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August  2021, 1(3): 170-185. doi: 10.3934/steme.2021013

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

Christopher C. Tisdell 1,2,,

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

* Correspondence: cct@unsw.edu.au; Tel: +61-2-93856792

Academic Editor: Roland Dodd

Received  June 2021 Revised  July 2021 Published  August 2021

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.

Keywords: 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. STEM Education, 2021, 1 (3) : 170-185. doi: 10.3934/steme.2021013
References:
[1]

Assessment and Reporting Authority (ACARA), Australian Curriculum: Mathematics Year 4. 2021. https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ Google Scholar

[2]

M. Beishuizen, C. Van Putten and F. Van Mulken, Mental arithmetic and strategy use with indirect number problems up to one hundred, Learning and Instruction, 7 (1997), 87-106.  doi: 10.1016/S0959-4752(96)00012-6.  Google Scholar

[3] A. Benjamin and M.B. Shermer, Mathemagics: How to look like a genius without really trying, Contemporary Books, Chicago, IL, 1993.   Google Scholar
[4]

K. Bronk, The exemplar methodology: An approach to studying the leading edge of development, Psychology of Well-Being: Theory, Research and Practice, 2 (2012), 5.  doi: 10.1186/2211-1522-2-5.  Google Scholar

[5] J.R. Brown, Philosophy of mathematics: The world of proofs and pictures, Routledge, New York, 1999.   Google Scholar
[6] F. Capra, The Science of Leonardo, DoubleDay Press, New York, NY, 2007.   Google Scholar
[7]

B. Casselman, Pictures and proofs, Notices of the AMS, 47 (2000), 1257-1266.   Google Scholar

[8]

Common Core State Standards Initiative, Grade 4, Number & Operations in Base Ten. 2021. http://www.corestandards.org/Math/Content/4/NBT/ Google Scholar

[9] M. Crotty, The foundations of social research: meaning and perspective in the research process, SAGE, London, 1998.   Google Scholar
[10]

Da Vinci, Leonardo, Codex Atlanticus. 1478-1519, Milan: Biblioteca Ambrosiana. https://www.codex-atlanticus.it/#/Detail?detail=820 Google Scholar

[11] Leonardo Da Vinci, Codex Madrid II, Biblioteca Nacional, Madrid, 2007.   Google Scholar
[12]

Da Vinci, Leonardo, Studies of geometry. RCIN 919145, 1509, Windsor, UK: Royal Collection Trust. https://www.rct.uk/collection/search#/12/collection/919145/studies-of-geometry Google Scholar

[13]

Day A. L., Case Study Research, in Research Methods & Methodologies in Education, 2nd ed. R. Coe, M. Waring, L.V. Hedges & J. Arthur Ed. 2017, pp. 114-121. Los Angeles, CA: SAGE. Google Scholar

[14]

Department for Education, National curriculum in England: mathematics programmes of study. 2021. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study Google Scholar

[15] R.W. Doerfler, Dead Reckoning: Calculating Without Instruments, Gulf Publishing Company, Houston TX, 1993.   Google Scholar
[16] A. Dowker, Individual differences in arithmetic: Implications for psychology, neuroscience, and education, Psychology Press, New York, 2005.   Google Scholar
[17]

J.M. Furner and E.A. Brewer, Associating mathematics to its history: Connecting the mathematics we teach to its past, Transformations, 2 (2016), Article 2.   Google Scholar

[18] M. Giaquinto, Visual thinking in mathematics, aaa, Oxford Univ. Press, Oxford, 2007.   Google Scholar
[19]

S. Goktepe and A. Ozdemir, An example of using history of mathematics in classes, European Journal of Science and Mathematics Education, 1 (2013), 125-136.   Google Scholar

[20] B. Handley, Speed mathematics: Secret skills for quick calculation, John Wiley & Sons, Hoboken, NJ, 2000.   Google Scholar
[21] B. Handley, Speed maths for kids, John Wiley & Sons., Milton, Qld, 2005.   Google Scholar
[22]

G. Hanna and N. Sidoli, Visualisation and proof: a brief survey of philosophical Perspectives, ZDM Mathematics Education, 39 (2007), 73-78.  doi: 10.1007/s11858-006-0005-0.  Google Scholar

[23]

Hatano, G., Foreword, in A The development of arithmetic concepts and skills: Constructing adaptive expertise. J. Baroody & A. Dowker Ed. 2003, pp. xi-xiii. Mahwah: Lawrence Erlbaum Associates. DOI: 10.4324/9781410607218 Google Scholar

