the PCA is a powerful tool for identifying students who are

at the risk of not passing AP Calculus AB or BC tests and thus helping teachers, parents and

students take the necessary measures in a timely manner.

Introduction

Assessments are a major practice in the K-12 educational system as well as post-

secondary. High school students are required to take more and more assessments to

demonstrate what they have learned. Most states require high school students to take either an

end of course (mathematics) exam and/or a graduation exam (with a portion being

mathematics) to complete a course or to graduate from high school (Teuscher, Dingman,

Nevels, & Reys, 2008). In addition, most colleges require students to take a mathematics

placement exam to direct students into the appropriate first year mathematics course. Even

though students are required to send official transcripts and take one of the college entrance

exams (e.g.; ACT, SAT), they are asked to demonstrate their knowledge of mathematics on

multiple assessments.

Math placement exams vary in mathematics content, the number and type of questions

(multiple choice, open-ended, etc.), use of calculators, and time limits. The number of

different mathematics placement exams used by institutions across the country continues to

increase. However, there are only two commonalities among these placement exams (1) the

focus of exam items is on content and procedures taught in remedial mathematics classes,

which satisfy general education requirements or serves as prerequisites such as college

algebra and precalculus; and (2) the results are not used to inform student or teachers of the

possible deficiencies in student knowledge.

This article reports research results on how high school students performed on the AP

Calculus AB or BC exam, the Mathematics portion of the standardized American College

Testing (ACT) and on the Precalculus Concept Assessment (PCA), a research developed

instrument based on college students’ common misconceptions of functions (Carlson,

Oehrtman, & Engelke, 2010) after completing four years of college preparatory mathematics

and AP Calculus.

Theoretical Framework

Precalculus Concept Assessment

The PCA is a 25-item multiple choice exam that helps researchers and instructors learn

what students think and understand about the foundational concepts of precalculus and

beginning calculus (see Carlson et al., 2010, for released items). The PCA was developed

based on research with collegiate level mathematics classes, and was piloted and revised over

the past 15 years.

The PCA is based on a taxonomy developed to determine student’s understandings and

reasoning of foundational concepts learned during precalculus (Carlson et al., 2010).

Although the PCA was not created to be used as a placement exam for Calculus, Carlson et

al. (2010) reported that 77% of college students who scored a 13 or higher on the PCA passed

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 788

a first semester calculus course with a C or better. The correlation coefficient for college

student PCA scores and their calculus grades was 0.47.

The PCA was validated with college students who enroll in College Algebra and

Precalculus. The study reported in this paper provides a different sample of students, those

who are in high school and enrolled in Advanced Placement (AP) Calculus. Students took the

ACT, PCA and the AP Calculus AB or BC exams during their high school years. Although

one might assume that precalculus at the college level is equivalent to precalculus at the high

school level, the PCA had not been used to analyze student thinking at the high school level.

Content of AP Calculus courses and exams

The AP Calculus AB course focuses on limits, derivatives, and an introduction to

integrals, which is typically taught in a first semester calculus course in college. The AP

Calculus BC course focuses on the topics studied in AB; however, more depth is given to

integration and students are introduced to sequences and series as well as parametric and

polar functions, which is typically taught in a two semester calculus sequence in college. The

AP Calculus exams award students with a score of one to five inclusive with five being the

highest score. On each of the AB and BC exams, those students who score four or five pass

the exam, and may use their scores to receive college credit for Calculus courses when they

enter college. It is evident that a strong foundation of precalculus is absolutely essential for

success in Calculus; therefore the PCA can be a valuable tool for identifying the students’

weaknesses in precalculus if administered to students prior to entering Calculus.

Content of ACT mathematics exam

The ACT mathematics exam is a 60-question, 60-minute test designed to measure the

mathematical skills students have typically acquired in courses taken by the end of 11th grade

(ACT, 2011). Students receive an overall score between one and 36 inclusive and three sub-

scores based on six content areas: pre-algebra (23%), elementary algebra (17%), intermediate

algebra (15%), coordinate geometry (15%), plane geometry (23%) and trigonometry (7%).

