Supporting Dynamic Conceptions of Multiplication

• In Event: 45.082 - Multiplicative Models of Situations Across Disciplines

Abstract

Objective: The conception of multiplication most commonly promoted in school mathematics is repeated addition: A collection of size n is repeatedly joined k times to itself to produce an accumulated collection of size k*n. Yet in many situations, a more dynamic view of multiplication suggests fruitful alternative interpretations of product. During the past decade, we have examined the feasibility of supporting elementary students in developing polysemic interpretations of multiplication in contexts involving products as results of dynamic processes.

Perspective: To explore ways of making dynamic interpretations of products intelligible to children, we have focused on dynamic movement in length, area and volume measurement. In length measure, children first employ unit iteration and travel-of-path to make sense of a measure, such as 12u, considering that such a measure means that the length measured is 12 times the length of the unit measure. We follow up with expressions such as “3 x 4u” as 3 iterations of the composite 4u, so that 12u can be interpreted as 3 times as long as 4u. In this way, products are not simply measures; they also represent multiplicative relations. We generate area measures by moving one length through another at a non-zero angle using physical materials of squeegees, paint, and a surface to act upon. The squeegees’ sweeping motion is elaborated as a measure by unit dissection of the resulting planar figure. In this view, “4u x 3u” names a new quantity, an area, whose measure of 12u2 means that the measured area is 12 times the area of the unit. This image of product is then generalized to volume measure, where an area is swept through a length to create a 3-dimensional quantity.

Method & Data Sources: We present results from several settings and lines of inquiry, in all cases following IRB-approved protocols. First, in clinical interviews, grade 3, 5, and 6 students interpreted the difference in meaning of expressions such as “3 x 4,” “3 x 4u,” and “3u x 4u.” The second involves whole-class investigations where in one grade 3 classroom students invented sweeping to generate and measure the volume of a cylinder. In another grade 5 classroom, a teacher invoked images of sweeping to help children make sense of the formula for the volume of a right rectangular prism.

Results: Overall, students developed understandings of (a) alternative interpretations of numerical products (e.g., “3 x 4u” as tripling a length versus “3u x 4u” as “dragging” 3u through 4u), (b) multiplication as an act relating continuous quantities (including fractional quantities), and (c) properties of multiplication (e.g., in area measurement, experiencing the commutative property through different actions – first sweeping one length versus the other). Students’ embodied interactions with materials served as important conceptual resources for making sense of these ideas and engaging in mathematical argumentation.

Significance: Although measurement models are often used to support students’ sense-making of multiplication, we argue that students need first to understand the meaning of these models in relation to multiplication. Our results suggest one such approach.

Authors

• Richard Lehrer, Vanderbilt University

• Marta Anna Kobiela, McGill University