Key Math Ideas Not Taught In School
There is a shape, a progression, to the typical school curriculum from Kindergarten through High School, at least in the United States, and this progression hasn’t changed much in the last 50 years or so. We’ve gotten used to this progression, this story arc, and it shapes our very perception of what math is. There is nothing particularly wrong with this, but it is useful at times to take a fresh look and question this arc.
In a recent TED talk, Arthur Benjamin does exactly this, by suggesting that the apex of the arc may want to be Statistics rather than Calculus. If I understand him correctly, he isn’t merely talking about the courses you take in college, he is talking about the material taught at lower levels that prepare people properly for Calculus or for Statistics. His argument, roughly, is that no matter how important Calculus is as a cultural heritage, and no matter how important it is for college students in the sciences and engineering, very few others will use Calculus in daily life, and many more would benefit from a grounding in statistics and probability, as a way to deal with risk and uncertainty.
I’m not interested, at least not here, in discussing the details of Benjamin’s idea further. My first reaction is two-fold: the first one is that it is great that somebody is doing this kind of thinking, though I predict that even such a relatively mild shift in how mathematics is taught (and seen) will undoubtedly provoke a massive amount of reaction arguing that the way we currently do things must not be altered lest Western Civilization collapse. The second one is that Benjamin is looking at relatively late stages in the curriculum – too late, I think – given that vast masses of children will be permanently turned off from any kind of enjoyment of mathematics well before any fork in the road appears where one way leads to Calculus and the other leads to Statistics.
I am starting a new series, with this post, to look at important mathematical ideas that currently have no place in the math curriculum. They aren’t necessarily complicated ideas, and the ideas aren’t necessarily alien to the current way of doing it. They are mostly ideas whose relevance to the math curriculum has been largely unexamined. The ideas that I will highlight mostly have their origin in the field of computer science, a branch of thinking that has revolutionized the way we look at computation and algorithms over the last 50 years, and which has helped shape the computer revolution that has utterly changed the face of the world, yet has left almost no trace in the mathematics curriculum (other than in the arguments whether to allow children to use calculators when doing their math homework)!
The ideas I want to highlight in this series are interesting and fascinating in their own right, I think, and I don’t want to leave the impression that I have a fully-formed curriculum in mind in which those ideas would fit. You may find it more useful to read the entries as background ideas aimed at teachers rather than content for a curriculum for children.
In the next few entries in this series we will be looking at the notion of invariants and transactions. Both are, at bottom, very simple ideas, which have become refined and well-articulated in the world of computer science. A number of well-known mathematical ideas and practices will look very different once viewed through this other perspective.