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Strategy ranges: describing change in prospective elementary teachers’ approaches to mental computation of sums and differences

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Abstract

This study investigated the sets of mental computation strategies used by prospective elementary teachers to compute sums and differences of whole numbers. In the context of an intervention designed to improve the number sense of prospective elementary teachers, participants were interviewed pre/post, and their mental computation strategies were analyzed. The analysis led to the identification of the strategy ranges used by the participants, as well as descriptions of changes pre/post in those strategy ranges. This article illustrates how strategy ranges, as an analytic tool, afford useful descriptions of the repertoires of mental computation strategies that individuals use.

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Notes

  1. It is beyond the scope of this paper to attempt to account for the differential changes in PTs’ strategy ranges, e.g., to explain why Zelda’s addition strategy range changed substantially, while her subtraction strategy range did not. The author’s dissertation includes two case studies of the number sense development of individual PTs. These case studies attempt to answer such fine-grained questions (Whitacre 2012).

  2. Note that, in theory, Inflexible need not imply MASA-bound. However, individuals who use only one strategy to perform an operation mentally do tend to be MASA-bound. Likewise, Heirdsfield and Cooper (2004) described accurate, inflexible mental calculators as relying on the MASAs specifically. It is not clear how to integrate invalid strategies into the picture of strategy ranges. This matter will require further attention.

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Acknowledgments

I am grateful to the editor and anonymous reviewers for their very thoughtful reading and helpful feedback. I thank Dr. Susan Nickerson for her efforts as the instructor of the course and for her mentorship. Finally, I thank the interview participants for their time and willingness to share their thinking.

Author information

Authors and Affiliations

  1. Florida State University, Tallahassee, FL, USA

    Ian Whitacre

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  1. Ian Whitacre
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Correspondence to Ian Whitacre.

Appendices

Appendix 1: Mental computation interview tasks

Instructions

For the following tasks, please find an exact answer mentally and explain your thinking. Each task involves a story about Bobo, who sells oboes.

(Addition Story)

Bobo’s oboe business is booming! However, he could use some help keeping track of his sales.

  • A1. Bobo sells 37 oboes 1 month and 52 the next month. How many oboes did he sell in those 2 months?

  • A2. Bobo sells 64 oboes 1 month and 87 the next month. How many oboes did he sell in those 2 months?

  • A3. Bobo sells 96 oboes 1 month and 157 the next month. How many oboes did he sell in those 2 months?

  • A4. Bobo sells 38 oboes 1 month and 99 the next month. How many oboes did he sell in those 2 months?

(Subtraction Story)

As everyone knows, oboes don’t grow on trees. Bobo has to spend money to make money.

  • S1. If Bobo buys an oboe for $34 and then sells it for $78, how much money does he make?

  • S2. If Bobo buys an oboe for $52 and then sells it for $178, how much money does he make?

  • S3. If Bobo buys an oboe for $45 and then sells it for $82, how much money does he make?

  • S4. If Bobo buys an oboe for $49 and then sells it for $125, how much money does he make?

Appendix 2

Coding scheme for mental addition strategies

Strategy

Description

MASA

The student used the mental analogue of the standard (USA) addition algorithm. Language such as “carry the one” often accompanied the use of this strategy. Students generally used non-place-value language

Right to Left

The student added place-value-wise from right to left but did not necessarily picture the digits aligned, as in the standard algorithm. In contrast to the MASA, the student used place-value language

Left to Right

The student added place-value-wise from left to right. Typically, she used place-value language

Aggregation

The student began with one of the two addends and added the other one on in convenient chunks, generally working from big to small and keeping a running subtotal

Giving

The student altered the problem such that part of one addend (usually a small number of ones) was added (“given”) to the other prior to finding their sum

Single Compensation

The student altered one of the two addends (usually rounding up or down to the nearest multiple of ten) prior to performing the addition. The student added the rounded numbers and then compensated for rounding

Double Compensation

The student altered both addends (usually rounding them up or down to the nearest multiple of ten) prior to performing the addition. The student added the rounded numbers and then compensated for rounding

Appendix 3

Coding scheme for mental subtraction strategies

Strategy

Description

MASA

The student used the mental analogue of the standard (USA) subtraction algorithm. Language such as “borrowing” often accompanied the use of this strategy

Right to le

The student subtracted place-value-wise from right to left but without visualizing the numbers aligned as in the standard algorithm

Left to Right

The student subtracted place-value-wise from left to right

Aggregation

The student either (a) began with the subtrahend and added onto it in convenient chunks until the minuend was reached or (b) began with the minuend and subtracted off the subtrahend in convenient chunks. The student kept a cumulative mental record of the amount added or subtracted. In the case of adding on to the subtrahend, this amount gave the difference. In the case of subtracting from the minuend, the result gave the difference

Minuend Compensation

The student altered the minuend (often rounding up or down to a multiple of ten) prior to performing the subtraction. The student found the difference between the subtrahend and rounded minuend and then compensated appropriately for rounding

Subtrahend Compensation

The student altered the subtrahend (often rounding up or down to the nearest multiple of ten) prior to performing the subtraction. The student found the difference between the minuend and rounded subtrahend and then compensated for rounding. Specifically, she compensated correctly: She added to the difference to compensate for having added to the subtrahend, or she subtracted from the difference to compensate for having subtracted from the subtrahend

Shifting the difference

The student added the same amount to, or subtracted the same amount from, both the minuend and subtrahend. She then found the difference between the rounded numbers

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Whitacre, I. Strategy ranges: describing change in prospective elementary teachers’ approaches to mental computation of sums and differences. J Math Teacher Educ 18, 353–373 (2015). https://doi.org/10.1007/s10857-014-9281-8

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