Academic journal article Australian Mathematics Teacher

# Algebra Students' Difficulty with Fractions: An Error Analysis

Academic journal article Australian Mathematics Teacher

# Algebra Students' Difficulty with Fractions: An Error Analysis

## Article excerpt

The National Assessment of Educational Progress (NAEP), a United States report, raises concerns regarding trends in student achievement over the past twenty years (NCES, 2000). The results indicate that students of age seventeen recurrently demonstrated a lack of proficiency with fraction concepts. An analysis of the 1990 NAEP mathematics achievement by Mullis, Dossey, Owen, and Phillips (1991) found that only 46 percent of all high school seniors demonstrated success with a grasp of decimals, percentages, fractions, and simple algebra. If algebra is for everyone, then a bridge must be built to span the gap between arithmetic and algebra. The building materials are conceptual understanding and the ability to perform arithmetic manipulation on whole numbers, decimal fractions, and common fractions.

Augustus De Morgan, writing in Study and Difficulty of Mathematics (1910) acknowledges that the learning of fractions is expected to "present extraordinary difficulties." This was true in the nineteenth century and it is still true today. Consider the following (p. 41):

```   What is 1/4 of 2/7 of a foot? What is 2/5 of 1/3 of 3/4 of
a foot? Into how many parts must 3/7 of a foot
be divided, and how many of them must be
taken to produce 14/15 of a foot? What is 1/3 + 1/7 of
a foot? and so on.
```

Is the above a logical and natural progression from operations on whole numbers? Examine the difficulty in finding the product of two common fractions. Multiplication is precisely defined as repeated addition, multiply 5 and 8 together and the product is either eight fives or five eights. Having become proficient at whole number multiplication with a solid understanding of the concept, consider the product of 2/3 and 3/5. The result is absurd (p. 34). For the sake of the learner the absurdity must be alleviated, the mystery of 1/3 x 1/7 = 1/21 must be resolved. When fractions should be taught, how fractions should be taught, and how competence with fractions affects the transition from arithmetic to algebra, are questions that mathematics educators and researchers have examined for the past century.

Kieren (1980) suggests that the instruction of rational numbers be postponed until the student has reached the stage of formal operations. He reasons that five concepts of fractional numbers must be both differentiated and connected to form a cogent rational number construct. The five ideas, (1) part-whole relationships, (2) ratios, (3) quotients, (4) measures, and (5) operators, represent "five separate fractional or rational number thinking patterns" (p. 134).

The rational number concept is rich and complex. Kieren (1980) asserts that the number of disjointed protocols a learner must control to form the rational number concept is extensive. Too often simply an algorithm has been taught, abandoning the student deep in the rational number construct. This provides no connection for understanding, and leaves the student clinging to a prescribed step-by-step set of instructions. If the algorithm is forgotten, then the learner must retreat to familiar protocols, which can be applied in the given situation. For example, the individual may try to apply a natural number protocol for fraction addition, adding both numerators and denominators, since addition of natural numbers arises from the natural activity of children (p. 102). Algorithms that are taught when the concept is beyond the learner's cognitive development force the learner to abandon his or her own thinking and resort to memorisation--doing without understanding. Lamon (1999) insists that the consequences of doing rather than understanding affect both a student's enjoyment of and motivation for learning mathematics (p. xi).

This article will investigate error patterns that emerge as students attempt to answer questions involving the ability to apply fraction concepts and perform operations on fractions. This analysis will provide a source that can assist teachers in detecting and correcting common mistakes students make when manipulating fractional numbers. …

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