## 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.

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## 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.

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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.

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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

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### Keywords

- Integers
- Children’s thinking
- Negative numbers
- Case study
- Developmental milestones
- Ordinal numbers