Mathematics Momentum to Community College Calculus Success

PhD Dissertation Defense: Mathematics Education

Michael Bostick

University of Wyoming

November 17, 2025

Overview

  • Background
  • Methods
  • Results
  • Discussion

STEM, Calculus and Community Colleges

Need for more STEM aspiring students and diversity in STEM.

STEM Aspiring Students at Community Colleges (CC)

Background

Student Demographics

Mathematics Placement

Mathematics Pathways

Learning Contexts

Pathways to Community College Calculus

Traditional Pathways to Community College Calculus

Traditional CC Pathways to Calculus

  • Strict math placement with test cut-scores.
  • Up to three developmental math courses.

Concerns

  • HS GPA is a better predictor of success than ACT Math or other placement scores (Bahr et al., 2019), but its impact decreases with time since HS (Hayward, 2020).
  • Efficacy of strict math placement test scores are questioned, holistic placement is recommended (Ngo et al., 2018).
  • Long developmental math course sequences have not proven effective (Xu & Dadgar, 2018).

Contemporary Pathways to Community College Calculus

Contemporary CC Pathways to Calculus

  • Math placement with test cut-scores.
  • Holistic placement options with HS GPA/transcript for some students.
  • Up to two developmental math courses.
  • Options to bypass developmental math classes by taking corequisite College Algebra.
  • Have these changes influenced calculus success for students at CCs?

Purpose of This Study

  • Fill the the gap in the research literature by studying how student characteristics, prior academic achievement and learning contexts influence success in CC calculus using a multilevel logistic regression model.

Methods

  • Theoretical Framework
  • Data and Model Selection
  • Multilevel Logistic Regression Model for Calculus Success

Theoretical Framework

Academic Momentum

  • Academic momentum conceptualizes continued academic achievement as a result of past course taking behaviors and academic achievements (Adelman, 2006).

  • This framework has been extended to momentum for CC student success (Wang, 2017) and STEM academic momentum (Wang, 2015; Zhang, 2022).

Academic Momentum to Calculus Success

  • Does mathematical momentum influence student calculus success in a rural CC system?

  • Considerations from this framework relate to academic preparation (HS GPA, ACT Math), enrollment pathways (concurrent enrollment participation and years since HS), student characteristics (demographics and number of math prerequisites), and learning contexts (section mean HS GPA, corequisite course availability).

  • This study investigated if students’ academic preparation by their enrollment pathways (RQ1), and student characteristics by learning contexts (RQ2) can predict success in CC calculus.

Research Questions

Research Question 1 (RQ1)

To what extent does student academic preparation (measured by HS GPA and ACT Math) predict success in CC calculus, and how are these effects moderated by time since high school graduation or participation in concurrent enrollment courses?

Research Question 2 (RQ2)

To what extent do student-level factors including demographics (biological sex and URM group status) and prior coursework (math prerequisite course completions) predict success in CC calculus, and how are these effects moderated by the availability of corequisite courses or by section mean HS GPA?

Research Question 1

Research Question 1 (RQ1)

To what extent does student academic preparation (measured by HS GPA and ACT Math) predict success in CC calculus, and how are these effects moderated by time since high school graduation or participation in concurrent enrollment courses?

RQ1 Model Terms

  • Concurrent prerequisite (cwc) (\(\gamma_{01}\)) - if student took concurrent enrollment prerequisite course, centered within sections (cwc).
  • HS GPA (cwc) (\(\gamma_{10}\)) - high school grade point average (GPA), centered within course section (cwc).
  • ACT Math (cwc) (\(\gamma_{20}\)) - ACT Math score, centered within sections.
  • Years since HS (cwc) (\(\gamma_{30}\)) - number of years since high school (HS), centered within sections (cwc).
  • HS GPA (cwc) × years since HS (cwc) (\(\gamma_{80}\)) - student level interaction of HS GPA (cwc) and years since HS (cwc).
  • ACT Math (cwc) × concurrent prerequisite (cwc) (\(\gamma_{90}\)) - student level interaction of ACT Math (cwc) and concurrent prerequisite (cwc).

