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Briefly: I am looking for research on the extent to which learning mathematics (let's say "college algebra" if we want to be specific) impacts problem solving skills, abstract reasoning, etc.

Less briefly: I teach a college algebra class in an american research university. Most of my students struggle with motivation, and want to know (for example) why they need to learn how to complete the square. My stock answer is to compare the classroom to a gym and to the instructor (me) as a personal trainer: the purpose of mathematics is then to strengthen their "analytic-reasoning muscles." The students accept this answer for the most part, but I feel a little guilty in that I don't actually know of any research that supports the argument that learning mathematics improves abstract reasoning or problem solving skills.

Possibly unnecessary: What do I mean by "abstract reasoning / problem solving skills"? That is a good question. I guess what I want to know is whether or not learning to complete the square is helps students learn how to solve problems beyond those that ask them to complete the square. I suspect this depends on what I am actually doing when I say that I am teaching them to complete the square. Am I giving them a recipe for them to follow? Or am I giving them something more to engage with?

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  • $\begingroup$ I think you have the wrong frame of reference. College algebra is not going to change their care in critical thinking. Heck, look at all the implicit assumptions made here by extremely well-trained mathies, that they don't even recognize they are making. Or the various logical fallacies (e.g. assuming problems are single variable in the social sciences, arguing from extreme cases rather than average, etc.) Or the inability to structure an argument for best analysis and communication. What college algebra will do is satisfy basic STEM needs (chemistry, algebra-based physics, etc.) It also gives $\endgroup$
    – guest
    Commented Sep 28, 2022 at 15:40

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The answer you are giving to your students overlaps with the so-called "Mental Discipline" theory (also "Theory of Formal Discipline") for justifying mathematics education. As you search for research related to your favored justification, this theory may help you focus in on more relevant work.

One researcher who has written about this is George Stanic. You can see one of his articles on the subject here. But this comes up in a few places in the detailed and wide-ranging Stanick and Kilpatrick edited work on the history of mathematics education (Stanic & Kilpatrick, 2003).

There are varied opinions on whether or not mathematics education disciplines your thinking in ways that extend beyond using math in the specific situations in which it is taught. Some consider it "debunked." However, it is an idea that is still the subject of research today, as exemplified here (Attridge & Inglis, 2013).

Cited:

Attridge, N., & Inglis, M. (2013). Advanced Mathematical Study and the Development of Conditional Reasoning Skills. PLoS ONE, 8(7), e69399. doi:10.1371/journal.pone.0069399

Stanic, G. M. A., & Kilpatrick, J. (Eds.). (2003). A history of school mathematics. National Council of Teachers of Mathematics.

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    $\begingroup$ Attridge & Inglis is not encouraging: "We found that students studying post-compulsory mathematics did change their reasoning behavior...this change appeared to be best described as development...towards a defective understanding." $\endgroup$
    – user173
    Commented Nov 24, 2014 at 1:26
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    $\begingroup$ @MattF. Right. Personally I don't hold Formal Discipline as a good motivation to teach mathematics, but I wanted to show that there are still researchers doing this work. $\endgroup$
    – JPBurke
    Commented Nov 24, 2014 at 1:28
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I think the evidence suggests arguing for college algebra differently.

Many students do not develop these sorts of skills in college at all, according to the book Academically Adrift. A key result from the book is that "45 percent of students show no significant improvement in the key measures of critical thinking, complex reasoning and writing by the end of their sophomore years." If the students are majoring in education, social work or business, it's especially likely that they're not developing these skills.

So I would sell college algebra without reference to general problem-solving skills or abstract reasoning, and preferably without completing the square. More directly, it helps to be able to answer:

  • How many widgets do we need to sell to make a profit?
  • Given my scores so far and the exam weights, what score do I need on the final to get an A or a B?
  • If my income tax rate is 30%, how much do I need to earn before tax to have $1,000 more after tax?

I would also want the class to focus on answering questions like these.

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    $\begingroup$ If this is the motivation, at some point isn't this just an economic question of whether you want to do your taxes, or pay someone else? Or hire a "math person" to figure out how many widgets to sell? At some point, it is probably wise to just say "I know myself, and it is better for me to farm these kinds of tasks out to other people". $\endgroup$
    – Steven Gubkin
    Commented Nov 22, 2014 at 23:26
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    $\begingroup$ My own answer would be to just get rid of these courses, and make required courses involve actual critical thought. $\endgroup$
    – Steven Gubkin
    Commented Nov 22, 2014 at 23:27
  • $\begingroup$ Based on these answers and comments, then it would seem to me that the point of learning maths up until GCSE should be for people to be able to answer questions like the widget or tax one? Or at least be able to comprehend such a question? If so, shouldn’t functional mathematics, which is a GCSE course here in the UK, be mandatory, rather than GCSE mathematics (which is currently more-or-less mandatory if you don’t want to majorly limit your job prospects)? Perhaps I should ask this as a question on this site? $\endgroup$
    – Adam Rubinson
    Commented Sep 29, 2022 at 10:10

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