An Experiment Inspired by Vladimir Arnold

I have a tiny book written by Vladimir Arnold Problems for Kids from 5 to 15. A free online version of this book is available in Russian. The book contains 79 problems, and problem Number 6 criticizes American math education. Here is the translation:

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle.
American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

Arnold has inflated expectations for kids. The book presents the problems according to the increasing order of difficulty, and this suggests that he expects kids under 10 to solve Number 6.

Arnold claimed that every student from Moscow would notice what is wrong with this problem. I can forgive his exaggeration, because I’ve met such kids. Anyways, I doubt that Arnold ever stumbled upon an average Russian student.

My own fundamental interest is in the state of American math education, so I decided to check his claim concerning American students. I asked my students to calculate the area of the triangle in the above puzzle.

Here are the results of my experiment. Most of them said that the answer is 30. Some of them said that it is 24. In case you’re wondering where the 24 is coming from, I can explain. They decided that a right triangle with hypotenuse 10 must have two other legs equal to 8 and 6.

Some of the students got confused, not because they realized that there was a trick, but because they thought the way to calculate the area of the right triangle is to take half the product of its legs. As lengths of legs were not given, they didn’t know what to do.

There was one student. Yes, there was one student, who decided that he could calculate the legs of the triangle from the given information and kept wondering why he was getting a negative number under the square root.

You decide for yourself whether there is hope for American math education. Or, if you are a teacher, try running the same experiment yourself. I hope that one day I will hear from you that one of your students, upon reading the problem, immediately said that such a triangle can’t exist because the altitude of the right triangle with the hypotenuse as the base can never be bigger than half of the hypotenuse.

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12 Comments

  1. Maria Roginskaya:

    If one just look from the point of Mathematical Logic on the statement “If a hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches then the area of the triangle is 30 inches squared” it is a true statement as “false statement implies anything” is true.

    However I must admit that it would never occure me given a rutine problem to verify the correctness of the premises, so if you’d asked me, I’d answer 30 as well. One can say that I lack in critical thinking – may be I do, but I think one have to draw line somewhere to be able to function.

  2. Jonathan:

    I could see students stalling on this only if they’d recently worked with circles (inscribing right triangles…) Otherwise, I wouldn’t expect any students, Russian or American, to notice the problem.

    But, you’ve got me thinking. Some little problems to sharpen: find the maximum length of the altitude to a hypotenuse in terms of the length of the hypotenuse. Minimum (just to keep them honest). Show that the shortest altitude in a right triangle must be the altitude to the hypotenuse.

  3. Denise:

    That will be a fun problem to give my math club students the next time we are together. But on standardized tests, there is a sort of unwritten contract between the test maker and student, to the effect that the student can accept problems at face value and crank out answers as quickly as possible. It would never occur to me to *think* about such a problem—and if I did, that would surely kill my score on the rest of the test.

  4. Misha:

    Maybe that’s why the Soviet Union had failed: too much critical thinking.

  5. Austin:

    I agree with Maria and Denise, but it still speaks badly of American math education that such a problem appeared on a standardized test, and for 10 years no less!

    You may have trouble believing this story; I myself find it hard to believe:
    When I was in (a California) high school, we got a standardized test with a problem that asked: “Which of the following CANNOT be a factor of 36x^2-85x+50?” I do not remember the actual quadratic of course, but otherwise, this was the exact wording of the question. There were four choices (all linear polynomials), at least two of which were not actually factors of the quadratic. I think we were supposed to use the rational roots test or something like that, which would eliminate only one of the answers as impossible. Heaven protect us from knowing too much!

  6. misha:

    Austin, we don’t need Heaven to protect us from knowing too much. We can protect ourselves very well. Just stop learning, stop thinking, stop teaching, and everything will be fine.

  7. Juliette Culver:

    I wonder if it could be in part linguistic – the first image that came into my head on reading the question was of a right-angled triangle with the altitude being a vertical edge perpendicular to the base (rather than am image of the hypotenuse as a horizontal base). In fact I almost didn’t see the word ‘base’. I can certainly imagine myself making having made the ’24’ mistake if I’d answered the question in a rush, or maybe that’s why I did my PhD in algebra rather geometry 🙂

    Perhaps in Russian the words somehow conjure up slightly different images?

  8. yakov nakhodkin:

    Tanya,
    Is there the book “Problems for Kids from 5 to 15.” in English?
    Thanks

  9. Tanya Khovanova:

    Yakov,

    I do not know.

  10. ete:

    tanya,
    could you plz provide an english translationl. it would be valuable to my kids and hopefully to others too.

  11. Dan Fox:

    It seems to me that Arnold’s problem should be understood as a criticism of American teachers more than of their students.

  12. Jon Adams:

    English translation available here
    http://jnsilva.ludicum.org/HMR13_14/Arnold_en.pdf