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Integers as Directed Quantities

  • Nicole M. Wessman-EnzingerEmail author
Chapter
Part of the Research in Mathematics Education book series (RME)

Abstract

Mathematics education researchers have long pursued—and many still pursue—an ideal instructional model for operations on integers. In this chapter, I argue that such a pursuit may be futile. Additionally, I highlight that ideas of relativity have been overlooked; and, I contend that current uses of translation within current integer instructional models do not align with learners’ inventions. Yet, conceptions of relativity and translation are essential for making sense of integers as directed quantities. I advocate for drawing on learners’ unique conceptions and actions about directed number in developing instructional models. Providing evidence of student work from my research, I illustrate the powerful constructions of relativity and translation as students engage with directed quantities.

Keywords

Conceptual models Integers Integer addition and subtraction Integer instructional models Integer operations Number line 

References

  1. Altiparmak, K., & Özdoğan, E. (2010). A study on the teaching of the concept of negative numbers. International Journal of Education in Science and Technology, 41(1), 31–47.  https://doi.org/10.1080/00207390903189179CrossRefGoogle Scholar
  2. Barrett, J. E., Sarama, J., Clements, D. H., Cullen, C., McCool, J., Witkowski-Rumsey, C., & Klanderman, D. (2012). Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study. Mathematical Thinking and Learning, 14(1), 28–54.  https://doi.org/10.1146/annurev.psych.59.103006.093639CrossRefGoogle Scholar
  3. Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645.  https://doi.org/10.1146/annurev.psych.59.103006.093639CrossRefGoogle Scholar
  4. Beatty, R. (2010). Behind and below zero: Sixth grade students use linear graphs to explore negative numbers. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 219–226). Columbus, OH: The Ohio State University.Google Scholar
  5. Bell, A. (1982). Teaching theories in mathematics. In A. Vermandel (Ed.), Proceedings of the 6th Conference of the International Group for the Psychology of Mathematics Education (pp. 207–213). Antwerp, Belgium: PME.Google Scholar
  6. Bishop, J. P., Lamb, L., Philipp, R., Schappelle, B., & Whitacre, I. (2010). A developing framework for children’s reasoning about integers. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 695–702). Columbus, OH: The Ohio State University.Google Scholar
  7. Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2014). Using order to reason about negative numbers: the case of violet. Educational Studies in Mathematics, 86(1), 39–59.  https://doi.org/10.1007/s10649-013-9519-xCrossRefGoogle Scholar
  8. Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2016). Leveraging structure: Logical necessity in the context of integer arithmetic. Mathematical Thinking and Learning, 18(3), 209–232.  https://doi.org/10.1080/10986065.2016.1183091CrossRefGoogle Scholar
  9. Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning: An analysis of children’s thinking and the history of mathematics. Journal for Research in Mathematics Education, 45(1), 19–61.  https://doi.org/10.5951/jresematheduc.45.1.0019CrossRefGoogle Scholar
  10. Bofferding, L. (2010). Addition and subtraction with negatives: Acknowledging the multiple meanings of the minus sign. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 703–710). Columbus, OH: The Ohio State University.Google Scholar
  11. Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194–245.  https://doi.org/10.5951/jresematheduc.45.2.0194CrossRefGoogle Scholar
  12. Bofferding, L., Aqazade, M., & Farmer, S. (2018). Playing with integer concepts: A quest for structure. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 3–23). Cham, Switzerland: Springer.CrossRefGoogle Scholar
  13. Bofferding, L., & Farmer, S. (2018). Most and least: Differences in integer comparisons based on temperature comparison language. International Journal of Science and Mathematics Education.  https://doi.org/10.1007/s10763-018-9880-4
  14. Bofferding, L., & Wessman-Enzinger, N. M. (2017). Subtraction involving negative numbers: Connecting to whole number reasoning. The Mathematics Enthusiast, 14, 241–262 https://scholarworks.umt.edu/tme/vol14/iss1/14Google Scholar
  15. Bofferding, L., & Wessman-Enzinger, N. M. (2018). Nuances of prospective teachers’ interpretations of integer word problems. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 289–295). New York: Springer.CrossRefGoogle Scholar
  16. Bolyard, J., & Moyer-Packenham, P. (2006). The impact of virtual manipulatives on student achievement in integer addition and subtraction. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 879–881). Mérida, Yucatán: Universidad Pedagógica Nacional.Google Scholar
