# Math should be uncanny.

When I read E.D.’s piece on maybe teaching less math in elementary school, I figure that since I studied math in college and still do some recreational math, I should have something to say about math education in America. But whenever I dip into literature on math pedagogy, I come out baffled. There are a lot of people who have worked really hard to figure out how to make an effective math curriculum, but to students it all still seems arbitrary. I am not at all sure what the math curriculum ought to be like. So here are a few humbly presented thoughts on learning math, from someone who’s always enjoyed mathematics and had really great math teachers in high school. (If it matters, I went to a public high school.)

I’ve always found it very handy that I have my basic times tables and division tables burned into my brain, and it seems like elementary school is a fine time to do this kind of rote learning. I’ve tutored kids that will use calculators for single-digit multiplication, and it always makes me sad. A lot of the weirdest stuff I see in math textbooks for elementary schoolers seems to be in exercises designed to teach addition and multiplication by the kind of extremely basic reasoning that elementary schoolers are capable of, and this strikes me as a recipe for frustration and lifelong math fear. Maybe they don’t need to understand what’s going on until later. I don’t think my brain was developed enough to get the kinds of abstractions that math is really about until middle school.

Our high school math courses are a whole other mess. The problem with high school geometry right now, at least in the textbooks I have seen, is that it veers wildly between a Euclidean framework and the function-on-the-Cartesian-plane setup that leads into calculus. I don’t know what should be done about this, but the Euclidean stuff gets so watered down that it’s practically useless. Unless the math teacher is very good, the students aren’t going to learn mathematical reasoning in the section on proofs; they’re going to learn how to play a weird and somewhat arbitrary fill-in-the-blanks game that’s based mainly on guessing what the teacher wants to see. And then, SATs aside, they’ll probably never see Euclid again, even in college. When I learned how to write rigorous proofs in college, I had to prove number-theory statements rather than geometric statements.

But if I can end on a weakly Poulosian point, the political pressures to make kids good at math usually comes out of a view of math as a collection of techniques that will help them become awesome scientists or engineers. Or hedge fund quants, I guess. So boring! Very few teenagers, if any, will have the kind of drive to learn calculus just because they hope to be engineers or accountants in a decade’s time. You get excited about math when you manage to catch even a small glimpse of what’s so weird and special about mathematics: the abstract realm where numbers live, shimmering and clear, connected to each other in so many strange webs; where perfect flashes of inspiration disperse the thickest mists of confusion; where mysterious creatures — functions, complex numbers, rings and fields — dwell up in the high far-off hills. Not that most students would put it this way. In high school, it’s more like: “Isn’t it *weird* that the golden ration shows up in the Fibonacci sequence?” or “Isn’t it *bizarre* that I can find an exact value for the area between these two curvy lines?” Small moments of wonder can be enough to make the whole endeavor worthwhile — but if the teachers don’t know what’s wondrous about math, how will the kids ever see it?

~~~~~

Update: I wrote all of this last night, and I’ve got two things I want to add. First, the comment from T. Greer is worth reading, since it gives an actual educator’s perspective. Second, the book that first evoked number-wonder in me was Math for Smarty Pants by Marilyn Burns. I read it so many times. Buy it for your kids.

Building blocks upon building blocks.

I think that there are, generally, two kinds of math.

The first kind of math is the kind that absolutely everybody needs… and it’s fairly rudimentary. The ability to add, subtract, multiply, and divide small numbers in ones head, the ability to add simple fractions (1/2 plus 1/3), and quick and dirty estimation skills. You know… balance a checkbook, bake a cake. Life skills.

The second kind of math is the kind that is also a kind of metaphysical discussion. These are axioms. We use these laws and these axioms to lead to proofs… and upon simple elegant proofs, we can build complex elegant proofs. And, from there, we can move to algebra and trig and from there we can start explaining the universe through calculus and, from there, oh my goodness we are just getting started.

I would say that I’d be surprised if more than 10% of the population would get any actual benefit from trigonometry. 5%, even… which is not to say that we ought not teach it. Indeed, the 5% of the population that does benefit from it goes on to provide many, many more dividends from having it taught to them than it costs to teach the 95% who won’t benefit particularly.

There seems, however, to be an assumption that we are doing people a disservice if we do not do everything we can to assume that they are part of the 5%… and, in practice, this seems to lead to such things as, let me quote you here, “Euclidean stuff gets so watered down that it’s practically useless”.

I don’t know what ought to be done, mind. My old school system had Basic, Regents, and Honors… which, I suppose, was good enough. They discussed math as metaphysics in Honors, they discussed drill and kill in Basic, and Regents was both hot and cold and ended up being lukewarm (Rev 3:16). The kids who saw the Divine in math tended up to Honors… and the kids who did not were scuttled off to the other two depending on their gifts. Maybe stuff like that will have to be good enough, if we want to make sure that everybody gets a crack at trig before they graduate.Report

Something like your distinction works for English class too: basic reading and writing skills vs. literature appreciation and rhetoric.

