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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)

J. M. ain't a mathematician
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Matt
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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 01:09
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    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 01:40
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    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 03:03
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    Nice question, plus one. Have you ever told any of your students what you wrote in your question, i.e. that being confused and stuck is an okay thing to do? I suspect your students feel stupid and frustrated and think they should be able to just write down a solution. Also: what is the deadline scale? At my uni there are deadlines for each homework and so much homework that it's not actually an okay thing to be stuck because unless you complete 60-80% of the homework you won't be allowed to take the exam. – Rudy the Reindeer Aug 15 '12 at 05:47
  • @Matt: if I were a teacher I'd try all to make things such easy that everybody may easily understand. – user 1591719 Aug 15 '12 at 10:10
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    @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
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    @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
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    @Matt That's great. Then I assume they should know that it's okay if they are stuck. I don't know what it's like at your uni but I'm usually constantly panicked about meeting homework deadlines. Also, no one has ever told me that it's ok to be stuck (I found out myself, only recently). Maybe you can just tell them again to remind them? : ) – Rudy the Reindeer Aug 15 '12 at 16:03
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    I wish I knew the answer myself. I'm teaching intro to abstract algebra this semester and one idea I am considering trying is incorporating in class proof writing group exercises, where I go around the class and discuss their ideas and, if necessary, where they've got stuck, particularly at the beginning of the semester. I'm looking to increase dialogue early in the semester so that there is a good channel of feedback between the students and I. Being able to identify a *specific* instance of where getting stuck is okay and how to overcome it may be more useful than general encouragement. – Michael Joyce Aug 16 '12 at 04:57
  • Students in first-year calculus courses think that any instructor who asks them to _understand_ the material (whether at an intuitive level or any other) is thereby committing malpractice. They "know" that they have a _right_ not to be asked to do that. – Michael Hardy Apr 05 '18 at 12:27

5 Answers5

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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.

Ronnie Brown
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    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](http://mathforum.org/kb/servlet/JiveServlet/download/206-1960117-6767154-560215/obtusity.pdf) where they were to prove that $e$ is irrational. – Dave L. Renfro Aug 15 '12 at 19:54
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    Incidentally, the method Ronnie Brown describes reminds me of the book [Bobby Fischer Teaches Chess](http://www.amazon.com/dp/0923891609) 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 04 '16 at 03:01
  • @YoTengoUnLCD: The link is to amazon.com's listing of the book, and it still seems fine where I am at (in the USA). Maybe this site is blocked where you are at? Incidentally, there are 419 customer reviews of the book -- quite a lot of comments for a book I didn't think many people would be aware of so many years later! – Dave L. Renfro May 04 '16 at 15:46
  • @DaveL.Renfro Could you share the results of said experiment? Did it "work"? I would be very interested in knowing how things turned out. – Aloizio Macedo Nov 09 '17 at 16:57
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    @Aloizio Macedo: The experiment pretty much fizzled, mainly because it was too time consuming to think of useful ways to do this. ("Useful" in the sense of being mathematically useful, rather than in developing editing skills.) That particular semester I was teaching 6 different classes (i.e. 6 different class preparations), and trying to do this wound up taking a back seat and I finally gave up. I still think it could be useful, but perhaps better for graduate students, whether teaching or research is their primary later goal. **(continued)** – Dave L. Renfro Nov 09 '17 at 18:08
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    For high school students, probably better would be to have them explain in front of a class or to others (during group work), which teachers already do. I think I was trying to reproduce some aspect of the adage that one learns something really well after having to teach it (in this case, having to understand and then clear up misunderstandings). – Dave L. Renfro Nov 09 '17 at 18:13
  • @DaveL.Renfro Thank you very much for sharing! – Aloizio Macedo Nov 09 '17 at 18:15
  • @RonnieBrown very nice answer. The method you described has been labeled scaffolding by Merrienboer, who has proposed it for a few decades now (initially for med students and now more broadly). His book is t.ly/0PZd , with the theory summarized in his 4cid model https://www.4cid.org/about-4cid – Ricardo Pietrobon Nov 07 '20 at 16:30
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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.

Kevin Arlin
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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. ain't a mathematician Aug 15 '12 at 02:39
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    @J.M.: ’Tis indeed, though to judge by my (pretty extensive) experience of foreign students, this is far from being an exclusively American problem. – Brian M. Scott Aug 15 '12 at 02:55
  • I agree that they just want the grades. One of the things I try to impress on them is that part of the grade is producing on a test, and you can't do that without having spent the time developing intuition in something similar to this way. But they don't like the non-instant gratification of this. You start struggling day 1 so that by week 5 you've developed enough to do well on the test. They want night before studying. Thanks for the Tao reminder. I had forgot about that. Maybe I'll assign it next time. – Matt Aug 15 '12 at 02:57
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    @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 03:04
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    @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 Arlin Aug 15 '12 at 03:24
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    If for nothing else, we can all admire @Brian's stamina. :D – J. M. ain't a mathematician Aug 15 '12 at 03:45
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    ... or small class sizes. – Robert Israel Aug 15 '12 at 06:29
  • @Robert: I’ve done it with classes of $25$ and even $30$, but more often with $15$ to $20$. Occasionally I was fortunate enough to have only around $10$. – Brian M. Scott Aug 15 '12 at 08:07
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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.