Note. cwc = centered within course sections, gmc = grand mean centered.

Research Question 2

Research Question 2 (RQ2)

To what extent do student-level factors including demographics (biological sex and URM group status) and prior coursework (math prerequisite course completions) predict success in CC calculus, and how are these effects moderated by the availability of corequisite courses or by section mean HS GPA?

RQ2 Model Terms

  • HS GPA (gmc) (\(\gamma_{01}\)) - section mean HS GPA, grand mean centered (gmc).
  • URM × HS GPA (gmc) - cross level interaction of URM and HS GPA (cwc).
  • Corequisite college \((\gamma_{02})\) - availability of corequisite courses at college of calculus section.
  • Mathematics prerequisites (cwc) (\(\gamma_{40}\)) - number of math courses taken at college prior to calculus, centered within sections (cwc).
  • Mathematics prerequisites (cwc) × corequisite college \((\gamma_{41})\) - cross level interaction of mathematics prerequisites (cwc) and corequisite college
  • Female \((\gamma_{50})\) - Biological sex, dummy coded 0=Male, 1=Female
  • URM (\(\gamma_{70}\)) - Student is from a Underrepresented Racially Minoritized (URM) group including Black, Latine, Native American.

Note. cwc = centeredwithin sections, gmc = grand mean centered.

Model Selection and Analytical Data Sample

Model Selection

Data Source and Analytical Sample

Sample Demographics

Multilevel Logistic Regression Model

Note. Path diagram to represent multilevel logistic regression model. Bold & italic font = grand mean centered (gmc), bold font = section mean centered (cwc) (Curran & Bauer, 2007). Red coefficients are for RQ1 and blue/green/bold black coefficients are for RQ2.

Multilevel Logistic Regression Model

Model Equation \[\scriptsize {\begin{aligned} & \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\ &\quad = \gamma_{00} + u_{0j} \\ &\quad + \mathbf{\gamma_{01}(\text{HSGPA})_{gmc}} \\ &\quad + \mathbf{\gamma_{02}(\text{Coreq College})} \\ &\quad + \mathbf{\color{red}{\gamma_{10}(\text{HSGPA})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{20}(\text{ACT Math})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{30}(\text{Year since HS})_{cwc}}} \\ &\quad + \mathbf{\color{blue}{\gamma_{40}(\text{Math Prereqs})_{cwc}}} \\ &\quad + \mathbf{\color{blue}{\gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College})}} \\ &\quad + \mathbf{\color{green}{\gamma_{50}(\text{Female})}} \\ &\quad + \mathbf{\color{red}{\gamma_{60}(\text{Concurrent Prereq})_{cwc}}} \\ &\quad + \mathbf{\color{green}{\gamma_{70}(\text{URM})}} \\ &\quad + \mathbf{\color{green}{\gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc}}} \end{aligned}}\]

Note. Multilevel logistic regression path diagram and model equation. The log odds of passing calculus is modeled as a linear combination of terms at the student level, section level, and cross level (Raudenbush & Bryk, 2001). Red coefficients are for RQ1 and blue/green/bold black coefficients are for RQ2.

Multilevel Logistic Regression Model

Model Variables

Model Variables and Specifications
Effect Parameter Level Centering
Intercept and Random Effect \(\gamma_{00}+u_{0j}\) -- --
HSGPA \(\gamma_{01}\) Section gmc
Coreq College \(\gamma_{02}\) Section none
HSGPA \(\color{red}{\gamma_{10}}\) Student cwc
ACT Math \(\color{red}{\gamma_{20}}\) Student cwc
Years since HS \(\color{red}{\gamma_{30}}\) Student cwc
Math Courses \(\color{blue}{\gamma_{40}}\) Student cwc
Math Courses (cwc) × Coreq College \(\color{blue}{\gamma_{41}}\) Cross-Level --
Female \(\color{green}{\gamma_{50}}\) Student none
Concurrent Prereq \(\color{red}{\gamma_{60}}\) Student cwc
URM \(\color{green}{\gamma_{70}}\) Student none
URM × HSGPA (gmc) \(\color{green}{\gamma_{71}}\) Cross-Level --
HSGPA (cwc) × Years since HS (cwc) \(\color{red}{\gamma_{80}}\) Student-Level --
Concurrent Prereq (cwc) × ACT Math (cwc) \(\color{red}{\gamma_{90}}\) Student-Level --

