Mathematical Knowledge Space

I've been thinking a lot about Linear Knowledge Space and Random Knowledge Space. (Incidentally, the term "Random" seems to connote haphazard and unconnected, and I don't think that's the point. If I come up with a better term, I'll put it out there.)

Linear Knowledge Space seems pretty self-explanatory. Start at the beginning and proceed to the end. This is certainly the most common kind of presentation of mathematics seen in schools. Orderly and hierarchical. Step 1, Step 2, etc. And don't dare miss any steps!

This view of mathematical knowledge is the one that permeates the culture. It is popularly believed that "math people"* are rigid and linear and excessively rational in their thoughts and actions. School mathematics often gets framed, or positioned, in these linear knowledge spaces. The conventional wisdom says that in order to learn section 12.5 of the textbook, you must have first mastered chapter 1-11 and sections 12.1-12.4. But, where is the evidence that this is necessary? Check any two textbooks for the same course (at any level) and you will see that there are differences. These differences could be minor, like the order of the topics, or they could be major, such as the philosophical stance on mathematics that the authors take. Regardless, differences exist. If mathematics was linear, there would be only one line!

History could be our guide here. Mathematics, like all other human endeavors (and mathematics is as susceptible to humans foibles as any other discipline) is written by the victor. If history of mathematics can teach us anything, it should be that mathematics was developed in fits and starts, through argument, experimentation, and collaboration. A question was posed and people tried to answer it. Newton and Leibniz knew an awful lot about Calculus and they didn't have any idea what a "limit" was. Weierstrass comes along (about 100 years later) and defines "limit" to everyone's satisfaction, despite a century of doing calculus without that definition. What is the first chapter of pretty much every major calculus textbook? Limits. There is absolutely no historical reason for it. We force students to learn linearly that which does not need to be linear. What we need is better questions to answer.

"But", you say, "But, certainly there are topics that have prerequisite knowledge." Yes, of course there are. It's pretty hard to learn how to multiply if you are sketchy on what it means to add. However, the analogy that learning mathematics is like building a tower is fundamentally flawed. Learning mathematics is more like building a city. Yes, there are towers. But there are expansions to the towers and there are other buildings, and some of those buildings have additions. And there's an infrastructure of roads, and maybe even tunnels, that connect the various parts of the city. Put another way, mathematics is less like a novel and more like short stories that take place in the same universe. Do you have to have seen episodes 1-99 of "Friends" before watching the 100th one? Certainly not. Could it help you understand something different about the characters if you have seen episodes 1-99? Quite possibly.

Bob and Ellen Kaplan, whom I consider mentors, have said that when teaching math, if you want the children to learn topic A then you should ask them about topic B where A is necessary. I think this is the "sweet spot" that instruction needs to hit. There are prerequisites, but many fewer than we think there are. Mathematical Knowledge Space isn't necessarily Linear, but it's not completely Random either. It doesn't matter if the instruction is face-to-face or online. What matters is that there is a compelling question, some strategies to answer it, and collaboration. What turns it from the answer to a specific problem into mathematical knowledge is connecting the specific answer to answers of other problems.

*Frankly, that there are "math people" and "non-math people" is a ridiculous notion.