Word Problems in Russia and America

Andrei Toom calls this an “extended version” of a talk he gave a few years ago at the Swedish Mathematical Society. At 159 pages [2010 updated version is 98 pages], I would call it a book. Whatever you call it, it’s a must-read for math teachers:

Main Thesis: Word problems are very valuable in teaching mathematics not only to master mathematics, but also for general development. Especially valuable are word problems solved with minimal scolarship, without algebra, even sometimes without arithmetics, just by plain common sense. The more naive and ingenuous is solution, the more it provides the child’s contact with abstract reality and independence from authority, the more independent and creative thinker the child becomes.

When we teach children to solve problems in school, we do not expect them to meet exactly and literally the same problems in later life. Mathematical education would be next to useless if its only use were literal. We want much more, we want to teach children to solve problems in general. In this respect traditional word problems are especially valuable, because to solve a word problem, you have to understand what is said there. This function of word problems is very poorly understood in America.

I suggested that the main educative value of word problems is that they serve as mental manipulatives, paving children’s road to abstract thinking. Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds.

Also check out Andrei Toom’s articles on math education and humanities.

Hat tip: Mathematics under the Microscope [but I can’t take you back to the original article, because apparently his new blog doesn’t include the old archives.]

10 thoughts on “Word Problems in Russia and America

  1. I think one of the most valuable aspects of word problems is that they help students see why they are studying math. I was always very frustrated that we nearly always ran out of time to do the interesting word problems at the end of a chapter.

    Now I realise it was all wrong. The chapter should have started with the word problems so that students could have a chance to think through the concepts. The algebra should come later, so that they know why they need it.

    >>”Especially valuable are word problems solved with minimal scolarship, without algebra, even sometimes without arithmetics, just by plain common sense.”

    Yes – some constructivism is essential…

  2. I wish word problems were a bigger part of all levels of math. Even in elementary school, I was “weird” that way.

    Constructivism is not quite what Toom means here — at least, I don’t think he has any intention of the students having to invent mathematics for themselves, without instruction or guidance. Rather, he recognizes that the algebra approach can degenerate into simply pulling out a formula and plugging in numbers. A non-algebraic solution often brings more insight into the mathematical relationships inherent in the problem.

  3. 1. I would argue that a non-algebraic solution that is achieved “just by plain common sense” – is a form of constructivism, and an essential one at that. We’re not talking about “having to invent mathematics for themselves” – by solving the problem in any way they see fit is constructing their own meaning of what is going on.

    2. The links to your CSS stylesheets are broken today… They point to “s.wordpress.com/ etc”

    3. Happy New Year 🙂

  4. 1. In that sense, I agree with you. That sort of constructivism happens in any learning experience — and it is exactly why I love math contests. I learn so much from having to think my way through each problem.

    2. I don’t know why the CSS does that, but it happens fairly often. Usually, hitting “Refresh” will bring the formatting back, but not always. Perhaps WordPress.com is growing too big for its britches?

    3. You, too, Zac! Our house is being overrun by girls: our littlest and middle daughters have each invited friends over to watch movies and giggle all night. So the men in my life are looking for any excuse to head out on the town…

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