Tuesday, February 13, 2007

A Formula For Success?

A question came up during one of our Cohort 2 sessions today when we were discussing the reading (the article discussed incorporating the student’s point of view into instruction.). It was something along the lines of, “How do we honor the student’s point of view on a math assessment where there is typically one “right” answer?” We didn’t get a chance to really discuss that question during the session, but I wanted to revisit it on the blog.

It won’t be a surprise to most of you that I want to focus on the concept of “one right answer.” The initial thought that came to my mind was that it depends a whole lot on what you’re assessing. If you’re assessing a particular math skill, taken out of context, then there may indeed be a “right” answer. Although even then I think in the long run the most important part might be their ability to describe the process, how and why they obtained the answer they did and why they think it’s “right.” Because unless they are presented with that exact same problem again – in some kind of context and where it actually matters – getting the right answer on that particular problem isn’t really all that important in the scheme of things.

But I think it’s more interesting to ask if that’s the type of assessment we really want to be doing. I’m not saying that we don’t need a way to evaluate whether they have mastered a particular skill, but I think the question that was posed was more along the lines of a unit test or semester final kind of assessment. And for that, I would suggest that the type of questions I think we should be asking might not have one “right” answer. For example, what if your assessment question was, “You’re designing a new school. What size and shape should the math classrooms be? Why?” To me, that is a much more interesting question and I think most of us would agree does not have one right answer. It also would require a whole lot of applied math skills, as well as research and critical thinking.

Which leads me to a larger question – why do we still insist on teaching mathematics as an isolated subject? As a set of skills, taken out of context, with the primary goal being to master the skills? Before you start throwing compasses at me, please realize I’m not saying this as someone who doesn’t like math. I happen to love math, and used to teach it, so this is not coming from someone who is math phobic or harboring deep-seated resentments to all things mathematical. It is said by someone who truly wonders if this is the best way to teach students about mathematics and how it applies to their world. Shouldn’t we be focusing more on the application of mathematics, not on the theory of mathematics in the abstract?

Let’s try an example. Actually, a quiz, since this is about math after all and math quizzes are something we’ve all experienced. It’s only four questions, and they’re all related, so don’t panic. (The very fact that so many of you smiled at the words “don’t panic” says something in and of itself.)
  1. Write down the quadratic formula from memory.

  2. Assuming you were able to complete number one, explain what a, b and c stand for in the quadratic formula.

  3. Assuming you were able to complete both number one and number two, explain –in detail - when, why, how and for what you would use the quadratic formula in a math class.

  4. Assuming you were able to complete numbers one through three, now explain – in detail – when, why, how and for what real-world situation you would use the quadratic formula.
Now, give this quiz to everyone who has graduated from high school in the United States between 1955 and 2005. (And, of course, please take it yourself.) The quadratic formula is covered in Algebra – and even Pre-Algebra where that is offered - and a very large percentage of high school graduates completed at least Pre-Algebra.

So, what percent of those high school graduates were able to answer all four questions correctly? (As an aside, I think percent is a math concept that we should have our students spend more time thinking about, as opposed to just computing.) I’m assuming your answer is close to 100%, correct? Especially considering that just about every day I read how much better schools were in the 1950’s (and 1960’s and 1970’s and 1980’s and well, just about anytime other than now). And certainly the answer was 100% among the highly-educated, clearly above-average-intelligence readers of The Fischbowl.

Now, let me be clear here that I think the quadratic formula is something that students might spend some time thinking about, investigating and applying. But not in isolation – not just so they can complete one through thirty-one odd. But in context, in a way where they are thinking deeply about solving a problem (as real-world as we can make it) that is important and relevant. They should be thinking about the quadratic formula because it would be helpful in solving a meaningful problem, not just because they need it for the next unit (or math class). Because what exactly is the point of teaching mathematical skills in isolation that the vast majority of adults will not be able to remember as soon as they are done with their last math class, much less even remember or be able to apply a few years later?

