Beliefs and Belongings in Mathematics
By David Bressoud @dbressoud
As of 2024, new Launchings columns appear on the third Tuesday of the month.
I believe that there is no discipline that elicits such strong negative reactions as mathematics. To admit that one is a mathematician is to invite comments on how difficult and frustrating others have found their experience of mathematics. This is especially true of those from under-resourced educational backgrounds. For those who wish to pursue a career in a STEM field, mathematics is an obstacle to be overcome, not just in terms of what must be learned but also the psychological toll that is taken, as documented so well in Talking about Leaving (Seymour and Hewitt, 1997) and Talking about Leaving Revisited (Seymour and Hunter, 2019).
A large part of the problem is the way mathematics is taught and mathematical proficiency is assessed. When mastery of mathematics is reduced to the ability to recognize which procedure is relevant to a particular problem and then to execute it quickly and flawlessly, it is not surprising that most students find this difficult and uninspiring. When, on top of this, society sends the message that mathematics is the purview of certain favored racial, ethnic, and gender classes, it is no wonder that persistence is low even among those who are fully capable of excelling in mathematics.
For these reasons, it is useful to know what students, especially underrepresented students, believe about mathematics and about their place within a mathematical community. Knowledge of student beliefs and sense of belonging can identify areas that need to be addressed if we are to promote success. It is with the goal of developing such an instrument that a group of mathematicians, mathematics education researchers, and psychologists—with a mechanical engineer thrown in for good measure—developed the College Mathematics Beliefs and Belongings Survey.
The article “The College Mathematics Beliefs and Belonging Survey” (Sidney et al. 2024) recently appeared in the International Journal of Research in Undergraduate Mathematics Education. It documents the process of developing this survey and presents evidence for its validity and reliability.
Figure 1: The correlation matrix for the 15 characteristics.
The survey, which can be found in Appendix B, pages 54–57, consists of fifteen clusters of statements expressing beliefs about mathematics or sense of belonging within a mathematical community, each cluster containing three to five statements. Students are asked to rate each statement on a Likert scale of 1 to 6 from strongly disagree to strongly agree. A neutral position is not possible.
The first set of clusters probes student understanding of the nature of mathematics. Four of these are strands of mathematical proficiency as described in the National Research Council’s publication Adding It Up (Kilpatrick et al., 2001). The fifth cluster in this set is based on Gray and Tall’s (1994) proceptual thinking, the ability to take a process and encapsulate it as a concept. For each cluster, I report the description given in the IJRUME article, and I include the one statement that loads most heavily within this cluster
1. Conceptual reasoning. Connecting and relating concepts across contexts, topics, and representations. “I make connections with previous topics I have learned when learning a new mathematical topic.”
2. Procedural reasoning. Carrying out procedures accurately and efficiently. “I am precise when doing calculation.”
3. Adaptive reasoning. Seeking to understand, explain, and justify. “I justify my solution when I solve a math problem.”
4. Strategic reasoning. Thinking flexibly and solving problems with multiple strategies. “I think about multiple different ways to solve a math problem and choose the best one.”
5. Proceptual reasoning. Simplifying, transforming, and reinterpreting mathematical objects and expressions. “I transform complex math problems to make them easier to solve.”
The second set of clusters looks at motivational beliefs about mathematics.
6. Logic beliefs. Mathematical learning is focused on formalism and abstraction. “Learning mathematics involves understanding abstract patterns and structures.”
7. Conventional beliefs. Mathematics is learning facts and calculating answers efficiently. “Mathematics is about definitions and theorems.”
8. Growth beliefs. Anyone can learn mathematics through productive struggle. “Mistakes are part of learning mathematics.”
9. Utility beliefs. Mathematics is useful and applicable. “Mathematics will be useful to me in the future.”
10. Self-efficacy beliefs. Confidence in one’s own mathematical ability. “I am confident in my ability to do mathematics.”
The final set of clusters looks at sense of belonging.
11. Community belonging. Feelings of inclusion and connection in the mathematical community. “I feel like I am part of the math community.”
12. Individual belonging. Feelings of acceptance and appreciation in a mathematical setting. “When in a math setting, I feel valued.”
13. Classroom belonging. Relating to classmates in a mathematical setting. “I find similarities in the ways my classmates and I solve problems.”
14. Collaborative belonging. Feeling one’s ideas are accepted when working with other to solve mathematical problems. “When working on math problems with others, my ideas and ways of thinking are valued.”
15. Exclusion (negatively correlated). Feelings of exclusion in a mathematical setting. “When in a math setting, I feel excluded.”
The final version of the survey also includes several open-ended questions.
References
Gray, E., & Tall, D. (1994). Duality, Ambiguity, and Flexibility: A “Proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116. https://doi.org/10.2307/749505
Kilpatrick, J., Swafford, J. O., & Findell, B. R. (Eds.). (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. https://nap.nationalacademies.org/catalog/9822/adding-it-up-helping-children-learn-mathematics
Seymour E. and Hewitt, N. M. (1997). Talking about Leaving: Why Undergraduates Leave the Sciences. Boulder, CO: Westview Press.
Seymour, E. and Hunter, A.-B., Editors. (2019) Talking about Leaving Revisited: Persistence, Relocation, and Loss in Undergraduate STEM Education. Cham, Switzerland: Springer Nature. https://link.springer.com/book/10.1007/978-3-030-25304-2
Sidney, P., Braun, B., Jong, C. et al. (2024). The College Mathematics Beliefs and Belonging Survey: Instrument Development and Validation. Int. J. Res. Undergrad. Math. Ed. (2024). https://doi.org/10.1007/s40753-024-00247-1
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org