[24] P. Innocenzi, The Innovators Behind Leonardo, Springer International Publishing, Cham, Switzerland, 2019.  doi: 10.1007/978-3-319-90449-8_14.  Google Scholar
[25]

A. Izsk, Teaching and Learning Two-Digit Multiplication: Coordinating Analyses of Classroom Practices and Individual Student Learning, Mathematical Thinking and Learning, 6 (2004), 37-79.  doi: 10.1207/s15327833mtl0601_3.  Google Scholar

[26]

Jahnke, H.N., The use of original sources in the mathematics classroom, in History in mathematics education, the ICMI study. J. Fauvel & J. van Maanen Ed. 2000, pp. 291-328. Dordrecht: Kluwer Academic. Google Scholar

[27]

U.T. Jankvist, A categorization of the pwhysq and phowsq of using history in mathematics education, Educ Stud Math, 71 (2009), 235-261.  doi: 10.1007/s10649-008-9174-9.  Google Scholar

[28] M. Kemp, Leonardo da Vinci: Experience, Experiment, and Design, Princeton University Press, Princeton NJ, 2006.   Google Scholar
[29]

C. Kettle, The Symbolic and Mathematical Influence of Diophantus's Arithmetica, Journal of Humanistic Mathematics, 5 (2015), 139-166.  doi: 10.5642/jhummath.201501.08.  Google Scholar

[30]

G. Kospentaris, P. Spyrou and D. Lappas, Exploring studentso strategies in area conservation geometrical tasks, Educ Stud Math, 77 (2011), 105-127.  doi: 10.1007/s10649-011-9303-8.  Google Scholar

[31]

Larsson, K., Connections for learning multiplication, in Proceedings from Symposium Elementary Mathematics Education: Developing mathematical language and reasoning. J. Novotná & H. Moraová Ed, 2015, pp. 202-211. Prague: Charles University, Faculty of Education Google Scholar

[32]

K. Larsson, K. Pettersson and P. Andrews, Studentso conceptualisations of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers, The Journal of Mathematical Behavior, 48 (2017), 1-13.  doi: 10.1016/j.jmathb.2017.07.003.  Google Scholar

[33] J.E Littlewood, A Mathematicianos Miscellany, Methuen & Co. Ltd., London, 1953.   Google Scholar
[34]

G.L. Marshall and B.S. Rich, The Role of History in a Mathematics Class, Mathematics Teacher, 93 (2000), 704-706.   Google Scholar

[35]

A. McIntosh, B.J. Reys and R.E. Reys, A proposed framework for examining basic number sense, For the Learning of Mathematics, 12 (1992), 2-44.   Google Scholar

[36]

Mental Calculation World Cup, 2021. https://en.wikipedia.org/wiki/Mental_Calculation_World_Cup Google Scholar

[37]

S. Russ, P. Ransom and P. Perkins, The experience of history in mathematics education, For the Learning of Mathematics, 11 (1991), 7-16.   Google Scholar

[38]

H.G. van der Ven Sanne, Marthe Straatemeier, R.J. Jansen Brenda, Sharon Klinkenberg and L.J. van der Maas Han, Learning multiplication: An integrated analysis of the multiplication ability of primary school children and the difficulty of single digit and multidigit multiplication problems, Learning and Individual Differences, 43 (2015), 48-62.  doi: 10.1016/j.lindif.2015.08.013.  Google Scholar

[39]

Santhamma, C., Vedic mathematics. lecture notes 1 - multiplication. 2021. Retrieved from http://mathlearners.com/vedic-mathematics/multiplication-in-vedic-mathematics/ Google Scholar

[40]

Siu, M. -K., The ABCD of using history of mathematics in the (undergraduate) classroom, in Using history to teach mathematics—an international perspective, V. Katz Ed. MAA notes, 2000, 51: 3-9. Washington, DC: The Mathematical Association of America. Google Scholar

[41] S.B. Smith, The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies Past and Present, Columbia Univ Press, New York, 1983.   Google Scholar
[42] J.W. Stigler, S. Lee and H.W. Stevenson, Mathematical knowledge of Japanese, Chinese, and American elementary school children, National Council of Teachers of Mathematics, Reston, VA, 1990.   Google Scholar
[43]

