also clear. Mathematically, this approach promotes an understanding of one particular aspect of rational number—the way rational numbers indicate parts of a whole. This part–whole subconstruct is one of the basic interpretations of rational numbers.

However, this introduction is grounded in additive thinking. It reinforces the very concept that students must change to master rational number. Children tend to treat the individual parts that result from a partition as discrete objects. The four pieces into which a pie is cut are just four pieces. Although the representation does have the potential to bring out the multiplicative relations inherent in the numbers—considering the shaded parts in relation to the whole—this is not what students naturally extract from the situations presented given their strong preconceptions regarding additive relationships.33

Recall that Wyatt, the fifth grader, asserted that 2/3 and 3/4 were the “same sized” number, supporting his erroneous claim with reference to pie charts. He explained that the picture showed they were both missing one piece. His lack of focus on the different relations that are implied in these two fractions is evident from his interpretation.

Alternative Instructional Approaches: Ratio and Sharing

For some time now, researchers have wondered whether alternative instructional approaches can help students overcome this misunderstanding. As Kieren34 points out, “… rather than relying on children’s well developed additive instincts we must find the intuitions and schemes that go beyond those that support counting. Whole number understandings are carefully built over a number of years; now we must consider how rational number understanding develops and is fostered.”

But what would such instruction look like? Over the last several years, a number of innovative approaches have been developed that highlight the multiplicative relations involved, a few of which are highlighted here. Kieren35 has developed a program for teaching fractions that is based on the multiplicative operations of splitting. As part of his approach he used paper folding rather than pie charts as its primary problem situation. In this approach, both the operator and measure subconstructs are highlighted. Confrey’s36 3-year developmental curriculum uses a number of contexts for ratio, including cooking, shadows, gears, and ramps.37 Streefland’s38 approach to teaching fractions is also driven by an emphasis on ratio. His basic image is of equal shares and quotients. In his procedure for teaching fractions, children are presented with realistic situations in which they are asked to share a quantity of something, such as chocolate bars or pancakes (e.g., five children sharing two bars). To represent these situations, children use a notation system that they devise themselves, which emphasizes proportional rather



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