Gendered Pathways: How Mathematics Ability Beliefs Shape Secondary and Postsecondary Course and Degree Field Choices

Do mathematics ability beliefs explain gender gaps in the physical science, engineering, mathematics, and computer science fields (PEMC) and other science fields? We leverage U.S. nationally representative longitudinal data to estimate gendered differences in girls' and boys' perceptions of mathematics ability with the most difficult or challenging material. Our analyses examine the potentially interacting effects of gender and these ability beliefs on students' pathways to scientific careers. Specifically, we study how beliefs about ability with challenging mathematics influence girls' and boys' choices to pursue PEMC degrees, evaluating educational milestones over a 6-year period: advanced science course completion in secondary school and postsecondary major retention and selection. Our findings indicate even at the same levels of observed ability, girls' mathematics ability beliefs under challenge are markedly lower than those of boys. These beliefs matter over time, potentially tripling girls' chances of majoring in PEMC sciences, over and above biological science fields, all else being equal. Implications and potential interventions are discussed.

Major retention: To assess whether students stayed in their intended major two years after high school, we used NCES-generated variables f2b15 and f2major2. F2b15 reflected students' intended major, through their retrospective response (two years after high school) to the question "When you began your post-secondary education, what field of study did you think you were most likely to pursue?" F2major2 recorded students' declared majors two years after high school, as designated below. f2b15 categories and f2major2 categories differed significantly from one another. In particular, one category for f2b15 combined natural sciences and mathematics. We therefore collapsed all of the categories for both f2b15 and m2major2 into two categories: • "PEMC + Bio" (F2B15 categories engineering or engineering technology, computer or information sciences, and natural sciences or mathematics; f2major2 categories biological and biomedical sciences, computer and information sciences or support technology, engineering technologies and technicians, mathematics and statistics, and physical sciences) and • "All Other Majors." Major retention was therefore coded as: • 1 = PEMC and/or Biology major abstainers (All Other Majors  All Other Majors); • 2 = PEMC and/or Biology stayers (PEMC/Bio  PEMC/Bio); • 3 = PEMC and/or Biology leavers (PEMC/Bio  All Other Majors); and • 4 = PEMC and/or Biology newcomers (All Other Majors  PEMC/Bio).

Declared Major:
We used ELS variables f2major2 to indicate major and f2b22 to indicate whether a major was declared. ELS major variables use the Classification of Instructional Program (CIP) 2digit coding scheme. Our categories reflect various National Center for Education Statistics reports on what constitutes STEM and non-STEM majors (e.g., Ginder & Mason, 2011).
• 1 = Undeclared/Not in a Degree Program (f2b22 categories not in a degree program and not yet declared); • 2 = Non-STEM (f2major2 categories area, ethnic, cultural, and gender studies; visual and performing arts; business, management, marketing, and related fields; communication, journalism, and communication technology; construction trades; education, English language, literature and letters; family, consumer, and human sciences; foreign languages, literature, and linguistics; legal professions and studies; mechanical and repair technologies and technicians; multi and interdisciplinary studies; parks, recreation, leisure and fitness studies; precision production; personal and culinary services; philosophy, religion and theology; pubic administration and social services; security and protective services; transportation and materials moving; other; and liberal arts, sciences, general studies, and humanities.); • 3 = PEMC (f2major2 categories computer/info sciences/support tech, engineering technologies/technicians, mathematics and statistics, and physical sciences); • 4 = Biological Sciences (f2major2 category biological and biomedical sciences); • 5 = Health Sciences (f2major2 category health professions/clinical sciences); • 6 = Social/Behavioral and Other Sciences (f2major2 categories agricultural/natural resources/related, architecture and related services, science technologies/technicians, psychology, social sciences). We combined Social/Behavioral and Other STEM sciences because of small n's in the Other STEM categories and an already complex multinomial logistic model.

