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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

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.

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/

[2]

M. BeishuizenC. 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. KospentarisP. 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. LarssonK. 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. McIntoshB.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. RussP. 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 SanneMarthe StraatemeierR.J. Jansen BrendaSharon 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. StiglerS. 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. BeishuizenC. 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. KospentarisP. 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. LarssonK. 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. McIntoshB.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. RussP. 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 SanneMarthe StraatemeierR.J. Jansen BrendaSharon 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. StiglerS. 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.

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