  17. Bruno, A., & Martinon, A. (1996). Beginning learning negative numbers. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 161–168). Valencia, Spain: PME.Google Scholar
  18. Carpenter, T. P., Fennema, E., Franke, M., Levi, L., & Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.  https://doi.org/10.1111/chso.12047CrossRefGoogle Scholar
  19. Carr, K., & Katterns, B. (1984). Does the number line help? Mathematics in School, 30–34.Google Scholar
  20. Carraher, D., Schliemann, A. D., & Brizuela, B. M. (2001). Can young students operate on unknowns? In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 130–138). Utrecht, The Netherlands: PME.Google Scholar
  21. Chiu, M. M. (2001). Using metaphors to understand and solve arithmetic problems: Novices and experts working with negative numbers. Mathematical Thinking and Learning, 3(2–3), 93–124.  https://doi.org/10.1111/chso.12047CrossRefGoogle Scholar
  22. Durell, F., & Robbins, E. R. (1897). A school algebra. New York, NY: Charles E. Merrill.Google Scholar
  23. Ernest, P. (1985). The number line as a teaching aid. Educational Studies in Mathematics, 16(4), 411–424.  https://doi.org/10.1111/chso.12047CrossRefGoogle Scholar
  24. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: D. Reidel Publishing.Google Scholar
  25. Galbraith, M. J. (1974). Negative numbers. International Journal of Mathematical Education in Science and Technology, 5(1), 83–90.  https://doi.org/10.1080/002073974005011CrossRefGoogle Scholar
  26. Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transitions from arithmetic to algebra. Educational Studies in Mathematics, 49, 171–192.  https://doi.org/10.1023/A:1016210906658CrossRefGoogle Scholar
  27. Gallardo, A. (2003). “It is possible to die before being born.” Negative integers subtraction: A case study. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the Joint Meeting of PME 27 and PME-NA 25 (Vol. 2, pp. 405–411), Honolulu, HI.Google Scholar
  28. Goldin-Meadow, S., Cook, S. W., & Mitchell, Z. (2009). Gesturing gives children new ideas about math. Psychological Science, 20(3), 1–6.  https://doi.org/10.1111/j.1467-9280.2009.02297.xCrossRefGoogle Scholar
  29. Heeffer, A. (2011). Historical objections against the number line. Science & Education, 20(9), 863–880.  https://doi.org/10.1007/s11191-011-9349-0CrossRefGoogle Scholar
  30. Henley, A. T. (1999). The history of negative numbers (Unpublished doctoral dissertation, South Bank University).Google Scholar
  31. Herbst, P. (1997). The number-line metaphor in the discourse of a textbook series. For the Learning of Mathematics, 17(3), 36–45.Google Scholar
  32. Iannone, P., & Cockburn, A. D. (2006). Fostering conceptual mathematical thinking in the early years: A case study. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 329–336). Prague, Czech Republic: PME.Google Scholar
  33. Janvier, C. (1985). Comparison of models aimed at teaching. In L. Streefland (Ed.), Proceedings of the 9th Conference of the Psychology of Mathematics Education (pp. 135–140). Noordwijkerhout, The Netherlands: International Group for the Psychology of Mathematics Education.Google Scholar
  34. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic books.Google Scholar
  35. Larsen, S., & Saldanha, L. (2006). Function composition as combining transformations: Lessons learned from the first iteration of an instructional experiment. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 790–797). Mérida, Yucatán: Universidad Pedagógica Nacional.Google Scholar
  36. Liebeck, P. (1990). Scores and forfeits. Educational Studies in Mathematics, 21(3), 221–239.  https://doi.org/10.1007/BF00305091CrossRefGoogle Scholar
  37. Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in “filling” in gaps in children’s extension of their number concept to include negative numbers. Educational Studies in Mathematics, 39, 131–147.  https://doi.org/10.1023/A:1003726317920CrossRefGoogle Scholar
  38. Loomis, E. (1857). Treatise on algebra. (12th ed. New York, NY: Harper & Brothers.Google Scholar
  39. Marthe, P. (1979). Additive problems and directed numbers. In D. Tall (Ed.), Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics Education (pp. 317–323). Coventry, England: PME.Google Scholar
  40. Marthe, P. (1982). Research on the appropriation of the additive group of directed numbers. In A. Vermandel (Ed.), Proceedings of the 6th Conference of the International Group for the Psychology of Mathematics Education (pp. 162–167). Antwerp, Belgium: PME.Google Scholar
  41. Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: Adapting and reinterpreting physical environments in the development of the fraction concept. Cognitive Science, 29(4), 587–625.  https://doi.org/10.1207/s15516709cog0000_15CrossRefGoogle Scholar
