Imagine that you are manning the door for a battle of the bands competition in the gym at your school. At the start of your shift, the two bands—the Positives and the Negatives—both have the same number of fans in the gym. But fans are constantly going into and leaving the gym as the two bands play into the night. Can you tell which band has more fans in the gym simply by observing who goes in and who goes out?

Pretend that, when your shift starts, 4 Positive fans go into the gym. A few minutes later, a group of 3 more Positive fans go into the gym. Then, 8 Negative fans go into the gym, 6 Negative fans leave the gym, 9 Positive fans leave the gym, and 7 Negative fans go into the gym. At this point in time, which band has more fans in the gym? Remember, the two bands are tied when your shift starts.

- 4 Positive fans go into the gym: the Positive band is up by 4 fans.
- 3 more Positive fans go into the gym: the Positive band is up by 7 fans.
- 8 Negative fans go into the gym: the Negative band is up by 1 fan.
- 6 Negative fans leave the gym: the Positive band is up by 5 fans.
- 9 Positive fans leave the gym: the Negative band is up by 4 fans.
- 7 Negative fans go into the gym: the Negative band is up by 11 fans.

Young children can solve this problem easily without any formal instruction in integers. In fact, formal instruction in integers will often trip them up as they try to remember various rules. All they need to do is understand the scenario and then apply some basic “common sense.” Within minutes, they are figuring out that 4P + 3P + 8N − 6N − 9P + 7N is equivalent to 11N. I typically introduce this notation as a convenient shorthand later in the lesson, but it’s not necessary.

What I’m trying to demonstrate is that students walk into my classroom with pre-existing cognitive models that they can use to understand integers. I’m not actually teaching them anything in this first lesson, and this activity should not be considered a math material as Papert defines it. If I guide the students at all, it is to help them raise things that they already know to the conscious level. For example, I might ask them to consider whether or not 5 Positive fans leaving the gym is equivalent to 5 Negative fans entering the gym when we only care about which band is ahead. Or I might ask them to derive their own rules for adding and subtracting pairs of integers. But children will start making those observations on their own; I’m simply nudging them to formulate and test their own hypotheses.

If I had to guess, the math materials that children are using to build these cognitive models come from their natural environments and not the classroom at all. It might come from keeping track of scores in sports or how many cookies each sibling has eaten over the course of the day. As a teacher and curriculum developer, I am noting where children appear to have highly developed cognitive models (a.k.a common sense or intuition), and then putting them in scenarios to apply and build on those models.

In Papert’s vision, teachers and curriculum developers would not be necessary. The world would be rich in math materials, and children would use those math materials to construct a rich set of cognitive models naturally through immersive experiences. In turn, those models would then make it easier for the children to assimilate new, more advanced concepts. The only reason why that isn’t happening today is because those math materials don’t exist.

But what about integers? Clearly, there must be sufficient materials in the environment to enable children to construct sophisticated cognitive models about integers because—as this unit demonstrates—they walk into my classroom with a pre-existing understanding of how to add and subtract integers. So, why are so many children struggling to add and subtract integers in school? I would say it’s because those cognitive models are so sparse. They may have sophisticated models that they can use to assimilate some concepts, but that happens so rarely in school that most children never even think to apply common sense in math class. I refer to this particular phenomenon as “math head,” and I spend months trying to reverse it. Also, when applicable cognitive models are sparse, it’s less likely that a given scenario will activate one.

In many standards-based math programs, the math material that children were expected to use to construct cognitive models about integers were red and yellow chips. But red and yellow chips never made sense to me. Most of us have little experience with zero-sum pairs in real-life, especially zero-sum pairs that magically materialize to make accounting easier. And why would we try to provide materials for children to construct cognitive models about integers when children are already in the process of constructing sophisticated models on their own? While it was intended to be hands-on and constructivist, this struck me as highly disrespectful. It feels like an artificial attempt to create a scalar model for integers when integers are such a natural fit for vectors. Papert extensively discusses how well-intentioned human-designed materials are often distorted by culture. Are we avoiding vectors because the prevailing culture says that vectors are too formal and abstract for children? Reality says otherwise.

