The Key to Understanding Fractions

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Among the many difficult mathematical concepts for children to understand, fractions ranks as one of the most difficult. The complex concept poses a lot of problems not only for children, but for adults as well. The only way to build a conceptual understanding of fractions is to build on children’s informal knowledge of partitioning and fair share. Throughout the paper I will discuss how we can use these simpler topics as a starting point in the uphill battle to understanding this extremely difficult concept. I will identify common partitioning strategies used by children, and shed light on the way these strategies can be used to tackle the remaining four subconstructs, as well as guide in the understanding of multiplication and division of fractions.

Five subconstructs make up a whole, which is used to explain all definitions of the fraction concept. “The understanding of fractions depends on gaining and understanding of each of these different meanings, as well as of their confluence” (Charalambous & Pitta-Pantazi, 2007, p. 295).

The ratio construct of fractions focuses on the comparison of two quantities of the same type. An important realization students need to make is that there is a direct relationship between two quantities. When one quantity in the relationship changes so does the other, therefore the relationship of the two quantities remains the same. If both numbers in the ratio are multiplied by any number other than zero the ratio remains the same. The operator construct can be explained as taking a set or region and mapping it into another set or region. The quotient construct allows fractions to be seen as division problems. These problems are usually situations where students must partition objects into an equal number of parts. They need to understand that the dividend is the number of parts in each share, while the divisor refers to the fraction name of the share. The measure construct is usually used with a number line where students must identify the length of a certain segment (Charalambous & Pitta-Pantazi, 2007). Finally, the fifth, and newly added is the part-whole construct. Before this was added as the fifth part to the whole, it was thought to be so embedded in the other constructs that it did not need to be specified on its own. “The part-whole subconstruct of rational numbers, along with the process of partitioning, is considered a fundament for developing understanding of the four subordinate constructs of fractions” (Charalambous & Pitta-Pantazi, 2007, p. 295). The part-whole subconstruct of fractions is a situation where a continuous area or a set of discrete objects is partitioned into equal parts. In this case, “the fraction represents a comparison between the number of parts of the partitioned unit to the total number of parts in which the unit is partitioned” (Charalambous & Pitta-Pantazi, 2007, p. 295). Students need to develop an understanding of the part-whole subconstruct before they can begin to understand those that follow. The key to developing an understanding of this concept is to build on students’ informal knowledge of partitioning. Knowing that sets and quantities can be partitioned into equal-sized parts, and understanding the importance of equal-sized partitions is crucial to recognising the part-whole relationship between the numerator and denominator in fractions. According to Kieren (1983), partitioning experiences may be as important to the development of rational number concepts as counting experiences are to the development of whole number concepts.

That being said, it is important to understand the strategies our students use to partition objects, so we can help them mature in their understanding of the fraction concept. Charles and Nason (2000) do a great job explaining and categorizing the strategies most young children use when partitioning. In their study they chose to focus on learning fractions in relation to the partitive...
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