Skip to main content
SpringerLink
Log in

Using order to reason about negative numbers: the case of Violet

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

This case study illustrates how a 2nd-grade child, Violet, used an ordinal view of number to reason about positive and negative integers and arithmetic involving integers. Violet’s ordinal view of number facilitated her ability to reason about and correctly solve some integer-related problems and constrained her solutions to others. We demonstrate how Violet’s thinking evolved over time while she extended the properties of whole numbers and addition and subtraction to the integers. Using this case study as a basis, we propose a series of developmental milestones that build toward one’s understanding of integers and integer arithmetic in an order-based way. We believe that understanding Violet’s order-based reasoning can help us listen to other children.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. We use the terms negative number and negative integer as synonyms in this paper because the child in this study consistently referred to integers less than zero as negative numbers. She did not have occasion to reason about noninteger values less than zero. We acknowledge that, mathematically speaking, the set of negative integers is a subset of negative numbers, but in this paper we do not discuss nonwhole-rational numbers or irrational numbers that are less than zero.

  2. We define an ordinal view of number more broadly than the more traditional definition of ordinal numbers as acquiring and using the words first, second, third, etc. to indicate position in a series.

  3. Note that although we refer to Violet’s model, we do not know what Violet’s model is. We use the phrase Violet’s model throughout the rest of the paper to mean our model of Violet’s conceptual model for integer addition and subtraction.

  4. We do not view her use of the fact 3 − 3 = 0 as an instance of a formal view of number because she did not indicate that that this was a generalized principle that held for all numbers. Her intuitive use of 3 − 3 = 0 was a number fact that she used to subtract 5 from 3 and that leveraged the unique position of 0.

  5. In fact, none of the 40 second graders interviewed in the larger study solved 5 +  = 3 correctly; 85 % of these second graders said that this problem was not possible to solve as written. One child thought that the box might be a negative number but was unsure.

  6. For the problem 5 −  = 8, only one of the 40 second graders was able to solve it correctly; 80 % of the children said that this problem was impossible to solve, with the remaining responses including “3” or other numbers. Again, one student thought that the answer could be a negative number, but was unsure.

  7. We do not know whether Violet had conversations with others about negative numbers between interviews 2 and 3, and, if she did, what the nature of those conversations was.

  8. We believe that Violet’s intuitive use of the relationship that adding 7 moves in the opposite direction as adding negative 7 provides a solid foundation for her to develop and make sense of the more formal definition of additive inverses in the future, namely 7 = − (−7) and 7 + (−7) = 0. In her strategy, we see Violet’s emerging ideas about additive inverses.

References

  • Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397.

    Article  Google Scholar 

  • Baroody, A. J., & Wilkins, J. L. M. (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. V. Copley (Ed.), Mathematics in the early years (pp. 48–65). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Behrend, J. L., & Mohs, L. C. (2006). From simple questions to powerful connections: A two-year conversation about negative numbers. Teaching Children Mathematics, 12, 260–264.

    Google Scholar 

  • Bishop, J. P., Lamb, L. C., Philipp, R. A., Schappelle, B. P., & Whitacre, I. (2011). First graders outwit a famous mathematician. Teaching Children Mathematics, 17, 350–358.

    Google Scholar 

  • Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2013). Obstacles and affordances for integer reasoning: An analysis of children’s thinking and the history of mathematics. Journal for Research in Mathematics Education., in press.

  • Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26, 499–531.

    Article  Google Scholar 

  • Carpenter, T. P., & Peterson, P. L. (1988). Learning through instruction: The study of students’ thinking during instruction in mathematics. Educational Psychologist, 23, 79–85.

    Google Scholar 

  • Clements, D. H., & Sarama, J. (2007). Early childhood mathematics learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461–555). Charlotte, NC: Information Age.

    Google Scholar 

  • Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83–94.

    Article  Google Scholar 

  • Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). Reston, VA: National Council of Teacher of Mathematics.

    Google Scholar 

  • Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21(3), 180–206.

    Article  Google Scholar 

  • Fuson, K. C., Smith, S. T., & Lo Cicero, A. M. (1997). Supporting Latino first graders’ ten-structured thinking in urban classrooms. Journal for Research in Mathematics Education, 28(6), 738–766.

