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Frustration in mathematical problem-solving: A systematic review of research
Reimagining multiplication as diagrammatic and dynamic concepts via cutting, pasting and rescaling actions
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 |
Recently, Tisdell [
References:
[1] |
Assessment and Reporting Authority (ACARA), Australian Curriculum: Mathematics Year 4. 2021. https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ |
[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. |
[3] |
A. Benjamin and M.B. Shermer, Mathemagics: How to look like a genius without really trying, Contemporary Books, Chicago, IL, 1993.
![]() |
[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. |
[5] |
J.R. Brown, Philosophy of mathematics: The world of proofs and pictures, Routledge, New York, 1999.
![]() |
[6] |
F. Capra, The Science of Leonardo, DoubleDay Press, New York, NY, 2007.
![]() |
[7] |
B. Casselman,
Pictures and proofs, Notices of the AMS, 47 (2000), 1257-1266.
|
[8] |
Common Core State Standards Initiative, Grade 4, Number & Operations in Base Ten. 2021. http://www.corestandards.org/Math/Content/4/NBT/ |
[9] |
M. Crotty, The foundations of social research: meaning and perspective in the research process, SAGE, London, 1998.
![]() |
[10] |
Da Vinci, Leonardo, Codex Atlanticus. 1478-1519, Milan: Biblioteca Ambrosiana. https://www.codex-atlanticus.it/#/Detail?detail=820 |
[11] |
Leonardo Da Vinci, Codex Madrid II, Biblioteca Nacional, Madrid, 2007.
![]() |
[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 |
[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. |
[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 |
[15] |
R.W. Doerfler, Dead Reckoning: Calculating Without Instruments, Gulf Publishing Company, Houston TX, 1993.
![]() |
[16] |
A. Dowker, Individual differences in arithmetic: Implications for psychology, neuroscience, and education, Psychology Press, New York, 2005.
![]() |
[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.
|
[18] |
M. Giaquinto, Visual thinking in mathematics, aaa, Oxford Univ. Press, Oxford, 2007.
![]() |
[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.
|
[20] |
B. Handley, Speed mathematics: Secret skills for quick calculation, John Wiley & Sons, Hoboken, NJ, 2000.
![]() |
[21] |
B. Handley, Speed maths for kids, John Wiley & Sons., Milton, Qld, 2005.
![]() |
[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. |
[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 |
[24] |
P. Innocenzi, The Innovators Behind Leonardo, Springer International Publishing, Cham, Switzerland, 2019.
doi: 10.1007/978-3-319-90449-8_14.![]() ![]() |
[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. |
[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. |
[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. |
[28] |
M. Kemp, Leonardo da Vinci: Experience, Experiment, and Design, Princeton University Press, Princeton NJ, 2006.
![]() |
[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. |
[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. |
[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 |
[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. |
[33] |
J.E Littlewood, A Mathematicianos Miscellany, Methuen & Co. Ltd., London, 1953.
![]() |
[34] |
G.L. Marshall and B.S. Rich,
The Role of History in a Mathematics Class, Mathematics Teacher, 93 (2000), 704-706.
|
[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.
|
[36] |
Mental Calculation World Cup, 2021. https://en.wikipedia.org/wiki/Mental_Calculation_World_Cup |
[37] |
S. Russ, P. Ransom and P. Perkins,
The experience of history in mathematics education, For the Learning of Mathematics, 11 (1991), 7-16.
|
[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. |
[39] |
Santhamma, C., Vedic mathematics. lecture notes 1 - multiplication. 2021. Retrieved from http://mathlearners.com/vedic-mathematics/multiplication-in-vedic-mathematics/ |
[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. |
[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.
![]() |
[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.
![]() |
[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. |
[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. |
[45] |
G. Thomas, How to Do Your Research Project: A Guide for Students., SAGE Publications Ltd., London, 2017.
![]() |
[46] |
C.C. Tisdell,
Schoenfeld's problem-solving models viewed through the lens of exemplification, For the Learning of Mathematics, 39 (2019), 24-26.
|
[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. |
[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. |
[49] |
E. Tufte, The visual display of quantitative information, Graphic Press, Cheshire, CT, 1983.
![]() |
[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. |
[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 |
[52] |
West, L., An Introduction to Various Multiplication Strategies. Thesis, 2011, Lincoln, NE: University of Nebraska at Lincoln. |
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/ |
[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. |
[3] |
A. Benjamin and M.B. Shermer, Mathemagics: How to look like a genius without really trying, Contemporary Books, Chicago, IL, 1993.
