Hierarchical Levels of Abilities that Constitute Fraction Understanding at Elementary School

Abstract

This article examines whether the 7 abilities found in a previous study carried out by the authors to constitute fraction understanding of sixth grade elementary school students determine hierarchical levels of fraction understanding. The 7 abilities were as follows: (a) fraction recognition, (b) definitions and mathematical explanations for fractions, (c) argumentations and justifications about fractions, (d) relative magnitude of fractions, (e) representations of fractions, (f) connections of fractions with decimals, percentages, and division, and (g) reflection during the solution of fraction problems. The sample comprised of 182 sixth grade students that were clustered into 3 categories by means of latent class analysis: those of low fraction understanding, those of medium fraction understanding, and those of high fraction understanding. It was found that low fraction understanding students were sufficient in fraction recognition and relative magnitude of fractions, those belonging to the medium category in fraction recognition, relative magnitude of fractions, as well as in connections with decimals, percentages and division and representations of fractions, while high fraction understanding students were sufficient in all 7 abilities. It was also found that these levels were stable across time; the hierarchical levels were the same across three measurements that took place. Possible implications for fraction understanding are discussed, and directions for future research are drawn.

Appendix

Items used to measure each ability:

Fraction recognition

Task 3

One of the following fractions differs from the others. Find that fraction and circle it.

$$ \begin{array}{ccccccccccccc}\hfill \frac{2}{7}\hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{3}{2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{14}{49}\hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{10}{35}\hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{4}{14}\hfill \end{array} $$

Definitions and mathematical explanations for fractions

Task 7

How many fractions exist in the interval 0–1. Explain your answer.

(Niemi, 1996c)

Argumentations and justifications about fractions

Task 11

For each of the following statements, circle T if it is true or F if it is false. You must also justify your answer. You should definitely justify your answer.

If I double both the numerator and the denominator of a fraction, then the formed fraction will have double the value of the initial one.

Justification:

……………………………………………………………………………………

(Lamon, 1999)

Reflection during the solution of fraction problems

Task 14

The following diagram represents a machine that outputs $\frac{2}{3}$ of the input quantity. What is the input quantity if the output quantity is equal to 12? You should definitely reason about your thinking and your answer.

(Charalambous & Pitta-Pantazi, 2007; Lamon, 1999)

Relative magnitude of fractions

Task 17

Order the fractions $\frac{1}{2}$, $\frac{4}{3}$, $\frac{2}{3}$, and $\frac{1}{4}$ starting from the smallest one.

…………………………………………………………………..

(Clarke & Roche, 2009)

Representations of fractions

Task 27

Marinos ate $\frac{1}{2}$ of a cake and Marina $\frac{3}{8}$ of the same cake. Construct a drawing to show what part each child ate and what part the two children ate together.

(Gagatsis et al., 2001)

Connections of fractions with decimals, percentages, and division

Task 30

Decide whether the following statements are correct or incorrect and circle C if they are correct and I if they are incorrect.

• 1. $\frac{2}{3}$ is the quotient of the division 2 ÷ 3.

  

• 2. $\frac{12}{7}$ is the quotient of the division 7 ÷ 12.

  

• 3. Three pizzas were evenly shared among six children. Each child got $\frac{3}{6}$ of the pizza.

  

(Kieren, 1993)