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:

A few weeks ago in Algebra II I had students fill out a series of questions… questions which was going to lead to a discussion about mathematics and intelligence. I cribbed this sheet from my friend and teacher extraordinare Bowman Dickson.

I didn’t capitalize on it immediately, but I think I can still get some good mileage out of this. The thing that brought me back to this sheet was that yesterday in Algebra II, I gave an assessment that students didn’t fare as well as thought they would.

With one section, today, I had a heart to heart with them about what I saw, and this disconnect, and I talked a lot about the difference between active learning and passive learning. I think I got through to them. And I said: take what I said to heart. Be an active learner. And I’m going to give you an assessment on the same material next week. Show yourselves that you are capable. Because I know you are — but you just need to learn and implement the right strategies to be able to do it and make a lasting change.

Or something like that.

So we’ve had the talk about the concrete things… and I think next week, after kids take the reassessment (and hopefully — HOPEFULLY! — do much better), it would be worth it to have a talk about the more abstract side of things… attitude. The way students approach math, think about math, think about intelligence.

I’d love for any ideas about how to structure/have this discussion. I’ll throw my class data below, but if I’m going to do this, I want whatever I plan to be as powerful as possible. I want it to really get kids to think about what learning is, and how important having a growth mindset is. I have a few thoughts, but nothing great. So any brainstorming you might have, awesome.

As linked to on Sonata Mathematique:

Yeah, I want that DOUBLE POSTER SIZE in my classroom.

Without further ado, here’s my (fascinating) class data…

ALL DATA COMPILED

DATA REDUCED TO AGREE/DISAGREE

It’s now more than halfway through the first quarter, and things are … messy.

I’m pretty much going through Calculus like I did last year, except for the fact that everything is so much easier because I have standards based grading down. [1] I know what works. While Calculus was hell for me first quarter last year, it’s cake for me now. So calculus is not messy. [2]

So while Calculus is going smoothly, I’m finding Algebra II to be messy. Not in terms of my kids. I love my Algebra II classes. But like last year — when I vowed to really focus on Calculus and leave my other courses well-enough alone — this year I vowed to focus on Algebra II and leave my other courses alone.

Specifically, I’m working on two major things: making groups and groupwork a norm, and having problem solving be a regular (and non-special) part of the curriculum. (As you can guess, the two go hand-in-hand.)

I haven’t written much about my inclusion of problem solving into the curriculum, but right now we’re doing a day of problem solving before each unit (related to the unit), I have slowly started including problem-solving problems in our home enjoyment (our supremely corny term for homework), I have been putting simple problem-solving problems on each assessment, and we so far have had a single problem set (something which I may or may not continue with). Still, I should be clear that most of my curriculum and my classes are traditional.

Now, if you’re a teacher who teaches more traditionally and uses a standard curriculum, you know that this a huge change. Because there’s a huge activation energy involved in switching teaching modes. For me, I kept on saying “next year, next year” and I never did. It’s daunting! And why screw around with something that works well?

And if you’re a teacher who teaches with lots of groupwork, and uses problem solving regularly, you probably remember the year you went through the transition. And how it got easier each subsequent year, as you picked up more tricks of the trade. Tacit knowledge.

And if you’re not a teacher, what the heck are you doing reading this blog? Seriously?!? GET OUTTA HERE!

Switching to this mode has played havoc with my emotions. You see, it’s not healthy and I try to avoid it, but my self-worth is tied up with how well I think I’m doing in the classroom. When I feel like I’m doing things well, I walk around like I own the world. I have confidence. My head is held high. And when I feel like I’m doing a poor job, my head hangs low. I question my desire to teach. I wonder what I’m doing in the classroom. And I’m depressed.

This year, I’m playing emotional ping-pong.

There are times when I feel like I’m killing it in Algebra II. These are usually days before each unit, where we spend the entire period working in groups and problem solving. I love watching kids think and discuss, and they’ve gotten how to work well in groups down. I’ve never had it work so seemlessly. It’s amazing. They’re independent. They’re identifying their own misconceptions and fixing them. I leave these classes wondering why it took me so long as a teacher to get to this point… I feel like my kids are finally and truly grappling, and I love that. (And I’m starting to do this successfully when we’re not problem solving… I made an “exponent lab” which was just 20 “simplify this” problems… and I was seeing great things when they worked together.)

And then there are times when I feel like I’m being killed. I have classes where I want to crawl under my desk and hide. Some of these classes happen the day after kids problem solve, and they present their solutions. Kids put their work on the board, or under the document projector, and present. Or if we don’t have time, I’ll have them put their work up, and I’ll talk through it. These classes have never worked for me. It’s like pulling teeth. Kids don’t know how to present. They don’t know how to engage if they’re in the audience. It takes forever. I don’t think anyone is getting much out of these days. [3] Or there are the more frequent regular classes (where we’re not doing problem solving), and I find I’m standing at the front of the classroom the entire class, cold calling and explaining. And it’s ugh. I feel ugh. There’s no spontaneity. It’s not fun. I don’t mix things up or have different ways of introducing/practicing material to break up class.

What’s interesting is that I feel my kids think that I’m doing a crappy job. I know they — in actuality — don’t think our classtime sucks. (I had my kids anonymously answer some questions, including the what two or three adjectives would you use to describe our classtime question.)

But even though intellectually I know that my kids don’t think I’m doing a crappy job teaching, it doesn’t change the fact that I feel they think I’m doing a crappy job teaching. It’s a slight distinction, but maybe others of you out there know what I’m talking about.

So as I said changing things is messy. Because you don’t know what works yet, and what doesn’t. It’s taking a risk. It requires more work. And you feel like you’re constantly flailing and failing. And that’s not a good feeling. Here’s a recent Facebook “convo”:

I know this is sort of rambling. I’m just trying to work through some things, but I still don’t know where things are going. Which is why there isn’t a real point to this. Just a state of affairs, from an emotional vantage point. I’m not looking for sympathy or advice. I just wanted to try to get my thoughts down — and just let you know that if you’re going through a similar transition, you’re not alone.

[1] I have a list of standards I can choose from, I have good exemplars of problems for each standard, I learned how to effectively introduce it, and I know how to set it up so I don’t die with all the extra work that comes along with reassessments.

[2] But yes, there are lots of things I could do to improve it. Always, always…

[3] I’ve talked with a teacher who does a lot of group work and presentations, and she gave me some excellent suggestions (revolving around using giant whiteboard) which I’m going to take on board.