Getting Students to Not Fear Confusion

Has anyone tried as an additional technique the "fill-in" method?

This is based on the tried and tested method of teaching called "reverse chaining". To illustrate it, if you are teaching a child to put on a vest, you do not throw it the vest and say put it on. Instead, you put it almost on, and ask the child to do the last bit, and so succeed. You gradually put the vest less and less on, the child always succeeds, and finally can put it on without help. This is called error-less learning and is a tried and tested method, particularly in animal training (almost the only method! ask any psychologist, as I learned it from one).

So we have tried writing out a proof that, say, the limit of the product is the product of the limits, (not possible for a student to do from scratch), then blanking out various bits, which the students have to fill in, using the clues from the other bits not blanked out. This is quite realistic, where a professional writes out a proof and then looks for the mistakes and gaps! The important point is that you are giving students the structure of the proof, so that is also teaching something.

This kind of exercise is also nice and easy to mark!

Finally re failure: the secret of success is the successful management of failure! That can be taught by moving slowly from small failures to extended ones. This is a standard teaching method.

Additional points: My psychologist friend and colleague assured me that the accepted principle is that people (and animals) learn from success. This is also partly a question of communication.

Another way of getting this success is to add so many props to a situation that success is assured, and then gradually to remove the props. There are of course severe problems in doing all this in large classes. This will require lots of ingenuity from all you talented young people! You can find some more discussion of issues in the article discussing the notion of context versus content.

My own bafflement in teenage education was not of course in mathematics, but was in art: I had no idea of the basics of drawing and sketching. What was I supposed to be doing? So I am a believer in the interest and importance of the notion of methodology in whatever one is doing, or trying to do, and here is link to a discussion of the methodology of mathematics.

Dec 10, 2014 I'd make another point, which is one needs observation, which should be compared to a piano tutor listening to the tutees performance. I have tried teaching groups of say 5 or 6, where I would write nothing on the board, but I would ask a student to go to the board, and do one of the set exercises. "I don't know how to do it!" "Well, why not write the question on the board as a start." Then we would proceed, giving hints as to strategy, which observation had just shown was not there, but with the student doing all the writing.

In an analysis course, when we have at one stage to prove $A \subseteq B$, I would ask the class: "What is the first line of the proof?" Then: "What is the last line of the proof?" and after help and a few repetitions they would get the idea. I'm afraid grammar has gone out of the school syllabus, as "old fashioned"!

Seeing maths worked out in real time, with failures, and how a professional deals with failure, is essential for learning, and at the reasearch level. I remember thinking after an all day session with Michael Barratt in 1959: "Well, if Michael Barratt can try one damn fool thing after another, then so can I!", and I have followed this method ever since. (Mind you his tries were not all that "damn fool", but I am sure you get the idea.) The secret of success is the successful management of failure, and this is perhaps best learned from observation of how a professional deals with failure.

Only part of this will be an attempt at an answer, because my first reaction was, bluntly, "fat chance." American students-and I see yours are American-have come to you via a system that's much better at turning talented students' ambitions towards high grades than towards deep understanding. Even in an upper-level math class, the majority of your students are not going to be mathematicians. Those who have arrived at the last year or two of their education without truly engaging are unlikely to be converted even by a master teacher, for whom the best opportunity was much earlier on.

All pessimism aside, what you can do depends a lot on how free you are in course design. If you give a course in which the grade is decided by whether weekly homework assignments and a couple exams come in with accurate solutions, your students will try to produce a decent simulation of an accurate solution as efficiently as possible, with some pleasant exceptions. Various (untested) ideas: Involve writing in your assignments, both when a student can and can't come up with a solution. In the former case, ask them to express carefully and fully what they've thought of, and what they've stumbled on. This will, naturally, often lead to more success. When they do succeed, ask them to write some thoughts about different variations of the problem, which they should invent themselves: why is this hypothesis necessary? Could I weaken it? What if I tweak this series slightly? You might show them this advice from Terry Tao, as well as his notes on valuing partial progress and on asking yourself dumb questions, to this end.

The general principle I'm proposing is that if you want students to spend time lost and confused, reward them for doing so and then telling you about it. I'd even consider grading better a student who couldn't prove the MVT from Rolle's Theorem but wrote down three different plausible, thorough attempts than one who just said "Define $g(x)=f(x)-\frac{f(b)-f(a)}{b-a}x-f(a).$ Rolle's applies to $g$ at $c$. MVT is satisfied there for $f$." The exams, naturally, wouldn't bear the same conditions, since nobody should get out of real analysis without being able to do that last.

One important thing that helped me to get through Intro To Real Analysis is doing some reading on logic and introduction to proofs. Learning some proofs techniques, what are the ways to attack a problem. That's what students never learn in Calculus and that's the main reason why it's hard to go from Calculus to Real Analysis.

So, what I would recommend is offering supplementary readings on that subject: logic and introduction to proofs. The book I used was S. Lay, Analysis with introduction to proofs. Logic and intro to proofs are the first few chapters, probably the best in the whole book (I didn't particularly care about the "analysis" part). I'm sure there are lots of other similar books and well but that's the one that helped me to make a good start with Baby Rudin.