  42. Martínez, A. A. (2006). Negative math: How mathematical rules can be positively bent. Princeton, NJ: Princeton University Press.Google Scholar
  43. Mathematics Education Researchers. [ca. 2017]. In Facebook [Group page]. Retrieved November 9, 2017, from https://www.facebook.com/groups/mathedresearchers/.
  44. Moreno, R., & Mayer, R. E. (1999). Multimedia-supported metaphors for meaning making in mathematics. Cognition and Instruction, 17(3), 215–248.  https://doi.org/10.1207/S1532690XCI1703_1CrossRefGoogle Scholar
  45. Moyer, P. S. (2001). Are we having fun yet? How teachers use manipluatives to teach mathematics. Educational Studies in Mathematics, 47, 175–197.  https://doi.org/10.1023/A:1014596316942CrossRefGoogle Scholar
  46. Murray, E. (2018). Using models and representations: Exploring the chip model for integer subtraction. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 231–255). Cham, Switzerland: Springer.CrossRefGoogle Scholar
  47. Murray, J. C. (1985). Children’s informal conceptions of integers. In L. Streefland (Ed.), Proceedings of the 9th Conference of the Psychology of Mathematics Education (pp. 147–153). Noordwijkerhout, The Netherlands: International Group for the Psychology of Mathematics Education.Google Scholar
  48. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  49. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Author. Retrieved from http://www.corestandards.org/the-standards.
  50. Nicodemus, R. (1993). Transformations. For the Learning of Mathematics, 13(1), 24–29.Google Scholar
  51. Nurnberger-Haag, J. (2007). Integers made easy: Just walk it off. Mathematics Teaching in the Middle School, 13(2), 118–121.Google Scholar
  52. Peled, I., & Carraher, D. W. (2008). Signed numbers and algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 303–328). New York, NY: Routledge.Google Scholar
  53. Pettis, C., & Glancy, A. W. (2015). Understanding students’ challenges with integer addition and subtraction through analysis of representations. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 165–172). East Lansing, MI: Michigan State University.Google Scholar
  54. Poirier, L., & Bednarz, N. (1991). Mental models and problem solving: An illustration with complex arithmetical problems. In R. Underhill (Ed.), Proceedings of the 13th Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 133–139). Blacksburg, VA: Christiansburg Printing Company.Google Scholar
  55. Reeves, C. A., & Webb, D. (2004). Balloons on the rise: A problem-solving introduction to integers. Mathematics Teaching in the Middle School, 9(9), 476–482.Google Scholar
  56. Saxe, G. B., Diakow, R., & Gearhart, M. (2013). Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principle representational context. ZDM Mathematics Education, 45(3), 343–364.  https://doi.org/10.1007/s11858-012-0466-2CrossRefGoogle Scholar
  57. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition; Number conceptions underlying the development of analysis in 17–19th century France and Germany. New York, NY: Springer.Google Scholar
  58. Schultz, K. (2017, November 7). Fantastic beasts and how to rank them. The New Yorker. Retrieved from https://www.newyorker.com/magazine/2017/11/06/is-bigfoot-likelier-than-the-loch-ness-monster.
  59. Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993). Positives about negatives: A case study of an intermediate model for signed numbers. Journal of the Learning Sciences, 3(1), 37–92.  https://doi.org/10.1207/s15327809jls0301_2CrossRefGoogle Scholar
  60. Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference—A longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389–408.  https://doi.org/10.1007/s10649-011-9305-6CrossRefGoogle Scholar
  61. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, MA: Cambridge University Press.  https://doi.org/10.1017/CBO9780511499944CrossRefGoogle Scholar
  62. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games – but not circular ones – improves low-income preschoolers’ numerical understanding. Journal of Experimental Psychology, 101(3), 545–560.  https://doi.org/10.1037/a0014239CrossRefGoogle Scholar
  63. Smith, L. B., Sera, M., & Gattuso, B. (1988). The development of thinking. In R. Sternberg & E. Smith (Eds.), The psychology of human thought (pp. 366–391). Cambridge, UK: Cambridge University Press.Google Scholar
  64. Steffe, L. P. (1983). Children’s algorithms as schemes. Educational studies in Mathematics, 14, 109–125.  https://doi.org/10.1007/BF00303681CrossRefGoogle Scholar
  65. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  66. Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428–464.  https://doi.org/10.5951/jresematheduc.43.4.0428CrossRefGoogle Scholar
