Combinatorial and quantitative reasoning: Stage 3 high school students’ reason about combinatorics problems and their representation as 3-D arrays

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH
Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro
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引用次数: 0

Abstract

Researchers have identified three stages of units coordination that influence a range of domains of student reasoning. The primary foci of this research have been students’ reasoning in discrete, non-combinatorial whole number contexts, and with fractions, ratios, proportions, and rates represented using length quantities. This study extends this prior work by examining connections between eight high school students’ combinatorial reasoning and their representation of this reasoning using 3-D arrays. All students in the study were at stage 3 of units coordination. Findings include differentiation between two student groups: one group had interiorized three-levels-of-units, but had not interiorized four-levels-of-units; and the other group had interiorized four-levels-of-units. This differentiation was coordinated with differences in how they reasoned to produce 3-D arrays. The findings from the study indicate how combinatorics problems can support quantitative reasoning, where combinatorial and quantitative reasoning are framed as a foundation for algebraic reasoning.

组合和定量推理:第三阶段高中生对组合问题的推理及其三维数组表示法
研究人员发现,单位协调的三个阶段会影响学生推理的一系列领域。这些研究的主要重点是学生在离散、非组合整数情境中的推理,以及使用长度量表示的分数、比率、比例和比率的推理。本研究通过考察八名高中生的组合推理与他们使用三维阵列表示这种推理之间的联系,对之前的研究进行了扩展。所有参与研究的学生都处于单元协调的第三阶段。研究结果包括两组学生之间的差异:一组学生将三级单位内部化,但没有将四级单位内部化;另一组学生将四级单位内部化。这种差异与他们制作三维阵列的推理方式的差异相协调。研究结果表明了组合问题如何支持定量推理,其中组合和定量推理是代数推理的基础。
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来源期刊
Journal of Mathematical Behavior
Journal of Mathematical Behavior EDUCATION & EDUCATIONAL RESEARCH-
CiteScore
2.70
自引率
17.60%
发文量
69
期刊介绍: The Journal of Mathematical Behavior solicits original research on the learning and teaching of mathematics. We are interested especially in basic research, research that aims to clarify, in detail and depth, how mathematical ideas develop in learners. Over three decades, our experience confirms a founding premise of this journal: that mathematical thinking, hence mathematics learning as a social enterprise, is special. It is special because mathematics is special, both logically and psychologically. Logically, through the way that mathematical ideas and methods have been built, refined and organized for centuries across a range of cultures; and psychologically, through the variety of ways people today, in many walks of life, make sense of mathematics, develop it, make it their own.
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