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Where I live and work, there is a widely-accepted and often-repeated claim that there are two kinds of students: "algebra people" and "geometry people". This claim sometimes gets expressed in different ways; sometimes it is articulated as a difference in ability ("The students who are good at algebra tend not to be good at geometry and vice versa") and other times as a difference in interest ("The students who like algebra tend not to like geometry and vice versa"). This belief is often conflated with claims about some students being "visual learners" (despite the fact that so-called "learning styles" do not exist).

On its surface, this belief seems to me to be almost certainly a myth -- at least when expressed in terms of ability. But it also seems like it could be empirically tested. If people who are good at algebra were bad at geometry and vice versa, that would surely show up through an analysis of the grades they get in school, or how students perform on standardized tests that include both kinds of content.

As far as interest goes, the claim has slightly more surface plausibility. I know plenty of adults who will testify that the only math class they ever enjoyed was Geometry, and others who will say with equal conviction that Geometry was the only math class they hated. But a few anecdotes is hardly compelling evidence. This, too, seems like it could be easily studied empirically with some kind of attitudinal survey.

So the question:

Have there been such studies? Is there any evidence for or against the claim that people tend to be either "geometry people" or "algebra people"?

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    $\begingroup$ Of course, it seems the "geometry" here is the classical sort. Abstract truly modern geometry involves much reasoning which is hardly visual. Sure, visual heuristics exist, but, algebra is what allows us to calculate the topological and geometrical features of higher dimensional spaces. $\endgroup$ – James S. Cook Mar 23 '16 at 3:01
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    $\begingroup$ Yes, I am referring to a traditional high school course in Euclidean geometry, such as is commonly taught in the United States after a first course in Algebra. $\endgroup$ – mweiss Mar 23 '16 at 3:05
  • $\begingroup$ What I've read suggests not that learning styles don't exist, but that they shouldn't be used in the way that they have sometimes been used. You can still be more or less inclined towards visual or algebraic arguments. $\endgroup$ – Jessica B Mar 23 '16 at 7:54
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    $\begingroup$ I picked up algebra without effort on my own and was able to test out of algebra 1 and learn algebra 2 independently. I had to be taught geometry and it was a great deal of work for me. I always thought that I was an algebra person and worked hard at becoming a geometry person also. I realize this is only anecdotal, but I would be interested in research along these lines. $\endgroup$ – Amy B Mar 23 '16 at 9:47
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    $\begingroup$ Once, at a department barbecue, we noticed that all of the algebra people ate their corn off of the cob like a typewriter, and all of the geometry people ate their corn off the cob in a spiral. $\endgroup$ – Andrew Mar 23 '16 at 14:45
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This is the story of an attempt to find some evidence for something "similar". Let me start with an anecdote.

Personal anecdote. Yeeeears ago, as a math student I was good at continues mathematics (e.g., analysis) and bad at discrete mathematics (e.g., algebra, graph theory). To be honest, I wasn't even able to accept the given proofs in graph theory as "real" proofs, let alone to attempt for one.

Why this is related to your question. I don't know if there is an established difference between continuous and discrete mathematics, but I feel the distinction between geometry and algebra as you mentioned is somehow related to it. If not, you can see the rest of my answer as an attempt to address a similar concern.

Research. Yeeears after my student time, I was a mathematics educator working in a math department where I had a chance to tell the anecdote above to my mathematician colleagues. Interestingly, many of them knew someone else with the same experience. Thus I decided to find some hard evidence to show there is something called "continuous thinking" and something called "discrete thinking" where some people are good at one but not necessarily at the other. A master student of mine "chose" this as the topic of her dissertation. We focused on understanding of proof in analysis and algebra. We saw a proving activity as a problem solving activity and accordingly we use some of the variables in research on problem solving (as introduced here by Kilpatrick) to study whether there is any difference in students' thinking of proofs in analysis and algebra.

Design of the experiment. We gave students several pairs of proofs. In each pair, one proof was from algebra and the other from analysis. Both proofs had the "same structure". Students didn't know that the pairs are purposefully related. They were reading the proofs and then answering some questions designed in the light of some of the variables (given by Kilpatrick).The variables were too many, we just tried some of them. Unfortunately, I couldn't remember now which variables we used, perhaps because of the result of the research.

Result. We didn't find any difference noticeable!

Conclusion. I still believe there is some differences there that we have failed to find!!

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Not quite the same question, but along the same lines is On the Persuasiveness of Visual Arguments in Mathematics. It looks at how convincing different mathematicians find visual proofs compared to written ones.

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I don't know if this has been studied. Here's how to do it.

I think the data required for this might very well be public in Finland. At least the final grades of pupils are public knowledge. I don't know about the situation elsewhere.

One could go to a school and ask for grades of mathematics courses.

There are two forms of the hypothesis.

The strong one is that geometry and algebra grades are negatively correlated. This is very unlikely, since practically all grades across subjects are positively correlated (e.g. history and mathematics), at least in the compulsory school levels.

The weak one is: After accounting for the average mathematics grade, the geometry and algebra grades are negatively correlated. This might even be true, but I would be surprised if the effect was large.

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    $\begingroup$ -1. This doesn't answer (either part of) the question. $\endgroup$ – Benjamin Dickman Mar 23 '16 at 6:52
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    $\begingroup$ @BenjaminDickman Though he has not answered the question, I wonder if has actually done a good job of defining what an answer may look like and how it could be studies. If some people are "algebra people" and others are "geometry people" shouldn't we expect a negative correlation between performance on these tasks (possibly within a restricted population of "math people")? If not, what exactly is the OP asking? Maybe the question needs to be clarified rather than this answer down voted. $\endgroup$ – Richard Mar 24 '16 at 9:51
  • $\begingroup$ @TommiBrander I believe that your strong argument has been used in support of a unitary IQ as opposed to the "multiple intelligences" theory. Your weak argument (correcting for average grades) would only work if the curriculum was split into many topics that were fed into the analysis, and a small negative correlation would indicate no effect. If all assessments were split into "geometry type" and "algebra type", you would by necessity end up with a -1 correlation! $\endgroup$ – Richard Mar 24 '16 at 10:00
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    $\begingroup$ @Richard I find that it is often more useful, in the application of mathematics, not to answer questions absolutely but instead to pose them correctly. I agree that Tommi has done this. Perhaps an edit to the OP with Tommi's contribution would be advisable? $\endgroup$ – Andrew Mar 24 '16 at 13:31

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