CR parallelogram

How did the student get from 0.8 times 1.6 to 8.0?

What tendency is this an example of? (Or is the mistake unique to the context?)

How would you test your theories?

Thanks to Chris Robinson for the work.

  • mpershan

    I’m usually against attributing mistakes to memorization issues or a lack of conceptual understanding. Since so many mistakes happen without explicit reasoning, but with fast, nearly automatic thought, I want to give explanations that explain why specific near-automatic instincts exist.

    In this case, my best bet is that this student is doing something like cross-multiplying. And I wonder if the decimal point is seen, in this associative, quick thinking, as akin to the fraction bar. (Notice that the decimal point is placed near the center of the two numbers. That seems significant to me.)

    I have to think about how I would test my theory. I’m not sure.

  • My first thought was that they read their sloppy 6 as a 0 and the multiplied everything ok, but failed on the decimal. That might be a stretch though.

    • Dave

      Exactly what I thought. 8 times zero (actually 6) is zero. 8 times 1 is 8. Draw the decimal in line with the ones above it (like in addition and subtraction).

  • Also, I know for me when I see anything times 0 or 1 my brain says “yay!” and spits out an answer, whereas 6 * 8 and double digit multiplication both involve thinking and juggling place value. So this student might successfully multiply, say, 0.1 * 2.3 (or at least get the right digits with the wrong decimal place). This “cross multiplying” (which is also what I noticed, Michael) may be an invented-remembered process because the “cross products” are the easy ones here, and get you out of worrying about place value.

    • mpershan

      Excellent comment.

  • Also, in terms of reasonableness, it’s unusual for students to experience multiplying base by height and getting an area that’s smaller than one of the dimensions, since students have little experience with side lengths less than one. So the student might be comforted by doing base * height = biggish number