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A complementary survey on the current state of teaching and learning of Whole Number Arithmetic and connections to later mathematical content

  • Xu Hua SunEmail author
  • Yan Ping Xin
  • Rongjin Huang
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Abstract

Whole Number Arithmetic (WNA) appears as the very first topic in school mathematics and establishes the foundation for later mathematical content. Without solid mastery of WNA, students may experience difficulties in learning fractions, ratio and proportion, and algebra. The challenge of students’ learning and mastery of fractions, decimals, ratio and proportion, and algebra is well documented. Most of this research has focused on either fractions, decimals, ratio and proportion, algebra, or WNA. There is a lack of research that addresses the connection between these relevant topics. Within WNA, most research focuses on counting, computation, or solving word problems. There is a lack of research that investigates connections within WNA. This special issue is intended to bridge this research gap by explicitly highlighting the conceptual knowledge of counting, calculations, and quantity relationships, as well as the structure of word problems within and beyond WNA.

Notes

Acknowledgements

This study was funded by the research committee of the University of Macau, Macao, China [MYRG2015-00203-FED].

References

  1. Askew, M. (2019). Mediating primary mathematics: Measuring the extent of teaching for connections and generality in the context of whole number arithmetic. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-1010-9/ (this issue).Google Scholar
  2. Baroody, A. J., Bajwa, N. P., & Eiland, M. (2009). Why can’t Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15(1), 69–79.Google Scholar
  3. Bartolini Bussi, M. G. (2015). The number line: A “western” teaching aid. In X. Sun, B. Kaur, & J. Novotná, (Eds.), Proceedings of the 23rd ICMI Study on ‘Primary mathematics study on whole numbers’. China, Macao: University of Macau, retrieved on February 20, 2016, from http://www.umac.mo/fed/ICMI23/proceedings.html.
  4. Bartolini Bussi, M. G., & Sun, X. H. (2018). Building a strong foundation concerning whole number arithmetic in primary grades: Editorial introduction. In M. G. Bartolini, Bussi & X. Sun (Eds.), Building the foundation: Whole numbers in the primary grades (pp. 3–18). Cham: Springer.Google Scholar
  5. Bartolini Bussi, M. G., Sun, X. H., & Ramploud, A. (2013). A dialogue between cultures about task design for primary school. In C. Margolinas (Ed.), Proceedings of ICMI Study 22 on task design in mathematics education (pp. 409–418). Oxford, United Kingdom. Retrieved February 10, 2019 from https://hal.archives-ouvertes.fr/hal-00834054v3.
  6. Beckmann, S., & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17–38.Google Scholar
  7. Beckmann, S., Izsák, A., & Ölmez, I. B. (2015). From multiplication to proportional relationships.In X. Sun, B. Kaur, & J. Novotná (Eds.), Proceedings of the 23rd ICMI Study ‘Primary Mathematics Study on Whole Numbers’. China, Macao: University of Macau, retrieved on February 20, 2016, from http://www.umac.mo/fed/ICMI23/proceedings.html.
  8. Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra (pp. 115–136). Dordrecht/Boston/London: Kluwer Academic Publishers.Google Scholar
  9. Björklund, C., Kullberg, A., & Runesson, K. U. (2019). Structuring versus counting: critical ways of using fingers in subtraction. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0962-0 (this issue).Google Scholar
  10. Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J. S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87.Google Scholar
  11. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspective (pp. 5–23). Berlin: Springer.Google Scholar
  12. Boaler, J. (2015). What’s math got to do with it? How teachers and parents can transform mathematics learning and inspire success. New York: Penguin Books.Google Scholar
  13. Bruner, J. S. (1966). Toward a theory of instruction (Vol. 59). Cambridge: Harvard University Press.Google Scholar
  14. Čadež, T. H., & Kolar, V. M. (2015). Comparison of types of generalizations and problem-solving schemas used to solve a mathematical problem. Educational Studies in Mathematics, 89(2), 283–306.Google Scholar
