It’s been a while since I’ve posted anything here – I’ve been pretty busy with work.

In earlier posts I’d mentioned interesting articles on mathematics by Bill Thurston, Terry Tao, and Paul Lockhart. Today I’m going to try and relate these articles to the topic of educating children in mathematics generally.

Let me start with Lockhart’s article. In that article, Paul takes the aggressive position that math is more like art than anything else; because we treat it in a rote fashion, we rob children of the beauty of the subject matter. I find this an interesting point because I think it perfectly demonstrates our collective bipolar attitude towards math. There are two simultaneous memes about mathematics out there:

1) Math is rigorous, precise, mechanical, devoid of spirituality, dry, cold, logical, spock-ish, etc.

2) Math is beautiful, creative, artistic, elegant, wondrous, God keeps its most amazing results in a little book, etc.

I assert that most people have had exposure to *both* memes; unfortunately, most people merely have evidence for the first meme, and not the second, and so the first one is perhaps the more commonly-held belief.

But my main point here is that both memes are one-dimensional: they fail to capture the entirety of mathematics, and (more importantly) they fail to capture the spectrum of reasons why children should learn math.

There are at least two dimensions to this problem. One dimension might represent the range of attractions of mathematics itself: the stereotypical engineer loves the pragmatic value of the Fast Fourier Transform or of error correcting codes; the stereotypical head-in-the-clouds number theorist loves ancient riddles about whole numbers (ok, yes, this is foreshadowing – turns out the lover of error-correcting codes now has a lot in common with the lover of arithmetic – but that’ll be a *much* later post!).

But the other dimension to this problem is that there’s a hierarchy of specialization: not every citizen needs to get a PhD in mathematics to be deemed mathematically literate. So, in effect, a math curriculum will by necessity partition the body of math knowledge into ~3 strata:

1) The math that is valuable for the general public (basic math literacy)

2) The math that is needed for certain vocations (say, for undergrad majors in engineering, science, etc)

3) The math that is needed for specialists in mathematics itself.

Paul makes a very compelling argument that, in partitioning math as above, we’ve inadvertently also removed all the interesting and beautiful stuff about math from the first and second tiers of this strata. I would generally agree with this point. But there is a flip side to this: focusing on the beautiful patterns, and on selling the wondrous-ocity of these patterns to kids. In my experience, middle-school kids can be quite pragmatic, too, and you want to appeal to that side as well.

Richard Feynman once wrote that he’d aimed his physics lectures at both the theorist and the experimentalist: he wanted to provide something for each personality type. Similarly, if you teach math purely as an art form, and fail to note the “unreasonable effectiveness of mathematics,” then some of your class will miss the point. BTW, I’m very confident that Paul personally provides something for everyone in his class, I am pointing this out mainly to reference what I think is a weakness in his thesis.

To expound on this point some more, let me suggest that math is in many ways like English. Consider the problem of teaching English to the masses. There are two parallel goals: basic proficiency with the language (both literacy and writing skills) and exposure to literature as an art. Great English teachers understand that reading and discussing great works of literature is a powerful and effective way of developing a child’s ability to understand the existing corpus and reason effectively about it. At the same time, diagramming sentences and “mechanically” writing 5 paragraph essays – these are great etudes, a focused bit of deliberate practice that (when properly motivated) hone a student’s craft. Great English teachers juxtapose both kinds of work throughout their curriculum.

The same is of course true for music: a great music education trains you in musicianship, in instrument-specific technique, in theory and in developing a rich understanding of the history of music and the body of work that has been done before. You study the great works, and understand why they are wonderful, and you strive to copy them, and then later to find your own musical voice.

So, big surprise: math education is like many other human intellectual endeavors, and getting great at it requires developing facility with the tools and techniques of the trade, while also deepening one’s conceptual understanding of the subject. This conceptual understanding is in turn achieved thru a combination of carefully studying specific examples in great depth, along with then abstracting out core principles, patterns, and connections to other topics. At various times, the subject of mathematics has been developed and expanded by folks who were motivated in dramatically different ways: some were motivated by very pragmatic, even mundane or quotidian problems; others were driven purely by aesthetic considerations, whether it was beauty, fun, farce, or perhaps even a combination of these and other driving forces.