All of these topics are highly correlated with the content of the PCA and if used with PCA,

can be employed as a diagnostic tool for predicting how students are likely to succeed in the

AP Calculus system.

Regression Analysis

Regression analysis includes techniques for modeling and analysis of several variables,

when attention is focused on the relationship between a dependent variable and one or more

independent variables. More specifically, regression analysis helps understand how the

typical value of the dependent variable changes when any of the independent variables is

varied, while the other independent variables remain fixed. Usually, regression analysis

estimates the conditional expected value of the dependent variable given the independent

variables (i.e. the mean (average) value of the dependent variable when independent variables

are kept fixed). Regression analysis is widely used for estimation and prediction. It is also

used to explore and comprehend the causal relationships that exist among the independent

variables in relation to the dependent variable. In this study, regression analysis is the primary

means of inquiry to explore the relationships between the AP Calculus AB and BC exam

scores, PCA results and ACT mathematics test sub-scores.

Research Questions

In light of the scope of the PCA, the ACT mathematics test as well as the AP Calculus AB

and BC exams, this study specifically seeks to answer the following research questions:

1) How are high school students’ PCA scores and AP Calculus AB or BC scores related?

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 789

2) Can students’ PCA scores predict their performance on the AP Calculus AB or BC

exams?

3) Can the prediction be improved when the ACT scores are available?

Methodology

In this study, the 16 different regression schemas are built upon three regression based

models, namely, Multiple Linear Regression Model, Multinomial Logistic Regression Model

and Cumulative Odds (CO) – Ordinal Regression Model.

Regression Models

Regression models were used in this study to predict students’ AP Calculus AB or BC

scores (i.e. an integer between 1 and 5 inclusive). Students’ AP Calculus AB or BC scores

were the dependent variables and their response to the 25 individual questions in the PCA,

each being a 1 (that represents a correct answer) or a 0 (that represents an incorrect answer)

were the independent variables. . In some of the regression models students’ ACT

mathematics scores were used as a second independent variable.

The Multiple Linear Regression Model. This model assumes that a linear relation exists

between the dependent and the independent variables where the random errors are assumed to

be independent and normally distributed random variables with zero mean and constant

standard deviation, (i.e., assumptions of normality, linearity, and homogeneity of variance are

met). The dependent variable is students’ AP Calculus AB or BC score and the independent

variables are the responses to the 25 PCA questions with or without the ACT mathematics

scores. Tthus, depending on the regression model, there are 25 (without the ACT mathematics

score) or 26 (with the ACT mathematics score) independent variables.

The Multinomial Logistic Regression Model. Multinomial logistic regression does not

require any assumptions of normality, linearity, and homogeneity of variance for the

independent variables (Kutner et al., 2005). Because this regression model is less stringent it

is often preferred to discriminant analysis when the data does not satisfy these assumptions.

Suppose the dependent variable has M nominal (unordered) categories. One value of the

dependent variable is chosen as the reference category and the probability of membership in

each of the other categories is compared to the probability of membership in the reference

category. For the dependent variable with M categories, this requires the calculation of M – 1

equations, one for each category relative to the reference category, in order to describe the

relationship between the dependent and the independent variables. Please note that

multinomial logistic regression model ignores the ordinal nature that might exist within the

levels of the dependent variable and treats each category in a similar manner.

The Cumulative Odds (CO) – Ordinal Logistic Regression Model. The CO – ordinal

regression model calculates the probability of being at or below category m of an ordinal

dependent variable with M categories (Kutner et al., 2005). Ordinal logistic regression is

different from multinomial logistic regression in that it takes into account the ordinal nature

inherent within the levels of the dependent variable, which might be useful in some cases.

For the two logistic regression models (multinomial or CO – ordinal) each of the AP

Calculus AB or BC scores had five levels (i.e. an integer between one and five inclusive). For

multinomial the logistic regression model, the last level (AP Calculus AB or BC score being

equal to 5) was selected as the reference category.

