Building Maps

I have two sons, both of whom are good at math. One is in grade five and at the top of his math class. The other would be in grade eight, except that he is freelancing his education this year, taking a break from the standard school setting. His grade seven math teacher said he was brilliant in math, but she couldn’t assign him a mark, because he refused to do any assignments. He recently started on IXL, a math app that teaches the Ontario curriculum through online questions and guided practice, and we’ve been looking at it together. It’s amazing to watch him. He can answer almost anything, without writing anything down, but he doesn’t follow the algorithms we teach in school. He just knows the answers. Or, when he doesn’t ‘just know’ the answer, he falls back on his understanding of how math works to figure out his own algorithm. It’s amazing to watch. But I’m struck by the gap between his understanding and his inability to follow the steps most people do. He says they don’t make sense to him.

I was particularly struck by this strange dichotomy because I’ve been teaching math to a group of students who have the opposite phenomenon. They can follow the rules of math, but seem to have no idea how math works. They are stuck in what we call “old math” as opposed to the “new math” that has so many people hot under the collar. These are students who can “borrow” and “carry” but struggle to add two single digit numbers using mental math. They know their timetables and can perform beautiful long division, but they don’t understand why long multiplication works. They don’t care why it works – they want to know if it’s right.

I asked my students this week why they think we take math tests and then why they think we learn math at all, and the answers were very revealing. They were very focused on grades, report cards, and knowing their levels. They did not seem to understand that the point is to learn how math works. I’ve been thinking a lot about this recently, and I’ve started talking to my students differently about math.  I think they are missing the point of learning math. They are missing the point because we are missing the point, and I’m starting to wonder where else we are missing this same point. In what other areas of life are we focused on getting the “right” answer rather than learning how life works?

I’ve started talking to my students about building a confident ‘math map’. We talked about how they each have a map in their heads that helps them know how to get around the school. Those who’ve been at the school a long time have a more complete map, while those who are new this year, during COVID, have a very limited map, because they haven’t traveled much around the school. We talked about how we could give visitors a map to get around, but that the visitors would have more trouble reading the map than the students who already know their way around. I told them we are building math maps in their heads, with different rooms for different concepts. I don’t want to have to hand them a map that tells them how to get around math concepts. I want them to build their own map of how the concepts work and fit together. The way to do this is by exploration, practice, and relevant guidance. Students at all levels, not just in the early primary years, need to work with concrete manipulatives. They need to try things out, take risks, and observe relationships between real life and math. We need to teach them that math is a language we use to talk about real things that happen. I’m excited about what I see happening already.

But what about outside of math? What about social skills and life skills and self-regulation and values? Could we build maps for those aspects of life instead of teaching individual skills as right and wrong, good and bad, conforming or shamed? What if, instead of teaching children what to do in different situations, we taught them how emotions work, how relationships work, and how society works? What if we taught children how to determine their own happiness instead of allowing society to form their ideas of success? Instead of giving them a blueprint for the building we’ve already approved, what if we gave them building materials and lessons in architecture? What would they build if we let them explore? What would they build if they weren’t so set on getting it right? What would they build if they could build anything they wanted? I wonder if we can even imagine the society that could be built using everything we now know about psychology, relationships, justice, and happiness. I can only imagine the progress that could be made if we allowed them to dream, to construct, and to change systems as their understanding grows. How do we open that door, just a crack? How do we change our conversation, to show them that the point of life is not to ‘get it right’ but to understand how the world works and use our understanding to make the most beautiful and authentic versions of ourselves possible? How could we start to change our own?