Well, haven’t posted in awhile – I’ve been meaning to write extensive posts about each of the following two links. But maybe it’s best to simply start the conversation by providing links to them:

First: Terry Tao has a great essay entitled “Does one have to be a genius to do maths?“. I think that this essay, along with its hyperlinks, should probably be required reading for anyone who’s interested in math.

Second: Paul Lockhart was a very popular math professor at UCSC, back when I was a grad student there. In particular, I remember him teaching a very down-to-earth, hands-on course in algebraic geometry, that I was too cool to take seriously, because back then I had exactly the type of lame attitudes towards math that he has since come to decry; I certainly view that class as a missed opportunity for me. Recently he wrote a book that has been referenced multiple times in Steven Strogatz’s recently-started math blog, a book which I assume¬† is a more fleshed-out version of his essay “A Mathematician’s Lament” (the book has the same title!).

I think that the Mathematician’s Lament article, and the follow-up piece by Keith Devlin here, provide an excellent back-drop for some points I’d like to make about math education. And even if I have nothing valuable to say about either Terry’s piece or Paul’s, I think you’ll gain much from simply reading both these pieces.

In my next post, I’ll say more about both pieces.

OK, so as many know, the common pattern for the three problems from the last post is called the “Pigeonhole principle”, and it’s deceptively simple: it says that if you have n+1 pigeons and you are trying to fit them into n pigeonholes, then there exists at least one pigeonhole with 2 pigeons in it. Sounds simple enough, but then how do you apply this? The trick is to identify and properly define the pigeons, and also define the pigeonholes. In this case, the buckets are the different possible remainder values modulo 11: there are of course 11 pigeonholes.
The 12 distinct integers are the pigeons. And so there exists a pigeonhole with two pigeons in it: since they are distinct integers, their difference is a multiple of 11 (of course, had they been the same number, that would also be true, since 0 is divisible by 11).

What about the problem with the trees? You could have 600,000 pigeonholes, one pigeonhole for each possible value of (number of pine cones in a tree); there are only 600K, because 600K is the max value for the number of pine cones, according to the problem (as my friend Bernard pointed out, this is a wild over-estimate). The pigeons in this problem are the different trees: put a tree in the pigeonhole corresponding to how many pine cones it has. Since there are 1M trees, you have way more than 600K, and so you’ve got many pigeons crammed in the same holes (at least 400K repeats, as Susan’s daughter Evelyn pointed out – great job, Evelyn!).

What about the last problem? This is a slightly tricky one. For now, I’ll give a hint: Let Timer 1 be the timer that goes off every x seconds, where x is irrational. If I knew that there was an integers k and m > k so that the k-th and m-th firings of Timer 1 happened within \mu seconds of each other, then this means that every (m-k) firings are exactly \mu seconds apart.

If we’re clever, we can get \mu < \epsilon, and so you could imagine traversing the face of a clock \mu seconds at a time (basically ignoring all firings of Timer 1 except the ones that occur at k, k+ (m-k), k + 2(m-k), etc, seconds). How could this help us solve the problem?

Some things to focus on in your solution:

  • what are the pigeonholes?
  • what are the pigeons?
  • why do we insist that x is irrational?

Math Craft


One of my favorite essays on math is an essay by Bill Thurston, entitled On proof and progress in mathematics. In that essay, he refers to the difficulty of even defining what mathematics actually is; as an aside, he offers a proposed definition of mathematics (that he himself is not 100% satisfied with): “the theory of formal patterns.” In a later post, I’ll talk more about Thurston’s article, but for now I want to mainly use that definition as a jump-off point for this post.

I love this definition because I think it goes a long way both towards identifying what mathematics is, as well as de-mystifying the process of doing mathematics. What do I mean by this? Well, for example, for people who are unfamiliar with math, the concept of proving a theorem sounds abstract and intangible; in contrast, identifying a pattern, and then clearly articulating what that pattern is, sounds like something you remember doing with relative ease in elementary school.

The other reason that I like that definition is that it dovetails with my math-as-craft metaphor. Folks in the software industry use the term pattern to describe and identify recurring software problems along with their solutions; their terminology was in turn inspired by the architect Christopher Alexander, who wrote “Each pattern describes a problem which occurs over and over again in our environment, and then describes the core of the solution to that problem, in such a way that you can use this solution a million times over, without ever doing it the same way twice”.