Primary Independent Variables: Mathematics Ability Beliefs
With respect to the values of each of the ability beliefs studied, the general, verbal, 10 th grade mathematics, and 12 th grade mathematics scales were all originally coded such that 1 indicated the least agreement and 4 indicated the most agreement. The growth mindset measure, bys88a, had an opposite coding structure. Thus, we reversed the growth mindset coding structure for our analysis, such that each of the variables below ranged from 1 (strongly disagree) to 4 (strongly agree).
Because we were interested in the potential interaction of gender with these ability beliefs in their effect on degree field of study, we created cross-product terms for each of the mathematics ability beliefs that seemed influential in our prior study. Factor analysis was used to create the general, verbal, 10 th grade mathematics, and 12 th grade mathematics scales. All variables loaded on each factor with eigenvalues over 2.0. Finally, we generated the gender*perceived ability interaction terms.
• Growth mindset used ELS variable bys88a. This variable measured participants' agreement with the following statement: "Most people can learn to be good at math." The variable was originally coded from 1 to 4 with 1 indicating higher agreement with the statement. We reverse coded the variable to match the coding on the items that made up the perceived ability under challenge measures, which were coded 1 to 4 with 4 indicating higher agreement with the statement. This variable was then standardized to match the perceived ability under challenge measures described below." The following scales measure perceived ability under challenge, in general, verbal, and mathematics domains, and are derived from multiple items, described below. Scales were estimated from the items by domain using factor analysis. All variables loaded on a single factor with a minimum eigenvalue of 1.0. Stata 14's factor analysis function automatically standardizes the values such that the mean = 0 and SD = 1.
• The general scale used the following ELS variables: bys89e, bys89j, bys89o, bys89s, and bys89v. These variables measured participants' agreement with the following statements in the 10 th grade: o "When I sit myself down to learn something really hard, I can learn it." o "When studying, I keep working even if the material is difficult." o "When studying, I try to work as hard as possible." o "When studying, I try to do my best to acquire the knowledge and skills taught." o "When studying, I put forth my best effort." • The verbal scale used the following ELS variables: bys89c, bys89f, and bys89m. These variables measured participants' agreement with the following statements in 10 th grade: o "I'm certain I can understand the most difficult material presented in English texts." o "I'm confident I can understand the most complex material presented by my English teacher." o "I'm certain I can master the skills being taught in my English class." • The 10 th grade mathematics scale used the following ELS variables: bys89b, bys89l, and bys89u. These variables measured participants' agreement with the following statements in 10 th grade: o "I'm certain I can understand the most difficult material presented in math texts." o "I'm confident I can understand the most complex material presented by my math teacher." o "I'm certain I can master the skills being taught in my math class." • The 12 th grade mathematics scale used the following ELS variables: f1s18b, f1s18c, and f1s18e. These variables measured participants' agreement with the following statements in 12 th grade: o "I'm certain I can understand the most difficult material presented in math texts." o "I'm confident I can understand the most complex material presented by my math teacher." "I'm certain I can master the skills being taught in my math class."

Analytic Models
The following models were used to conduct the analyses referenced in the 2015 article, which includes more information on the variables were used in each step of the models. There is only one distinction of note between the 2015 paper and our new analyses: the earlier paper measured 10 th grade ability with respect to performance on the most difficult verbal and mathematics ability test sections; the present paper uses broader and commonly used mathematics and reading ability measures. Otherwise, the variables are consistent. Full models are shown below, and correspond respectively to Tables 4-7 from the 2015 article, with the exception of this substitution in the mathematics and reading score variables. 1 The figures generated in this analysis were based on the full model titled "Declared major field (2006), (women= reference)."

Supplementary Analyses and Figures
To state our findings with clear interpretability, we used Stata 14 commands to generate predicted probabilities for categorical outcomes like major choice. To generate the predicted probabilities explained starting on line 287, about men and women's chances of each major with all other predictors at their mean values, we used the margins command in Stata 14, including its use to derive the average marginal effect increase for men, in interaction with their growth mindset. For the figures that follow and the accompanying discussion (starting around line 293), we used the prtab and prgen from the spost9 legacy suite of user-written commands for use in Stata (Long & Freese, 2014a, 2014b.