Teaching prospective teachers about fractions: historical and pedagogical perspectives

Sustainability Development in Mathematics Education—A Case Study of What Kind of Meanings Do Prospective Class Teachers Find for the Mathematical Symbol “23”?

“…Fractions are those rational numbers which are written in the form m/n, in which m and n are integers (n = 0). This kind of definition is seen in many textbooks, followed by explanations of what the symbols mean and how to manipulate with them [12]. The argument is true also in Finnish mathematics textbooks, especially in grades 7-9.…”

“…Stewart [24] also referred to the multifaced construct of fractions in introducing the following subconstructs in her thesis: part-whole relationship: the ratio of the part to the whole; measurement: the position of a number on the number line; operator: a transformation; quotient: division; and ratio: relationship between two quantities. Park, Güler, and McGrory [12] approached fractions in their research framework from the point of four subconstructs to the development of fractions found in history: part-whole: conceptualizing a part of a whole as a new unit; measurement: finding fractions from whole numbers through measurement and proportions, addressing the need for a common unit of measurement for two quantities; division: finding the algebraic solution for an equation A X = B where A and B are whole numbers and A is nonzero; and set-theoretical: defining rational numbers as a set of ordered pairs consisting of whole numbers. The different meanings of the fraction concept introduced above give quite a good picture of how intricate this concept is [12,23,24]; it can be assigned various meanings according to the context in which the concept is used.…”

“…The research (e.g., [8][9][10][11]) shows that fractions are one of the most challenging areas in school mathematics. Pupils, prospective class and subject teachers, and sometimes even experienced teachers, have trouble with fractions, especially understanding fractions as numbers that extend the whole number system to rational numbers (e.g., [12,13]). To understand this extension, one must have a clear picture of the different meanings of fractions.…”

“…Typically, textbooks describe the ratio of a part to the whole as an area model (e.g., as a pie model), and a pie model for "2/3" is read for example as two thirds of the pizza. In addition, the part-whole approach may apply when teachers use area models divided into equal parts, or mention that a part of a collection or object can be expressed as a fraction with equal partitioning and counting [12].…”

“…According to Park, Güler and McGrory [12], the roots of the definition of rational numbers are based on a set-theoretical approach. Rational numbers Q are defined as follows: m n m, n ∈ Z, n = 0 (1)…”