How to Help Students Understand Mathematical Proofs

[matqkks]

I am having difficulty with students understanding mathematical proofs. Even if they do understand them in the class they cannot reproduce them in the final examination. Majority of the students avoid doing the proof questions and concentrate on the numerical or technique application of mathematics.
I have looked through so many books on mathematical proofs but they all offer the same remedy which does not work for the students I teach. Mostly it is like reading an essay with symbols.
Has anyone experienced with a sort of flow chart method or something similar (boxing important results or tools used in the proof)?
How can I get students to appreciate a mathematical proof and the importance of it?
Is there a more systematic or better layout method that I could use in my classes?
Thanks for any replies in advance.

 

[Hootenanny]

The best method that worked for me, when I was an undergrad, was the tutor would often give a proof for a special case of some theorem. We would then be set the task of proving the full theorem, either for homework or in class. Usually these special cases was quite easily generalised, so it would depend on which theorems you teach and whether you can come up with a special case that can be generalised fairly easily. However, this does force students to try and understand the method of the proof, rather than mechanically learning it.

I try to use it as often as I can when I tutor and it usually works well.

 

[Stephen Tashi]

I think you must describe what kind of proofs your class is doing.

There is no single standard for what constitutes a mathematical proof. Proofs written for math journals are a different matter than proofs written as exercise. Proofs written as exercises have standards that vary with the instructor and the course material. For sake of ease of grading, many classes want proofs written in a very concise, barely literate form. In contrast, a few instructors want proofs written as essays using complete sentences, correct grammar and correct spelling.

I'm trying to write some advice about doing proofs since I promoised micromass I would post one. It would be helpful to me to discuss the details of your students' problems.

 

[matqkks]

Thanks for your reply. The proofs are for a class in linear algebra and they are first year undergraduate students in the UK. They have not covered proofs previously apart from simple proof by induction.
As soon as they read the proposition they don't have an idea which method of proof to use. Proving things such as A implies B is impossible to a number of students. I have explained at length that they can assume A and should try to deduce B but to no avail.
One or two students suggested that sign posting previous results which are to prove the present proposition would help.Hope this helps to clarify my question.

 

[daveb]

If the students don't have much practice following lines of logic, no flowchart or technique is probably going to work, except with a few of the brightest students. I used to do those Dell logic problems all the time as a kid for fun, so when I got to my first proof class, I understood the necessity and concept behind following lines of logic to a conclusion. I credit that dorky hobby for my propensity to understand and perform proofs.

 

[Stephen Tashi]

matqkks,

I think the starting point for doing a proof must be to understand the definitions involved. There are several techniques that students are taught in the liberal arts (and elementary math classes) that are lead them astray when they approach mathematical defitions. The ones that come to mind are

1. "When you read a definition, rephrase it in your own words."
2. "To understand the meaning of a statement, analyze the meaning of the individual terms in it."

It does help to rephrase some mathematical definitions, but the way it should be done is:
"When you read a definition, rephrase it as a logical equivalence between two statements."

Many people, including some mathematicians, have the philosophical view that mathematical objects exist (in some sense) independently of the words we write about them. This leads some heated exchanges in the bull-session type of threads on the forum that deal with controversies such as "Is multiplication repeated addition", "Is .999... = 1" , "Are dx and dy actual numbers?". Having one's own private concept of mathematical objects is a permissible philosophy of life, but it can't be employed in a proof. One part of the curriculum should be to teach students that the mathematical approach to definitions (in a course involving proofs) must be legalistic and hair splitting - regardless of the fact that this may be repulsive to free spirited independent thinkers. Students should be warned against rephrasing mathematical definitions using their own words.

Students are taught from an early age (even in math classes) that we must define "terms" and their training teaches that a definition applies to a nouns or a short phrase. This is a misleading way to approach mathematical definitions. For example the definition of "The limit of f(x) as x approaches a is equal to L", can't be analyzed by trying to interpret "x approaches a" in any literal manner. The phrase "x approaches a" suggests a process taking place in time or in steps. It's a nod to historical and intuitive ways of thinking about limits but it doesn't convey anything that could be used in a proof.

Students should be discouraged from approaching mathematical definitions as definitions of "terms". Instead they should be taught that a definition (however casually it is phrased in the textbook) must interpretable as a logical equivalence between two statements. For example, in calculus, we don't define "limit", we define an entire statement: "The limit of f(x) as x approaches a is equal to L". This statement is defined by saying it is equivalent to another statement "For each epsilon greater than zero ... ". When definitions are phrased as logical equivalences, it is clearer how to use them in proofs.

Rewriting definitions as logical equivalences is a good elementary exercise and there are many elementary exercises that involve interpreting definitions. Even an exercise in rephrasing something as simple as "A group is a set of ..." to "G is a group if and only if..." is useful. This type of exercise may also highlight the sloppy practice some textbooks have of saying "V is a vector space if..." when they mean "V is a vector space if and only if...".

The important thing is that the above points about definitions should be emphasized throughout the entire course. (and I happen to think that repetition is the only effective form of emphasis.) Let me know what you think.

I agree with daveb that using logic is a critical skill, but I'll post on that later. First, there is more to say about definitions - special situations that cause students trouble.

