# Style of teaching/learning mathematics: by proofs of theorems only?

• yucheng
In summary, this method involves forcing students to prove the theorems themselves. This method has been widely criticized for being too difficult and not suitable for beginners.
yucheng
I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone....

@fresh_42 ?

yucheng said:
I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone....

@fresh_42 ?
Somehow I don't think that the method you are referring to was named @fresh_42.

Math100, WWGD, fresh_42 and 2 others
When I was young, they temporarily changed teaching one of the subjects to be learnt by doing something - a set of tests or experiments - and coming to your own conclusions from the results, which sounds similar. I was in the class where they tested this before it was to be adopted, our teacher being one of the group who pushed for this method.

Even back then (aged 13 or 14) I spotted that if you got the wrong result, you "learned" the wrong answer. And then you would fail the exams.

I've never been keen on this approach to learning.
Imagine teaching someone to drive on a busy road with no explanation of what the car controls and pedals did...

"This is one of three modes of student learning in mathematics described in this article by Frank Quinn; it is the least powerful, most fragile, and most error-prone of the three."
Looks like others have spotted the problem of this approach as well.
Interestingly my school abandoned this method after experimenting on our class.

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vanhees71, kuruman and PeroK
PhDeezNutz
kuruman said:
Somehow I don't think that the method you are referring to was named @fresh_42.

Definitely not.

DrJohn said:
When I was young, they temporarily changed teaching one of the subjects to be learnt by doing something - a set of tests or experiments - and coming to your own conclusions from the results, which sounds similar. I was in the class where they tested this before it was to be adopted, our teacher being one of the group who pushed for this method.
This sounds like copying physics: make a couple of experiments, guess a law, and check that law. Just to be clear: I, too, think that this is stupid. I'm not surprised that it magnificently failed. Mathematics is deductive. You cannot pretend it is not no matter how many inductive experiments you perform. I'd rather derive a proof that it has to fail than inductively testing it.

My personal position, in case someone is looking for a name, is that ...
Code:
For n=1 to 1,000
Loop Until 80% got it
Select Example E from Algorithm A(n)
Solve E
End Select
Test On 80%
End Loop
End For
... can hardly be called "teaching mathematics". It is teaching calculations that they actually do in math classes, counting in the end! And by the way, only in math classes. They teach physics in physics classes, biology in biology classes, and chemistry in chemistry classes. Only in math, they still confuse counting with mathematics.

As always in real life, I think the truth lies between these extremes. I'm particularly opposed to the attitude that children are dumber than they are. You have to challenge them and not hammer algorithm after algorithm into their heads. No surprise that most of them hate what they erroneously think is mathematics.

The usual curriculum goes: We don't have negative numbers. EOY. And here are the negative numbers. We don't have irrational numbers. EOY. And here are the square roots. We don't have negative roots. EOY. And here are imaginary numbers ... What's an invalid input of the current algorithm is sold as non-existent until a new algorithm comes that deals with it. However, the newly invalid inputs are again sold as non-existent until ...
Who could still take such teachers seriously?

yucheng said:
It's called the Moore method
I had a couple classes in graduate school that used this method. At that level or in advanced undergrad classes, I believe it is appropriate, but not before students have the mathematical maturity to be able to construct the proofs.

Math100, PhDeezNutz, vela and 1 other person
I thought the most efficient way of teaching math is "front-of-class teaching" + doing a lot of problem-solving by the students, a method which fell out of favor in the didactics community some decades ago. My (obviously subjective) observations about the abilities of "freshmen" at the university seem to indicate a complete failure of the "new methods" with their over-emphesis of what the eupemistically call "competence orientation", which however means just that students are trained to solve a lot of standardized problems without deeper understanding of the foundations the corresponding solution strategies rest on.

More recently the didactics community seems to come to different conclusions. One, pretty obvious from a common-sense point of view is that indeed, you remember best about a subject what you learnt first, i.e., if you use the reconstructive method (i.e., the funny idea to invent 3000y of mathematical research within 12-13 years at high school by the students themselves) all the mistakes you inevitably make in the process burn in to the brains best.

PhDeezNutz, martinbn and PeroK
vanhees71 said:
front-of-class teaching
We call it "chalk and talk". The educationalists among us hate it.

I would argue that the only way to learn to solve tough problems is to solve tough problems, and this has to be done on your own. A good teacher can help guide (including selecting good tough problems) but ultimately the student gets out what he puts in. The educationalists among us hate this too.

apostolosdt, Mark44, PhDeezNutz and 4 others
We call it "chalk and talk". The educationalists among us hate it.

