Fractions are often challenging for students. Describe the three characteristics of fractions described in the Skill Hierarchy section that the authors believe contribute to the complexity of fractions
What will be an ideal response?
Students may find fractions challenging because the entire range of mathematical operations previously discussed in the book is applicable to fractional numbers. Fractions do not represent an extension of a familiar skill. While early instruction on whole numbers covers counting by groups of one or more than one, that instruction does not usually help students generalize to numbers that are less than one. In operations containing fractions, the one-to-one and one-to-many correspondences involve fractional numbers. While students can quickly learn to count 2, 4, 6, 8 for 2 x 4, counting 2/3, 4/3, 6/3, 8/3 for 2/3 x 4 is not easy. The problem 2/3 x 4/7 is even less comprehensible because students have no experience counting 4/7ths times. Another major difficulty with fractions is incompatibility of units. Whole numbers represent a simple type of equivalent unit, so they may be added and subtracted in any combination. In contrast, fractional numbers do not represent one type of unit: all thirds represent equivalent units and all fourths represent equivalent units, but thirds are not the same as fourths.
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