  15. Carpenter, T. P., Fennema, E., Franke, M., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth: Heinemann.Google Scholar
  16. Carpenter, T. P., Moser, J. M., & Romberg, T. A. (Eds.). (1982). Addition and subtraction: A cognitive perspective. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  17. Ching, B. H.-H., & Nunes, T. (2017). The importance of additive reasoning in children’s mathematical achievement: A longitudinal study. Journal of Educational Psychology, 109, 477–508.Google Scholar
  18. Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333(6045), 968–970.Google Scholar
  19. Cooper, J. (2019). Mathematicians and teachers sharing perspectives on teaching whole number arithmetic: Boundary-crossing in professional development. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0995-4 (this issue).Google Scholar
  20. Ding, M. (2016). Opportunities to learn: Inverse relations in U.S. and Chinese textbooks. Mathematical Thinking and Learning, 18(1), 45–68.Google Scholar
  21. Ding, M., & Auxter, A. E. (2017). Children’s strategies to solving additive inverse problems: a preliminary analysis. Mathematics Education Research Journal, 29(1), 73–92.Google Scholar
  22. Elementary Mathematic Department. (2005). Mathematics teacher manual, Grade 1(Vol.1). Beijing: People Education Press.Google Scholar
  23. Freiman, V., Polotskaia, E., & Savard, A. (2017). Using a computer-based learning task to promote work on mathematical relationships in the context of word problems in early grades. ZDM Mathematics Education, 49(6), 835–849.Google Scholar
  24. Fuson, K., & Li, Y. (2009). Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking. ZDM Mathematics Education, 41, 793–808.  https://doi.org/10.1007/s11858-009-0183-7.Google Scholar
  25. Gelman, R., & Butterworth, B. (2005). Number and language: How are they related? Trends in Cognitive Sciences, 9(1), 6–10.Google Scholar
  26. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12(3), 306–355.Google Scholar
  27. Givvin, K. B., Stigler, J. W., & Thompson, B. J. (2011). What community college developmental mathematics students understand about mathematics, Part II: The interviews. The MathAMATYC Educator, 2(3), 4–18.Google Scholar
  28. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 276–295). New York: Macmillan.Google Scholar
  29. Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87(1), 18.Google Scholar
  30. Hino, K., & Kato, H. (2019). Teaching whole-number multiplication to promote children’s proportional reasoning: a practice-based perspective from Japan. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0993-6 (this issue).Google Scholar
  31. Hitt, F., Saboya, M., & Zavala, C. C. (2017). Rupture or continuity: The arithmetic-algebraic thinking as an alternative in a modelling process in a paper and pencil and technology environment. Educational Studies in Mathematics, 94(1), 97–116.Google Scholar
  32. Horner, R. D., & Baer, D. M. (1978). Multiple-probe technique: A variation of the multiple baseline. Journal of Applied Behavior Analysis, 11, 189–196.Google Scholar
  33. Houdement, C., & Tempier, F. (2019). Understanding place value with numeration units. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0985-6 (this issue).Google Scholar
  34. Howe, R. (2019). Learning and using our base ten place value number system: theoretical perspectives and twenty-first century uses. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0996-3 (this issue).Google Scholar
  35. Huang, R., Zhang, Q., Chang, Y. P., & Kimmins, D. (2019). Developing students’ ability to solve word problems through learning trajectory-based and variation task-informed instruction. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0983-8 (this issue).Google Scholar
  36. Kaur, B. (2015). The model method—A tool for representing and visualising relationships. In X. Sun, B. Kaur, & J. Novotná, (Eds.), Proceedings of the 23rd ICMI Study on primary mathematics study on whole numbers. China, Macao: University of Macau, retrieved on February 20, 2016, from http://www.umac.mo/fed/ICMI23/proceedings.html.