PPJ
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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. ain't a mathematician Aug 16 '12 at 03:04
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    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
  • Looking back on my experience teaching an upper-division linear algebra class which was largely formalizing proofs of stuff they already knew from lower-division linear algebra (until getting into things like Jordan canonical form, normal operators, etc.) - I think if I'd known about natural deduction proof systems at the time, I would have presented an overview of that. It would have been so helpful when addressing student confusion over the structure of a proof to say "here we're using $\exists$-elim" or so. – Daniel Schepler Nov 01 '17 at 17:51
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I have done a lot of office hours, along the way, on math, physics, and quantitative economics. My thoughts:

1) Students need to get over 'math phobia'. Many students fear being 'wrong', because that is seen as failing, a character (or deeper) defect, or whatever. Because math has exact answers, they fear that.

2) Rather, they need to see it is a 'puzzle', that, like a crossword puzzle, is entertaining (hopefully) or at least not judgemental. Lots of people don't finish crossword puzzles, but they try and enjoy the process.

3) In anything, it is important to realize that the struggle is the journey. I had a professor who always told students "for a student, confusion is a state of grace".

These are vague, but it is really more psychological than anything else, in my experience. People don't feel as stressed trying to learn to play tennis or golf, or trying to rock climb, etc.; they know that it is a long path to mastery, and the first few attempts are the hardest. But, with math, they give up - fearing they will never succeed.

I have done a lot of my teaching one-on-one, and always ask the student to try the next step and I will help them without judging them (a safety net), as well as reinforcing every time they make a move in the right direction (even if it is actually wrong). With learning a sport they get their own positive feedback, because they know their goal - to hit the ball, straight, for over 200 yards. And when they hit a golf ball well (which they know if they did), even if it is infrequent, it is both something they apprehend and feel good about.

If I had one really big piece of advice, however, that comes from lots of time as a TA, it is this: don't let your own ego get in the way. Too often I find that the problem is not the student but the professor. Everyone knows that you know the answer. But, I believe, the art of teaching is knowing what the student doesn't know now that they need to know next. Too often I see a simple idea made into a complex question that violates this. For example, physicists are often terrible at this: while introducing Newton's laws, they quickly move to problems of a frictionless plate on ice subject to a force, with and object on that frictionless plate subject to another force, and want the student to find the position of the object over the ice as a function of the displacement of the bottom plate...not really helpful.

There is a perhaps apocryphal story of Einstein helping a child with algebra. When the kid complained he (actually, I believe it was a she) couldn't enough of the problems right, Einstein responded along the lines of 'if you think you have trouble solving most of the problems, don't worry - you would be shocked to see how few of the problems I work on that I can't answer'.

eSurfsnake
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(In addition to other good answers, and echoing some parts...) I think it's not so much that students fear confusion, but see confusion as a failure. Not failure to understand (because "understanding" is not their goal), but failure to be set up to get the grade they want (in many or most cases...)

Unless we can figure out how to reward students, in terms that they can appreciate, for engagement sufficient to be confused, and then work through it, they'll have no motivation to do so or think that way. The "it's good for you" argument has limited impact, if it's not manifestly/visibly helpful, by their (often-misguided) criteria.

There is also an unfortunate undercurrent of a belief system that "sufficient formalization", a.k.a., "a sufficient system of rules", or "study of first-order predicate logic", will remove the complicated, amorphous need for thinking in a new way, with an awkward transition to that new way. In my observation, this is (unfortunately) attractive because it allows students to keep the mathematics "external" to them, excusing them from thinking much about it. "Just learn the rules", as opposed to "understand why so-called rules are intended to be one choice of codification of aspects of reality".

And, indeed, quite a few confusions do concern convention, rather than mathematical fact. Of course few people can intuit convention, no matter how reasonable. Being confused about choices of convention is fine, and does not bear much reflection. Being confused about facts of course does deserve further reflection. Helping students distinguish the two sorts of issues is perhaps one of our main jobs, I think.

I would go so far as to claim/observe that the epsilon-delta "definition" of "limit" is itself only one possible convention making (more) precise a primitive, colloquial sense of "limit". In particular, many of the features are matters of convention, so cannot be "deduced", and this sort of complication deserves explicit recognition.

paul garrett
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