So, what to do about it?
My preferred solution would be to have A-level marking require them to get it right, to reinforce what school teachers tell them. If the teaching wasn't undermined by the assessment, they'd be more likely to take the logical structure seriously in the first place, and form good habits rather than bad. Unfortunately, that's outside my control.
Next up is a heartfelt plea to teachers to at least insist on it being done in class/homework, and explicitly mark down when it isn't done right. That's partly because then it won't be my problem any more (which is always a good thing), but more because it's harder to re-shape a habit than form one. Also, it's a pity if those who don't go on further with maths don't come to appreciate this very central idea of how to construct a valid argument. I am skeptical about transferable skills at the best of times, but would like to think that the ability to construct a correct argument and spot a fallacious one is something that people could take away from a mathematics course.
OK, let's assume I can't just pass the buck either to schoolteachers or my colleagues. What can I do about it?
I've got some ideas which I plan to flesh out. Some are just exhortations, some are ideas for exercises which I hope will help.
First, the general advice to get them thinking about it the right way.
Before anything specific, remind them that when they are writing maths, they are supposed to be giving somebody a reason to believe the answer, not just giving them the answer. And explain that that's all we mean by a proof: it's the answer to the question "Why should I believe that?" Quite a few seem to go into a flat spin when the see the instruction 'Prove that...', because they don't actually know what they're being asked for, but they think it's something both mysterious and difficult.
Next, two ways of explaining how to see what might be missing in their working.
- Suggest that they verbalize what they are writing down. Thanks to @DrBennison for this, it's a way of making the previous item concrete.
- A slight variation on the previous: tell them they should read it out loud when they're finished. It should make sense!
I suspect the the problem is exacerbated by the fact that the appropriate direction for the implications depends on what one is trying to establish. There are two common types of problem involving chains of inference that students have to solve:
- Find the values of \(x\) that satisfy some condition, usually an equation \(E(x)=0\) (or an inequality, \(E(x)>0\)).
- Given statement \(A\), show that statement \(B\) follows.
Therefore the solution set of \(E(x)=0\) is \(S\).
The only problem is that this chain of inferences doesn't establish that those are actually the solutions to \(E(x)=0\), only that \(S\) contains the solution set of \(E(x)=0\); we have proved that \(E(x)=0 \Rightarrow x \in S\), and what we need to show is that \(E(x)=0 \Leftrightarrow x \in S\).
So we need to keep track of the relationships and (usually) make sure that they're really equivalences, and what the consequence is when they aren't. (Have we potentially lost solutions? Or gained spurious ones? And what do we have to do about that?)
Similarly, in the second case, it seems very common for students to start off with what they want to prove, \(B\), and find a chain of inference from \(B\) to \(A\), rather than vice versa. I see this in both attempts to prove that two expressions are equal, and in attempts at proof by induction, which I already grumped about.
Well, that's restated the problem and considered it in a bit more detail, and as a mathematician I'm tempted to regard that as a job well done and go off and have a cup of tea or think about something else for a bit. But I really want some progress: if not a solution, at least a solution strategy.
I think both ends of the problem need something done: the detail of just what the relationship is between successive statements in a solution, and the broader picture of just which relationships are required in a particular case.
I think that the following set of ideas for exercises will go some way towards addressing both problems. Now all I have to do is come up with some good, appropriately graded particular cases and try them out on a real live class of students.
- Pairs of statements (equations or inequalities): which implies which?
- Correct arguments with the connectives missing: fill them in.
- Fixable arguments with some incorrect connectives: fix them.
- Examples of "solving" equations where each statement implies the next, but the final solution set is wrong: where did it go wrong, and what can we do about it?
- Samples of invalid arguments: explain what is wrong.