The Moonspeaker:
Where Some Ideas Are Stranger Than Others...
AN ESSAY INTO MATHEMATICS
I will state my views here up front. The two worst taught courses at any level of education are mathematics and the completely misnamed "english." Both subjects are critical to every other subject we need to learn from the earliest grades, but at this point curricula for them seem set up specifically to alienate students from nearly minute one. They have also both developed a bifurcation between applied and theoretical/abstract areas of study and research. This is probably not coincidental in light of the ongoing neoliberal wars against the humanities, and the constructed panic over "STEM" fields and students' supposedly poor performance in them. I say "supposedly" not because I don't agree students are struggling, which would not be consistent with my starting point in this essay. Rather, it is not at all evident what students are struggling with or doing poorly at in these bouts of theatrical handwaving. High performance in most "standardized tests" is dictated by understanding the technique for writing them, not memory and understanding of specific subject matter. So poor performance on those tests may not be telling us much else besides that students are bad at writing that sort of test. Meanwhile, sex role stereotypes are so pervasive in education that girls and women are expected to do well in english and be articulate while supposedly never producing any great literature. Boys and men are expected to be automatically better at mathematics because they dominate the field, which of course they do because under these conditions girls and women are repeatedly routed out of it unless they are profoundly stubborn and get honest and accurate academic advising.
Connecting mathematics and english in this way may seem counterintuitive, but I am quite serious about it. Mathematics and english are equally about language, starting from simple building blocks that must be memorized by rote to get started, moving on to patterns that once defined may be used to understand and create new language. Learners need to understand why a particular method applies and when, otherwise they will struggle to apply what they are learning in any test, let alone life outside the classroom. Mathematics is an extremely abstracted language, deliberately designed not to talk about obvious things in its most professionalized forms. This is a genuine feature of mathematics. Its foundations in arithmetic start from a means of counting applicable to anyone or anything, and it turned out this is a profoundly clever and useful thing to have. The trouble is both mathematics and english are taught in a manner apt to obscure their day to day uses, so many students come to hate and resent a subject they are told is relevant, yet seems completely irrelevant anyway. Between cognitive dissonance and the grim knowledge how keeping them in class grappling with this stuff is an expression of adult power over them, it is a happy wonder so many students manage to learn and love these subjects anyway.
With that set out, I am now going to turn more specifically to mathematics, because attempts to reform the subject's curriculum are as perennial there as any, but they have been entangled lately in trying to make them "culturally specific." Some of the articles I have read claiming to describe what these curriculum changes consist of strike me as rank nonsense, even deliberate parody, if not purposeful misrepresentation. The basic practice of having students work with familiar objects in the arithmetic instruction phases seems like it should be obvious – which means it isn't, and so had to be made explicit. There are some amazing and brilliant projects in which Indigenous elders and teachers work together to explore the mathematics expressed in such ordinary daily objects like baskets. These projects bring out another curious aspect of so much mathematical instruction, especially in the early grades: even students considered members of the mainstream community where they live rarely receive instruction showing how what they are learning connects to other things in their real lives, from trigonometry to carpentry, to statistics in playing games. Considering that right now we are in the midst of an era of total contempt for any work entailing or potentially entailing physical labour, this is not at all coincidental. This is also an era of contempt for the arts, many of which include considerable use of, yes, mathematics. Not just basic arithmetic, mathematics too.
