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Equivalence is critically important across mathematical domains. In particular, the equivalence of expressions and equations is important to understanding and justifying algebraic transformation, and to interpreting the meaning of solution sets. Yet little is known about how college students conceptualize what it means for expressions and equations to be equivalent. In this study we analyze 198 student responses to open-ended items asking students about the meaning of equivalence, both generally in mathematics, and in the context of expressions and equations specifically. Data was collected in 18 different courses, ranging from elementary to linear algebra. Reflexive thematic analysis was used to generate themes to describe overarching patterns in the data. At the same time, narrative analysis was employed to better understand student conceptions in more rich and nuanced ways that explored how different codes might interact and describe more complex ways of thinking. Analysis revealed substantial ambiguity around terminology, including for the terms “expression” and “equation”, “solving”, and “same answer”. This was often related to ambiguity, inconsistency, or non-normative framings of certain key conceptions related to equivalence, including: 1) conceptions of expressions vs. equations as objects, including how their equivalence criteria differ; and 2) conceptions of numerical equality vs. non-numerical equivalence relationships. These various themes were interrelated in important and interesting ways that reveal a critical gap in our understanding of how existing instruction impacts students’ conception formation in the context of equivalence of expressions and equations. The majority of students demonstrated ambiguity or inconsistency in treating expressions and equations as distinct objects with distinct equivalence criteria. Students also overwhelmingly provided numerical equality definitions and examples, even in contexts (e.g., equivalent equations) where standard equivalence criteria describe non-numerical equivalence relationships. This points to a critical need to modify existing algebra instruction to better support conception formation around the equivalence of algebraic expressions and equations.
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Is equivalence just “Same Number”? College students’ numerical vs non-numerical conceptions of equivalence.
As coursework extends beyond arithmetic, students encounter forms of equivalence beyond numerical equality. This study examines how college students characterize equivalence in mathematics generally, and specifically for equations and expressions. Responses from 198 students were analyzed and coded as numerical or non-numerical. Across questions, most responses appealed to numerical criteria, treating equivalence as equality to a single number, with higher prevalence of non-numerical criteria correlating with higher-level courses. We observed multiple variants of both numerical and non-numerical criteria, including converting algebraic expressions to numbers to justify equivalence. Findings highlight the need for educators to support students in distinguishing types of equivalence and their associated criteria.
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- Award ID(s):
- 1760491
- PAR ID:
- 10673477
- Editor(s):
- Cook, S; Katz, B P; Melhuish, K
- Publisher / Repository:
- Proceedings of the 28th Annual Conference on Research in Undergraduate Mathematics Education.
- Date Published:
- Format(s):
- Medium: X
- Location:
- Alexandria, VA.
- Sponsoring Org:
- National Science Foundation
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