Reimagining multiplication as diagrammatic and dynamic concepts via cutting, pasting and rescaling actions

Christopher C. Tisdell; Christopher C. Tisdell

doi:10.3934/steme.2021013

STEM Education

2021, Volume 1, Issue 3: 170-185. doi: 10.3934/steme.2021013

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Reimagining multiplication as diagrammatic and dynamic concepts via cutting, pasting and rescaling actions

Christopher C. Tisdell ^{1,2
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1.
School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia
2.
Institute for Teaching and Learning Innovation (ITaLI), University of Queensland, Brisbane Qld 4072, Australia

Received: 01 June 2021 Revised: 01 July 2021
Article

Recently, Tisdell [48] developed some alternative pedagogical perspectives of multiplication strategies via cut-and-paste actions, underpinned via the principle of conservation of area. However, the ideas therein were limited to problems involving two factors that were close together, and so would not directly apply to a problem such as 17 × 93. The purpose of the present work is to establish what diagrammatic and dynamic perspectives could look like for these more complex classes of multiplication problems. My approach to explore this gap is through an analysis and discussion of case studies. I probe several multiplication problems in depth, and drill down to get at their complexity. Through this process, new techniques emerge that involve cut-and-paste and rescaling actions to enable a reimagination of the problem from diagrammatic and dynamic points of view. Furthermore, I provide some suggestions regarding how these ideas might be supplemented in the classroom through the employment of history that includes Leonardo Da Vinci's use of conservation principles in his famous notebooks. I thus establish a pedagogical framework that has the potential to support the learning and teaching of these extended problems from diagrammatic and dynamic perspectives. groups.
- mathematics education,
- multiplication,
- conservation of area,
- cutting,
- pasting,
- rescaling
Citation: Christopher C. Tisdell. Reimagining multiplication as diagrammatic and dynamic concepts via cutting, pasting and rescaling actions[J]. STEM Education, 2021, 1(3): 170-185. doi: 10.3934/steme.2021013

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Abstract

Recently, Tisdell [48] developed some alternative pedagogical perspectives of multiplication strategies via cut-and-paste actions, underpinned via the principle of conservation of area. However, the ideas therein were limited to problems involving two factors that were close together, and so would not directly apply to a problem such as 17 × 93. The purpose of the present work is to establish what diagrammatic and dynamic perspectives could look like for these more complex classes of multiplication problems. My approach to explore this gap is through an analysis and discussion of case studies. I probe several multiplication problems in depth, and drill down to get at their complexity. Through this process, new techniques emerge that involve cut-and-paste and rescaling actions to enable a reimagination of the problem from diagrammatic and dynamic points of view. Furthermore, I provide some suggestions regarding how these ideas might be supplemented in the classroom through the employment of history that includes Leonardo Da Vinci's use of conservation principles in his famous notebooks. I thus establish a pedagogical framework that has the potential to support the learning and teaching of these extended problems from diagrammatic and dynamic perspectives. groups.

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