I deeply believe that knowledge is constructed in relationship to what we already know. Understanding means constructing a web of connections, ideas, experiences, etc.  So I am certainly a constructivist in mathematics, although I tend to see individual children as learning through not only their own experience, but through engagement with others, and through social positioning. I also believe that students with disabilities learn in this same way, since it is outrageous to assume that the learning process is completely different for certain people. This is not an idea that everyone agrees with.

This week I was reviewing some new materials for my class of pre-service teachers. The project is called Children’s Mathematical Learning, and it is a collection of free on-line videos of children doing math problems, focusing on children’s thinking. You can see the videos here WWW.CMLVideos.com and learn more about the project here WWW.CMLProject.com, including a low-cost PDF book that connects Common Core Mathematics Standards directly to math ed research, particularly constructivist sources. There is a lot of great, well thought out stuff here, and I thank David Feikes, Keith Schwingendorf, and Jeff Gregg for creating it.

However, when I watched the video Additive and Multiplicative Thinking I was faced with my pet peeve with applying constructivist ideas to children. In the video a variety of second graders are asked “How many dots” on a dot array like this:

The task is designed to elicit students strategies for arrays, particularly they approach the task additively or multiplicatively. The first child counts by ones. Other children count by 2, 4, and 5s.  This distinction between additive and multiplicative thinking is a critical one in mathematics. As the authors discuss in their accompanying PDF, students need lots of experience with grouping in order to develop multiplicative thinking. Such a complex shift, from thinking about ones to thinking about groups of ones, is critical in almost every area of mathematical development (Fosnot & Dolk, 2002).

In the video, after showing us the girl who counts by ones, the narrator states, “This child is an additive thinker. She counts the dots one at a time.” After we see the students who are grouping, the narrator says, “The last three children are ready to be introduced to multiplication. They are multiplicative thinkers.”

Several issues are present here, all of which are critical in intervention and working with students with disabilities. First, are there such things as multiplicative thinkers and additive thinkers? Are you one kind of thinker in mathematics?

Here is what the narrator said:

“This child is an additive thinker. She counts the dots one at a time.”

Here is how I wish that all researchers and teachers would phrase that:

“This child was thinking additively. She counted the dots one at a time.”

Learners are not 100% additive thinkers who suddenly transform into 100% multiplicative thinkers. Many adults approach fractions additively, rather than multiplicatively. Are they not ready for multiplication? No! The relationship between strategies, such as counting by groups, and concepts, such as unitizing, are a two way street. We move back and forth between the two as we construct meaning with new operations and kinds of numbers.

Second, I wouldn’t make a pronouncement about a child’s conceptual level by a single task. The girl may have counted because she thought that was what the interviewer wanted, not because she was incapable of counting in groups.

Unfortunately for students with disabilities, the educational system seems to excel at testing them, and then making pronouncements about their capabilities. If constructivist pedagogies are applied in special education without a critical eye to false labeling, we will soon see IEPs that describe children as an “additive thinker.” In CGI, we use the term direct modeling to describe a particular strategy of representing each object to solve a problem. I have heard educators describe children as “direct modelers.” Again, this was never the intention of the term.  Direct modeling is a description of a strategy, not a child.

These labels matter because, as the video states, they can affect children’s access to instruction. Only the “multiplicative thinkers” are “ready for multiplication”?

I often hear the phrase, “constructivism is not a theory of teaching, it is a theory of learning.” I agree. And I will push back whenever constructivism becomes a theory of labeling.