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The most commonly stated reason for why mathematics should be a required condition for graduating is }to improve problem-solving skills". Usually it's taken for granted that taking a mathematics course does improve one's ability to solve problems. Does anyone know of any studies that either back that up or contradict it?

Edit: I would also be interested in studies backing up claims that taking a math course improves logical reasoning, especially for mathematics courses for non-majors.

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    $\begingroup$ I'm surprised to hear that cited as the justification -- in my humble opinion, learning mathematics, especially at an undergraduate level, is less about developing problem-solving skills and more about honing logical acuity. For example, I am not infrequently asked which courses exactly one should take to develop skill at the Putnam (presumably one of the most elementary examples of mathematical problem-solving), to which I hem and haw and then say to just take some standard classes so you know the material and then go work some problems instead. What I think a mathematics education IS good... $\endgroup$
    – Arnav Tripathy
    Commented Feb 8, 2011 at 15:10
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    $\begingroup$ ...for, at least at the very basic level that we're talking about when we discuss "courses required for graduating", is being able to easily see flaws in reasoning, gaps in arguments, things that follow 'tautologically', and so forth. $\endgroup$
    – Arnav Tripathy
    Commented Feb 8, 2011 at 15:11
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    $\begingroup$ Anna: I would be surprised if there was any high-quality research on this, simply because "problem-solving" as a general skill -- as opposed to mathematical problem-solving, which has some overlap, but is definitely distinct -- is very hard to define or measure across-the-board. You might try specific situations or something like Raven's Progressive Matrices -- which I would suspect would correlate decently with mathematical training -- but you're running into a definition problem, especially as "problem-solving" is nowadays being used as a sort of catch-all term in education-speak. $\endgroup$
    – dvitek
    Commented Feb 8, 2011 at 17:10
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    $\begingroup$ I have heard that math majors tend to do well on the LSAT, though I suppose this is more a test of logical reasoning than problem solving. $\endgroup$
    – Kevin Ventullo
    Commented Feb 8, 2011 at 18:24
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    $\begingroup$ Generally the sort of mathematics courses which are required for graduating are calculus courses where memorizing a list of rules are all that is required. I do not think such courses contribute to problem solving or logical reasoning skills at all. They do contribute to intellectual stagnation and an attitude that you should believe what authorities tell you. Of course not all calculus courses are like this - I would not be in grad school if I had such a course as a high school student. For some reason it seems like most of them are this way though. $\endgroup$
    – Steven Gubkin
    Commented Feb 9, 2011 at 1:51

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(I don't think my answer directly answers the question, but I'm hoping it would be useful.)

I assume that when you say "problem solving" you mean mathematical "problem-solving as a skill" ("being able to obtain solutions to the problems other people give you to solve," Schoenfeld, 1992).

I was unable to find any studies that answer the question "Does taking an ordinary undergraduate mathematics course improve one's ability to solve (mathematical) problems?" (where ordinary means the instruction is not explicitly targeted at improving problem solving skills).

But there have been studies that show that undergraduates taking certain "problem-solving courses" experienced "marked shifts in [their] problem solving behavior" (e.g., Schoenfeld, 1987, p. 207).

As I understand it, researchers in mathematics education usually don't consider questions of the type "does the ordinary way of teaching improve this skill/understanding?" important (where "ordinary" is usually referred to as "traditional"). They usually consider it more valuable to ask questions of the type "what way of teaching will improve this skill/understanding?"

A good reference is

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.

which uses some material from

Schoenfeld, A. H. (Ed.). (1987). Cognitive Science and Mathematics Education. New Jersey: Erlbaum.

Chapter 2 (Foundations of cognitive theory and research for mathematics problem-solving, by E. A. Silver) and Chapter 8 (What's all the fuss about metacognition? by A. H. Schoenfeld) of the 1987 Schoenfeld book are particularly useful.

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