Consider the following two problems:

(A) Square 2 has sides that are longer than those of square 1 by a factor of 1.1. Compare the areas of the squares.

(B) Square 2 has sides that are longer than those of square 1 by 10%. Compare the areas of the squares.

I find that many college students majoring in the sciences get A right, but for B they compute $0.10^2=0.01$.

What is going on here? Does K-12 education in the US not ever teach students to convert a fractional difference to a ratio? (In the lower grades, it seems like the word "ratio" is for some reason coupled to notation like "2:3," and some students never realize that this connects to fractions.) Does the symbol % just trigger a sequence of operations in the student's brain, distracting from the larger context? Obviously the people who get $0.01$ aren't thinking about whether their answer makes sense. A contributing issue may be that they get little experience with problems in which the answer is a ratio.

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    $\begingroup$ There's an even larger issue going on. When they answer B with .01, is there no further thought that a longer side should produce a larger area? It seems to me there is a (missing) skill of how to check one's answers. That seems a lost art. $\endgroup$ – JTP - Apologise to Monica Jun 2 '14 at 15:18
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    $\begingroup$ Have you ever watched one of them working these out? $\endgroup$ – JPBurke Jun 2 '14 at 15:59
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    $\begingroup$ Check out, e.g., Schoenfeld on sense making. Yes, this topic is covered in US K-12 education; yes, many students do not understand it and end up covering it repeatedly (compare syllabi at public schools for elementary/middle school and "remedial" or "developmental" mathematics classes at community colleges). Yes, sometimes it is an issue of mindlessly carrying out computations without thinking about the underlying meaning. If you observe this phenomenon repeatedly, you could ask students to show their work, or try engaging with one of them around their thinking using a clinical interview. $\endgroup$ – Benjamin Dickman Jun 2 '14 at 16:47
  • $\begingroup$ @JPBurke: Have you ever watched one of them working these out? Yes. It doesn't seem to occur to them to check whether their results make sense, and if I ask them whether their results make sense, often they think they do. Usually I resort to an example such as a candy bar that costs \$1, another candy bar that costs \$1.10, and having them divide the two numbers to see what they get. At that stage, they act surprised at the suggestion and may be uncertain about which number to divide by which. I have them try it both ways on a calculator and observe that the % difference is about the same. $\endgroup$ – Ben Crowell Jun 2 '14 at 17:34
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    $\begingroup$ If I had to guess (and I wouldn't put this in an answer) there is a lot that could be going on here. They could just be thinking of what procedures they have used in the past, and what procedures are they reminded of based on the form of the question, as you suggested. I doubt it's the % symbol causing difficulty, since they're using the same procedure in both cases. Here's something I strongly suspect based on ongoing research: scale factor is a very strong concept in proportional thinking for teachers and students. $\endgroup$ – JPBurke Jun 2 '14 at 19:06

Some of this comes down to the recipes the students overlearn and teachers (myself included) try to make too efficient. For example, when I tutored kids who arein high school, they were taught "something multiplied by a factor x" as multiplying the something by x, so they'd almost never mess up a problem like the first one. On the other hand, when, in the second problem they see "are longer than those by 10%" they use their recipe to translate the percent to .1 and then think to square it because they can't think of anything else to do.

Something that is absent from many of these recipes is to draw some kind of visual representation of the problem. If they draw a square, extend the sides, and then try to find the area, I'd be surprised if the second kind of mistake would ever happen.

While I don't have firm research to back up these two observations, I have had many students I've tutored talk about these kinds of problems in terms of recipes and I have also helped them overcome their struggles by having them draw pictures.

  • $\begingroup$ +1 Visualising problems helps create a notion of whether a solution 'makes sense' or deserves further scrutiny. $\endgroup$ – JvR Jun 3 '14 at 12:33

Not sure if this is what's going on with your class, but I've found that when many students have the same incorrect answer that you can't explain, it's because they were all copying from each other and the original was never very good. At least, for homework I've seen that quite a few times. You may want to consider giving students similar questions with different numbers, which helps curtail this problem.

  • $\begingroup$ Reasonable comment, but I've worked with a number of students individually on this, and seen the same mistake many times over the years. $\endgroup$ – Ben Crowell Jun 4 '14 at 4:05
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    $\begingroup$ Perhaps they are taking the difference of the two square lengths, not realizing that exponents can't be applied linearly. It's my belief that a lot of students that have had calculus still have fundamental flaws in their understanding of basic algebra, and apply order of operations rules incorrectly when they solve simple problems. $\endgroup$ – farrenthorpe Jun 4 '14 at 6:12

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