## Learning and Mathematics

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### Learning fractions - Mack (1990)

In this article, Nancy K. Mack of the University of Pittsburgh considers how informal knowledge (such as dividing a pizza) can be used to enhance formal knowledge (such as one's understanding of fractions). Informal knowledge in this context means applied knowledge, whether correct or incorrect, developed by the individual and used to solve problems in real-life situations. Mack also explores how formal algorithmic and procedural knowledge may interfere with the use of informal knowledge.

Article:

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, pp. 16-32.

Overview:

Initially, students' informal knowledge is disconnected from their symbolic knowledge. Students tend to use their informal knowledge to solve problems related to real life, such as how much pizza would be left if one started with 2 pizzas and ate 1/8 of one pizza, and their formal knowledge for problems regarding concrete or symbolic representations, such as the result of two minus one-eighth.

Teaching rote procedures before acknowledging the nature of the concepts on which these build causes students to apply procedures blindly. It also interferes with their use of potentially helpful informal knowledge. Many students have misconceptions about fractions because they attempt to apply the rules for whole number arithmetic to their work with fractions.

Students can build successfully on their informal knowledge to construct meaning from formal representations, although a clear relation must exist between the two for this to happen.

Building on informal knowledge often results in a developmental progression that differs from the traditional sequence for teaching the concepts involved. For example, more complex topics, such as problems involving the regrouping or conversion of fractions, are generally taught at the end of a unit. Mack's research indicates that students can and possibly should handle these concepts much sooner.

"Five themes characterized the nature of the students' informal knowledge of fractions and the ways they were able to build on informal knowledge during instruction:

1. Students' informal solutions involved partitioning units;

2. students' informal knowledge of fractions was initially disconnected from their knowledge of fraction symbols, procedures, and concrete representations;

3. students could build on informal knowledge when they could match problems represented symbolically to problems presented in the context of real-world situations they understood;

4. students often encountered interference from knowledge of rote procedures when attempting to solve problems represented symbolically and in real-world situations; and

5. limited transfer of knowledge was observed even in students who were able to build on informal knowledge in other contexts" (p. 20-21).
"One situation where students' responses suggested this initial disconnection between their informal and symbolic knowledge involved comparing fractional quantities... each student was asked a question like, 'Suppose you have two pizzas of the same size, and you cut one of them into six equal-sized pieces and you cut the other one into eight equal-sized pieces. If you get one piece from each pizza, which one do you get more from?' All students responded that they would get more from the pizza cut into six pieces... Each student also was asked a question like 'Tell me which fraction is bigger, 1/6 or 1/8.'... Four of the students who first solved the real-world problem and three students who first worked with the symbolic representation responded, 'One eighth is bigger'" (p. 22).

"All students encountered some situations where they successfully used their informal knowledge on their own to give meaning to fraction symbols and procedures. In general, this occurred when students first were presented problems in the context of real-world situations and then were presented symbolic problems that were closely related. In instruction I frequently had to move back and forth between problems represented symbolically and problems in the context of real-world situations before students successfully related fraction symbols and procedures to their informal knowledge" (p. 23).

"All students soon began sharing [the] responsibility [of matching representations to their informal knowledge] as they related fraction symbols and procedures to informal knowledge in meaningful ways. On their own initiative, all eight students explained their solutions in terms of problem contexts that were meaningful to them" (p. 25).

"[Knowledge of rote procedures frequently interfered] with students' attempts to give meaning to fraction symbols and procedures in two ways: 1) Knowledge of procedures often kept students from drawing on their informal knowledge of fractions even for problems presented in the context of real world situations; 2) initially, students often trusted answers obtained by applying faulty procedures more than those obtained by drawing on informal knowledge" (p. 27).

-- Summarized by Mia Ong Wenbourne and Andrea Hall