LEARNING TO TEACH DIVISION OF FRACTIONS MEANINGFULLY

Alfinio Flores

Arizona State University

Alfinio@asu.edu

Erin E. Turner

University of Texas-Austin

erineturner

We describe how two teachers learned to teach division of fractions for conceptual understanding in terms that would be meaningful for students, and discuss their success and limitations in learning to teach a hard topic on their own in relation to their knowledge of mathematics and pedagogical content knowledge.

Each teacher was teaching a combined fourth-fifth grade. The teachers had a double role, as the subjects of study and also as active participants in the research. Sources of data are a baseline interview, a questionnaire, the notes and materials developed by the teachers, and a final interview. Data for the learning process are the materials written by the teachers.

Both teachers had a solid understanding of the fraction concepts they teach in 4th and 5th grade. They had explicit conceptual understandings of procedures such as finding equivalent fractions, adding fractions, finding common denominator, adding, subtracting and multiplying fractions. They knew how to use concrete representations to do division of fractions in terms of measurement or sharing, without having to use rules. Both were familiar with the “invert the second fraction and multiply” procedure, but did not know why this rule works. The two teachers share some of the characteristics, but not all, of teachers with profound understanding of fundamental mathematics. Their knowledge of elementary mathematics is still not a unified body of knowledge. In the case of division of fractions their repertoire of situations and story problems was limited to the measurement interpretation of division. They are acquainted with basic ideas such as the use of reciprocals, the identity nature of 1, units of different kinds, and the inverse nature of multiplication and division. However, these basic ideas do not seem to guide their mathematical activity.

The connection from previous understanding of division of fractions in terms of measurement or sharing to gaining understanding of the reason of why the “invert and multiply” method works occurred through three different approaches. First, they used common denominators. For fractions with the same denominator, they noticed that the denominator will not appear as part of the answer. Later, they understood how finding common denominators and dividing relates to the original fractions, and they made the connection to the algorithm of multiplying by the reciprocal explicit. Second, they explored the inverse relation of multiplication and division, first with whole numbers ands then with fractions. Third they used examples to illustrate that when you divide by a number, it is the same as multiplying by the inverse.

Using their previous understanding of division of fractions with concrete representations and the measurement interpretation of division, they learned to explain the “multiply by the reciprocal” algorithm, via fractions with common denominators. However, they did not make completely explicit the role of reciprocals and their connection to the inverse relation between multiplication and division. Both their knowledge of mathematics and concerns about teaching influenced teachers approach and explained their success and limitations. It is possible for teachers like these to learn to teach for conceptual understanding. However, it is striking how long the process was.

References

Kennedy, M. M., Ball, D. L. & McDiarmid, C. W. (1993). A study package for examining and tracking changes in teachers’ knowledge. East Lansing, MI: The National Center for Research on Teacher Education.

Ma, Liping. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(1), 4-14.