What is the goal of math education?

There's been quite the buzz of late over this article, A Better Way to Teach Math, published in the NY Times' Opinionator Blog last week. If you haven't read it, the author discusses the idea that maybe math achievement doesn't have to be distributed along a bell curve at all, and that we're actually just not teaching math in a way that allows most students to succeed. The method highlighted in the article is a curriculum called JUMP Math. According to its website, "JUMP Math is a charitable organization working to create a numerate society." I certainly have no beef with their mission. The emphasis of their method seems to be confidence-building and breaking each mathematical procedure down into its most basic component pieces and "assess[ing] each student's understanding at each micro-level before moving on."


There are many claims made in the article that I agree with, and many ways in which I applaud the JUMP program. There is a huge achievement gap in math and I agree that "for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice." In the end, I am a proponent of any program that effectively levels the playing field and allows all students to reach their potential, mathematical and otherwise. However, these words - "potential", "achievement", etc. - are riddled with bias and my fear is that programs like this one pander to our current paradigm of math education instead of questioning its rather tenuous premises. What are some of those premises? Standardized testing as a measure of numeracy. The AP obsession. The glorification of calculus as the be-all and end-all of high school math.


The author states, "In every math class I've taken, there have been slow kids, average kids, and whiz kids. It never occurred to me that this hierarchy might be avoidable ... Can we improve the methods we use to teach math in schools - so that everyone develops proficiency? Looking at current math achievement levels in the United States, this goal might seem out of reach." My immediate response to that is: When we measure "achievement" as a single proficiency score between zero and 100, then of course the scores are going to fall along some kind of a bell curve. That is the nature of such simplified quantitative data. In some ways, it seems like our system is set up to produce high-achievers, middle-achievers, and low-achievers.


There is currently a lot of amazing brainpower being devoted to developing strategies for helping kids succeed in the current system - Khan Academy and JUMP Math are two examples. I wonder where we'd be if there were similar amounts of brainpower devoted to shifting the paradigm of math education and creating an actual, tangible resource bank that is in line with the paradigm shift. In my own little math edutopia, math classes would look a lot like the ones presented in Lockhart's A Mathematician's Lament. Students would do mathematics as mathematicians do  - by collaborating, by posing natural questions, and by attempting to answer them. Mathematics is meant to be critiqued and refined just as a piece of creative writing is, and the art of proof is meant to be taught as such (an "art") and not misrepresented as an exact science. This is of course oversimplified summary and I encourage you to read the Lament. It's a beautiful piece of writing that may just change the way you think about education. 


My esteemed colleague at Broken Airplane (who I also have the privilege of working with every day) makes a great point: Sure, Lockhart's Lament sounds great and provides lots of food for thought, but where's the stuff? Where's the curriculum, the activities, the books full of usable tangible things? Until he's got the goods to back them up, his ideas are somehow destined to take a back seat to the current system (for which there are a plethora of really effective resources).


In the end, this tension between skills-based math and inquiry-based "pure" math exists because we haven't yet decided what the goal of math education really is. Why is it that we make our kids study math for at least twelve of their formative years? Is it so that they can be good little calculus students in college and maybe even good engineers? Or is it so that they can develop an intellectual appreciation for inquiry and patterns and proof and abstraction, ultimately applying that creativity and critical reasoning to the endeavor of their choice? If it is the former, then breaking down every mathematical concept into skills-based components is certainly the way to go. If it is the latter, then doing so might just obfuscate the very beauty of math that we are trying to impart.


It is my impression that a lot of us are trying to strike a balance between the two. We want to prepare our students for college-level mathematics and engineering because that is our duty, but we also want them to experience why it is that we fell in love with math. What I find, though, is that I wind up betraying that second goal so that I can adequately cover all of the content that I feel compelled to. Of course there must be some ideal balance between the two, but it seems to me that right now the pendulum has swung much too far in the skills-based direction. In my humble opinion, this is because (a) it's much easier to assess, and (b) it's much easier to teach. [One could argue that (b) is a direct corollary of (a).]


My question is: are these two goals mutually exclusive? Can one both help students develop a great skills-based mathematical toolkit while simultaneously creating a classroom where students really become little mathematicians? Am I missing something? What do you do in your classroom to strike a balance?

Festival del Sol - Cuckoo for Calculus!

I teach at a project-based school, yet in 12th grade math I rarely do an actual "project." Of course, the meaning of that word is completely subjective and I don't mean to say that I don't do anything interesting or creative in my classes, just that I don't try to stuff content into a contrived project just for its own sake. However, this past week was our annual "Festival del Sol" and each class was expected to exhibit something. I had been racking my brain for a a calculus project all year - one in which students would truly learn the content through the project - and couldn't come up with anything. (The closest I've come to this was the "Great Calculus Challenge" where we dropped a block off the roof of the school - see previous post about that one.)

So, I stopped stressing out about it and figured that I'd give my kids a "break" for a couple of weeks with the following project: Pick any concept or problem that you've enjoyed this year, write a short technical paper explaining the concept / problem, and figure out a cool way to present the concept / problem at Exhibition. At some students' suggestion, we called this project "Cuckoo for Calculus!" (To my surprise, no one volunteered to dress up as the crazy Cocoa Puffs bird.) You can see the actual project handout and specifications for the write-up here.

I rationalized spending two and a half weeks on this by telling myself:

1. My kids are learning a ton of math this year, most of which they'll probably forget anyway - so why not spend time delving deeper into a topic they enjoyed with the hopes that they might actually remember it?
2. It's probably worth doing something fun and rejuvenating that might ward off the inevitable post-spring break Senioritis.

I did several projects last year as an 11th grade teacher, yet this was the first one the kids were somewhat excited about - and I'll admit, that felt rad. I think that the student choice element was key, as was the fact that my students are generally motivated and enjoy the class. Many students chose to return to a problem from a past challenge set, which was kind of cool. (My office mate commented that we try to get kids to work on these cool problems, and some of them get 'em and some don't, and then too often the problems just "die" and we never return to them.) You can browse through the challenge sets here.

There were two distinct pieces to this project: the write-up, and the exhibition product. I'll talk about the latter now, because it's easier. Basically, I got some really creative products. Some of my favorites were:

A giant Tower of Hanoi game made of a PVC base and handmade pillow "discs"

The background poster for
the 3-D product rule

A physical representation of the
proof of the 3-D product rule
(from a challenge set)

An artpiece demonstrating the "picture proof" that any triangle constructed
with the diameter of a circle and any point on the circumference is a right triangle.
The piece opens up to a full circle in order to demonstrate the proof, which is
inspired by the famous problem from Paul Lockhart's A Mathematician's Lament. 

A giant pop-up book explaining derivative shortcuts with
the help of "Deric the Differentiating Duck"

This one was exceptional - a painstaking model of the notorious
(among my students) Ferris Wheel / Water Cart problem
presented by Goonies and made of pipe cleaners, complete with a
diving ballerina and a Lady Gaga-esque emcee.

Two pirates present the box-method for the chain rule
with nested boxes, ending in a treasure chest with gold coins for
those who are successfully able to take the derivative of a
complicated composite function.

A comic (a la xkcd) presenting the challenge problem involving pirates and a secret
language for communicating the identity of a card using only four other cards.

Pretty fun, huh? I loved seeing the responses of people who came to check out my kids' projects - in general, they were impressed with their creativity and with their understanding of the material. Even though math isn't necessarily the most exhibit-able subject, it's fun for the kids to get to show off their fancy math once in awhile (whether or not they're at a "project-based" school).