In the battle of the bands competition scenario, I’m not attempting provide material so that children can construct new cognitive models. I’m providing a scenario that experience tells me most children will be able to assimilate into pre-existing models. I would argue that the learning that occurs is experiential but not personalized. I also recognize that, in Papert’s ideal world, the work that I’m doing is unnecessary if not counterproductive. But we don’t live in Papert’s ideal world. We live in a world where math materials are sparse and most children have not had the opportunity to construct rich sets of models on their own. How do I help those students? I gather and create as many materials as I can, and I encourage children to apply pre-existing models. Am I helping or hurting? I’m not sure. Like most of us, I’m just doing what I think is best.

What I’m trying to demonstrate is that students walk into my classroom with pre-existing cognitive models that they can use to understand integers. I’m not actually teaching them anything in this first lesson, and this activity should not be considered a math material as Papert defines it. If I guide the students at all, it is to help them raise things that they already know to the conscious level. For example, I might ask them to consider whether or not 5 Positive fans leaving the gym is equivalent to 5 Negative fans entering the gym when we only care about which band is ahead. Or I might ask them to derive their own rules for adding and subtracting pairs of integers. But children will start making those observations on their own; I’m simply nudging them to formulate and test their own hypotheses.

If I had to guess, the math materials that children are using to build these cognitive models come from their natural environments and not the classroom at all. It might come from keeping track of scores in sports or how many cookies each sibling has eaten over the course of the day. As a teacher and curriculum developer, I am noting where children appear to have highly developed cognitive models (a.k.a common sense or intuition), and then putting them in scenarios to apply and build on those models.

In Papert’s vision, teachers and curriculum developers would not be necessary. The world would be rich in math materials, and children would use those math materials to construct a rich set of cognitive models naturally through immersive experiences. In turn, those models would then make it easier for the children to assimilate new, more advanced concepts. The only reason why that isn’t happening today is because those math materials don’t exist.

But what about integers? Clearly, there must be sufficient materials in the environment to enable children to construct sophisticated cognitive models about integers because—as this unit demonstrates—they walk into my classroom with a pre-existing understanding of how to add and subtract integers. So, why are so many children struggling to add and subtract integers in school? I would say it’s because those cognitive models are so sparse. They may have sophisticated models that they can use to assimilate some concepts, but that happens so rarely in school that most children never even think to apply common sense in math class. I refer to this particular phenomenon as “math head,” and I spend months trying to reverse it. Also, when applicable cognitive models are sparse, it’s less likely that a given scenario will activate one.

In many standards-based math programs, the math material that children were expected to use to construct cognitive models about integers were red and yellow chips. But red and yellow chips never made sense to me. Most of us have little experience with zero-sum pairs in real-life, especially zero-sum pairs that magically materialize to make accounting easier. And why would we try to provide materials for children to construct cognitive models about integers when children are already in the process of constructing sophisticated models on their own? While it was intended to be hands-on and constructivist, this struck me as highly disrespectful. It feels like an artificial attempt to create a scalar model for integers when integers are such a natural fit for vectors. Papert extensively discusses how well-intentioned human-designed materials are often distorted by culture. Are we avoiding vectors because the prevailing culture says that vectors are too formal and abstract for children? Reality says otherwise.

In the battle of the bands competition scenario, I’m not attempting provide material so that children can construct new cognitive models. I’m providing a scenario that experience tells me most children will be able to assimilate into pre-existing models. I would argue that the learning that occurs is experiential but not personalized. I also recognize that, in Papert’s ideal world, the work that I’m doing is unnecessary if not counterproductive. But we don’t live in Papert’s ideal world. We live in a world where math materials are sparse and most children have not had the opportunity to construct rich sets of models on their own. How do I help those students? I gather and create as many materials as I can, and I encourage children to apply pre-existing models. Am I helping or hurting? I’m not sure. Like most of us, I’m just doing what I think is best.

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