Monday, August 3, 2009

Teaching math backwards

As I have mentioned I will be teaching my first math class this year-6th grade (not counting student and substitute teaching). I want to drop the textbook as much as possible, but have to admit that I am a bit intimidated to commit to not using it at all. I see textbooks as a crutch for teachers because I think that they are terrible. But textbooks do make it easier to plan and teach and I am not sure I can prep without it every day.

Today I was challenged by two blog posts to give it a realistic try. The first is by Nancy Stewart about 8 principals that she will use in teaching her special education math class. She mentions ditching the textbook which is encouraging just to hear that others are taking the same step. She also gives some good ideas on how to at the same time make math more real world and personally relevant to students.

She also mentions one of my favorite bloggers Ira Socol and links to this post. Math teachers you must read this post as Ira paints alternative ways to teach math and turns much of my thinking upside-down. He shows examples of using sports, construction, cooking, and money to make math authentic and meaningful. I am sure he would argue that my times table idea is a waste of time.

The most challenging thought to me was a comment (you must read the comments) by Homer the Brave:

"Start with philosophy. Teach kids about logical systems. Teach them how to understand a provable statement and how to spot a fallacy. Then say, 'We're going to now apply this same set of rules about philosophy to math.' Then teach algebra. The details of arithmetic will then follow, imbued with purpose and meaning.

So basically we teach math backwards arithmetic, algebra, and then philosophy. I personally was an excellent math student in school because I was very good at memorizing and working algorithms. It was not until taking classes about teaching math in college that I understood the philosophy and reasoning behind the math. Homer argues that we should start with the philosophy (logic) and then move to algebra with arithmetic last. This is very new to me but does make sense.

Of course this change would have to happen at the district curriculum level. The easy cop out for me is that I must teach to the standards assigned to me. But I can at the same time as teaching the standards, teach the logic and philosophy behind the math. I can teach authentically without the textbook as much as possible. I am up for the challenge. How about you?


  1. Long ago (long, long ago), I heard a Cornell University professor say that we taught everything backwards. "Kids come to kindergarten absolutely ready to study physics, geometry, and storytelling," he said, "and we force them into symbolic systems coding none are really ready for."

    How sad and how true.

    I hated math in school, but I was great at math ideas. And I could go from being a high school math failure to a star architectural engineering student. But that's because I needed what Homer (one of my favorite thinkers) recommends - that math not be taught as a discrete disconnected set of skills, but as a philosophy which - down the line - utilizes a specific language.

    I once watched an Algebra teacher fumble the question "why do we need this?" She resorted to that ridiculous claim, "for the next class." I stood up and said, "Algebra is an idea. An idea for finding something unknown from things which are known. If you get this way of thinking - lots of problems get easier to solve." Then I said, "Numbers really don't matter in algebra, it's the system of detective work that matters." The teacher told me I was wrong - that "Algebra is all about numbers."

    Oh well. I tried.

    - Ira Socol

  2. @Ira
    Thanks for sharing. I like your definition of a
    Algebra. I think many math teachers do not really understand the philosophy and reasoning behind what they teach, but like myself are good with algorithms.

    My question is can elementary students think abstractly enough for algebra? That is the difficulty I see for many middle schoolers is the lack of development of abstract thinking. How does the cognitive development of abstract thinking skills fit into the math sequence?

  3. Different kids, of course, develop differently, but I think we need to work toward making abstract thinking available before we insist that kids make sense of "nonsense formulas."

    My son "the math major" talks about using programming to introduce functions, because they will see "actual" results. That's one idea. I think another is introducing the abstractions of algebra through notions of 'predicting the future' - "If you were making $5 a day raking a neighbor's lawn, and you got a raise to $6 a day how much more quickly could you afford that guitar?"

    There's no "magic moment." Abstraction is very hard for some people. Try explaining the value of Twitter to a large group of educators, or the value of mobile phones in the classroom. They reject these ideas immediately because they can not "see" the results in a concrete way. So, I've seen the need to mix instruction in math abstractions in primary school and in universities.

  4. Michael,
    I'll be going on six years teaching high school math this fall. Your enthusiasm is inspiring! Rather than being the bearer of "experience from down in the trenches," I want to instead encourage you to give it your best! Your students will surely benefit and you will learn a lot about math and its applicability along the way. I look forward to learning from you as you (I hope!) blog about this experience along the way.

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