Model Equation \[\scriptsize {\begin{aligned} & \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\ &\quad = \gamma_{00}+ u_{0j} \\ &\quad + \mathbf{\gamma_{01}(\text{HSGPA})_{gmc}} \\ &\quad + \mathbf{\gamma_{02}(\text{Coreq College})} \\ &\quad + \mathbf{\color{red}{\gamma_{10}(\text{HSGPA})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{20}(\text{ACT Math})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{30}(\text{Year since HS})_{cwc}}} \\ &\quad + \mathbf{\color{blue}{\gamma_{40}(\text{Math Prereqs})_{cwc}}} \\ &\quad + \mathbf{\color{blue}{\gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College})}} \\ &\quad + \mathbf{\color{green}{\gamma_{50}(\text{Female})}} \\ &\quad + \mathbf{\color{red}{\gamma_{60}(\text{Concurrent Prereq})_{cwc}}} \\ &\quad + \mathbf{\color{green}{\gamma_{70}(\text{URM})}} \\ &\quad + \mathbf{\color{green}{\gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc}}} \\ &\quad + \mathbf{\color{red}{\gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc}}} \end{aligned}}\]

Note. The intercept \(\gamma_{00}\) represents the grand mean log-odds of passing, and the random effect \(u_{0j}\) captures section-specific deviations from this grand mean. \(\gamma\) coefficients represent the expected change in log-odds of passing CC calculus for a (unit) change in the variable. Grand mean centered = gmc, centered within sections = cwc. Red coefficients are for RQ1 and blue/green/bold black coefficients are for RQ2.

Results

  • Academic Preparation and Enrollment Pathways (RQ1)
  • Student Characteristics and Learning Contexts (RQ2)

Research Question 1

RQ1 - To what extent does student academic preparation (measured by HS GPA and ACT Math) predict success in CC calculus, and how are these effects moderated by time since high school graduation or participation in concurrent enrollment courses?

Statistically Significant Terms for RQ1:

  • HS GPA (cwc) \(\scriptsize{(γ_{10}=1.608,p<0.001)}\)
  • HS GPA (cwc) x years since HS (cwc) \(\scriptsize{(γ_{80}=-0.798,p<0.001)}\)
  • ACT Math (cwc) x concurrent prerequisite (cwc) \(\scriptsize{(γ_{90}=0.203,p<0.05)}\)

Research Question 1 - Coefficients

\[{ \large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{10}(\text{HSGPA})_{cwc}}} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{20}(\text{ACT Math})_{cwc}}} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{30}(\text{Year since HS})_{cwc}}} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{60}(\text{Concurrent Prereq})_{cwc}}} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc}}} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc}}} \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[\large {\begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{10}(\text{HSGPA})_{cwc}}} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[\large {\begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{10}(\text{HSGPA})_{cwc}}} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{30}(\text{Year since HS})_{cwc}}} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc}}} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[\large {\begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{20}(\text{ACT Math})_{cwc}}} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{60}(\text{Concurrent Prereq})_{cwc}}} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \mathbf{\color{red}{\gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc}}} \\[0.5em] \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

RQ1 - HS GPA (cwc)

  • ACT Math (cwc) (\(\gamma_{20}=0.053, p=0.128\)) was not a statistically significant predictor of success.

  • HS GPA (cwc) (\(\gamma_{10}=1.608, p<0.001\)) was a statistically significant predictor of success with a large positive effect size.

  • For a one-standard deviation increase in HS GPA (0.39 GPA) compared to peers, the odds of passing increased by 87%.

  • The positive effect of HS GPA (cwc) was moderated by the number of years since HS.

