One of my students sent me a Slate article, yet another piece of tripe with an attention-grabbing, gag-inducing headline: “How To Fix Math Education in High School and College.” Barf.
And the article is short and doesn’t really say how to fix math education in high school and college. So there ya go. But my student asked me for my thoughts. And I gave myself 20 minutes to compose a response. I had to give myself a time limit because I know myself. I’d obsess, second guess, and then think: well, that’s not precisely right, and then get diverted to go into this or that tangent, and never actually send it. And if I did, I wouldn’t be happy with it and it would be maybe 5 pages of things I wouldn’t be happy with.
So I did it under time constraints. And I figured I’d share it here. It is not precisely what I believe, and it is a lot of broad strokes. And it certainly is choppy (because I didn’t having time to proof). But here you go…
Hi [Stu],
I think this article brings up a lot of good points, and I know at all the math conferences I attend and all the conversations I have with math teachers (at Packer and around the country), these are the discussions we are having.
When it gets down to it, there are two claims that I think are worth discussing.
First, that our kids are being pushed on a “calculus” track, while the real action and usefulness is elsewhere. I do think that there is this standardized curriculum in high schools, where kids are being put on a track where calculus is the pinnacle of their math studies. It’s not just Packer, but everywhere in the US. And I think that is not always the appropriate track — and we could come up with alternatives. We could have multiple tracks, culminating is statistics, discrete math and number theory, alternative geometries, or something interdisciplinary. Of course there are about a zillion things in the way, including staffing (who would teach these courses, how would they get paid, when would they have time to write the curriculum which would have to be something untraditional) and colleges (which look for calculus on a transcript, or so I’ve been told… I don’t really know much about that world). But I think most math teachers would say that calculus is just one possible, and not always the best, ending to a high school math career (depending on who the kid is and what the kid’s interests are in math). Very deep-seeded cultural, social, institutional, and even political barriers get in the way of revolutionizing what math is taught and how it is taught. On the other hand, I disagree with the argument that calculus should not be pushed because it doesn’t have as much “practical” “applied” use to most people. If we only cared about pushing the things that would be useful for students in the real world, why teach Shakespeare and Pynchon and hydrogen bonds and what makes a rainbow — if most students aren’t going to be working in a lab or becoming writers or critics? I think there’s a value to calculus for the sake of it being calculus, for it showing (for many, the very first time) the abstractness and beauty that a few simple ideas can bring to the table — and how these simple ideas can be stretched in crazy and amazing ways. (Given that a student has the algebra tools to accomplish it.) But to be clear, I honestly believe most math curricula in high school aren’t solely bent on helping kids understand calculus. If that were the case, I could come up with a curriculum where we elminiate geometry, and combine Algebra I and Algebra II into a 1.5 year course… and students would have the background to do calculus afterwards. That’s not the goal. The goal is building up ways of thinking, putting tools in your mathematical toolbelt, and leading up to abstraction and reasoning… with the hope that the structure, logic, and incredible beauty and creativity of it all come tumbling out. Now whether that actually happens… let’s just say it’s not easy to accomplish. We teachers don’t get students as blank slates, and we aren’t always perfect at executing our vision under the constraints we have.
Second, there is the claim that ” that schools should focus less on teaching facts—which can be easily ascertained from Google—and more on teaching them how to think.” I think most teachers would agree with that. But then the article goes on to claim: “mathematical education will be less about computation—we’ve got calculators for that!—and more conceptual, like ‘understanding when you need to do integrals, when you need to do a square root.’ This is a much bigger issue and it can’t be simplified into these two sentences. There is a large discussion going on in the math education community about the use of graphing calculators, and if they can be the panacea for math education. That students who struggle with basic algebra can still explore and discover using their calculators. I half-agree with that. Pattern-finding is great. It invites creativity and expression, this sort-of calculator-based discovery-learning. But if the calculator is used as a black-box, and we don’t know what it’s calculating for us, or how we could calculate what it’s doing (but just much slower, and possibly with different algorithms), we’re in trouble. If you can find patterns in pascal’s triangle, but you can’t prove them or at least have some plausible argument as to why they exist, then you’re just finding patterns. It’s cool, but has very little depth. If you let a calculator factor for you (the new ones can! like wolfram alpha!), but you don’t know what it’s doing, then I fear math can easily turn into magic, where the magician is the calculuator. And that’s one thing I big thing I worry about as a teacher: math being seen as a bag of magic tricks, where there is no logic or structure to it. And if the calculator is the magician, and the student is the audience, the student might marvel at the trick, be excited by whatever pattern is found, but never really understand what makes it all hang together. That’s why you hear me harping on understanding so much. And why when you found the power rule pattern, you did the first step, but the real learning came when you went off to prove it. It stretched your mind, and you spent a long while working it out. You wanted to understand the pattern, the logic, the conjecture. When technology helps with understanding, I LOVE IT. When technology helps generate questions, I LOVE IT. But when it replaces understanding, I’m a bit more wary.