  37. Kaur, B. (2019). The why, what and how of the ‘Model’ method: A tool for representing and visualising relationships when solving whole number arithmetic word problems. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-1000-y (this issue).Google Scholar
  38. Klein, F. (1924). Elementarmathematik vom höheren Standpunkte aus: Arithmetik, algebra, analysis [Elementary mathematics from a higher standpoint: Arithmetic, algebra, analysis] (Vol. 1, 3rd edn.). Berlin: Springer.Google Scholar
  39. Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children’s understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and in the United States. Cognition and Instruction, 26, 195–217.Google Scholar
  40. Liu, F., Xu, F., & Geary, D. C. (1993). A comparative study of the cognitive factors affecting Chinese and American children’s numerical skill. Psychological Science, 16, 22–27.Google Scholar
  41. Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New York: Routledge Taylor & Francis Group.Google Scholar
  42. Martin, M. O., & Mullis, I. V. (2013). TIMSS and PIRLS 2011: Relationships among reading, mathematics, and science achievement at the fourth grade—Implications for early learning. Chestnut Hill: TIMSS & PIRLS International Study Center, Boston College.Google Scholar
  43. Marton, F. (2015). Necessary conditions of learning. New York: Routledge.Google Scholar
  44. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht/Boston/London: Kluwer.  https://doi.org/10.1007/978-94-009-1732-3_5.Google Scholar
  45. Matar, M., Sitabkhan, Y., & Brombacher, A. (2013). Early primary mathematics education in Arab countries of the Middle East and North Africa. Bonn: Deutsche Gesellschaft fur Internationale Zusammenarbeit (GIZ) GmbH.Google Scholar
  46. Mellone, M., Ramploud, A., Di Paola, B., & Martignone, F. (2019). Cultural transposition: Italian didactic experiences inspired by Chinese and Russian perspectives on whole number arithmetic. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0992-7 (this issue).Google Scholar
  47. National Assessment of Educational Progress result, NEAP (2015). Retrieved March 15, 2018, from http://www.nationsreportcard.gov/reading_math_2015/#mathematics/acl?grade=4.
  48. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: Author.Google Scholar
  49. National Research Council and Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.Google Scholar
  50. Nunes, T., Dorneles, B. V., Lin, P.-J., & Rathgeb-Schnierer, E. (2016). Teaching and learning about whole numbers in primary school. Dordrecht: Springer.Google Scholar
  51. Ostad, S. (1998). Developmental differences in solving simple arithmetic word problems and simple number-fact problems: A comparison of mathematically normal and mathematically disabled children. Mathematical Cognition, 4(1), 1–19.Google Scholar
  52. Peltenburg, et al. (2012). Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure. Educational Studies in Mathematics, 79(3), 351–369.Google Scholar
  53. Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of pre-algebra understandings. In A. P. Shulte & A. F. Coxford (Eds.), The ideas of algebra, K-12 (pp. 78–90). National Council of Teachers of Mathematics, 11906 Association Dr., Reston, VA 22091.Google Scholar
  54. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogy in the mathematics classroom. Science, 316, 1128–1129.Google Scholar
  55. Savard, A., & Polotskaia, E. (2017). Who’s wrong? Tasks fostering understanding of mathematical relationships in word problems in elementary students. ZDM, 49(6), 823–833.Google Scholar
  56. Sun, X. H. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76(1), 65–85.Google Scholar
  57. Sun, X. H. (2013). The structures, goals and pedagogies of “variation problems” in the topic of addition and subtraction of 0–9 in Chinese textbooks and reference books. Paper presented in Eighth Congress of European Research in Mathematics Education (CERME 8), Feb. 6–10, 2013, Antalya, Turkey.Google Scholar
  58. Sun, X. H. (2015). Chinese core tradition to whole number arithmetic. In X. Sun,, B. Kaur, & J. Novotná, (Eds.), Proceedings of the 23rd ICMI Study on primary mathematics study on whole numbers’ (pp. 140–148). China, Macao: University of Macau, retrieved on February 20, 2016, from http://www.umac.mo/fed/ICMI23/proceedings.html.
