To investigate student views of transfer, we included a portion of it as part of our focus group, asked questions of the workshop students (a class where we had seen evidence of transfer), and interviewed graduating seniors from 2003. Currently, not all of this evidence has been analyzed.
The main skills that students saw transfer from their mathematics training were habits of mind. These habits broke into four distinct areas: argumentation, problem solving, abstraction, and communication. Many will argue that all classes try to teach these skills, and we would be the first to agree. What we see the mathematics classes doing differently (and in particular the workshop course) is that we attempt to focus on these skills as an end, rather than as part of a specific content. In particular, as mathematics content is very abstract, it may be easier focus on these skills more directly.
In any case, many of our students claim to feel that they think differently from other students because of their training. For example, student W in the focus group said,
I just think that we (mathematics majors) are molded into having the same methods of thinking and modalities and ways of working through the problem and I think that distinguishes us from other people. Most of the people I know who were math or science people, which I think is along the same way of thinking. Yet even now when I am with all people who are all science majors I see a difference. I don’t know how to explain it. People are like, “Oh you weren’t a science major,” and I say, “no I was a math major” and they say, they notice, “that’s probably why you think a little bit differently.”
This feeling of thinking a little differently was even noticeable in the students that had only completed their first year of the mathematics major. A list of their comments on the anonymous survey about transfer shows this.
Curiously, the discussion of transfer in the focus group was initiated by a question about mathematical writing and why students hadn’t listed that among the three things that were most important in the workshop course. The student responses were that writing went along with the thought processes (that they had listed), but then immediately student L comments on how the writing had “contributed to my other classes.” She goes on to say, “it is just a lot more subtle, this is how I communicate ideas.” Later she specifically mentions English as a class. The students see the mathematics as valuable for helping them structure their writing. Student J then pointed out, “writing kind of comes along with the thought process. Once you learn to reason, putting it on paper is a little bit different, but as long as the reasons following, and you are not writing too, too much, it kind of follows the thought process.” Thus, the students see the writing as tying together with the thought process. As a proviso, however, despite the fact that student L writes mathematical Haikus, students overwhelmingly see this transfer as related to argumentative writing.
A similar theme to writing was borne out in the graduating senior interviews and in the interviews with students in the workshop course. Some of the students in the workshop course also saw the writing skills as transferring. Four of the students on the anonymous evaluation questions suggested that writing in other classes had improved, and in the five class interviews, two of the students mentioned writing skills and a third mentioned her speech class as being a place she felt had improved by looking at mathematical argumentation.
The other skill students explicitly mentioned overwhelmingly was problem solving or reasoning skill. Student J talks about using these skills in his father’s construction business (as though they didn’t exists as strongly before becoming a math major), and student M mentions using them in golf. One of the main skills mentioned by students L, J, and W is the methodical nature of mathematical problem solving or reasoning. L mentions the need to “have all of their steps,” and how she looks at poems “methodically,” which might not be a good thing.
One last skill that comes across in several student statements from the focus group is the ability to look at abstract ideas. In one sense students linked this to the notion of dealing with ideas that you don’t quite understand yet. I would phrase this more as dealing with uncertainty. One tool the abstraction of mathematics appears to help with is going forward with uncertainty, by treating what one doesn’t know as something abstract that will become clear later.
In the student interviews from the workshop course, two of the students explicitly saw transfer beyond math classes, one saw it, but had to stretch to see it, one saw transfer only within mathematics, and one didn’t address the transfer issue at all. Of the two that mentioned using the math elsewhere explicitly, both mentioned writing first. Student AF mentions that he had never seen how to use an outline properly before thinking about pre-planning in problem solving. He felt that might not have been able to learn this any other way besides in a math class. The other student, ES, also mentions writing as the one thing that he knows he takes beyond math and science classes.