So there are my very quickly typed two cents. They might not make a whole lot of sense, but they just sort of poured out. My thoughts change in subtle ways on these issues all the time, so ask me again in a few months and I might have switched some of my thinking.
Best,
Mr. Shah
To be honest, I’m posting this as part of my desire to archive my evolution as a teacher. You’re welcome to comment, and have discussions, if you so wish, but I probably won’t engage too much. I’m tired.
In other news, explaining why I’m so tired, I spent the last week and half writing narrative comments on all my students. I think they are better this year than in years past (each year I try to improve a tiny bit), so maybe if I have the time and desire, I’ll post about my process. But who knows, school is like a train and time just keeps whooshing by. I can’t believe a quarter is already done. It feels like we just started, and I barely have scratched the surface of my kids. (Right at this moment, that is. You know, by Thursday or Friday it’s going to have felt like this year is turning into a piece of taffy that keeps getting stretched out, the end getting further and further away while my grip on reality is getting as delicate as the taffy is getting thin.)
PS. On the views of math:
Continuous Everywhere but Differentiable Nowhere 
I wish I could give my students such a thoughtful and articulate response. I wonder what your thoughts on the role of computer programming to learn mathematics are. It seems like a way to sidestep some of the ‘black box’ issues that arise when a student has the magical calculator to do the work for them. If they created the magic thing, they are less likely to regard it as magical. That being said, programming is a massive time-sink, and is foreign to many math instructors.
In any case, well done and thanks for sharing!
Great response Sam. Like you I am concerned about too much reliance on using the machines to do the calculating. I’m all for using the tools to HELP but not at the expense of the students learning the concepts in the first place.
I’m all for problem solving and critical thinking, but believe that much of this is only possible with a good grounding of basic concepts, and more importantly, a good number sense. I don’t believe students can develop a number sense if they use a calculator to do all the calculating for them.
Cheers,
Chris.
That article is very popular, unfortunately. The general point Benjamin makes (and which you disagree with) is that we shouldn’t be focused on calculus because it’s not as useful for most people. He completely ignores that calculus is actually fundamental to math in a way statistics is not. I actually think student should learn statistics before college and that it is greatly de-emphasized in the US. But calculus is necessary to even understand what it means to take a number to an irrational power, or understand why the area of a circle is pi * r^2 (it’s of course possible to avoid the formalities of calculus, but morally one cannot avoid the limiting arguments that underpin everything). In fact, we often talk about the length of some curve, or the area of some curvilinear shape, but these notions do not make sense without knowing some calculus!
Yet Benjamin says students won’t find calculus useful. This is like saying students won’t find reading a novel useful (less and less people are reading them nowadays… they are more likely to be reading magazines or webpages!), so we should avoid discussing and analyzing famous novels!
Calculus is fundamental to several branches of mathematics, but not all. One could argue that mathematical logic is far more fundamental, as is set theory, but we don’t teach these to everyone (nor do I think we should).
I think that almost everyone would benefit from a better understanding of probability and statistics, and only a few from a better understanding of calculus. That said, my high-school-aged son is currently in a calculus class and has not yet had a decent course in probability or statistics. He has done some reading on logic (though from a somewhat watered-down philosophy text that we picked out of a garage-sale free box). He’s gotten a little Bayesian probability in a machine-learning context, but not enough to really be useful for anything.
Since he is currently thinking of becoming a computer scientist, I’ll certainly be making sure that he gets discrete math, probability, and statistics, but I won’t push him in calculus. On the other hand, he’s also interested in learning modern physics, and for that he’ll certainly need multivariable calculus and diff eq. If he wants those courses while in high school, I’ll pay for them, but I’m going to insist on discrete math and strongly encourage (Bayesian) statistics.