Note. Predicted probability of passing calculus for +-2 standard deviations of HS GPA (cwc) (1 SD = 0.393 GPA) holding all other variables constant at mean values. 95% confidence interval error bar for the predicted value of pass is shaded blue.

RQ1 - HS GPA (cwc) x years since HS (cwc)

  • The number of years since HS was not statistically significant alone \((\gamma_{30}=-0.055, p<0.559)\), but moderated the effect of HS GPA.
  • The student-level interaction term HS GPA and years since HS \((\gamma_{80}=-0.798, p<0.001)\) was statistically significant with a negative effect.
  • Students who were out of high school two years longer than their peers received no predicted benefit from HS GPA on the odds of passing.

Note. Predicted probability for passing calculus is shown for student level interaction HS GPA (cwc) x years since HS (cwc). HS GPA (cwc) is on the horizontal axis plotted across +-2 standard deviations (1 SD = 0.393 GPA), with slider for +-2 SD for the moderating effect of years since HS (cwc) (1 SD = 1.086 years), holding all other variables at mean values. 95% confidence interval error bars are shaded blue.

RQ1 - ACT Math (cwc) x concurrent enrollment prereq (cwc)

  • While the main effects of ACT Math \((\gamma_{20}=0.053, p=0.128)\) and concurrent enrollment prerequisites \((\gamma_{60}=-0.474, p=0.131)\) were not statistically significant, their interaction was, with a positive effect \((\gamma_{90}=0.203,p<0.05)\).
  • Students who took concurrent enrollment courses receive a positive benefit of a higher ACT Math score in predicting calculus success.
  • For students who took concurrent prerequisite the same or less than their peers, ACT Math had little effect on their odds of success.
  • Students who took more concurrent prerequisite than their peers (+1 SD = 0.307), an increase in ACT Math by 1 SD (3.15) increased their odds of success by 44%.

Note. Predicted probability for passing calculus is shown for student level interaction ACT Math (cwc) x concurrent enrollment prereq (cwc). ACT Math (cwc) is on the horizontal axis plotted across +-2 standard deviations (1 SD = 3.15 ACT), with slider for +-2 SD for the moderating effect of concurrent enrollment prereq (cwc) (1 SD = 0.307), holding all other variables at mean values. 95% confidence interval error bars are shaded blue.

Research Question 2

RQ2 - To what extent do student-level factors including demographics (biological sex and URM group status) and prior coursework (precalculus course completions) predict success in CC calculus, and how are these effects moderated by the availability of corequisite courses or by mean HS GPA within a course section?

Statistically Significant Terms for RQ2:

  • Number of math prerequisites (cwc) \(\scriptsize{(γ_{40}=0.3053,p<0.05)}\)
  • Number of math prerequisites (cwc) × corequisite availability \(\scriptsize{(γ_{41}=-0.5524,p<0.01)}\)
  • Section mean HS GPA (gmc) \(\scriptsize{(γ_{01}=2.9118,p<0.001)}\)
  • Section mean HS GPA (gmc) × URM \(\scriptsize{(γ_{71}=-2.7377,p<0.05)}\)

Research Question 2 - Coefficients

\[{\large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \mathbf{\gamma_{01}(\text{HSGPA})_{gmc}} \\[0.5em] &\quad + \mathbf{\gamma_{02}(\text{Coreq College})} \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \mathbf{\color{blue}{\gamma_{40}(\text{Math Prereqs})_{cwc}}} \\[0.5em] &\quad + \mathbf{\color{blue}{\gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College})}} \\[0.5em] &\quad + \mathbf{\color{green}{\gamma_{50}(\text{Female})}} \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \mathbf{\color{green}{\gamma_{70}(\text{URM})}} \\[0.5em] &\quad + \mathbf{\color{green}{\gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc}}} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \\[0.5em] \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[{\large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \mathbf{\color{blue}{\gamma_{40}(\text{Math Prereqs})_{cwc}}} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[{\large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \gamma_{01}(\text{HSGPA})_{gmc} \\[0.5em] &\quad + \mathbf{\gamma_{02}(\text{Coreq College})} \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \mathbf{\color{blue}{\gamma_{40}(\text{Math Prereqs})_{cwc}}} \\[0.5em] &\quad + \mathbf{\color{blue}{\gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College})}} \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[\large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \mathbf{\gamma_{01}(\text{HSGPA})_{gmc}} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \gamma_{70}(\text{URM}) \\[0.5em] &\quad + \gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

