A common way to teach students how to add and subtract integers is to use zero pairs. If negative integers are represented by red chips and positive integers by yellow chips, then one red chip and one yellow chip is a zero pair: they have a sum of zero. Since a zero pair has a sum of zero, students learn that they can add or remove zero pairs from a pile of chips without changing the value of the pile, and that this can make the addition and subtraction of integers easier.

Using red and yellow chips and zero pairs arose from the desire to help students develop mental models for adding and subtracting integers. Mental models are theories that we generate to help us explain our experiences and the world around us. As Eric Mazur, Professor of Physics at Harvard, pointed out during the panel discussion I attended on Tuesday, knowledge transfer is only the first stage of learning; after that, the individual still has to make his or her own meaning out of that knowledge.

Unfortunately, the model used to represent integers (red and yellow chips) has been conflated with the mental models that students develop in their heads. Just because I use red and yellow chips to model integer addition and subtraction does not mean that the mental model that I construct for myself has anything to do with red and yellow chips.

Mental models are not transferred, they are constructed. The reason why we model integer addition and subtraction with red and yellow chips is not to give students a mental model, it is to provide them with a set of experiences that we hope will lead them to construct an effective mental model for themselves. As teachers, we have no way of knowing what that mental model might look like. All we can do is place students in situations where they can apply and test the mental model they are developing, and give them feedback they can use for revision and extension.

I'm not a fan of the red and yellow chips. First, using red and yellow chips doesn't really build on any existing mental models or experiences that most students have. How often have students encountered and interacted with zero pairs in real-life? Second, I don't think that most mathematicians use zero pairs when thinking about integer addition and subtraction. We should always be wary when we expect beginners to think one way while experts think another way. A beginner's thinking may be more primitive or simplistic, but in general, it should not be different.

I prefer an approach for learning about integer addition and subtraction that I developed during the Florida recount in the 2000 presidential election. I've since changed the scenario to a battle of the bands held at a school gymnasium.

Imagine that there are two popular bands playing, one led by Sue Negative and the other by Joe Positive. You are manning the door. At the start of the event, there are an equal number of Negative and Positive fans. You don't know how many fans of each there are, just that the numbers are equal. Throughout the event, fans enter and leave. Your job is to keep track of which band has more fans in the gymnasium.

I prefer this scenario because it does build on existing mental models. My nieces and nephews watch each other like hawks and always have a running tally in their heads of who has gotten more treats over the course of a party or family outing. It also uses integers in a scenario where integers represent vectors. As you man the door, you are tracking changes to quantities and relative quantities, not the actual quantities themselves. This is how we encounter integers most often in real-life, and I believe that this is how most mathematicians think about positive and negative numbers. Contrast this with the red and yellow chip scenario where integers represent scalar quantities.

But the scenario is not the mental model; the scenario simply provides concrete experiences that students use to develop their own mental model. The teacher should help students probe those mental models as they are being developed to uncover misconceptions and increase flexibility and robustness. If given a sequence of entrances and exits, how does a student figure out which band has more fans in the gymnasium? Does she always process sequentially or has she realized that it is more efficient to group the movement of the Negative fans and the Positive fans separately and then combine them in the end? Does another student recognize that, if all you care about is which band is ahead in terms of fans, three Negative fans entering the gymnasium has the same effect as three Positive fans leaving the gymnasium?

Thinking about how to guide students as they construct mental models is a big part of what we do at Vertical Learning Labs.

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