I would argue that the only way to learn to solve tough problems is to solve tough problems, and this has to be done on your own. A good teacher can help guide (including selecting good tough problems) but ultimately the student gets out what he puts in. The educationalists among us hate this too.
I had an analysis professor who started with his left hand on the leftmost corner of the huge blackboard, switched hands in the middle, and finished on the rightmost corner. Carriage return.

Reminds me somehow of "proof by von Neumann"

We basically learned the subject by working our way through the book.

However, problem-solving might be useful for physicists, but not necessarily for mathematicians. Learning stupid algorithms (as in school) has nothing to do with mathematics.

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Math100
I had a professor who had the chalk in his right hand and the eraser in his left, following it a foot or two behind. Very efficient.

I might make a similar argument for mathematics. The way you get good at doing proofs is by doing proofs.

fresh_42
We call it "chalk and talk". The educationalists among us hate it.

I would argue that the only way to learn to solve tough problems is to solve tough problems, and this has to be done on your own. A good teacher can help guide (including selecting good tough problems) but ultimately the student gets out what he puts in. The educationalists among us hate this too.
Yes, as I said, it must be a combination of "chalk" (at universities called "lectures") and a lot of active (!!!) problem-solving by the students (at universities called "tutorials", "recitations", or something like that).

Just to expect the students to reinvent 1000s of years of research in math (about 400 years of physics) by themselves is unrealistic. Also, as said before, you remember best, what you learn first, including everything that has been explained in some wrong way.

My prime example is to introduce quantum mechanics with "old quantum theory" (Einstein's wrong photon picture as localized particles, Bohr's wrong theory of atoms, cementing the idea of classical trajectories of electrons around the nucleus, and all that), which fell now out of favor even with the physics didactics community, for precisely this reason.

PhDeezNutz
fresh_42 said:
However, problem-solving might be useful for physicists, but not necessarily for mathematicians. Learning stupid algorithms (as in school) has nothing to do with mathematics.
By problem solving they don't mean just rutine algorithmic problems. But also hard problems where you have to figure things out. After all mathematics is about solving problems. One shouldn't think that just because a problem is called a theorem and the solution a proof there isnt problem solving.

AndreasC and vanhees71
A lot of wisdom in this thread. Expecting someone to rediscover thousands of years of progress on their own……..not feasible…..at all.

As much as I love George Carlin the notion of “questioning everything” and “thinking for yourself” hinders progress past a certain point.

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BillTre, fresh_42, vanhees71 and 1 other person
fresh_42 said:
Mathematics is deductive
But that's not how math is generally developed... Of course there is a substantial deductive part, but just look at, say, Euler. His methods were highly "inductive", and he built half of modern math like that. You can't just start from a bunch of axioms and derive random things... I remember Riemann said something along the lines "if only I knew the theorems, proving them would be easy".

I think that suggests there should be a balance between learning the deductive, proof writing part and engaging in the practice of mathematical discovery. Right now the balance is probably massively in favor of the first part. Every advanced textbook has exercises like "prove x". But how do you know you should try proving x to begin with if no one asks you to? Attempts to teach mathematical discovery sound like a good idea.

vanhees71 and PeroK
PhDeezNutz said:
A lot of wisdom in this thread. Expecting someone to rediscover thousands of years of progress on their own……..not feasible…..at all.

As much as I love George Carlin the notion of “questioning everything” and “thinking for yourself” hinders progress past a certain point.
I don't think that's exactly the idea behind this. As in, I don't think someone following that method is intending to get you to rederive everything on your own. There is actually a book I know doing more or less that, but for astrophysics:

https://www.amazon.com/dp/0521467837/?tag=pfamazon01-20

The idea is not that there is no content and guidance whatsoever, but that you work on targeted problems relevant to the concepts that have to be taught, and you acquire the content from the solutions to the exercises. That way, the definitions and concepts are motivated, and after all many people believe you learn mathematics better by solving problems. Either way, I don't know how well this method works because I haven't tried it.

Either way not every method applies to everyone. If the students are not strong and mature enough it definitely won't work at all. Otherwise, it may be powerful.

Maybe I have a very superficial understanding of the method and made a knee jerk comment. Maybe it wasn’t meant to be taken to that extreme end. I’ll read it more in depth.

AndreasC
yucheng said:
I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone....

@fresh_42 ?
How does a teacher force students to do math proofs and theorems. Does he or she stand over them with a bullwhip. Does he or she say prove this theorem or I'll flunk you.
Any intelligent student would have to ask, what exactly is your job in all this. do you just give papers with proposed theorems theorems out in the beginning of class. I like to think I could learn from any teacher, but I would be at a loss to learn from this one.

vanhees71
I had a professor who had the chalk in his right hand and the eraser in his left, following it a foot or two behind. Very efficient.

