Abstract
This chapter examines the classroom implementation of a theoretical framework for the teaching and learning of mathematics—called DNR-based instruction in mathematics—focusing on characteristics of the implementation of DNR to help learners transition between proof schemes. Three episodes from a professional development program for middle and high school teachers are analyzed to reveal the teaching behaviors of an expert DNR instructor. Complexities highlighted include (1) how the instructor balanced the intended mathematical content with the learners’ current understandings, (2) the interplay of the questions whether a result holds and why it holds, and (3) how the instructor created intellectual need for new ideas. As a by-product, learning outcomes of this effort are also examined.
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Notes
- 1.
This is the counterpart to the aforementioned non-referential symbolic proof scheme.
- 2.
A detailed discussion of causality is beyond the scope of this chapter. Here, it suffices to think of causality as a strong understanding of why a result holds.
- 3.
All names are pseudonyms.
- 4.
Here and elsewhere, quotes from TR are modified slightly for clarity and brevity.
- 5.
Although this problem has some depth, the reader can understand the discussion merely by considering how many squares are white versus black on an n × n quilt.
- 6.
Recall that we do not make and substantiate claims about the actual transition between proof schemes; rather, we report TR’s model (as reported or inferred).
- 7.
Proving a statement by showing that it holds in different cases, whose union is the whole set under consideration.
- 8.
The proof involved directly counting the number of black squares in a generic even or odd quilt and then taking the difference for successive quilts.
- 9.
One year after the first, with a few follow-up sessions in between.
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Harel, G., Fuller, E., Soto, O.D. (2014). DNR-Based Instruction in Mathematics: Determinants of a DNR Expert’s Teaching. In: Li, Y., Silver, E., Li, S. (eds) Transforming Mathematics Instruction. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-04993-9_23
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