Swetz, F., Using problems from the history of mathematics in classroom instruction, in Learn from the masters, F. Swetz, J. Fauvel, O. Bekken, B. Johansson & V. Katz Eds. 1995, pp. 25- 38. Washington, DC: The Mathematical Association of America. Google Scholar

[44]

Swetz, F., Problem solving from the history of mathematics, in Using history to teach mathematics—an international perspective. V. Katz Ed. MAA notes, 2000, 51, pp. 59-65. Washington, DC: The Mathematical Association of America. Google Scholar

[45] G. Thomas, How to Do Your Research Project: A Guide for Students., SAGE Publications Ltd., London, 2017.   Google Scholar
[46]

C.C. Tisdell, Schoenfeld's problem-solving models viewed through the lens of exemplification, For the Learning of Mathematics, 39 (2019), 24-26.   Google Scholar

[47]

C.C. Tisdell, Tic-Tac-Toe and repeated integration by parts: alternative pedagogical perspectives to Lima's integral challenge, International Journal of Mathematical Education in Science and Technology, 51 (2020), 424-430.  doi: 10.1080/0020739X.2019.1620969.  Google Scholar

[48]

C.C. Tisdell, Why do nfasto multiplication algorithms work? Opportunities for understanding within younger children via geometric pedagogy, International Journal of Mathematical Education in Science and Technology, 52 (2021), 527-549.  doi: 10.1080/0020739X.2019.1692933.  Google Scholar

[49] E. Tufte, The visual display of quantitative information, Graphic Press, Cheshire, CT, 1983.   Google Scholar
[50]

Tzanakis, C. and Arcavi, A., Integrating history of mathematics in the classroom: An analytic survey, in History in mathematics education. J. Fauvel & J. van Maanen Ed. 2000, pp. 201-240. The ICMI Study. Dordrecht: Kluwer Academic Publishers. Google Scholar

[51]

Verschaffel, L., Greer, B. and De Corte, E., Whole number concepts and operations, in Second handbook of research on mathematics teaching and learning. F.K. Lester Jr Ed. 2007, pp. 557-628. Charlotte, NC: Information Age Publishing Inc Google Scholar

[52]

West, L., An Introduction to Various Multiplication Strategies. Thesis, 2011, Lincoln, NE: University of Nebraska at Lincoln. Google Scholar

show all references

References:
[1]

Assessment and Reporting Authority (ACARA), Australian Curriculum: Mathematics Year 4. 2021. https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ Google Scholar

[2]

M. Beishuizen, C. Van Putten and F. Van Mulken, Mental arithmetic and strategy use with indirect number problems up to one hundred, Learning and Instruction, 7 (1997), 87-106.  doi: 10.1016/S0959-4752(96)00012-6.  Google Scholar

[3] A. Benjamin and M.B. Shermer, Mathemagics: How to look like a genius without really trying, Contemporary Books, Chicago, IL, 1993.   Google Scholar
[4]

K. Bronk, The exemplar methodology: An approach to studying the leading edge of development, Psychology of Well-Being: Theory, Research and Practice, 2 (2012), 5.  doi: 10.1186/2211-1522-2-5.  Google Scholar

[5] J.R. Brown, Philosophy of mathematics: The world of proofs and pictures, Routledge, New York, 1999.   Google Scholar
[6] F. Capra, The Science of Leonardo, DoubleDay Press, New York, NY, 2007.   Google Scholar
[7]

B. Casselman, Pictures and proofs, Notices of the AMS, 47 (2000), 1257-1266.   Google Scholar

[8]

Common Core State Standards Initiative, Grade 4, Number & Operations in Base Ten. 2021. http://www.corestandards.org/Math/Content/4/NBT/ Google Scholar

[9] M. Crotty, The foundations of social research: meaning and perspective in the research process, SAGE, London, 1998.   Google Scholar
[10]

Da Vinci, Leonardo, Codex Atlanticus. 1478-1519, Milan: Biblioteca Ambrosiana. https://www.codex-atlanticus.it/#/Detail?detail=820 Google Scholar

[11] Leonardo Da Vinci, Codex Madrid II, Biblioteca Nacional, Madrid, 2007.   Google Scholar
[12]

Da Vinci, Leonardo, Studies of geometry. RCIN 919145, 1509, Windsor, UK: Royal Collection Trust. https://www.rct.uk/collection/search#/12/collection/919145/studies-of-geometry Google Scholar

[13]