The dependent variables were again the 25 PCA items used as categorical variables

(factors). The ACT mathematics test scores could be used as both categorical and ordinal

variables. When the ACT mathematics test scores were used as categorical variables (factors),

each level inherent within the score was a separate independent variable; when they were

used as ordinal variables (covariates), they constituted a single independent variable.

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 790

Participants

At the end of a school year, 193 high school students from two high schools in a mid-

western town were administered the PCA to assess their understandings and reasoning

abilities prior to entering AP Calculus (Teuscher & Reys, in press). Of the 193 students who

took the PCA and enrolled in AP Calculus AB or BC the following year, 143 students took

the AP Calculus exam at the end of the school year; 80 of these students were enrolled in the

AP Calculus AB course while the remaining 63 were enrolled in AP Calculus BC course.

The AP Calculus exam is scored and then students are given a grade of one to five.

Typically students who receive a four or five grade receive college credit for at least one

semester of calculus. Those students who take the AP Calculus BC exam receive a BC grade

and an AB sub-grade. It is possible that a student who takes the BC exam many not receive a

four or five on the BC exam, but receives a four or five on the AB portion of the test, which

can be interpreted as having passed the calculus AB exam, but not the BC exam.

Prediction Process

The models that used the PCA results to predict the AP Calculus AB of BC scores are

based on the three regression models (Multiple Linear Regression, Multinomial Logistic

regression and the CO – Ordinal Logistic Regression). The predictors in all three regression

models were the actual results of each of the 25 questions in the PCA test, (i.e. each test

question was associated with one of two categorical values, 1 if it was answered correctly and

0 if it was answered incorrectly).

The ACT mathematics score can theoretically take ordinal values between 1 and 36

inclusive (ACT, 2011). In statistics, higher level variables can always be downgraded to

lower level ones, such that, a metric scale variable can be downgraded to an ordinal or a

nominal variable; this process sometimes requires defining categories within the data and/or

creating discrete values based on the continuous scale variables (Kent, 2001). The ACT

mathematics score already takes discrete values, which is reason alone why it can be treated

to be ordinal or nominal as well. While it is theoretically possible for a student to score

between one and 36 inclusive (ACT, 2011), this is usually not the case in practice; for

instance the scores of a group of high school students attending the same school may exhibit

a certain pattern. The scores of the group of students subject to our analyses were between 19

and 34 and none of the students scored 22. This is another justification for the fact that ACT

mathematics score can be treated as an ordinal or categorical variable.

Dependent Variable

Independent Variable(s)

AP Calculus

PCA

AP Calculus

PCA and ACT mathematics sub-scores

Table 1. The variables used for the three linear regression models to predict students’ ACT

mathematics and AP Calculus (AB or BC) scores.

Two different linear regression models used students’ PCA results with or without the

ACT mathematics scores to predict students’ AP Calculus AB or BC scores; when the ACT

mathematics scores were used, they were treated as ordinal metric variables. The variables

used for the linear regression models used in this study are summarized in Table 1.

The Logistic Regression models (Multinomial and Ordinal) predict a categorical or an

ordinal dependent variable using categorical predictors as factors with or without ordinal

variables as covariates. These two models were employed to predict the AP Calculus AB or

BC scores separately using students’ PCA results with or without the ACT mathematics

scores; the ACT mathematics scores were used as categorical variables (predictors) or as

ordinal variables (covariates). The logistic regression models used in this study are

summarized in Table 2.

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 791

Model

Specification Dependent

Variable

Independent Variable(s)

Categorical Variables

(Factors)

Ordinal Variables

(Covariates)

A

AP Calculus AB or

BC Score

PCA

B

AP Calculus AB or

BC Score

PCA Score and

ACT mathematics Scores

C

AP Calculus AB or

BC Score

PCA Score

ACT mathematics

Scores

Table 2. The logistic regression models (multinomial or CO-ordinal) used to predict

students’ AP Calculus AB or BC scores.