Some folks would answer that question by saying that even if people don’t remember the specifics of the math, that learning it helped them to “think logically.” Now, I’ve heard that an awful lot, but I’ve never read any research that proves that. (I’m not saying the research doesn’t exists, just that I haven’t seen it.) But even if the research does exist, what I’m suggesting we think about would accomplish that purpose as well, without the completely out of context, learn the skill to pass the test and then forget it approach. If we approach mathematics – and mathematics assessment – in this way, then I think it’s much easier to see how the student’s point of view becomes more relevant. (Because wouldn’t point of view be more relevant when discussing what the size and shape of a new math classroom would be?)

Now I know this is tricky territory, and I don’t want to get bogged down in the “math wars,” so please take this in the context of thinking about what we believe the essential learnings are for our students. But also please realize that I am not stating that we need to make math easier, or watered down. In fact, this is a whole lot more difficult than the way we’ve been teaching math. Because we’re not just asking them to memorize the formula and plug in the right numbers, we’re asking them to understand. And that’s a whole different ballgame.

This was written rather spur of the moment, so I probably haven’t given this the full attention it deserves, and probably have left some holes in the above argument. So add your comments below, but please try to focus on the bigger picture here, and not some isolated horror story of some clerk who can’t make change at a convenience store. Because, really, is making change a skill that’s all that important in the 21st century? Yes, I think they should be capable of figuring it out, and the approach I suggest above would actually help with that problem, but at this point I’m more interested in making change (in education) than in making change (at the store).

Oh, and for those of you who were wondering . . .

Image Citation: Maths humour, originally uploaded by Rich Watts.


  1. Good grief, Karl! You wrote a long-winded post about math...and I stayed with you the whole way. Nicely done. Coming from a student that struggled with math, I love the idea of applying real-world applications. I don't know necessarily that I would feel more successful, but certainly seeing the immediacy of math would have made me ...do I dare say... care about it more?

    Your argument in the context of the question regarding what we're asking kids to do is so essential. Point in case. For the past 3 periods, I have had my seniors doing what I consider very constructivist activities: theater of the oppressed, applying literary criticism theories to the novel, etc. but at the beginning of class to keep the kids honest regarding the reading assignments, I gave them a knowledge-based quiz. So, we've been doing this now for Ch. 5-9. On the last quiz, I asked them to write down questions they still had for me about this science fiction novel. You know what the resounding comment was? It sounded similar to: I'm wondering why we haven't discussed the novel as a class and why we're focusing on such nic-picky details? I loved it! They actually wanted to discern meaning rather than just know the facts that they could get on wikipedia, SparkNotes, PinkMonkey and the likes.

    So, from one long-winded friend to another, I do think we need to consider how we ask kids to show us what they know and how we approach the essence of teaching our content.

    You're always pressing us to rethink, reconsider, and if need-be reteach...in new ways. Thanks, Karl!

  2. You both write a lot for a person to read. I like the idea of applications to the real world but some of the basics that you need in algebra are still teaching you to learn the sounds that the letters make or understand the structure of a sentence. You cannot teach someone to read a novel if they do not know the structures of a sentence. I see there are application for a lot of math, but at some of the lower levels you need to learn the basics.

  3. I agree to some extent, James, but if 95% of adults couldn't tell you the sounds the letters made, wouldn't that say something about how effective previous instruction had been?

  4. If we see ourselves only teaching students how to come up with the right answer, then the logical assessment will only have one "right answer.
    If I presume (based on the type question you feel is more interesting) to understand the change in education you would like to see, ultmately, the line between learning and assessment blurs or even disappears. Count me in!

  5. you sound so much like my math teacher now, he isnt the sort to teach us something and jsut leave it at that.
    he will go out of his way to make sure each and every member of his class understands WHY it is, not jsut that it is.
    he has taken his idea that far that he trusts students with his email address to ask questions anytime, and to those without internet conections, he has given out his mobile phone number with the instruction, text me anytime you need help, and ill call you back asap to help you with it.

    and you know the best thing, hes also the head, meaning he has ALOT more things to do beside teach math to Alevel students.

    but its people like that (and to some extent the blogger) that actualy help people learn