My point about the uselessness of Euclidean stuff was that the way geometry is taught in high school doesn’t make sense from

eitherperspective. Really: I have seen tests where the “proof” sections are fill-in-the-blank. (Exercise for the reader: read book I of Euclid, then compare it to the the section on triangles in the textbooks that North Carolina public schools use.)I don’t want to say that every student should be taking trigonometry or calculus. But I do think that a lot of students in the main portion of the math-skills bell curve* have a math phobia that they don’t need to have, and that maybe math wouldn’t be so painful for so many if we thought about the aesthetic value of math alongside the practical/scientific value. Basically, don’t treat the second kind of math as if it’s just like the first kind of math.

*I really have no idea if math skills show up as normally distributed.Report

One epiphany I had yesterday after reading the previous post:

I think there’s too much focus on “how would I use this in real life?”, because the actual applications of higher math in real life are useful but boring. I think there might be more success focusing on “how would I use this if I were Johnny Danger, Action Hero?”Report

That sounds like a really great idea for a book. Arithmetic for negotiating with the arms dealer, a little basic cryptography for code-breaking, some trig for resetting the coordinates of the ICBM before the launch timer runs out…

Did anyone else see PBS’s “Square 1” series when they were growing up? They had a Dragnet parody called “Mathnet” where the detectives used math to solve crimes. That’s where I first learned about the Fibonacci sequence. Maybe that’s where I got the idea that math would be useful.Report

I loved

Mathnet, but maybe I was a little too old, with the boringness of math already ingrained in my skull as a given, for it to take hold.ReportI love this idea. Would you object if I outright stole it? With a dedication, of course.Report

No objections from me! I’d really love to see a book like this on the shelf someday.Report

From William Bradford:

“…and still do some recreational math.”As a history/anthropology major that statement made my head hurt.

With that said, as a parent of two kids under 16 I am appalled by their math training. There is no emphasis on basic rote memorization of addition, subtraction, multiplication and division tables like we had when I was growing up.They are taught to still use their fingers well up to middle school. It’s insane.

As an adult who was never good at math when I was younger, my salvation has been Excel. Using it for work and real-world applications now makes me understand a lot of things better than when i was doing them on paper. My math skill has grown considerably since i began trying to organize numbers for my company 5 years ago. Application is pretty helpful to those who aren’t naturally talented with numbers.Report

I’d argue that Excel is one of the very few computer applications that has really, genuinely saved time and made my life easier. Notepad/TextEdit is another.Report

I can’t make a to-do list anymore without craving a validation list and a pivot table.Report

The Maya did some pretty amazing things with math using their fingers and toes.Report

The Mayans had a fully-developed base-20 numeral system, including 0:

http://en.wikipedia.org/wiki/Maya_civilization#MathematicsReport

Your headline is perfect. There’s something arcane and enticingly weird about the possibilities of math, at least when you realize there’s more to it than textbook problems—Cantor’s infinities, the Pythagorean cult, the invention of zero as a concept. I don’t know how exactly a teacher might harness that, but it seems like someone could. (I mean, someone besides the Wu-Tang Clan, bless their hearts.)Report

Graphic novels like Logicomix could be a good start. (See Wunderkammer mag for a review.) My brother’s not a math guy, and he enjoyed it almost as much as I did.Report

That looks great. Thanks for the links.Report

Anyone interested in math or math pedagogy ought to read Paul Lockhart’s book, A Mathmatician’s Lament. http://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative/dp/1934137170 The “By God They Don’t Subject My Kids to Years of Learning the Times Tables Any More” group will, of course, hate it. But Lockhart’s point, that Mathematics is Art and needs to be taught as such, is worth paying attention to.Report

Similarly, here’s a 25 page pdf article by the same author on the same subject: http://www.maa.org/devlin/LockhartsLament.pdfReport

I’ll keep my eyes open for the book. I’ve read the shorter article, and I recommend it highly. Especially the section on high school geometry.Report

The book is a modestly extended version of the article — the last sections offer some examples of how he believes math should be taught. You can get some of the flavor of it from Steven Strogatz’s recent pieces in the times — the first ones draw fairly heavily from the book. http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/ is the first.Report

I never got a good appreciation of math until after my first round of college. By way of biography, I had completed multivariate college calculus (3rd semester generally) by my high school graduation. I haven’t read too many math books since then, but one of Asimov’s book rekindled my flames. John Derbyshire’s book “Prime Obsession” was also excellent, although I had to return it to the library before I could finish it.

