Materials and Cognitive Models: Integers

In 2000, I was developing a curriculum unit on integers while election officials were busy counting hanging chads in Florida. As counties reported updated tallies, and George Bush and Al Gore gained and lost votes, I realized that the nation was witnessing integers in action. A few years later, I updated the Florida election recount scenario in the unit for something less dated: a battle of the bands competition.

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.

1. 4 Positive fans go into the gym: the Positive band is up by 4 fans.
2. 3 more Positive fans go into the gym: the Positive band is up by 7 fans.
3. 8 Negative fans go into the gym: the Negative band is up by 1 fan.
4. 6 Negative fans leave the gym: the Positive band is up by 5 fans.
5. 9 Positive fans leave the gym: the Negative band is up by 4 fans.
6. 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.

Mindstorms

Seymour Papert is an MIT professor of applied math and education who worked with Jean Piaget and co-invented the Logo Programming Language. In 1980, he published Mindstorms: Children, Computers, and Powerful Ideas, a book that profoundly influences thinking about STEM education today.

In Mindstorms, Papert makes the case that we find the study of mathematics difficult only because it is foreign and formal. He argues that, if we were immersed in an environment rich in appropriate materials, then we would construct our own mathematical understanding intuitively simply by interacting with the world around us. It would be like learning French while growing up in France rather than trying to learn French through the unnatural process of foreign-language instruction in a classroom.

Papert describes how his early love of automobiles and differential gears helped him make sense of equations later in school. Equations made sense to him because he was able to integrate them into his cognitive model for gears… a model he had constructed through years of play. Learning is easier when we have models that can help us relate to and explain new concepts. Fitting new information into pre-existing cognitive models is a process that Piaget called assimilation.

“Slowly I began to formulate what I still consider the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can’t, anything can be painfully difficult. […] What an individual can learn, and how he learns it, depends on what models he has available. This raises, recursively, the question of how he learned those models.”

Papert does not suggest that children should start studying math at an earlier age. Nor does he suggest that all children should play with gears. Gears worked for him because he fell in love with them. Instead, Papert believes the key is creating a world rich in materials so that all children can construct their own models by pursuing their own interests. And not just one or two models, but hundreds of them. In a material-rich environment, learning math would be like learning to walk or talk… personalized, experiential, and without formal instruction. Most of us learn to walk and talk easily and naturally because our environments are littered with the materials we need to construct the necessary models.

So, how do we create a world rich in mathematical materials? In many ways, this is a classic chicken-and-egg dilemma. Which comes first… the materials that enable children to grow up as math natives or the math natives capable of creating the appropriate materials?

“The computer is the Proteus of machines. Its essence is its universality, its power to simulate. Because it can take on a thousand forms and can serve a thousand functions, it can appeal to a thousand tastes. This book is the result of my own attempts over the past decade to turn computers into instruments flexible enough so that many children can each create for themselves something like what the gears were for me.”

Papert suggests that the computer may make up for our material deficit by functioning as a general-purpose material that children can craft into whatever form they need. But he also recognizes that creating new materials and adding them to the landscape is not enough. The lack of mathematical materials in our environment has shaped our culture for hundreds of generations. We have long-held and self-perpetuating beliefs about who can learn math and what learning math looks like. That culture and our beliefs can contaminate and warp any materials we might create… even materials created by children themselves. To counteract this culture, the computer must also function as a carrier of cultural “germs”, insulating and protecting young minds from the current culture and instilling a new one.

Thirty-five years later, the influence of Mindstorms can be seen in three educational trends: (1) the emphasis on personalized and experiential learning; (2) the creation of more mathematical materials to add to the landscape; and (3) the concentration of mathematical materials in after-school centers and maker spaces. But will these trends succeed where Papert’s Proteus of machines has so far failed? I’m not sure. Papert’s vision of change felt more abrupt and disruptive to me. The current path we are on feels more incremental: each generation provides a slightly richer environment so that the next generation can provide an even richer one… until we reach a tipping point. Will our self-perpetuating “mathophobic” culture allow incremental progress or is the current love affair with robotics and 3D printers another fad that will simply blow over? I’ve decided to hedge my bets.