So far as I can tell, even elementary school students are rarely told such helpful goalsetting information as that by the end of the calendar year they will have their times tables down flat. Or a bit later, how they will be able to work out the height and width of objects beyond their reach to measure directly with a ruler or weighted string. Secondary students don't necessarily find out when they begin instruction in algebra they'll be able to apply it to other subjects they may be more interested in, such as chemistry. Or that by the end of the semester they'll be able to derive Newton's method for finding and calculating derivatives, which is an entryway to calculus, if they want to use it. How can we reasonably expect students to persevere in confusing or difficult coursework without an understanding of how their courses go together or why mathematics isn't just dumb symbol manipulation? I don't see any way we can, and students certainly deserve better. I don't imagine a teacher spending a few minutes explaining how even what seems the craziest and most abstruse mathematics can end up being unexpectedly useful, like boolean algebra, or describe seemingly mundane things like crochet work. With luck besides bad, at least one student will be brave and skeptical enough to demand to know how. If not, I would suggest an instructor needs to have at least one real life application for students to look into or even do themselves for each technique – and no doubt actual math teachers working day to day know all this and do so when they can. Alas, my own experience is with mostly rather jaded and exhausted instructors who plodded doggedly through the required curriculum. It's not wholly their fault, they never talked about their subject in a manner similar to the excellent book Where Mathematics Comes From, which argues theoretical mathematics is a process of extending patterns and testing to see if they hold up from one area of mathematics to another. Where the analogizing process produces nonsense or unexpected results, there is usually something interesting to learn, as in the famous case of the parallel postulate; some people are really good at this and love it. No, I understand such teachers are dealing with unfair conditions. Yet even under the best of conditions, something is wrong, there is an information drop between levels, with particularly awful impacts on students as they go on to trade school or university.
During my own rocky transition from high school mathematics to undergraduate mathematics, I did not have time to sort out what was awry in my earlier mathematics education. Where I attended school, there was no such course as "precalculus" even for students like myself, who opted for the course stream intended for those going to university. Yet, as it turned out, it is precalculus courses and texts that covered the material I needed. It is in fact entirely possible to successfully learn much of the basics of differential and even integral calculus without it, as I did, but what a person inevitably learns in that case is more on the order of recipes than principles. However, mathematics cannot be pursued using rote learning the way early arithmetic can. But, learning recipes matched to specific types of questions is very good for certain types of standardized tests. It is also very good for mystification, and preventing understanding of why a person would ever want to learn how to estimate the answer to a question. It was not until much later I truly came to appreciate mathematics is a series of abstractions built on the basic arithmetic principles applied to routine tasks. Hence there is a genuine, traceable by a person without extensive mathematical training to trace the abstractions from counting to money handling, distribution, measurement, sorting, and containers. Part of how I got to this point was stumbling first upon a book about the many possible proofs of the right angle theorem and subsequently the trigonometric equations. It was mind blowing to discover these are powerful tools for understanding derivatives and integrals. They are among the first "crossover functions" we learn, the ones opening the way to another level of mathematical abstraction and to practical applications.
I remember being particularly annoyed by being directed to apply a Taylor series expansion in grade eleven math, when no instruction on what these were or how to use them was provided. I don't remember them even being covered in the textbook we were using, and by then my classmates and I did have the survival sense to try looking it up. The Taylor series was there, printed inside the back cover. But there was no associated chapter or section so far as we could tell. Later I learned the Taylor series would have been useful indeed for learning logarithms on different bases than 10, and the trusty and fascinating natural logarithms. These were still problems years on, as Vi Hart's wonderful video series of "doodling in math class" videos illustrate so vividly. I admit to a special fondness for Hart's explanation of logarithms and exponents, because it leads beautifully into exploring number systems on different bases, which itself leads to great examples amenable to support by reading analogue clocks. Then again, I get the stubborn impression many younger students have not learned to read analogue clocks, because it is so rare for even classroom clocks to show the right time anymore, let alone the public clocks in train stations and similar places.
In high school my class did received instruction in conic sections. Unfortunately, they were not framed as the first stages of working with polynomials, so we didn't learn how they enabled people to study them before today's familiar polynomial representations were invented. I remember wondering rather desperately why anybody spent time driving themselves up the wall working these things out and then trying to practice them. Perhaps it would not have consoled any of us when we couldn't get our work to come out right to learn these sections and formulae are used in architecture, machine building, astronomy, and navigation. Nevertheless, it would have been worth knowing about at least the astronomical uses, tying together our mathematics classwork more explicitly to our physics classes. By the time I got to university, instructors assumed students must already know these connections, especially if they had taken the prerequisites. This should be a safe assumption, of course. Later still, it seems university instructors are reluctant to state outright that by around the third year courses, students are mostly studying ways to solve specific types of special cases. This makes sense, by the third year students are beginning to specialize if they are continuing to take mathematics for a special field, they'll need those techniques for approximations.
