Monday, April 25, 2011

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?


  1. Allison,
    Sorry for the huge copy and paste, but this bears repeating:

    "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."

    I've been trying to think about what this would look like if we had examples of people thinking aloud (focusing on the NCTM Process Standards) as they did mathematics. Here are two of my first, clunky attempts -

  2. I think the goal should always be to develop people who can think mathematically. The algorithmic or computational approach of traditional mathematics classrooms does not accomplish this goal IMHO. Instead of people who can do mathematics, we have people who can follow algorithms (some of the time) and who largely forget the mathematics they've learned in schools.

    The people who do think mathematically either had a math teacher who took the time to show how the mathematics they were learning was embedded in the real world, or are so gifted in their understanding of mathematical algorithms that they are able to apply it to their daily life.

  3. Skills-based versus beauty-based teaching are not mutually exclusive. Mathematical beauty is intimately tied with the details. I don't think you can talk about Cantor's diagonal argument in the abstract. You need to get your hands dirty a bit to really understand the proof, and hence understand why its beauty.


  4. "Why is it that we make our kids study math for at least twelve of their formative years?" This question should be posed more generally and applied to every subject. What is in fact the goal of education?

    Khan's Academy and the JUMP program attempt to do better within the present system of math education. You advocate the next step of improvement is to better that system by changing the goals and practice of math instruction. But this is impossible unless the structure of the education system has been upgraded.

    The goal of education - perhaps utopian - should be to help every kid find and develop his or her natural abilities. Kids should be given a chance to try their hand in many subjects - and learn the basics along the way. But then they must be allowed to focus on what they like best and dig deeper in the subjects of their choice. And if they do not choose mathematics they should be left alone - it is not worth the effort to attempt to convince them that math is beautiful. Why not let them find beauty elsewhere?

  5. @delta_dc: The link you posted was not working. Can you post a new one? I'm really curious to see what you've come up with!

    @Alexander: Great point. You say that the "structure of the education system" must be upgraded. Can you elaborate?

  6. I agree that the structure of our education system must change, and I believe that it is quite possible train basic technical skills and knowledge (which I'll subsequently call "content") along with creating a space and time for rich exploration. Here are some suggestions:

    * separate training in content and exploration into distinct components of the curriculum

    * divide the content component into a series of small modules; allow students to work at the modules at their own pace, so that students each have enough time to master each module

    * eliminate grades, which are counterproductive to learning (they increase anxiety enormously, decrease motivation, and focus students on a false goal); instead, require mastery of each content module, and simply report on which modules have been mastered

    * ideally students work at their own pace, and are promoted to higher grade levels regardless of the number of content modules they have mastered; if they are exploring something and they suddenly find that they need some content that they have not yet studied, that is the best motivation to go back and master it, which they will then do with pleasure and enthusiasm (in this system you might have a Grade 10 student who has only completed up to Grade 5 level math content modules, and that is OK)

    There is more, but this comment is already rather long.

  7. Sorry to be adding to this thread so late, but I just found your blog (thanks to Quantum Progress).

    I agree with the Anonymous above who says that skill-based and beauty-based teaching of math are not in opposition. You seem to imply that teaching math for engineering is somehow sullying the math.
    I might have agreed with you in my teens, when I planned to became a pure mathematician, and I got as far as my MS in math before I realized that the math I loved was done computer science departments, not math departments. Over the past 40 years I have become gradually more and more applied. I still like some of the math I loved as a teen, but I've gotten a much greater appreciation for math that is also useful.

    Quaternions are pretty math, but it is even neater that they provide a numerically stable way to deal with rigid transformations in 3-space, which can be used to superimpose models of proteins and figure out how to manipulate robots.

  8. I attended a JUMP math presentation a few weeks back (some notes on my takeaways are here). I wouldn't put JUMP in with Khan academy or other status quo mathematics education initiatives. I think the essential vision of JUMP is not very concerned with test scores and the other pitfalls of traditional education, but on empowering students to actually enjoy mathematics. Unfortunately, in order to get traction in schools it is often presented in the context of improving student performance within the existing paradigm. One lesson from this is that it is difficult to present real educational alternatives (which I think JUMP could be a part of). Instead, new ideas tend to get absorbed into old ways of doing things, the truly different approaches that they present get lost, and they become yet one more failed educational experiment - another example that serves to strengthen the status quo.

  9. Hi, Allison,

    I came across your blog via David Wees, and as a fellow mathematics educator I thought you might be able to help in spreading the word about an educational TV show for preteens about math that we're putting together. "The Number Hunter" is a cross between Bill Nye The Science Guy and The Crocodile Hunter -- bringing math to children in an innovative, adventurous way. I’d really appreciate your help in getting the word out about the project.

    I studied math education at Jacksonville University and the University of Florida. It became clear to me during my studies why we’re failing at teaching kids math. We're teaching it all wrong! Bill Nye taught kids that science is FUN. He showed them the EXPLOSIONS first and then the kids went to school to learn WHY things exploded. Kids learn about dinosaurs and amoeba and weird ocean life to make them go “wow”. But what about math? You probably remember the dreaded worksheets. Ugh.

    I’m sure you know math is much more exciting than people think. Fractal Geometry was used to create “Star Wars” backdrops, binary code was invented in Africa, The Great Pyramids and The Mona Lisa, wouldn’t exist without geometry.
    Our concept is to create an exciting, web-based TV show that’s both fun and educational.

    If you could consider posting about the project on your blog, I’d very much appreciate it. Also, if you'd be interested in link exchanging (either on The Number Hunter site, which is in development, or on which is a well-established site with 300,000 page views a month) please shoot me an email. We're also always looking for input and ideas from other math educators!

    Thanks in advance for your help,


  10. The basic objective of Education - maybe utopian - ought to be to help each child discover and create his or her regular capabilities. Children ought to be given an opportunity to attempt their hand in numerous subjects - and take in the rudiments along the way. Be that as it may then they must be permitted to concentrate on what they like best and delve deeper in the subjects of their decision. Furthermore on the off chance that they don't pick science they ought to be allowed to sit unbothered - it is not worth the exertion to endeavor to persuade them that math is wonderful.