![]() |
[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. |
[5] |
J.R. Brown, Philosophy of mathematics: The world of proofs and pictures, Routledge, New York, 1999.
![]() |
[6] |
F. Capra, The Science of Leonardo, DoubleDay Press, New York, NY, 2007.
![]() |
[7] |
B. Casselman,
Pictures and proofs, Notices of the AMS, 47 (2000), 1257-1266.
|
[8] |
Common Core State Standards Initiative, Grade 4, Number & Operations in Base Ten. 2021. http://www.corestandards.org/Math/Content/4/NBT/ |
[9] |
M. Crotty, The foundations of social research: meaning and perspective in the research process, SAGE, London, 1998.
![]() |
[10] |
Da Vinci, Leonardo, Codex Atlanticus. 1478-1519, Milan: Biblioteca Ambrosiana. https://www.codex-atlanticus.it/#/Detail?detail=820 |
[11] |
Leonardo Da Vinci, Codex Madrid II, Biblioteca Nacional, Madrid, 2007.
![]() |
[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 |
[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. |
[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 |
[15] |
R.W. Doerfler, Dead Reckoning: Calculating Without Instruments, Gulf Publishing Company, Houston TX, 1993.
![]() |
[16] |
A. Dowker, Individual differences in arithmetic: Implications for psychology, neuroscience, and education, Psychology Press, New York, 2005.
![]() |
[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.
|
[18] |
M. Giaquinto, Visual thinking in mathematics, aaa, Oxford Univ. Press, Oxford, 2007.
![]() |
[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.
|
[20] |
B. Handley, Speed mathematics: Secret skills for quick calculation, John Wiley & Sons, Hoboken, NJ, 2000.
![]() |
[21] |
B. Handley, Speed maths for kids, John Wiley & Sons., Milton, Qld, 2005.
![]() |
[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. |
[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 |
[24] |
P. Innocenzi, The Innovators Behind Leonardo, Springer International Publishing, Cham, Switzerland, 2019.
doi: 10.1007/978-3-319-90449-8_14.![]() ![]() |
[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. |
[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. |
[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. |
[28] |
M. Kemp, Leonardo da Vinci: Experience, Experiment, and Design, Princeton University Press, Princeton NJ, 2006.
![]() |
[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. |
[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. |
[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 |
[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. |
[33] |
J.E Littlewood, A Mathematicianos Miscellany, Methuen & Co. Ltd., London, 1953.
![]() |
[34] |
G.L. Marshall and B.S. Rich,
The Role of History in a Mathematics Class, Mathematics Teacher, 93 (2000), 704-706.
|
[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.
|
[36] |
Mental Calculation World Cup, 2021. https://en.wikipedia.org/wiki/Mental_Calculation_World_Cup |
[37] |
S. Russ, P. Ransom and P. Perkins,
The experience of history in mathematics education, For the Learning of Mathematics, 11 (1991), 7-16.
|
[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. |
[39] |
Santhamma, C., Vedic mathematics. lecture notes 1 - multiplication. 2021. Retrieved from http://mathlearners.com/vedic-mathematics/multiplication-in-vedic-mathematics/ |
[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. |
[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.
![]() |
[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.
![]() |
[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. |
[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. |
[45] |
G. Thomas, How to Do Your Research Project: A Guide for Students., SAGE Publications Ltd., London, 2017.
![]() |
[46] |
C.C. Tisdell,
Schoenfeld's problem-solving models viewed through the lens of exemplification, For the Learning of Mathematics, 39 (2019), 24-26.
|
[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. |
[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. |
[49] |
E. Tufte, The visual display of quantitative information, Graphic Press, Cheshire, CT, 1983.
![]() |
[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. |
[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 |
[52] |
West, L., An Introduction to Various Multiplication Strategies. Thesis, 2011, Lincoln, NE: University of Nebraska at Lincoln. |





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