I am not sure what the most practical solution to this mess in the way mathematics is taught would be. Inevitably, the challenge at hand is a multifactor one and it is shared by other subjects, including its cousin english. There are perverse incentives driving instructors in most elementary and secondary schools to try to cram recipes into students' heads fast and effectively enough to pass certain types of examinations. They are actively denied a budget for materials and time to coordinate with other instructors so to support students in undertaking smallscale art or construction projects to assist their learning. The situation may be better for students in private schools, although that is not a given. he result is a growing proliferation of businesses providing auxiliary or remedial mathematics instruction for those parents or individuals who can afford it. Selfstarters can spend fruitful time working along with open courseware lectures, and for anyone interested and able to commit the time, they will be wellserved by even just a sampling of those recorded by renowned mit professor Gilbert Strang. There are free mathematics textbooks written by real instructors at all levels, including excellent ones dealing with the allimportant precalculus needed by both university and tradeschool bound students. The tools, the content, the willing people are all there.
In the end the major barrier is the authoritarian nature of anglophone society, which filters down to the way anything considered critical to social order, including education. Hence education at every level is currently run on hierarchical principles, which makes it extremely difficult to respond in a timely and effective way to the needs of a specific educational population. It doesn't matter whether the authoritarian structure is run by the government or a private company, the result is the same. Responsiveness is removed, and it is unusual for creative and effective approaches to build on initial success. All too often, students are literally unable to answer the question, "Why are you learning this bit of math?" except to say, "The teacher made me do it," or "If I don't I won't graduate." Trammelled by such experience, it is little wonder later in life so many former students decry most of their schoolwork, not just mathematics, as little more than make work in an expensive system of babysitting. Thankfully, contrary to what we are encouraged to think by the mainstream media though, those former students aren't generally demanding the whole thing be shut down. But everyone is quite fed up with new coats of paint put on the same shown to be pointless and ineffective structures in our schools, colleges, and universities.
 For those wondering on what basis I make this claim, I will set out my argument briefly here. Mathematical practice involves finding ways to change the shape of the equation to make it easier to solve in that case, and how to calculate solutions of equations for specific ranges of numbers. The arithmetic symbols and operations are the basic framework, the phonemes; how they are allowed to go together constitute the grammar of mathematical language. Hence computer programmers can create algorithms to compose sentences fairly easily just as they can code up mathematical algorithms. Coding those algorithms so that they always produce correct and meaningful results is not so simple, however.
 I will continue to use this poor label having registered my overall objection to it, and not keep using scare quotes or similar. For some more specific comment on this, there is a thoughtpiece for that, in fact there are two of them: A Little English History, and Another Attempt at a Little English History. The earlier piece is revised for clarity.
 A great article to find out more about this is Indigenous BasketWeaving Makes An Excellent Digital Math Lesson, by Veselin Jungic, writing for theconversation.com (23 january 2019). Jungic discusses Tla'amin functional mathematics, and includes several relevant links to further stories. Protip when trying to look up more via search engine: any search result starting with a bloviation about Indigenous trauma is garbage.
 Lakoff, George and Núñez, Rafael E. Where Mathematics Comes From: How The Embodied Mind Brings Mathematics Into Being. New York: Basic Books, 2000.
 Vi Hart does not seem to have her own web domain anymore, perhaps in part because she is busy these days working at microsoft. The most dependable way to track down her videos is to search for them on vimeo or youtube.
 Of course, the challenges of teaching mathematics well go far further back than my dubiously spent youth. For instance, Augustus de Morgan published On the Study and Difficulties of Mathematics with the Society for the Diffusion of Useful Knowledge in 1831. The book was so useful it remained in print and with minor updates until at least 1902.
 The internet archive mirrors the majority of mit's open courseware offerings, including several of Strang's courses.
