I spent the early part of the week at the NCTM Research Conference. One of the symposia I attended was called “Contrasting Perspectives on Multiplication, Area, and Combinatorial Problems” (Izsak, Jacobson, Tillema, & Lehrer, 2018). One of the presenters, Dr. Erik Jacobson from Indiana University, shared a mapping he created between three models of fraction multiplication and three common contexts for such problems.
The three models were as shown below. In the overlap model, students use one factor to partition and shade a shape vertically, the other factor to partition and shade the shape horizontally, and then find the product by expressing the double-shaded part as a fraction of the whole shape. There is only one referent for all three fractions in the problem — each factor and the product is considered in relation to the whole shape.
In the part-of-a-part model, students use one factor to partition and shade a shape in one direction (vertically in my example below), use the second factor to partition and shade only the already shaded part in the other direction, and then find the product by expressing the double-shaded part as a fraction of the whole shape. In this case there are two referents. The first factor and the product are considered in relation to the whole shape, but the second factor is considered in relation to only part of the shape.
In the length-area model, each factor is interpreted as a fraction of the length of one of the shape’s sides, and the product is the fraction of the whole shape that a rectangle with those side lengths takes up. In this case there are three referents, with each fraction in reference to exactly one of them: The horizontal length of the shape, the vertical length of the shape, and the area of the shape.
Dr. Jacobson went on to explain the ways different multiplication problem contexts match or do not match these differing numbers of referents. I didn’t manage to scribble down or retain his whole mapping, but there was one part that stuck with me: fraction-of problems (which the presenter called unit-conversion problems) can’t be coherently mapped onto the overlap model because of a mismatch in referents.
For example, consider this problem: Katie has ⅔ of a pizza. She eats ¾ of that. How much of the pizza did she eat? There are two referents in this problem. The ⅔, as well as the requested final answer, are fractions expressed in terms of the whole pizza. But the ¾ is expressed in terms of the ⅔ pizza that Katie starts with. This maps onto the part-of-a-part model. The overlap model is a poor fit because it only has one referent rather than two.
This point really resonated for me, because during my most recent bout of curriculum development, one of the many lessons I wrote was about using fraction-of thinking to solve problems like the pizza problem above. The previous edition of our curriculum emphasized the overlap model for solving those problems. I campaigned, successfully, to switch to the part-of-a-part model in this most recent edition. Dr. Jacobson’s point was essentially my argument, although I did not have the vocabulary to express it well at the time. My first thought upon hearing it come out of someone else’s mouth was one of satisfied validation.
This, as per usual, was immediately followed by some self doubt, because I also remembered the arguments against making the change. Some of our field test teachers, as well as some other staff members, didn’t see the benefits as strong enough to justify changing a tried-and-true lesson that kids tended to struggle with at first but ultimately understand. So even though I now had clearer vocabulary and arguments to justify the switch, remembering the debate led me to ask myself: But does that mismatch really matter to kids? Is that extra bit of shading really going to have detrimental effects to their later understanding?
I’ve been pondering this the last few days, and finally came to a (likely temporary) conclusion after revisiting some literature on manipulatives. Why do we have kids handle concrete objects in early mathematics classes? The main argument is that these objects help kids to connect their concrete knowledge from their own experiences to the abstract mathematical knowledge we hope they learn (Uttal, Scudder, & DeLoache, 1997). In my view, models like the ones shown above are meant to do the same thing. And if kids can’t directly connect the models to the problem contexts (which in turn are supposed to be connecting mathematics to their real-world experiences), then I don’t think the models are serving that purpose.
I don’t doubt that kids can successfully solve fraction-of problems using the overlap model. But when the model doesn’t map onto the problem context — or perhaps more importantly, when the problem context can’t be mapped onto the model — the models are probably serving as just another kind of procedure for them to execute. It may be a more meaningful procedure than a numerical algorithm, but it’s still a procedure that exists separately from the problem. The connection between the concrete and the abstract isn’t being made. I don’t buy that the overlap model will promote misconceptions, per se. But I do think it does less work in terms of helping kids wrap their heads around fractions with multiple referents and how that relates to real-world problems.
The answer to whether or not the change from the overlap to the part-of-a-part model was a good one, then, depends on what we were most concerned about kids taking away from the lesson. Considering that kids derive the numerical algorithm a few days later, I’m glad I campaigned for the shift to a model that might better support concrete-to-abstract connections. That seems more important than giving them procedure they can execute.
Izsak, A., Jacobson, E., Tillema, E., and Lehrer, R. (2018, April). Contrasting Perspectives on Multiplication, Area, and Combinatorial Problems. Symposium at the National Council of Teachers of Mathematics (NCTM) Research Conference in Washington, D.C.
Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37–54.