ABSTRACT OF

CONNECTING OPERATION CONCEPTS ACROSS THE PRE-K - 8 GRADE BANDS

Many secondary school students and adults exhibit persistent misconceptions about the nature of multiplication and division with rational numbers. Signed numbers present difficulties, as well, particularly when subtracting or multiplying. Many are prone to merely memorize rules and fail to see how various operations apply to real life situations. For example, when asked to choose the appropriate operation to calculate the unit price given that 0.75 pound costs $12, these students choose multiplication rather than division.

This paper presents the view that such difficulties are caused by students’ lack of rich mental models which can accommodate operations in a variety of domains--the set of integers, of rationals, as well as the whole numbers. Instead, they attempt to reason based on early concepts such as multiplication viewed as repeated addition, subtraction as take away, and division as sharing. Misgeneralizing from these inadequate models, they make inferences such as "multiplication always makes bigger" and "you can’t divide by a fraction."

The logical solution, then, is to present extensions of arithmetic operations in the context of models which students will find accessible. For instance, multiplication involving fractions can be motivated by considering boxes of cakes: 3 boxes of 4 cakes is 3 x 4 = 12 cakes, so 1/2 box of 4 cakes is 1/2 x 4 = 2 cakes. If a small box contains only 1/2 of a cake, so 1/2 of a small box is 1/2 x 1/2 = 1/4.

A number of educators have proposed additional models which can help students make sense of rational and integer operations--area models, repeated subtraction of a fractional divisor, linear models. When these models are used in familiar contexts and closely linked to the formal symbols, students can learn to associate correct, adequate models with these troublesome operations.

A dilemma remains, however: middle school students often seem to lack the prerequisite experiences with various whole number models which these extended models need as a foundation. For instance, students may think of subtraction exclusively as take away, or division exclusively as sharing. These students will have difficulty accepting

-2 - -5 as the directed distance from -5 to -2 because they have never used subtraction to find the distance between 7 and 10. Similarly, they will not recognize that repeatedly removing groups of 1/2 candy bar from 6 candy bars to find the number of groups is a division problem. Thus, elementary school teachers and textbook writers must offer a broad range of problems and concrete settings to students when they first learn to add, subtract, multiply, and divide. It is not sufficient to teach subtraction by using counters to solve take away problems--students must also have the opportunity to consider how far the temperature has fallen if it started at 17 degrees and now is at 3 degrees.

Thus, the standards need to explicitly point out to early elementary educators the vastly important foundation they are laying, and need to give explicit guidance regarding the models that students should encounter in these early years, models that will "grow" with the students as they grow in their mathematical understanding.