    Article  Google Scholar 

  • Glaeser, G. (1981). Epistemologie des nombres relatifs [Epistemology of signed numbers]. Recherches en Didactique des Mathematiques, 2(3), 303–346.

    Google Scholar 

  • Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11(1), 26–32.

    Google Scholar 

  • Henley, A. T. (1999). The history of negative numbers. Unpublished dissertation. South Bank University, London.

  • Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K–12 curriculum. In National Council of Teachers of Mathematics & Mathematical Sciences Education Board (Ed.), The nature and role of algebra in the K–14 curriculum: Proceedings of a national symposium. Washington, DC: National Academies Press.

    Google Scholar 

  • Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.

    Google Scholar 

  • Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Lamb, L. C., Bishop, J. P., Philipp, R. A., Schappelle, B. P., Whitacre, I., & Lewis, M. (2012). Developing symbol sense for the minus sign. Mathematics Teaching in the Middle School, 18(1), 5–9.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

    Google Scholar 

  • National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author.

    Google Scholar 

  • National Research Council (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

  • Peled, I. (1991). Levels of knowledge about negative numbers: Effects of age and ability. In F. Furinghetti (Ed.), Proceedings of the 15th international conference for the Psychology of Mathematics Education (Vol. 3, pp. 145–152). Assisi, Italy.

  • Peled, I., Mukhopadhyay, S., & Resnick, L. B. (1989). Formal and informal sources of mental models for negative numbers. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13th international conference for the Psychology of Mathematics Education (Vol. 3, pp. 106–110).

    Google Scholar 

  • Schifter, D., Russell, S. J., & Bastable, V. (2007). Number and operations, Part 3: Reasoning algebraically about operations, casebook, Developing Mathematical Ideas Series. Parsippany, NJ: Dale Seymour.

    Google Scholar 

  • Schubring, G. (2005). Conflicts between generalization, rigor, and intuition: Number concepts underlying the development of analysis in 17–19th century France and Germany. New York: Springer.

    Google Scholar 

  • Seife, C. (2000). Zero: The biography of a dangerous idea. New York: Penguin.

    Google Scholar 

  • Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). Albany, NY: State University of New York Press.

    Google Scholar 

  • Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307. doi:10.1016/S0732-3123(02)00075-5.

    Article  Google Scholar 

  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.

    Book  Google Scholar 

  • 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: Erlbaum.

    Google Scholar 

  • Thomaidis, Y., & Tzanakis, C. (2007). The notion of historical “parallelism” revisited: Historical evolution and students’ conception of the order relation on the number line. Educational Studies in Mathematics, 66(2), 165–183.

    Article  Google Scholar 

  • Thompson, P. W. (1982). Were lions to speak, we wouldn’t understand. Journal of Mathematical Behavior, 3(2), 147–165.

    Google Scholar 

  • Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19(2), 115–133.

    Article  Google Scholar 

  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Whitacre, I., Bishop, J. P., Lamb, L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. (2011). Integers: History, textbook approaches, and children’s productive mathematical intuitions. Proceedings of the 33 rd annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV.

  • Whitacre, I., Bishop, J. P., Lamb, L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. L. (2012). Happy and sad thoughts: An exploration of children’s integer reasoning. Journal of Mathematical Behavior, 31, 356–365.

    Article  Google Scholar 

  • Wilcox, V. B. (2008). Questioning zero and negative numbers. Teaching Children Mathematics, 15(4), 202–206.

    Google Scholar 

  • Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Thousand Oaks, CA: Sage.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Georgia, Athens, USA

    Jessica Pierson Bishop

  2. San Diego State University, San Diego, USA

    Lisa L. Lamb, Randolph A. Philipp & Bonnie P. Schappelle

  3. Florida State University, Tallahassee, USA

    Ian Whitacre

Authors
  1. Jessica Pierson Bishop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lisa L. Lamb
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Randolph A. Philipp
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ian Whitacre
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bonnie P. Schappelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jessica Pierson Bishop.

Additional information

This material is based upon work supported by the National Science Foundation under grant number DRL-0918780. Any opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bishop, J.P., Lamb, L.L., Philipp, R.A. et al. Using order to reason about negative numbers: the case of Violet. Educ Stud Math 86, 39–59 (2014). https://doi.org/10.1007/s10649-013-9519-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-013-9519-x

Keywords

Search

Navigation