\[\large \begin{aligned} &\quad \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) \\[0.5em] &\quad = \gamma_{00} + u_{0j} \\[0.5em] &\quad + \mathbf{\gamma_{01}(\text{HSGPA})_{gmc}} \\[0.5em] &\quad + \gamma_{02}(\text{Coreq College}) \\[0.5em] &\quad + \gamma_{10}(\text{HSGPA})_{cwc} \\[0.5em] &\quad + \gamma_{20}(\text{ACT Math})_{cwc} \\[0.5em] &\quad + \gamma_{30}(\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{40}(\text{Math Prereqs})_{cwc} \\[0.5em] &\quad + \gamma_{41}(\text{Math Prereqs})_{cwc} \times (\text{Coreq College}) \\[0.5em] &\quad + \gamma_{50}(\text{Female}) \\[0.5em] &\quad + \gamma_{60}(\text{Concurrent Prereq})_{cwc} \\[0.5em] &\quad + \mathbf{\color{green}{\gamma_{70}(\text{URM})}} \\[0.5em] &\quad + \mathbf{\color{green}{\gamma_{71}(\text{URM}) \times (\text{HSGPA})_{gmc}}} \\[0.5em] &\quad + \gamma_{80}(\text{HSGPA})_{cwc} \times (\text{Year since HS})_{cwc} \\[0.5em] &\quad + \gamma_{90}(\text{Concurrent Prereq})_{cwc} \times (\text{ACT Math})_{cwc} \end{aligned}\]

Note. \(^{*}p<0.05\), \(^{**}p<0.01\), \(^{***}p<0.001\),

RQ2 - Number of Math Prerequisites (cwc)

  • Neither Biological sex \((\gamma_{50}=-0.003, p=0.989)\) or URM \((\gamma_{70}=-0.215, p=0.362)\) were statistically significant.
  • The number of math prerequisite courses completed was statistically significant with a positive effect \((\gamma_{40}=0.305, p<0.05)\).
  • Students who had taken one more prerequisite course at the CC than their peers received a 36% increase in their odds of success.
  • The number of math prerequisite courses completed was moderated by the availability of corequisite courses.

Note. Predicted probability of passing calculus for +-2 standard deviations for number of math prerequisites (cwc) (1 SD = 1.03 courses), holding all other variables constant at mean values. 95% confidence interval error bar is shaded blue.

RQ2 - Math Prerequisites (cwc) x Corequsite Availability

  • The college-level context variable corequisite availability was not statistically significant alone \((\gamma_{02} =-0.189,p=0.430)\), but it had a negative moderating effect on the number of prerequisite math courses.
  • The positive effect from number of prerequisite math courses \((\gamma_{40}=0.305, p<0.05)\) was negated for students who attended colleges where corequsite math courses were available \(\gamma_{41}=-0.552, p<0.01)\).
  • The effect on the odds of passing from number of prerequisite math courses was reduced by 42% for students who had corequsite math courses available.

Note. Predicted probability for passing calculus is shown for cross level interaction Math Prerequisites (cwc) x Corequsite Availability. Math Prerequisites (cwc) is on the horizontal axis plotted across +-2 standard deviations (1 SD = 1.03 courses), with button for for the moderating effect of Corequisite Availability, holding all other variables at mean values. 95% confidence interval error bars are shaded blue.

RQ2 - Section Mean HS GPA (gmc)

  • Section Mean HS GPA was statistically significant with a large positive effect \((γ_{01}=2.912, p<0.001)\).
  • Students who enroll in course sections with high mean HS GPA have increased odds of passing calculus.
  • For a 1 standard deviation increase in section mean HS GPA (0.192 GPA), the odds of passing increased by 75%.
  • The positive effect from section mean HS GPA was moderated for students who were URM.