Results

As it might be expected, students enrolled in AP Calculus BC scored higher (mean of

17.51 and standard deviation of 3.18) than students in AP Calculus AB (mean of 15.69 and

standard deviation of 3.21) on the PCA (a total score possible was 25) prior to entering AP

Calculus. Eighty-one percent of the students in this study who took one of the AP Calculus

exams pass it with a four or five.

A positive Pearson correlation existed between students’ PCA scores and the AP Calculus

exam grades and it was statistically significant (r = 0.40, p = 0.000). This can be interpreted

as students who receive a high PCA score are likely to receive a high AP Calculus exam

grade. Then again, a positive Pearson correlation existed between students’ PCA scores and

the ACT mathematics test scores and it was statistically significant (r = 0.28, p = 0.02). This

can be interpreted as students who receive a high PCA score and/or a high ACT mathematics

test score are likely to receive a high AP Calculus exam grade. The AP Calculus AB scores

were available for 80 students; the mean score was 4.00 and the standard deviation was 0.95.

Whereas the AP Calculus BC scores were available for 63 students; the mean score was 4.13

and the standard deviation was 0.96.

The AP Calculus AB scores were predicted using the two multiple linear regression

models given in Table 1 and the Pearson correlations were calculated between the actual and

predicted values. The actual values of students’ ACT mathematics scores were also used to

assess whether or not their inclusion while predicting the AP Calculus AB and BC scores

would in fact improve the prediction. The results indicate strong positive correlations and are

summarized in Table 3 which can be interpreted as follows: AP Calculus AB scores can be

predicted with 48% (100 × 0.692 = 48%) accuracy when using students PCA results alone or

75% (100 × 0.872 = 75%) accuracy when using students’ PCA results along with their ACT

mathematics test scores.

Model

AP Calculus AB Scores

Predicted from the PCA

Scores

AP Calculus AB Scores Predicted from

the PCA and Actual ACT mathematics

Test Scores

Pearson Correlation

0.69

0.87

N

80

48

M (SD)

3.74 (0.83)

3.84 (0.91)

Table 3. AP Calculus AB test scores predicted using the multiple linear regression models.

The AP Calculus AB scores were predicted using the three distinct model specifications

for the multinomial logistic regression models given in Table 2 and the Pearson correlations

were calculated between the actual and predicted values. The results indicate strong positive

correlations and are summarized in Table 4. Model specifications B and C yielded perfect

correlations with 100% accuracy in predicting the AP Calculus AB test scores. The results

summarized in Table 4 can be interpreted as follows: AP Calculus AB scores can be

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 792

predicted with 91% (100 × 0.952 = 91%) accuracy when using students’ PCA results alone or

100% (100 × 12 = 100%) accuracy when using students’ PCA results along with their ACT

mathematics scores using the ACT mathematics scores as factors or covariates depending on

the model.

Model Specification

A

B

C

Pearson Correlation

0.95

1.00

1.00

N

80

48

48

M (SD)

3.80 (0.99)

3.99 (0.99)

3.97 (0.95)

Table 4. AP Calculus AB test scores predicted using the multinomial logistic regression

models.

The AP Calculus AB scores were also predicted using the three distinct model

specifications for the CO – ordinal logistic regression models given in Table 2 and the

Pearson correlations were calculated between the actual and predicted values. The results

indicate strong positive correlations and are summarized in Table 5. Model specifications B

yielded a perfect correlation with 100% accuracy in predicting the AP Calculus AB test

scores. The results summarized in Table 5 can be interpreted as follows: AP Calculus AB

scores can be predicted with 40% (100 × 0.632 = 40%) accuracy using the PCA results alone;

with 100% (100 × 12 = 100%) accuracy using both the PCA results and ACT mathematics

scores as factors or with 70% (100 × 0.842 = 70%) accuracy using PCA results and ACT

mathematics scores as covariates.

