Conceptual and Procedural Understanding: Prospective Teachers’ Interpretations and Applications

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The preparation of prospective secondary mathematics teachers often revolves around working to improve knowledge of mathematics for teaching and understanding the conceptual development and trajectories of mathematics. “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems” (NCTM, 2014, p. 42). Prospective teachers need to be prepared to teach concepts along with procedures (Ball, Thames, & Phelps, 2008) particularly with the implementation of the Common Core State Standards (CCSS). In their own experience as a learner of mathematics, however, many prospective teachers come with procedural understandings of mathematics and many struggle to understand the underlying concepts and why those procedures work. Challenging prospective teachers to examine their own understandings of mathematical concepts and their preconceived ideas of good mathematics instruction becomes an important aspect of mathematics teacher preparation.

In this study, prospective secondary mathematics teachers were asked to read Principles to Actions’ section on Conceptual Understanding and Procedural Fluency (NCTM, 2014). Having individually defined conceptual and procedural understanding in their own words, they were asked to apply those understandings to determine how a student might solve a percentage problem with a conceptual and with a procedural understanding. Prospective teachers’ definitions and student solutions were examined to answer the question: In what ways do prospective secondary mathematics teachers define conceptual understanding and procedural understanding and subsequently apply those definitions to solve a percent problem?

Using the prospective teachers’ own definitions of these two terms, the researchers compared the definitions with how each prospective teacher distinguished between the types of understandings when applied to the given percent problem. Data showed some disconnect between definitions and applications. Additionally, responses of prospective teachers to the percentage problem could have been either conceptual or procedural based upon varying aspects of student solutions.


North American Chapter of the International Group for the Psychology of Mathematics Education Annual Meeting (PMENA)


Tucson, AZ