Note. Predicted probability of passing calculus for +-2 standard deviation of section mean HS GPA (gmc) (1 SD = 0.192 GPA), holding all other variables constant at mean values. 95% confidence interval error bar for the predicted value of pass is shaded blue.

RQ2 - HS GPA (gmc) x URM

  • URM status was not a statistically significant predictor of success alone \((\gamma_{70}=-0.215,p=0.362)\).
  • The cross-level interaction variable GPA (gmc) x URM was statistically significant with a negative effect \((\gamma_{71}=-2.734, p<0.05)\).
  • Students who are URM did not share in the positive effect from attending a calculus section with a higher mean HS GPA.
  • For students who are URM, the effect of mean HS GPA on the predictive odds of passing decreased by 94%.

Note. Predicted probability for passing calculus is shown for cross level interaction HS GPA (gmc) x URM. HS GPA (gmc) is on the horizontal axis plotted across +-2 standard deviations (1 SD = 0.192 GPA), with button for for the moderating effect of URM, holding all other variables at mean values. 95% confidence interval error bars are shaded blue.

Discussion

  • Key Results
  • Research Questions
  • Implications
  • Limitations and Directions for Future Research

Summary of Key Findings

  • HS GPA (cwc) was strongest predictor of success, but years since HS (cwc) weakened its effect.

  • ACT Math (cwc) was not a statistically significant predictor of success, but students who took concurrent enrollment courses compared to peers benefited from having a higher ACT score.

  • The number of math prerequisite courses taken (cwc) had a positive effect on success, except for students who had access to corequisite courses.

  • Section mean HS GPA (gmc) was a strong predictor of success, but students who were URM did not receive any benefit from enrollment in a calculus section with a high mean HS GPA.

Academic Preparation and Enrollment Pathways (RQ1)

Predictive Power of HS GPA versus ACT Math

  • HS GPA consistently is a better predictor of academic achievement, including college mathematics success, compared to tests scores (Bahr et al., 2019; Ngo et al., 2018).
  • Combined measures of ACT/SAT Math and HS prior achievement could be best predictor (Sonnert et al., 2020).

Time Since HS Graduation

  • Time since HS was not found to be a statistically significant predictor of calculus success. Prior research has found time since HS as a significant predictor for entry level college precalculus mathematics (Perkins et al., 2023).

  • Time since HS had a statistically significant moderating effect on HS GPA, agreeing with prior studies (Hayward, 2020; Meeter, 2023).

Academic Preparation and Enrollment Pathways (RQ1)

Concurrent Enrollment Pathways

Math Placement for STEM-Aspiring Students

Student Characteristics and Learning Contexts (RQ2)

Student Characteristics

The Importance of Prior CC Coursework with the Existence of Alternatives

  • There was a negative statistically significant moderating effect on the number of math prerequisites for students who attend colleges with corequisite courses available.

  • Prior studies have shown no harm is done in shortening the length of the prerequisite course sequence (Park & Ngo, 2021; Xu & Dadgar, 2018).

  • Corequisite courses have been found to be an effective way to reduce the number of prerequisite mathematics courses (Ran & Lin, 2022; Schudde & Ryu, 2025; Stone-Johnstone, 2023).

Student Characteristics and Learning Contexts (RQ2)

Peer Composition Effects and Equity

Implications

Institutional and Policy Implications

  • Wyoming CCs are making good progress: concurrent enrollment, corequisite courses, increase in alternative placement, reduction in developmental courses.

  • Wyoming CCs all use ACT Math for math placement, only two use HS GPA/Transcripts.

  • Increase diversity of math courses to improve success (Bowman et al., 2023).

  • Studies have shown not all students have equitable access to dual/concurrent enrollment (Xu et al., 2021).

Limitations and Future Research

Questions?

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Appendix

Summary statistics for the dataset used in the model, model result details, diagnostics, and assumption testing are available here:

Multilevel Logistic Regression Model Dashboard