In the end I don’t think you can have a proper appreciation for things unless you have some experience. There seems to be a false dichotomy that if you don’t enjoy something that you can’t learn it and learn to appreciate it, as if enjoyment were a prerequisite for learning. It is very difficult to enjoy most any activity until you get the basics down.Report

How should we teach math to the half of all children who are below average in math ability?Report

I’d say recognizing that there is such a thing as child who will never be above-average in math proficiency would be a very useful start!Report

Freddie’s right — I’ve known people for whom algebra was just totally unnatural, to say nothing of calculus. Like I said in a different comment, I am not really sure where the average is.

Steve, I don’t have a good answer for your question. Sorry!Report

That would be a gross misrepresentation of data. Certainly we can say half the children will have an easier (or more difficult) time acquiring any sort of knowledge. That is merely the logical possibilities by definition. However, we can not use that to state that half will have trouble acquiring requisite knowledge. We can state with relative certainty that 97% of children in the United States will learn the alphabet, despite the logical postulates that half will have an easier time doing so than the other half. Likewise we can say 80% of people will not acquire the knowledge to perform Leibniz transformations, despite the logical postulates that half will have more difficulty doing so than the other half.Report

Suppose “ability to do math” were both one-dimensional and possible to measure accurately, say using the same scale as IQ where average is 100. There’s going to be a limit to how precisely it can be measured (note that I’m using ‘accurately’ and ‘precisely’ with their technical meanings here). Let’s define “101” as the lowest score that we can be certain reflects above-average prowess, and “99” as the highest score that we can be certain reflects below-average prowess. A significant number of students are going to score exactly 100, leaving fewer than half to be 99 or lower, that is, below average. And even that’s making unrealistic assumptions about the precision of testing. In the real world, we’d define average as something more like “between 95 and 105”, leaving even fewer students in the ranges below.

And that’s assuming that mathematical ability can be represented accurately with a single number. In real life, things are more complex (and here I don’t mean “part real and part imaginary”.) Some students have great facility with numbers but find the abstractness of algebra difficult. Some do fine with algebra but lack the ability to visualize geometric objects (and vice versa) Others are quite successful until calculus defeats them, while still others can grasp enough calculus to use it as a tool, but will never understand it at the deeper levels that mathematicians require. (Limits are hard; the naive use of infinitesimals that Bishop Berkeley objected to is much simpler.) There are going to be students who are below average in each of these, but since the sets aren’t co-extensive, their intersection is again going to be less than half.

At any rate, much as “half are below average” sounds tough-minded and realistic, it doesn’t stand up to any real scrutiny. I think there’s a lesson there.Report

Your argument is totally shorn from any context; here in America, we aren’t anywhere near accepting that some large number of students simply are not going to achieve at the level of mathematics that we are expecting all students to. Indeed, both our rhetoric and our legal policy requires

100%competence. 100% compliance is not a mature goal, and it is the mark of a society that is far more interested in talking tough than in actually making practical gains on vexing problems. But that’s America– image and symbolism over substance, always.Report“Always,” dude? Or “over the last several decades”?Report

There are very specific methods for testing for abilities, math and its subset skills included.

That there are kids who will never master algebra is obvious; that there are kids who will never master Chaucer, equally obvious.

What is really sad as that we focus on math/language arts and leave out a host of other skills; particularly the mechanical, musical, visual, and kinetic.

As the mother of two kids with extreme learning differences — top 2% in math, bottom six in some parts of language arts — I can speak volumes to the woes of teaching to the average instead of teaching to the child.

But I have noticed one important thing: many kids ‘outgrow’ their learning difficulties in their early 20’s, my assumption is that the change is related to the development of the frontal cortex. For my two kids, that’s when their lives as ‘well-rounded’ students began.

Patience is a virtue, particularly with education and the habits of life-long learning.Report

Speaking of uncanny math, it doesn’t get much uncannier than this:

http://en.wikipedia.org/wiki/Chaitin%27s_constantReport

I learned a lot of things in school that I did not fully appreciate at the time, and even when I was told why they were important, I didn’t really get it. Later in life, I’ve been surprisingly moved by discovering why what I was taught actually mattered. No 16-year-old is ever going to empathize with King Lear, or understand the practical importance of solving systems of linear equations, or really appreciate the subtle tactics of ski-racing. But there’s no better time to teach these things – if you wait until people need the knowledge to teach it to them, they’ll be too busy and their habits of mind will probably be too set. So we really have to teach kids to write essays on fileal ingratitude, solve silly little systems of 3 equations, and run gates (well, that ones not so important unless you grow up in Truckee), knowing that they won’t really understand the full significance of these things until they’re much older. This is why school is boring, and I don’t think there’s much we can do about it.Report