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?