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Mental processes underlying a mathematics teacher’s learning from student thinking

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Abstract

Teachers’ knowledge of students’ mathematical thinking is a growing area of research in mathematics education. The literature has reported plentiful evidence of the interplays between teachers’ mathematical knowledge and their knowledge of students’ mathematical thinking. The present study builds on this body of work to explain how such interplays occur. Over a semester, I worked in partnership with a secondary school mathematics teacher on cycles of task design, interactions with a student, and in-depth reflection on the student’s thinking about linear programming. Adopting and extending Piagetian constructs of assimilation and accommodation, I describe several key mental processes that illuminate the teacher’s learning of mathematics and of the student’s mathematical thinking. I conclude with a discussion of the study’s empirical and theoretical contributions to understanding teachers’ mathematical learning in relation to student thinking.

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Notes

  1. Note that some quantities’ relationships involved in a constant speed motion are nonlinear.

  2. The detailed conversations and decision-making during the co-planning sessions are beyond the scope of this article. In Results, I am only able to describe the resulting task design decision (along with the rationale) as a collective product of our discussions and focus the discussion on Taylor’s and Ms. K’s thinking around those tasks. I use the pronoun “we” in Results to indicate the collaborative nature of the co-planning sessions and consider the choice of tasks to be our joint decision.

  3. Here I have to shift to discuss SOM from my perspective rather than Ms. K’s. All researcher activities during data analysis must be the researcher’s constructions of SOM of the observed participants, which means my claims about Ms. K’s knowledge change are merely my inferences based on her talk. Regarding the third type of model (i.e., a researcher’s SOM of a teacher’s SOM of a student), Wilson et al. (2011) also defined it as third-order model.

  4. I chose to present only Cycle 1 and Cycle 3 because they generated a coherent story of Ms K’s learning about Taylor’s thinking that could lead to rich themes about her reasoning processes that constituted the major findings of the present paper. I omitted Cycle 2 (a follow-up session on ATSB) and Cycles 4–6 because they conveyed overlapping themes. Cycles 4–6 also addressed different task scenarios, and given the space constraint and the complexity of the thinking involved, I decided not to include additional tasks.

  5. This method is underlain by the corner point principle, which states that if there is a unique optimal solution to a linear programming problem, it must lie at a vertex of the feasible region. To obtain the optimal solution, one needs to evaluate the objective function (in this case, Revenue (videos, books) = 75 \(\times\) books + 95 \(\times\) videos) at all corner points of a feasible region, comparing their outputs, and identify the optimal value pair.

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Funding

This work was supported by the National Science Foundation under grant number DRL-1350342 and DUE-1920538. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. This article is based on the author’s dissertation research completed under the guidance of Dr. Kevin C. Moore at the University of Georgia. I also thank Dr. Oi-Lam Ng for her insightful feedback on previous versions of the article.

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    Biyao Liang

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Liang, B. Mental processes underlying a mathematics teacher’s learning from student thinking. J Math Teacher Educ (2023). https://doi.org/10.1007/s10857-023-09601-7

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