As part of the National Science Foundation’s Fractions Project (Children’s Construction of the Rational Numbers of Arithmetic), researchers have been studying how children develop an understanding of fractions. The implications of their findings are still being studied, but it is clear that children often develop limited concepts about fractions from traditional textbook-based instruction. The project identified four of these limited concepts:

1) Numerators and denominators are viewed independently in making meanings for fractions,

2) Counting is a fool-proof method for determining fractional amounts,

3) Equal parts are parts that look identical to each other, and

4) There is no connection between division and fractional representation of quantities.

National Science Foundation researchers are taking these findings into account as they work toward designing new curricula. They suggest, first of all, that instruction in fractions should be grounded in children’s understanding of whole numbers.

Working in pairs, third- to fifth-grade children in the project begin by using one of three specially-designed computer programs in which they manipulate “sticks” of different lengths. They are asked questions such as : what stick would result if the 6-unit stick were repeated seven times?” and, “How many 8-unit sticks would be needed to make a 24-unit stick?”

Students answer these questions using their knowledge of whole-number addition, multiplication and division. Researchers then introduce fraction language into the activity by noting that because the 8-unit stick could be repeated three times to make a 24-unit stick, an 8-unit stick is 1/3 of a 24-unit stick. In this way, researchers try to suggest that instead of thinking of 1/3 as one out of three equal parts, students think of 1/3 as the quantity which when repeated three times equals one whole. It should be understood, they stress, that fractions can be repeated any number of times to produce any number of units (1/3 can be repeated to produce any number of thirds).

Researchers report that being able to copy, join, cut, erase and mark objects on the computer screen is an effective and liberating experience for children. Using computers, children in the project were able to engage in estimating, predicting, testing and revising without leaving a permanent record of their actions. These activities, moreover, were not accompanied by any numerical data unless the student requested it. Thus, children were able to rely heavily on visual estimates — which, evidence shows, strengthens their concept of fractional quantities.

Despite the success the project had in developing effective fractions concepts in the experimental group, researchers are still faced with problems in developing practical applications from this research.

They find it difficult to:

1) adapt their new understanding to teaching heterogeneous classes of 25 or more children,

2) develop non-computer activities to accomplish what computer activities did in the project,

3) deal with the limitations of physical materials and classroom management problems, and

4) take into account district objectives for content coverage and assessment, while trying to remain true to the constructivist philosophy recommended by the National Council of Teachers of Mathematics.

### Classroom application

These difficulties were faced by two researchers, Beatriz S. D’Ambrosio and Denise Spangler Mewborn, University of Georgia, when they worked with one fourth-grade teacher in an attempt to implement classroom activities that would utilize the knowledge emerging from the Fractions Project.

Initially, D’Ambrosio and Mewborn tried to simulate computer activities with paper folding or by using concrete objects such as paper clips, M & M’s, and pattern blocks. However, paper folding was not precise enough and none of the objects facilitated manipulation of fractions. For this reason, D’Ambrosio and Mewborn began working with groups of five to eight students on the one available computer. But it soon became apparent that the quality of the learning experience was severely compromised when this many students tried to use the computer at one time.

This classroom experiment demonstrated, as well, that considerably more time is needed to develop adequate conceptual understanding of fractions than the one week usually devoted to the introductory chapter on fractions. In this study, three weeks of 45-minute lessons were not enough to expose and challenge all the preconceived, limited notions students had about fractions. Because of time constraints, the teacher was unable to pursue chance comments made by students and unable also to follow up on students’ work with important lines of questioning. Nevertheless, encouraging children to verbalize their thinking, however difficult and time-consuming this may be for the class, is important because it provides irreplaceable opportunities to gain insight into students’ thinking.

### Teachers ask different types of questions

A striking difference between the project and the regular classroom was in the type of questions teachers asked. In the Fractions Project, teachers asked questions to elicit responses that would reveal students’ developing understanding. However, because of the pressure to cover a predetermined amount of material, the classroom teacher tended to ask evaluative questions to determine whether she could move on to the next pre-planned topic or whether further instruction was needed.

D’Ambrosio and Mewborn conclude that substantial work in actual classroom settings is needed if the teaching/learning roles are to be adapted to help students develop better understanding of fractions.

“Children”s Construction of Fractions and Their Implications for Classroom Instruction”, Journal of Research in Childhood Education, Volume 8, Number 2, Summer 1994 pp.150-161.

**Published in ERN March/April 1995, Volume 8, Number 2 **