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I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro to Real Analysis.

Since this was essentially these student's first "real" math class, they haven't really learned how to study for or learn this type of thing. I've continually emphasized throughout the summer that they need to put in more work than just doing a few homework problems a week.

Getting a feel for the definitions and concepts involved takes time and effort of going through proofs of theorems and figuring out why things were needed. You need to build up an arsenal of examples so some general picture of the ideas are in your head.

Most importantly, in my opinion, is that you wallow in your confusion for a bit when struggling with problems. Spending time with your confusion and trying to pull yourself out of it (even if it doesn't work!) is a huge part of the learning process. Of course asking for help after a point is important too.

Question: What is a good way to convince students that spending time lost and confused is a reasonable thing and how do you actually motivate them to do it?

Anecdote: Despite trying all quarter to explain this in various ways, I would consistently have people come in to office hours who had barely touched the homework because "they were confused". But they hadn't tried anything. Then when I talk around an answer to try to get them to do certain key parts on their own or get them to understand the concept involved, they would get frustrated and ask "so does it converge or not?!"

It is incredibly hard to shake their firm belief that the answer is the important thing. Those that do get out of this belief seem to get stuck at writing down a correct proof is the important thing. None seem to make it to wanting to understand it as the important thing. (Probably a good community wiki question? Also, real-analysis might be an inappropriate tag, do what you will)

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From experience, trying to write a proof without understanding it is a bad idea, and you'll forget said argument right away. On the other hand if you "understand" but can't write down a complete proof maybe there's something else going on there. – Jose27 Aug 15 '12 at 1:09
Very often students will say they understand, or that they see something "intuitively", but they really don't. Part of the battle is to understand that you don't understand. – Robert Israel Aug 15 '12 at 1:40
Most students are lazy (I know I am/was): 'i'm confused' is often shorthand for 'i couldn't be bothered'. I'm not sure which is better. – Kris Aug 15 '12 at 3:03
@Matt (is this going to ping me or the commenter?) At the beginning of the class I wrote up a sheet, distributed it, and even went over it in class on how "I would take the class" in which I explained that being confused and stuck was a necessary part of learning the material. – Matt Aug 15 '12 at 15:59
@Chris'sister Of course its good for teachers to try to be clear and help the students to understand. But I think this is the wrong attitude. If you're teaching abstract concepts, then until the student puts in the work to understand it for themselves no amount of watching the teacher present perfectly clear explanations will help. To me the role of a teacher is mostly to motivate the student and help the student learn how learn on their own. – Matt Aug 15 '12 at 16:03
up vote 44 down vote accepted

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. 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.

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This is nice to see suggested explicitly. – Jon Bannon Aug 15 '12 at 16:59
This is a version of something I tried in a small class of very talented high school students in 1997. After grading several assignments involving proofs that I found very difficult to follow what they wrote, I decided to "turn the tables" on them. I handed out proofs -- sometimes little more than poorly written hints, other times almost an acceptable proof, depending to how difficult it was -- that needed to be cleaned up. Here's an example where they were to prove that $e$ is irrational. – Dave L. Renfro Aug 15 '12 at 19:54
Incidentally, the method Ronnie Brown describes reminds me of the book Bobby Fischer Teaches Chess that appeared back when Fischer was on top of the chess world. – Dave L. Renfro Aug 15 '12 at 19:59
I should add that the method was not for the specially talented but for the whole class. But as in a crossword puzzle, some of the bits missed out were more subtle to fill in than others. We even used this method as part of exams and tests. – Ronnie Brown Aug 15 '12 at 21:12
@DaveL.Renfro Hi Dave, I'm afraid to say your link no longer works. Is there any way you could provide a new one? Your method seems really interesting. – YoTengoUnLCD May 4 at 3:01

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.

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"...a system that's much better at turning talented students' ambitions towards high grades than towards deep understanding." - and that's a bloody shame. – J. M. Aug 15 '12 at 2:39
@Matt: I always based well over half of the grade in theory courses on (approximately) weekly homework, all of which I graded closely and on all of which I commented heavily. Yes, this is brutally time-consuming and frustrating, but it did at least force them to deal with the material steadily throughout the course. (I also made it clear that I’d award partial credit for intelligent attempts that could not be successful as well as for partial correct solutions.) – Brian M. Scott Aug 15 '12 at 3:04
@BrianM.Scott I certainly didn't mean to imply anything about foreign schooling, rather than to restrict my commentary to where I'm most experienced. Your policy of grading all homework personally is impressive. – Kevin Carlson Aug 15 '12 at 3:24
If for nothing else, we can all admire @Brian's stamina. :D – J. M. Aug 15 '12 at 3:45
... or small class sizes. – Robert Israel Aug 15 '12 at 6:29

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.

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Considering "even in an upper-level math class, the majority of your students are not going to be mathematicians." in Kevin's answer, I'm not sure they'll readily take to reading about logic and proofs in addition to the calculus they're already moaning about... – J. M. Aug 16 '12 at 3:04
I asked one graduated student who had moved into computer programming what course he found most useful. To my surprise, he replied: "Your course in analysis, as it gave me an idea of rigour." It is too easy to assume average people are uninterested in logic and proofs. But millions work happily away at Sudokus, which of course requires logic! One research method I learned from a supervisor I formulated as: "If MGB can try one damn fool thing after another, then so can I!", and I have followed this method ever since. The point is: "Try anything, but then test it. The crazy idea might work. " – Ronnie Brown Aug 16 '12 at 10:19

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