I’m afraid if you think I am arguing calculus is fundamental in the way set theory is “fundamental” then you miss the entire point of my argument. Please look over it again.
Perhaps ironically, your son won’t be able to study much Bayesian statistics without eventually having to apply some calculus at some point.
Oh, I agree that calculus is important for Bayesian statistics, and I think that my son will end up taking a pretty full load of calculus before he goes to college (probably through multi-variable and diff eq). But I don’t think that calculus is the most essential math for everyone to take. I’d rather see more probability and even combinatorics in high school, even at the expense of less calculus.
I recommend reading
http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap
http://www.artofproblemsolving.com/Resources/articles.php?page=discretemath
Thank you for the links. I am not advocating some kind of strange dichotomy between continuous and discrete mathematics. Indeed, if anything, I am advocating the importance of learning continuous mathematics (like calculus) even for those interested primarily in discrete math.
The ideas one has to understand to understand calculus are extremely useful in studying discrete math and conversely too. If your son doesn’t learn multivariable calculus (you said he would because he might consider physics, but that’s missing my point) because he (or you) think it is not useful in computer science, then he will get a nasty shock when he progresses much further and realizes it is useful for computer science– often discrete quantities are difficult to work with and thinking of them as continuous quantities instead can greatly simplify matters (a good book that doesn’t attempt to artificially separate areas of math is “concrete mathematics” by Knuth et al. “Concrete” refers to a blending of continuous and discrete. For example, to compute a large factorial, it is often better to use something like Stirling’s approximation, which is based on carefully approximating a certain sum with an integral).
My original point, however, does not have to do with this. It has to do with the simple fact that your son has learned a variety of factoids (like the formulas for area of a circle, volume of a sphere, etc, and properties of exponents for all real numbers etc). And he does not understand them unless he understands the essential concepts typically introduced in calculus. There are many bright young people that just memorize these things without understanding they are actually provable facts (I’ve even had some capable students tell me that the area of a circle is pi*r^2 “just because”!). I think if we are speaking as educators, rather than drillers of practical skills, we have to find it problematic that many children learn basic facts such as these without ever learning that there is a systematic way to prove them.
There’s a reason the ancients spent a lot of time working out these things (without calculus in its modern form, which made it all the harder), and it was only until Fermat’s time that the ideas behind probability were developed. That is what I mean by “fundamental”; not in the sense of foundational to the logical structure of mathematics, but in the sense of “anyone that is familiar even with the slightest bit of math has thought about this, even in a minor way”.
I don’t think we disagree on what should be taught in high school (because I don’t really have any firm opinions on that). But I think people advocating discrete over continuous, or probability/statistics over calculus, often miss that it is not the theorems per se that is important, but the way of thinking. And the calculus way of thinking (which involves comparing quantities that get larger or smaller, or ratios thereof, and approximating quantities and taking the limit) is just fundamental to a lot of mathematics in a way that probability is not. And even in probability (some would say *particularly* ), the calculus way of thinking is key, once one moves beyond the basics and either tackles harder theory or tries to put the basics into practice (for example, computing a binomial probability for large exponents).
I’m not saying caculus is useless to computer scientists. I have a PhD in CS and considered going into analysis of algorithms with Knuth as my adviser. I know (or knew) the uses for calculus in discrete math, and calculus is not central to combinatorics. Useful, yes, central, no. The main reason for using calculus to make approximations in combinatorics is that integration is a hell of a lot easier than summation for a lot of cases.
I am not claiming that probability is central to mathematics. Indeed I got an MS in math and PhD in computer science without any formal training in probability and statistics. What I’m claiming is that probability and statistics are more *important* for most people to know than mathematics is. People need enough math to understand probability (about Algebra 1), but unless they are continuing in a STEM field they don’t need much more.
Now, my son is planning to go into a STEM field, so he’ll be getting a lot more math that minimum, and calculus is important for him. I think it is important that calculus be taught in high schools, since we need people trained in STEM fields. But I think it is a major failing of our educational system that most people have so little understanding of probability that they can’t understand lotteries or risk assessment. I have heard from high school teachers that the tiny amounts of probabilities included in the textbooks and standards are usually the first thing that gets dropped from the curriculum, when it is probably the most important thing for most students to learn in their math classes.