## Abstract

We report on interview results from a classroom teaching experiment in a Number and Operations course for prospective elementary teachers. Improving the number sense of this population is an important goal for mathematics teacher education, and researchers have found this goal to be difficult to accomplish. In earlier work, we devised a local instruction theory for the development of number sense, which focused on whole-number mental computation. In this study, the local instruction theory was applied to the rational-number domain, with the help of a framework for reasoning about fraction magnitude, and it guided instruction in the content course. We interviewed seven participants pre- and post-instruction, and we found that their reasoning on fraction comparison tasks improved. The participants made more correct comparisons, reasoned more flexibly, and came to favor less conventional and more sophisticated strategies. These improvements in number sense parallel those that we found previously in mental computation. In addition to the overall results, we highlight two cases of improvement that illustrate ways in which prospective elementary teachers’ reasoning about fraction magnitude can change.

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## Notes

We acknowledge that finding a common denominator can be a useful approach to fraction comparison tasks. We further acknowledge that this procedure can be performed with conceptual understanding and can work hand in hand with reasoning about fraction magnitude. However, we find that this procedure is widely used but rarely understood.

Smith’s (1995) original framework is based on the reasoning of K-12 students.

This scheme is a revised version of that of Smith (1995, pp. 45–47). In the presentation here, we organize fraction comparison strategies by perspective, and we order the perspectives in accord with the hypothetical learning trajectory: Transform, Parts, Reference Point, and Components. Note that this ordering applies broadly on the level of the perspectives. Naturally, the emergence of particular strategies does not follow a strict ordering.

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## Appendix: Fraction comparison strategies
^{Footnote 3}

### Appendix: Fraction comparison strategies

### Transform perspective

In general, we code strategies as *Transform* when one or both fractions are converted to an alternate form and the student’s activity/explanation appears to be procedural, as opposed to being grounded in a Parts interpretation.

*Reduce to Lower Terms.* The student compares the two fractions by first reducing one or both of them to lower terms. If the transformation results in the fractions being identical, the student concludes that they are equal. If the two fractions reduce to different fractions, the student compares these by some other strategy, such as the Numerator Principle.

*Raise to Higher Terms.* The student compares the two fractions by first raising one or both of them to higher terms. If the transformation results in the fractions being identical, the student concludes that they are equal. If the transformation results in different fractions, the student compares these by some other strategy, such as the Denominator Principle.

*Convert to Common Denominator.* The student compares two fractions by transforming one or both so that they have a common denominator. The student may then employ the Numerator Principle or some other strategy to make the comparison.

*Convert to Common Numerator.* The student compares two fractions by transforming one or both so that they have a common numerator. The student then employs the Denominator Principle to make the comparison. If the student applies a Parts interpretation, the perspective should be coded as *Parts.*

*Cross Multiply.* The student compares two fractions by first multiplying the numerator of each fraction by the denominator of the other fraction. If these products are equal, the student concludes that the fractions are equal. If the products are not equal, the student selects the fraction whose numerator was a factor in the greater product as the larger of the two fractions.

*Convert to Decimals.* The student compares two fractions by converting them to decimal form. The conversion may be accomplished by long division, recall, or some other method. The student then compares the decimal numbers to determine which of the given fractions is greater.

*Convert from Improper to Mixed.* The student compares two improper fractions by first transforming both of them to mixed numbers. The student then employs a different strategy to compare the remaining fractional parts of the two mixed numbers (In our data, mixed numbers were between 1 and 2).

#### Parts perspective

*Numerator Principle.* Given a comparison in which the denominators are equal, the student selects the fraction with the greater numerator as the larger of the two fractions. The student’s explanation addresses the fact that the denominators are equal and applies a Parts interpretation.

*Denominator Principle.* Given a comparison in which the numerators are equal, the student selects the fraction with the lesser denominator as the larger of the two fractions. The student’s explanation addresses the fact that the numerators are equal and applies a Parts interpretation.

*Compare Complements.* The student compares two fractions by comparing their complements and concludes that the fraction with the smaller complement is greater. The student’s explanation makes applies a Parts interpretation and makes explicit the logic that a smaller complement implies a greater fraction.

*Denominator Dominance.* Given a comparison in which neither the numerators nor the denominators are equal, the student selects the fraction with the lesser denominator as the larger of the two fractions. The student’s explanation applies a Parts interpretation to the comparison of the denominators but does not account for the fact that the numerators are not equal.

*Numerator Dominance.* Given a comparison in which neither the numerators nor the denominators are equal, the student selects the fraction with the greater numerator as the larger of the two fractions. The student’s explanation applies a Parts interpretation to the comparison of the numerators but does not account for the denominators being unequal.

*Combination.* In some cases, students combine the strategies above. For example, a student might compare 7/9 with 6/11 by noting that ninths are larger than elevenths and that seven is more than six, so that 7/9 is greater. This strategy essentially combines the Numerator Principle and the Denominator Principle into a two-part argument. We code such a strategy as Parts Combination and note the particular strategies that were combined.

#### Reference Point perspective

*Straddle*. The student compares two fractions by reasoning that one fraction is greater than a benchmark fraction and the other is less than that same benchmark fraction. The student concludes that the fraction that is greater than the benchmark is the greater of the two fractions.

*Distance Below.* The student compares two fractions by comparing them to a benchmark fraction. In cases where both fractions are less than the benchmark, the fraction that is closer to the benchmark is the larger of the two fractions. A special case of Distance Below occurs when proper fractions are compared to 1. We code this as the *Residual Strategy.*

*Distance Above.* The student compares two fractions by comparing them to a benchmark fraction. In cases where both fractions are greater than the benchmark, the fraction that is further from the benchmark is the larger of the two fractions.

#### Components perspective

*Larger Components.* The student compares two fractions by choosing the one with larger components as greater (e.g., 5/8 > 2/3 because 5 and 8 are larger numbers than 2 and 3).

*Additive Within.* Given a pair of proper fractions, the student compares the fractions by comparing the differences between the numerator and denominator in each. The student selects the fraction with the smaller numerator–denominator difference as the greater fraction. If the differences are equal, the student concludes that the fractions are equal.

*Additive Between.* Given a pair of proper fractions, the student compares the fractions by comparing the differences between the numerators (one numerator minus the other) and the differences between the denominators. If the differences are equal, the student concludes that the fractions are equal (We only saw this strategy employed in the case of a common difference).

*Multiplicative Within.* The student compares two fractions by comparing the numerator–denominator ratios in each. The student need not explicitly discuss ratios, but necessarily makes multiplicative comparisons between the numerators and denominators. In cases of proper fractions, the student concludes that the fraction with the greater numerator–denominator ratio is greater. If the ratios are equal, the student concludes that the fractions are equal. Multiplicative comparisons involved in this strategy may be exact or approximate.

*Multiplicative Between.* The student compares two fractions by comparing the numerator–numerator and denominator–denominator ratios. The student need not explicitly discuss ratios but necessarily makes multiplicative comparisons between numerators and between denominators.

*Iteration.* The student compares two fractions by comparing multiples of them, using the same multiplier for both. For example, 3/16 is compared to 13/60 by multiplying both fractions by 5, which yields 15/16 and 65/60. The student concludes that the fraction that results in a greater product is the greater of the two fractions.

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Whitacre, I., Nickerson, S.D. Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude.
*J Math Teacher Educ* **19**, 57–77 (2016). https://doi.org/10.1007/s10857-014-9295-2

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DOI: https://doi.org/10.1007/s10857-014-9295-2

### Keywords

- Prospective elementary teachers
- Fractions
- Number sense
- Local instruction theory