Day A. L., Case Study Research, in Research Methods & Methodologies in Education, 2nd ed. R. Coe, M. Waring, L.V. Hedges & J. Arthur Ed. 2017, pp. 114-121. Los Angeles, CA: SAGE. Google Scholar

[14]

Department for Education, National curriculum in England: mathematics programmes of study. 2021. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study Google Scholar

[15] R.W. Doerfler, Dead Reckoning: Calculating Without Instruments, Gulf Publishing Company, Houston TX, 1993.   Google Scholar
[16] A. Dowker, Individual differences in arithmetic: Implications for psychology, neuroscience, and education, Psychology Press, New York, 2005.   Google Scholar
[17]

J.M. Furner and E.A. Brewer, Associating mathematics to its history: Connecting the mathematics we teach to its past, Transformations, 2 (2016), Article 2.   Google Scholar

[18] M. Giaquinto, Visual thinking in mathematics, aaa, Oxford Univ. Press, Oxford, 2007.   Google Scholar
[19]

S. Goktepe and A. Ozdemir, An example of using history of mathematics in classes, European Journal of Science and Mathematics Education, 1 (2013), 125-136.   Google Scholar

[20] B. Handley, Speed mathematics: Secret skills for quick calculation, John Wiley & Sons, Hoboken, NJ, 2000.   Google Scholar
[21] B. Handley, Speed maths for kids, John Wiley & Sons., Milton, Qld, 2005.   Google Scholar
[22]

G. Hanna and N. Sidoli, Visualisation and proof: a brief survey of philosophical Perspectives, ZDM Mathematics Education, 39 (2007), 73-78.  doi: 10.1007/s11858-006-0005-0.  Google Scholar

[23]

Hatano, G., Foreword, in A The development of arithmetic concepts and skills: Constructing adaptive expertise. J. Baroody & A. Dowker Ed. 2003, pp. xi-xiii. Mahwah: Lawrence Erlbaum Associates. DOI: 10.4324/9781410607218 Google Scholar

[24] P. Innocenzi, The Innovators Behind Leonardo, Springer International Publishing, Cham, Switzerland, 2019.  doi: 10.1007/978-3-319-90449-8_14.  Google Scholar
[25]

A. Izsk, Teaching and Learning Two-Digit Multiplication: Coordinating Analyses of Classroom Practices and Individual Student Learning, Mathematical Thinking and Learning, 6 (2004), 37-79.  doi: 10.1207/s15327833mtl0601_3.  Google Scholar

[26]

Jahnke, H.N., The use of original sources in the mathematics classroom, in History in mathematics education, the ICMI study. J. Fauvel & J. van Maanen Ed. 2000, pp. 291-328. Dordrecht: Kluwer Academic. Google Scholar

[27]

U.T. Jankvist, A categorization of the pwhysq and phowsq of using history in mathematics education, Educ Stud Math, 71 (2009), 235-261.  doi: 10.1007/s10649-008-9174-9.  Google Scholar

[28] M. Kemp, Leonardo da Vinci: Experience, Experiment, and Design, Princeton University Press, Princeton NJ, 2006.   Google Scholar
[29]

C. Kettle, The Symbolic and Mathematical Influence of Diophantus's Arithmetica, Journal of Humanistic Mathematics, 5 (2015), 139-166.  doi: 10.5642/jhummath.201501.08.  Google Scholar

[30]

G. Kospentaris, P. Spyrou and D. Lappas, Exploring studentso strategies in area conservation geometrical tasks, Educ Stud Math, 77 (2011), 105-127.  doi: 10.1007/s10649-011-9303-8.  Google Scholar

[31]

Larsson, K., Connections for learning multiplication, in Proceedings from Symposium Elementary Mathematics Education: Developing mathematical language and reasoning. J. Novotná & H. Moraová Ed, 2015, pp. 202-211. Prague: Charles University, Faculty of Education Google Scholar

[32]

K. Larsson, K. Pettersson and P. Andrews, Studentso conceptualisations of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers, The Journal of Mathematical Behavior, 48 (2017), 1-13.  doi: 10.1016/j.jmathb.2017.07.003.  Google Scholar

[33] J.E Littlewood, A Mathematicianos Miscellany, Methuen & Co. Ltd., London, 1953.   Google Scholar
[34]

G.L. Marshall and B.S. Rich, The Role of History in a Mathematics Class, Mathematics Teacher, 93 (2000), 704-706.   Google Scholar