Model Specification

A

B

C

Pearson Correlation

0.63

1.00

0.84

N

80

48

48

M (SD)

3.76 (1.07)

3.97 (1.06)

3.94 (1.01)

Table 5. AP Calculus AB test scores predicted using the CO – ordinal regression models.

The AP Calculus BC scores were predicted using the two multiple linear regression

models given in Table 1 and the Pearson correlations were calculated between the actual and

predicted values. The results indicate strong positive correlations and are summarized in

Table 6. The results can be interpreted as follows: The AP Calculus BC scores can be

predicted with 57% (100 × 0.752 = 57%) accuracy using the PCA results alone; or 95% (100

× 0.972 = 95%) accuracy using the PCA results along with the ACT mathematics test scores.

Model AP Calculus AB scores

predicted from the PCA scores

AP Calculus AB scores predicted

from the PCA and the actual ACT

mathematics scores

Pearson Correlation

0.75

0.97

N

63

25

M (SD)

4.15 (0.71)

3.92 (0.92)

Table 6. AP Calculus BC test scores predicted using the multiple linear regression models.

The AP Calculus BC scores were predicted using the three distinct model specifications

for the multinomial logistic regression models given in Table 2 and the Pearson correlations

were calculated between the actual and predicted values. The results indicate strong positive

correlations and are summarized in Table 7. Model specifications B and C yielded perfect

correlations with 100% accuracy in predicting the AP Calculus BC test scores. The results

can be interpreted as follows: AP Calculus BC scores can be predicted with 69% (100 × 0.832

= 69%) accuracy using the PCA results alone; or 100% (100 × 12 = 100%) accuracy using the

PCA results along with the ACT mathematics scores which are used as factors or covariates

depending on the model.

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 793

Model Specification

A

B

C

Pearson Correlation

0.83

1.00

1.00

N

63

25

25

M (SD)

4.10 (0.98)

3.89 (0.93)

3.89 (0.93)

Table 7. AP Calculus BC test scores predicted using the multinomial logistic regression

models.

The AP Calculus BC scores were predicted using the three distinct model specifications

for the CO – ordinal logistic regression models given in Table 2 and the Pearson correlations

were calculated between the actual and predicted values. The results indicate strong positive

correlations and are summarized in Table 8. Model specifications B and C yielded perfect

correlations with 100% accuracy in predicting the AP Calculus BC test scores. The results

can be interpreted as follows: The AP Calculus BC scores can be predicted with 49% (100 ×

0.702 = 49%) accuracy using the PCA results alone; or 100% (100 × 12 = 100%) accuracy

using the PCA results along with the ACT mathematics scores as factors or covariates

depending on the model.

Model Specification

A

B

C

Pearson Correlation

0.70

1.00

1.00

N

63

25

25

M (SD)

4.23 (0.94)

3.89 (0.93)

3.89 (0.93)

Table 8. AP Calculus BC test scores predicted using the CO – ordinal regression models.

Please note that each of the 16 regression models summarized above as well as the

Pearson correlation values reported were statistically significant at the 0.01 level.

Discussion

Assessments are the standard for which teacher and institutions judge students’

knowledge. With the No Child Left Behind ACT of 2001 (NCLB, 2001) K-12 students are

taking assessments each year to demonstrate adequate yearly progress. However, the majority

of these exams were not developed with the intention of providing students or teachers with

feedback on deficiencies in student’s knowledge. The college mathematics placement exams

were also not developed with the end goal of assessing relevant and connected concepts that

are foundational for calculus.

The results of this study provide evidence that the PCA may be an exam that could be

used for multiple settings across high schools and colleges in the United States. The PCA was

found to be significantly correlated with the AP Calculus AB and BC exams and

correspondingly students’ PCA scores were a statistically significant predictor of the AP

exam scores. The results verify that multiple linear, multinomial logistic and CO-ordinal

logistic regression models can successfully be used in one or more of these predictions. As

for the generalizability of the results obtained, the mean and standard deviation values

calculated for each of the actual and predicted AP Calculus AB or BC scores were very close

meaning that the results were indeed generalizable.