Christopher Alexander’s goal was to propose a sort of aesthetic philosophy of architecture that evoked the crafts of an earlier age: a great carpenter knew, for example, a variety of different joins, and each of these joins was a pattern that could be used in myriad contexts to solve a variety of individual problems. A master carpenter was first and foremost a problem-solver who identified possible solutions, and was able to weigh the trade-offs to each solution, and pick an appropriate solution based on the intended function of the object being made (a wheel barrel? a cask? a table?)

Much of mathematics can be described in exactly this way: in solving a math problem, it’s important to recognize an underlying principle (or pattern) that give that problem its structure.

But this is too abstract, let me give an example. Here are three problems, that on the surface, are dramatically different; I will illustrate how a common principle is key to the solution of the problem:

1) Show that, if you have a set of 12 distinct integers, then this set contains a pair of integers whose difference is divisible by 11.

2) Show that if you have 1,000,000 pine trees in a forest, and no single tree has more than 600,000 pine cones on it, then there must be two trees in the forest that have the exact same number of pine cones.

3) Suppose that I have a timer that goes off every x seconds, where x is an irrational number. Suppose that I have another timer that goes off at exactly y seconds after the beginning of each minute. Show that, if I wait long enough, I can guarantee that both timers go off within half a second of each other (in fact, you can ensure that there is a time when the timers go off within \epsilon seconds of each other, for arbitrarily small \epsilon.

I encourage you to try and solve each of these problems. In my next post, I’ll present a pattern that provides a solution to each of these problems. Basically what I’m saying is: when a mathematician is faced with solving one of the above three problems, she pulls out the same tool from her tool-belt as she does for the other two problems.

Now, why did I mention all this? If you are willing to suspend disbelief, and take me on my word (for now, I’ll provide a full justification later) that there is a common pattern here, and so my analogy with carpenters has some validity, then you could begin to see how practicing using these tools is the way to gain facility, and eventual mastery.

This is a theme that I hope to come to again and again in these posts.

My idea here is to start blogging about what a parent might want to know around how to have their kids be excellent at mathematics. For me, excellence in math means the following:

  • Passion for the subject: I firmly believe that every kid starts out being interested in the world and how it works. Generally, over time,¬† parents and teachers bore them to death, paralyze them with fear, and deny them any truly rewarding experiences around math. All the while telling them that they *should* get good at it. Or alternately, we tell them that there’s nothing wrong, they are just not “mathematically oriented”, and that they should focus on being artists or whatever. Meanwhile, all artists are mathematicians, for the correct definition of mathematics.
  • Mastery of the tools of mathematics: the concept of the naturally gifted genius is pretty much the wrong metaphor. I know plenty of “normal people” who have extraordinary facility in mathematics. I’m an average joe myself, but I’ve far surpassed any reasonable expectation I had of myself, by dint of keeping my eyes open and (on occasion) working hard. The correct metaphor for mathematics is that of a craftsman: by a cyclical process of facing new problems, working steadily, acquiring tools/techniques, and stepping back every once in awhile to consolidate and attempt to generalize on what you learn, you become a master over time.
  • Living connection to the subject: use it or lose it. Here the correct metaphor is something like running. You could brag about having run a marathon years ago, but if you haven’t run in the last 3 months, are you actually a runner? Probably not. Same goes with mathematics – there are (sadly) many math professors who haven’t done math for years; they aren’t currently in shape, but are getting by via memories of their glory days, regurgitating cool stuff they learned in their youth. Not to say that this can’t be valuable, from a pedagogical and/or historical perspective, but this isn’t what I would call excellence in mathematics, per se.

I have a vague notion of writing a book on the topic of helping parents guide their kids from pre-school to – well, at least thru college mathematics. I basically plan to use this blog as a whiteboard space for exploring themes along the lines of this book. Hopefully, I’ll learn more about this topic from my exchanges with you folks, and the final book will be better for it. And actually, I don’t care about ever publishing a book, but I do think that there is a lot of knowledge that I and many others have on this topic, that nevertheless is known to only a small sub-population. If I expand that sub-population by any margin, this project will have been a major win.