[35]

A. McIntosh, B.J. Reys and R.E. Reys, A proposed framework for examining basic number sense, For the Learning of Mathematics, 12 (1992), 2-44.   Google Scholar

[36]

Mental Calculation World Cup, 2021. https://en.wikipedia.org/wiki/Mental_Calculation_World_Cup Google Scholar

[37]

S. Russ, P. Ransom and P. Perkins, The experience of history in mathematics education, For the Learning of Mathematics, 11 (1991), 7-16.   Google Scholar

[38]

H.G. van der Ven Sanne, Marthe Straatemeier, R.J. Jansen Brenda, Sharon Klinkenberg and L.J. van der Maas Han, Learning multiplication: An integrated analysis of the multiplication ability of primary school children and the difficulty of single digit and multidigit multiplication problems, Learning and Individual Differences, 43 (2015), 48-62.  doi: 10.1016/j.lindif.2015.08.013.  Google Scholar

[39]

Santhamma, C., Vedic mathematics. lecture notes 1 - multiplication. 2021. Retrieved from http://mathlearners.com/vedic-mathematics/multiplication-in-vedic-mathematics/ Google Scholar

[40]

Siu, M. -K., The ABCD of using history of mathematics in the (undergraduate) classroom, in Using history to teach mathematics—an international perspective, V. Katz Ed. MAA notes, 2000, 51: 3-9. Washington, DC: The Mathematical Association of America. Google Scholar

[41] S.B. Smith, The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies Past and Present, Columbia Univ Press, New York, 1983.   Google Scholar
[42] J.W. Stigler, S. Lee and H.W. Stevenson, Mathematical knowledge of Japanese, Chinese, and American elementary school children, National Council of Teachers of Mathematics, Reston, VA, 1990.   Google Scholar
[43]

Swetz, F., Using problems from the history of mathematics in classroom instruction, in Learn from the masters, F. Swetz, J. Fauvel, O. Bekken, B. Johansson & V. Katz Eds. 1995, pp. 25- 38. Washington, DC: The Mathematical Association of America. Google Scholar

[44]

Swetz, F., Problem solving from the history of mathematics, in Using history to teach mathematics—an international perspective. V. Katz Ed. MAA notes, 2000, 51, pp. 59-65. Washington, DC: The Mathematical Association of America. Google Scholar

[45] G. Thomas, How to Do Your Research Project: A Guide for Students., SAGE Publications Ltd., London, 2017.   Google Scholar
[46]

C.C. Tisdell, Schoenfeld's problem-solving models viewed through the lens of exemplification, For the Learning of Mathematics, 39 (2019), 24-26.   Google Scholar

[47]

C.C. Tisdell, Tic-Tac-Toe and repeated integration by parts: alternative pedagogical perspectives to Lima's integral challenge, International Journal of Mathematical Education in Science and Technology, 51 (2020), 424-430.  doi: 10.1080/0020739X.2019.1620969.  Google Scholar

[48]

C.C. Tisdell, Why do nfasto multiplication algorithms work? Opportunities for understanding within younger children via geometric pedagogy, International Journal of Mathematical Education in Science and Technology, 52 (2021), 527-549.  doi: 10.1080/0020739X.2019.1692933.  Google Scholar

[49] E. Tufte, The visual display of quantitative information, Graphic Press, Cheshire, CT, 1983.   Google Scholar
[50]

Tzanakis, C. and Arcavi, A., Integrating history of mathematics in the classroom: An analytic survey, in History in mathematics education. J. Fauvel & J. van Maanen Ed. 2000, pp. 201-240. The ICMI Study. Dordrecht: Kluwer Academic Publishers. Google Scholar

[51]

Verschaffel, L., Greer, B. and De Corte, E., Whole number concepts and operations, in Second handbook of research on mathematics teaching and learning. F.K. Lester Jr Ed. 2007, pp. 557-628. Charlotte, NC: Information Age Publishing Inc Google Scholar

[52]

West, L., An Introduction to Various Multiplication Strategies. Thesis, 2011, Lincoln, NE: University of Nebraska at Lincoln. Google Scholar

Figure 1.  Diagram for Example 1
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Figure 2.  Diagram for Example 2
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Figure 3.  Diagram for Example 3
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Figure 4.  One example of conservation in Codex Madrid II captured by Leonardo
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Figure 5.  Another example of conservation in Codex Madrid II captured by Leonardo
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