### Construct a Sum: A Measure of Children’s Understanding of Fraction Size (Behr, Wachsmuth, & Post, 1985)

Behr, M. J., Wachsmuth, I., & Post, T. R. (1985). Construct a Sum: A Measure of Children’s Understanding of Fraction Size. *Journal for Research in Mathematics Education*, 16(2), 120-131. Retrieved from http://www.jstor.org/stable/748369

The authors used a “Construct a Sum” task with 4th and 5th graders (part of the Rational Numbers Project) to assess their conceptual understanding of rational numbers (including their abilities to order fractions and identify instances of fractional equivalence).

“An understanding of fraction size is important for children in performing computations and solving problems that involve rational number ideas” (p. 120).

Children’s understanding of fraction size may be assessed via the following skills: (Wachsmuth, Behr, & Post, 1983)

- Order and equivalence
- Estimating the location of a fraction on a number line
- Completing an operation with fractions

“The Second National Assessment of Educational Progress (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980) found that only 24% of the 13-year-olds and 37% of the 17-year-olds in the sample were able to correctly estimate the sum of 12 13 and 7 8 given the choices of 1, 2, 19, and 21. The two most frequent responses were 19 and 21” (p. 120).

***

The authors used a “Construct a Sum” task with 4th and 5th graders (part of the Rational Numbers Project) to assess their conceptual understanding of rational numbers (including their abilities to order fractions and identify instances of fractional equivalence).

“A successful solution within the constraints required at least: 1. Knowledge that each fraction addend must be less than 1 and that the numerator must consequently be less than the denominator 2. Knowledge that each fraction addend represents a quantity greater than zero and that the combination of two such quantities results in a third that is greater than either 3. Knowledge that if one addend is small relative to 1, the other must be large, and vice versa 4. Ability to construct a trial addend, estimate the size of the interval between that addend and 1, and estimate and construct, from the whole numbers given, either the largest fraction less than the interval or the smallest fraction greater than the interval”

“The deviation from 1 of each child’s responses was computed as a percent. The arithmetic mean of the three deviations for each child was used to define performance categories of high, middle, and low scores. A high score meant that the average deviation was less than or equal to 10%; a middle score, that the average deviation was between 10 and 30%; and a low score, that the average deviation was greater than or equal to 30%. The criteriawere set on a pragmatic basis without theoretical guidelines about what constituted high, middle, or low scores.”

Strategies deduced: Correct Reference Point Comparison (CR), Mental Algorithmic Computation (MC), Incorrect Reference Point Comparison (IR), Mental Algorithmic Computation Based on an Incorrect Algorithm (MCI), Gross Estimate (G)

“The high scorers almost uniformly used estimating procedures in the solution process. These procedures generally referred to some intermediate “reference point” (Trafton, 1978). The high scorers also displayed an ability for the spontaneous and flexible application of fraction order and equivalence concepts. It appears that a combination of skills in estimation and a firm grasp of order and equivalence notions are a prerequisite to success on the task.”

“We hypothesize an interactive relation between order and equivalence on the one hand and estimation on the other. That is, an improved understanding of order and equivalence results in more accurate estimation, which in turn results in a higher level of perception of rational number size.”

“Given the amount of instructional time, the degree of special attention, the extensive use of manipulative aids, and the amount of time devoted to developing rational number concepts, it is surprising that 20 of the 41 responses (49%) given by these students in the middle of Grade 5 — those responses in the categories of mental computation based on an incorrect algorithm, gross estimate, and other — reflected a process of fraction addition that was based on the incorrect algorithm of adding numerators and denominators or on some other procedure reflecting little or no comprehension of fraction addition or rational number size.”

“a high level of understanding for rational number order and equivalence appears necessary to the ability to give estimates of rational numbers and rational number sums. A “workable” combination of (a) order and equivalence and (b) estimation is fundamental to the development of a viable quantitative concept of rational number.”

Possible pre-post test questions:

- Construct a fraction close but not equal to 1, and then construct another closer still
- Construct a fraction closer to 1 than 5/6, 7/8, or…
- Construct a fraction not equal to 1/2 but closer to 1/2 than 3/7, 3/8,…
- Construct a fraction greater than 1 but closer to 1 than 7/8, 5/9, or…
- Construct a fraction greater than 1/2 but closer to 1/2 than 3/8, 2/5, or ….

“Such problems would force the child to think about fraction size in relative terms, either relative to a single reference point or relative to several reference points.”