  59. Sun, X. H. (2018). Uncovering Chinese pedagogy: Spiral variation—The unspoken principle of algebra thinking used to develop Chinese curriculum and instruction of the “two basics”. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt & B. Xu (Eds.), Invited lectures from the 13th International Congress on Mathematical Education (pp. 651–669). Cham: Springer.Google Scholar
  60. Sun, X. H. (2019). Bridging whole numbers and fractions: Problem variations in Chinese mathematics textbook examples. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-01013-9 (this issue).Google Scholar
  61. Sun, X. H., Chambris, C., Sayers, J., Siu, M. K., Cooper, J., Dorier, J., et al. (2018). What and why of whole number arithmetic: Foundational ideas from history, language, and societal changes. In M. G. Bartolini Bussi & X. H. Sun (Eds.), Building the foundation: Whole numbers in the primary grades (pp. 91–124). New York: Springer.Google Scholar
  62. Sun, X. H., Kaur, B., & Novotná, J. (Eds.)., (2015). Primary mathematics study on whole numbers: ICMI Study 23 Conference Proceedings, June 3–7, 2015 in Macau, China. University of Macau. Retrieved from http://www.umac.mo/fed/ICMI23/proceedings.html.
  63. Sun, X. H., Neto, T., & Ordóñez, L. (2013). Different features of task design associated with goals and pedagogies in Chinese and Portuguese textbooks: The case of addition and subtraction. In C. Margolinas (Ed.), Proceedings of ICMI Study 22 on task design in mathematics education (pp. 409–418). Oxford, United Kingdom. Retrieved February 10 from https://hal.archives-ouvertes.fr/hal-00834054v3.
  64. Thanheiser, E., & Melhuish, K. (2019). Leveraging variation of historical number systems to build understanding of the base-ten place-value system. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0984-7 (this issue).Google Scholar
  65. Venkat, H., & Mathews, C. (2019). Improving multiplicative reasoning in a context of low performance. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-0969-6 (this issue).Google Scholar
  66. Verschaffel, L., Greer, B., Van Doren W., & Mukhopadhyway (Eds.). (2009). Words and worlds: Modelling verbal descriptions of situations (pp. 21–38). Rotterdam: Sense Publishers.Google Scholar
  67. Vondrová, N., Novotná, J., J., & Radka Havlíčková, R. (2019). The influence of situational information on pupils’ achievement in additive word problems with several states and transformations. ZDM Mathematics Education.  https://doi.org/10.1007/s11858 (this issue).Google Scholar
  68. Xie, X., & Carspecken, P. (2008). Philosophy, learning and the mathematics curriculum. Dialectical materialism and pragmatism related to Chinese and U.S. mathematics curriculum. Rotterdam: Sense Publishers.Google Scholar
  69. Xin, Y. P. (2012). Conceptual model-based problem solving: Teach students with learning difficulties to solve math problems. Rotterdam: Sense Publishers.Google Scholar
  70. Xin, Y. P. (2015). Conceptual model-based problem solving: Emphasizing pre-algebraic conceptualization of mathematical relations. In E. A. Silver & P. A. Kenney (Eds.), More lessons learned from research: Useful and useable research related to core mathematical practices (pp. 235–246). Reston: National Council of Teachers of Mathematics (NCTM).Google Scholar
  71. Xin, Y. P. (2019). The effect of a conceptual model-based approach on ‘additive’ word problem solving of elementary students struggling in mathematics. ZDM Mathematics Education.  https://doi.org/10.1007/s11858-018-1002-9/ (this issue).Google Scholar
  72. Xin, Y. P., Liu, J., & Zheng, X. (2011a). A cross-cultural lesson comparison on teaching the connection between multiplication and division. School Science and Mathematics, 111(7), 354–367.Google Scholar
  73. Xin, Y. P., Zhang, D., Park, J. Y., Tom, K., Whipple, A., & Si, L. (2011b). A comparison of two mathematics problem-solving strategies: Facilitate algebra-readiness. The Journal of Educational Research, 104, 381–395.Google Scholar

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© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.University of MacauMacao SARChina
  2. 2.Purdue UniversityWest LafayetteUSA
  3. 3.Middle Tennessee State UniversityMurfreesboroUSA

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