I might make a similar argument for mathematics. The way you get good at doing proofs is by doing proofs.
I had a physics professor in my senior year who used to teach that way. One day, he did that so quickly that a student reacted with a loud “ftoo!” The professor turned to the class and calmly said: “Please, don’t spit.”

vanhees71 and AndreasC
To me the Moor method can be useful for school students or the first years at university, but for the more advance topics I cannot see how it can work. How does one learn Wiles' proof of Fermat using this method?

vanhees71
Axiom 1: if fermat is false, then there exists a semistable non modular elliptic curve.
Axiom 2: all semistable elliptic curves are modular.

problem: prove fermat is true.

vanhees71 and fresh_42
martinbn said:
To me the Moor method can be useful for school students or the first years at university, but for the more advance topics I cannot see how it can work. How does one learn Wiles' proof of Fermat using this method?
It actually tends to be kind of the opposite. It's better for students who have mathematical maturity, otherwise they just get completely stuck, or they come up with wrong things but without being able to see why they are wrong. The specific example (Wiles' proof) is a bit weird, people don't usually learn that. In general the idea would be to provide only the bare essentials but break hard problems into more manageable chunks that students can work out.

vanhees71 and mathwonk
AndreasC said:
It actually tends to be kind of the opposite. It's better for students who have mathematical maturity, otherwise they just get completely stuck, or they come up with wrong things but without being able to see why they are wrong.
Moor would point out all mistakes.
AndreasC said:
The specific example (Wiles' proof) is a bit weird, people don't usually learn that.
Well, I have taken a course specificaly on that.
AndreasC said:
In general the idea would be to provide only the bare essentials but break hard problems into more manageable chunks that students can work out.
I know, but why is it unacceptable to acknowledge that there are clever ideas, from which one can learn. Instead of trying to come up with a proof yourself.

vanhees71
martinbn said:
I know, but why is it unacceptable to acknowledge that there are clever ideas, from which one can learn. Instead of trying to come up with a proof yourself.
Who said it's unacceptable?

vanhees71
AndreasC said:
Who said it's unacceptable?
The very idea of the method. Moor puts a statement on the board and waits for the students to prove it. He does not move on until it is done, and he only shows what is wrong with their attempts but doesn't give hints.

martinbn said:
The very idea of the method. Moor puts a statement on the board and waits for the students to prove it. He does not move on until it is done, and he only shows what is wrong with their attempts but doesn't give hints.
It's a method for teaching some things, not a method for teaching every thing, or a declaration that some other method is unacceptable...

Regardless, you can even teach some of the clever things you say by pointing the students to the right direction by the choice of questions.

mathwonk and fresh_42
The only effective " invariant " in teaching( edit: In my experience)is being both knowledgeable and well-prepared for class, however trite. It all stems from that.

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apostolosdt, vanhees71 and PhDeezNutz
martinbn said:
To me the Moor method can be useful for school students or the first years at university, but for the more advance topics I cannot see how it can work.

martinbn said:
Moor would point out all mistakes.

martinbn said:
Moor puts a statement on the board
Let's get the name straight -- it's the Moore method, named after the topologist Robert Lee Moore. See https://en.wikipedia.org/wiki/Moore_method. The name is not a reference to North African Arabs.

Repeating what I said before, the first time I experienced a class that used this method was in an upper-division math class in college.

martinbn said:
The very idea of the method. Moor (sic) puts a statement on the board and waits for the students to prove it. He does not move on until it is done, and he only shows what is wrong with their attempts but doesn't give hints.
Seems to me that showing that some work is wrong is a pretty powerful hint.

martinbn said:
The very idea of the method. Moor puts a statement on the board and waits for the students to prove it. He does not move on until it is done, and he only shows what is wrong with their attempts but doesn't give hints.
As I experienced the Moore method (as used by one of Moore's students), the class is asked to prove or disprove statements. They don't know in advance whether a statement should be proven or disproven. They get experience in giving counterexamples.

In my opinion, the Moore method places a great demand on the instructor's ability to organize the subject matter so the class proceeds at an appropriate pace. It doesn't suit instructors who may have an encyclopedic knowledge of their field but don't have a vision of it as a sequence of inquiries and deductions. Perhaps there are fields of study where nobody has such a vision. And there may be mathematical results created by some genius that nobody else can understand as a step-by-step sequence of inquiries performed by mere mortals.

In addition to the general objections that have been raised against the method in this thread, there are some mundane ones. I think Moore (R. L. Moore) applied the method to teaching point set topology ,which is not everyone's favorite mathematics and did not use terminology completely matching that used by modern texts. Also (the web says) that R. L. Moore supported segregation and refused to admit black students to his classes. So a method associated with his name is tarnished by that.

Yes, " Moore Spaces" , in Topology, are named after him :

And , as per Stephen Tashi's post, we have the issue of whether we separate the artist's/author's work from their personal lives.

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