  67. Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19(2), 115–133.  https://doi.org/10.2307/749406CrossRefGoogle Scholar
  68. Tillema, E. S. (2012). What is the difference? Using contextualized problems. Mathematics Teaching in the Middle School, 17(8), 472–478.  https://doi.org/10.5951/mathteacmiddscho.17.8.0472CrossRefGoogle Scholar
  69. Tsang, J. M., Blair, K. P., Bofferding, L., & Schwartz, D. L. (2015). Learning to “see” less than nothing: Putting perceptual skills to work for learning numerical structure. Cognition and Instruction, 33, 154–197.  https://doi.org/10.1080/07370008.2015.1038539CrossRefGoogle Scholar
  70. Ulrich, C. L. (2012). Additive relationships and signed quantities (Doctoral dissertation, University of Georgia).Google Scholar
  71. Ulrich, C. L. (2013). The addition and subtraction of signed quantities. In R. Mayes & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 127–141). Laramie, WY: University of Wyoming.Google Scholar
  72. Vergnaud, G. (1982). Cognitive psychology and didactics: Signified/signifier and problems of reference. In A. Vermandel (Ed.), Proceedings of the 6th Conference of the International Group for the Psychology of Mathematics Education (pp. 70–76). Antwerp, Belgium: PME.Google Scholar
  73. Vig, R., Murray, E., & Star, J. R. (2014). Model breaking points conceptualized. Educational Psychology Review, 26(1), 73–90.  https://doi.org/10.1007/s10648-014-9254-6CrossRefGoogle Scholar
  74. Wallis, J. (1685). A treatise of algebra, both historical and practical shewing the soriginal, progress, and advancement thereof, from time to time, and by what steps it hath attained to the height at which it now is: with some additional treatises…Defense of the treatise of the angle of contact. Discourse of combinations, alternations, and aliquot parts. London, UK: John Playford, for Richard Davis.Google Scholar
  75. Wessman-Enzinger, N. M. (2015). Developing and describing the use and learning of Conceptual Models for Integer Addition and Subtraction of grade 5 students. Normal, IL: Proquest.CrossRefGoogle Scholar
  76. Wessman-Enzinger, N. M. (2018a). Descriptions of the integer number line in United States school mathematics in the 19th century. Mathematical Association of America Convergence: Convergence. Retrieved from https://www.maa.org/press/periodicals/convergence/descriptions-of-the-integer-number-line-in-united-states-school-mathematics-in-the-19th-century.
  77. Wessman-Enzinger, N. M. (2018b). Grade 5 children’s drawings for integer addition and subtraction open number sentences. Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2018.03.010Google Scholar
  78. Wessman-Enzinger, N. M. (2018c). Integer play and playing with integers. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 25–46). Cham, Switzerland: Springer.CrossRefGoogle Scholar
  79. Wessman-Enzinger, N. M. (in press). Consistency of integer number sentences to temperature problems. Mathematics Teaching in the Middle School.Google Scholar
  80. Wessman-Enzinger, N. M., & Bofferding, L. (2014). Integers: Draw or discard! game. Teaching Children Mathematics, 20(8), 476–480.  https://doi.org/10.5951/teacchilmath.20.8.0476CrossRefGoogle Scholar
  81. Wessman-Enzinger, N. M., & Bofferding, L. (2018). Reflecting on the landscape: Concluding remarks. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 289–295). Cham, Switzerland: Springer.CrossRefGoogle Scholar
  82. Wessman-Enzinger, N. M., & Mooney, E. S. (2014). Making sense of integers through story-telling. Mathematics Teaching in the Middle School, 20(4), 203–205.Google Scholar
  83. Wessman-Enzinger, N. M., & Tobias, J. (2015). Preservice teachers’ temperature stories for integer addition and subtraction. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 289–296). Hobart, Australia: PME.Google Scholar
  84. Wheeler, D., Nesher, P., Bell, A., & Gattegno, C. (1981). A research programme for mathematics education (I). For the Learning of Mathematics, 2(1), 27–29.Google Scholar
  85. Whitacre, I., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Bagley, S., & Schappelle, B. P. (2015). ‘Negative of my money, positive of her money’: Secondary students’ ways of relating equations to a debt context. International Journal of Mathematical Education in Science and Technology, 46(2), 234–249.  https://doi.org/10.1080/0020739X.2014.956822CrossRefGoogle Scholar
  86. Whitacre, I., Schoen, R. C., Champagne, Z., & Goddard, A. (2016). Relational thinking: What’s the difference? Teaching Children Mathematics, 23(5), 303–309.Google Scholar

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Authors and Affiliations

  1. 1.School of EducationGeorge Fox UniversityNewbergUSA

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