Hi Kalid - I’ve enjoyed your articles here and this one really forced me to pause and consider what goes into the success of people like George Polya or Martin Gardner in explaining complicated ideas to “the uninitiated”, so to speak. To me it seems like their writing always shows a talent for approaching topics as stories to be told - faithfully, accurately, but also in a way that engages the reader’s sense of “setting”, “characters”, “plot development” and “resolution”.
And it makes sense that nudging a student/reader’s brain into listening-to-stories mode would help so much with comprehension and retention. Storytelling is universal: every known culture on the planet passes on stories. Stories engage both the language center and parts of the brain involved in predicting other people’s actions. For thousands of years they’ve been a focus of social activity; people bond over anecdotes. They capture a sense of raw possibility (leading to what-ifs, branching story arcs, alternate endings, prologues, etc.) by building language structures that map to events so that the listener can come to grips with different possibilities by shuffling those structures around, composing the pieces together in a potentially huge number of different ways.
Well, we do an extremely similar thing in mathematics, don’t we? Choose a setting for the story: draw assumptions from observations of something we want to model, or from results whose stories we’ve already told. Cast and develop the characters: which lemmas, principles, intuitions, conjectures and solid, long-standing theorems will get involved? Carry the plot forward: describe how each step follows from the last, how the interplay between characters unfolds, letting the reader fill in routine details or ones that are more fun to imagine independently of the writer. Conclude: lead to the consequence(s) of what was assumed at the start.
I think it would be very worthwhile, at least in some areas of teaching, to approach mathematics as a giant, still-unfolding story to be told.
For instance, there are so many ways to tell the story of calculus and I wish one of them in particular could be known to high school students. To roughly outline what I mean: At first we had the set {0,1} where multiplying any two of its elements results in something that belongs to the set (it’s closed under multiplication). And that’s nice because x*y = 1 if and only if x = 1 and y = 1, so treating 1 as “true” and 0 as “false” we can capture the notion of logical “and” as multiplication. There’s another operation, addition, that allows us to count one thing. We can say x + y = 1 if either x or y is 1… but 1 + 1 doesn’t belong to this set so the idea comes up that we’d like to give names to expressions like 1 + 1 + … + 1 and write calculations in terms of those names. Closing {0,1} under the operation of addition we get the set ℕ = {0,1,2,3,…} of natural numbers where 2 is the name of 1 + 1, 3 is the name of 1 + 1 + 1 and so on.
Now ℕ is a nice set because it’s closed under both multiplication and addition - so that gives plenty of useful ways to think about whole number quantities of things. Of course, annoyingly enough, 1 - 2 doesn’t belong to ℕ and so we end up wanting to close ℕ under the subtraction operation to get ℤ = {…,-2,-1,0,1,2,…}, the set of all integers. (Where -1 is a name for {0 - 1, 1 - 2, 2 - 3, …}, -2 is a name for {0 - 2, 1 - 3, 2 - 4 …}, 1 is now considered a name for {1 - 0, 2 - 1, 3 - 2, …}, etc.) The integers serve us well, until whole numbers aren’t enough for our purposes and we want to chop up quantities very finely so that we can use arithmetic to say things about small pieces adding up to whole pieces. So we make ℚ, the set of rational numbers, out of ℤ in a way similar to how we made ℤ out of ℕ: say two ‘pairs’ of integers are considered to be the same element of ℚ if they represent the same fraction, so for example 1/2 is a name for {1/2, 2/4, 3/6, …}. The rational numbers serve us even better than the integers because they give us a lot of control over how large or small our quantities are.
Is it enough to close ℤ under division? For many purposes, sure, but now that we’re dealing with such fine-grained numbers we’ve become interested in challenges like calculating the slope of a curve at a point. But it’s not hard to draw a curve which has slope √2 at one of its points, and √2 can’t be written as a fraction! We can get as many rational numbers as we like, each one closer to the “instantaneous slope” than the number coming before it, by drawing better and better secant lines - expressing the slope as the limit operation applied to a sequence of rational numbers. Then to calculate with numbers like √2, we need the real numbers ℝ to be closed under the operation of taking a limit.
There could be so much to gain from collecting good learning references and having them (along with comments, questions, notes and such) laid out in a format where they naturally “branch off” from the appropriate conceptual story arcs. I wish something like that already existed (seriously, the internet has already been around for how long now?) and would love to contribute what I could.
I am wondering if it is too much to ask that an elementary school teacher, who is a generalist, can manage to teach math in an engaging way? I know your site usually attracts people who are at a more advanced level of math, but it nags me that at the elementary level, kids are not getting the sensitive handling required to cultivate the right attitude toward math - a curious, exploratory, experimental experience rather than rote learning. If we “lose” these kids even before they hit third grade, what then of the future of math in our country?