Last term some colleagues and I were involved in a set of workshops to help our first-year students learn to read and write mathematical proofs more effectively. Nobody who’s taught university mathematics will need much explanation of why this is an issue; what we’re still working on is finding a good way to address it.
In particular, this term we’ve introduced our students to the technique of “self-explanation”, and to the excellent wee booklets produced by the Mathematics Education Centre in Loughborough. Self-explanation is essentially a process of going through a proof step-by-step, asking explicitly how each step follows from the previous one and filling any gaps or implicit assumptions en route. The Loughborough team have found some surprisingly robust effects on students’ proof comprehension when they were introduced to the technique, and they were good enough to supply us with a stack of their booklets to distribute among our students at the start of the year. As part of our appraisal of the project, and to supply Loughborough with evidence of the impact of their work, we included four questions about self-explanation in our feedback sheet at the end of term, of which the first two were the following.
- Did you read the self-explanation booklet? If so, did you ﬁnd the idea easy to understand and use in practice?
- For what activities did you ﬁnd self-explanation to be useful?
On the whole, the responses were as positive as one could reasonably expect from first-year students during the tears-before-bedtime phase of term, and my own impression during the workshops was that self-explanation did give students a useful framework for their attempts to understand and to assess the validity of proofs. It may not have got some of them to the stage of being able to assess validity accurately, but it did seem to make it easier for us and them to determine where their understanding had broken down.
Scattered among the forty or so responses we received, though, were some remarks that make me wonder how much of the message about proofs is getting through.
One such remark provided the title of this post:
The self-explanation technique has helped me to understand and explain what is going on in the proof but it is more literature than mathematics.
(This recalls a conversation, recently overhead on a bus, between two students at a fairly prestigious institution. A second-year mathematics student was explaining to his friend that he wasn’t enjoying his degree course because “I really like numbers but this course seems to be all about letters”. Once again, the gap appears between the arithmetical and procedural activities presented as “maths” in school and the process of logical argument understood as “maths” at university; and, as ever, our students see the school version as definitive.)
The remark that really stood out, though, was made about half a dozen times by different students in slightly different words:
I didn’t read the self-explanation booklet. It was useful for proofs.
What are we to make of this? One explanation is that the students who didn’t read the booklet but nevertheless found it useful were deploying it as a kind of talisman: laying it on the desk beside a proof, muttering the appropriate conjurations, and finding understanding suddenly dawn. (There are, in my experience, students who expect lecture notes to work like this.) Two more interesting explanations are available, though.
One explanation is that these students, having been given a brief oral sketch of self-explanation in class, saw no need to consolidate this by reading the booklet and the examples contained in it — that they got the idea, unproblematically, first time (or at least felt that they had). What this reveals, I think, is the sense among some students that to read anything is onerous, so rather than spending a few minutes following the advice and reading the booklet in case there’s something worthwhile there, one is better off carrying on without it and taking the risk.
The other explanation is that — as so often — the students are not answering the question that we thought we’d posed. Bearing in mind that this is the generation who are accustomed to interpreting everything as a test, possibly they saw the second question not as a continuation of the first, but as an attempt to assess their knowledge about self-explanation: not “what did you find SE useful for?” but “what in general is the purpose of SE?” The response, then, is an attempt to show that they were paying attention even although they hadn’t read the booklet. I wonder how many other well-intentioned attempts to teach “metacognitive” skills meet the same fate…
In summary, then, we wanted to help our students to read complex arguments more effectively. We tried to help them to do this by a sort of bootstrapping: supplying short, clearly written instructions explaining how to read complex arguments. For some students this may have worked. However, as far as we can tell, some students didn’t see the connection between this and mathematics; some may have thought they were following our advice but were in fact doing something different because they didn’t read the instructions we supplied; some may have perceived the task as training them to give the correct response when asked what self-explanation was for; and a few may have been deploying the instructions as a form of magic.
I don’t think my job has much in common with literature, but I’m increasingly sure that it has even less to do with mathematics.