While predicting the AP Calculus AB and BC scores, using the ACT mathematics scores

as factors or covariates improved the results of the prediction; particularly using the actual

ACT mathematics scores as ordinal variables (or factors) while performing logistic regression

yielded very strong positive and sometimes perfect correlations between the actual and the

predicted values.

These findings are consistent with research reported by Carlson et al. (2010) who found

that the PCA was a predictor of college students ability to receive a passing grade in calculus

at the college level. The PCA was specifically created to provide feedback to instructors on

what their students understand and do not understand about functions. Instructors could use

the results from the PCA to determine what prior knowledge or more importantly what

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.

PME-NA 2011 Proceedings 794

misconceptions students have when entering a course that may cause them to not understand

and grasp the new material they encounter. The PCA could also be used to provide instructors

with diagnostic feedback on the specific precalculus topics that students did not understand

during their precalculus classes and then make modifications to their curriculum for future

classes.

A considerable amount of time and taxpayer money is spent every year on students who

retake calculus in college because they are not able to pass the college mathematics

placement test. There are also a vast number of students who drop out of calculus in college

and change their majors simply because of having the prejudice that they are unable to

succeed in mathematics; in fact this is not a new problem and it has not been solved as of yet

(Ma et.al., 1999). Thus, an early detection system could be part of the solution and be of

assistance to students, parents and teachers to take the necessary measures early (i.e. when

the students are still in high school). This is why a powerful tool like the PCA can be used to

identify students who need to spend more time on precalculus and are likely to have a hard

time in AP Calculus class or college level calculus, by predicting their AP Calculus AB or

BC test scores even before they enter the AP Calculus system.

However, it must be noted that the timing of the PCA test is an important factor to

produce the results which will enable the prediction of the scores on the AP Calculus AB or

BC tests. The PCA test should ideally be administered immediately after completing the

precalculus content courses prior to the students starting the AP Calculus AB or BC courses.

In closing, it is important to realize that without a purpose, assessments will become

something that students do and not something that is useful to them or instructors. The PCA is

a practical focused examination that can provide students and instructors with important

feedback to improve students’ understandings of the common mathematical topics that are

necessary for students to be successful in calculus.

References

ACT. (2011). ACT Test Prep - Math Test Description. Retrieved February 11, 2011 from

http://actstudent.org/testprep/descriptions/mathdescript.html.

Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A

tool for assessing students' reasoning abilities and understandings. Cognition and

Instruction, 28(2), 113-145.

Kent, R. (2001). Data Analysis and Data Construction for Survey Research, 1st Edition,

PALGRAVE, N.Y.

Kutner, M.H., Nachtsheim, C.J., Neter, J., Li, W. (2005). Applied Linear Statistical Models,

5th Edition, McGraw-Hill Irwin.

Ma, X., Willms, J.D. (1999). Dropping out of advanced mathematics: How much do students

and schools contribute to the problem? Educational Evaluation and Policy Analysis,

21(4), 365 - 383.

NCLB. (2001). Public law no. 107-110. Retrieved December 16, 2009, from

http://www.ed.gov/policy/elsec/leg/esea02/index.html.

Peng, C.-Y. J., Lee, K. L., & Ingersoll, G. M. (2002). An introduction to logistic regression

analysis and reporting. The Journal of Educational Research, 96(1), 3-14.

Teuscher, D., Dingman, S. W., Nevels, N., & Reys, B. J. (2008). Curriculum standards,

course requirements, mandated assessments for high school mathematics: A status report

of state policies. Journal of Mathematics Education Leadership, Fall, 50-55.

Teuscher, D., & Reys, R. E. (in press). Rate of change: AP calculus students' understandings

and misconceptions after completing different curricular paths. School Science and

Mathematics.

Wiest, L. R., & Lamberg, T. (Eds.). (2011). Proceedings of the 33rd Annual Meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education. Reno, NV: University of Nevada, Reno.