 

[matqkks]

I have repeated in class that students should learn the definitions and be able to apply them in any context and that they should be at their finger tips.
I have not tried your equivalent statements in terms of symbols apart from the obvious limit connection to epsilon. Is layout of the proof critical for their understanding?

 

[JSuarez]

Well, I'm not going to offer any positive suggestions or even try to cheer you up so, if you want, you may stop reading right here.

I taught for almost 20 years; taught Linear Algebra and Calculus, pretty much like you are doing now and also more advanced courses, in Logic, Set Theory and Differential Geometry. Furthermore, I taught mathematics (and physics) students, but also "gas-station" courses for students in other fields (engineering, economics, etc.).

I quit years ago, not because I didn't like teaching, but for reasons that are irrelevant here.

Now, regarding your situation, one that I also found myself many times, I come to the conviction that very few students are able to learn how to do mathematical proofs (or, as you say, even start in the wrong direction). You yourself described the situation very accurately:

Even if they do understand them in the class they cannot reproduce them in the final examination. Majority of the students avoid doing the proof questions and concentrate on the numerical or technique application of mathematics.

From my experience(s), from years past when I was young and foolish (at least one of them is no longer true), thought that I could devise some teaching scheme that would open up their heads to the "beauty" of mathematics and the virtues of "rigorous" thinking, I will say this: nothing will work. One more thing, I also though (again, foolishly) that, having studied engineering before mathematics, I was in a very good position to show them how "useful" proof-style thinking would be to them; for the life of me, I am now really unable to explain how could I have been so damn daft!

So, at the peril of contradicting myself, what can I tell you?

First, if you are teaching mathematics students (from little hints in your posts, I don't think you are), you must lean on them. Hard. No more advice on this case: a mathematics student that is unable to learn to do Linear Algebra proofs by himself has made an error of choice; he/she should be made to correct it as soon as possible.

Second, the "gas-station" case. Here, I now defend a position that, years ago, I would repudiate intensely; but I know better now. It is simply this: these students don't have any use for mathematical proofs, and these should be taught only as some sort of option (or not at all) that doesn't impact in their completion of the course (of course, they may be used to separate the ones that really deserve the grade or, like in the above, made an error of choice; they should be in mathematics).

 

[Stephen Tashi]

matqkks,

I have not tried your equivalent statements in terms of symbols apart from the obvious limit connection to epsilon. Is layout of the proof critical for their understanding?

I don't know what you mean by layout of the proof. You haven't said what style you expect students to use when writing proofs. To write a "high class" proof, one that is a literate essay, it is critical that students know how to phrase each definition as a logical equivalence between two statements. If your students write proofs in a very abbreviated style then who knows what's important! Those styles tend to have their own rules. Students can get away with doing a step and stating the reason as "definition of..." without really understanding how the definition implies the step. There merely need to have a strong intuitive feeling that the definition is associated with manipulation they did. (Perhaps, for elementary courses, having that intuitive feeling is an adequate level of comprehension.) Since I'm using this thread to develop a document of advice about proofs, I'll continue to prattle on about definitions.Definitions vs Assumptions, A Difficult Distinction

Students are taught that mathematics involves definitions, assumptions and theorems. Instructors expect students to understand the distinction among these 3 types of statements. Unfortunately, presentations of mathematics blur the distinctions. For example, elementary treatments of the real numbers may simply present a list of "properties" and not distinguish which properties are defintions, which are assumptions and which are theorems. (We should make it clear to students that they are expected to have difficulty with approaches to mathematics that use legitimate logic. Their training in elementary mathematics does not use it!)

Even in advance mathematical presentations, it is difficult to make a distinction between a definition and an assumption. For example, suppose we are writing a proof involving a mathematical object ( e.g. the real numbers, the complex numbers, a ring, a group etc. ) and we wish to justify the fact that the multiplication operation is associative ( i.e. (ab)c = a(bc) ). The natural way to think of the justification is that it is an assumption ( or equivalently a "law", "axiom", "postulate" or some other synonym for "assumption"). However, in most cases, the fact that multiplication is associative is stated as part of the definition of the mathematical object. For example, the definition of a group might be stated as "A group (G,*) is a collection of objects G together with an associative binary operation '*' defined on G. There is an identity element and ... etc."

In a rigorous presentation of a mathematical topic, stand-alone assumptions (those not embedded in definitions) are merely convenient abbreviations for complicated statements. After all, suppose you are writing a presentation of a mathematical object X (e.g. a group, a ring, a topological space) and you introduce Assumption 1. How shall the reader understand the scope of what Assumption1 states? If the reader assumes it is always true when discussing objects of type X, it effectively becomes part of the definition of what an object X is. If the reader is not suppose to assume Assumption 1 is always true, then he must be informed when Assumption 1 is to be assumed! This can be done in various ways. For example, a theorem might say "If x is an object of type X and x satisfies asssumption 1...".
Or we might introduce new terminology for a special type of object X which, by definition, satisfies Assumption 1. (For example, 'group' can be specialized to "abelian group", "topological space" can be specialized to "hausdorff space", "vector space" can be specialized to "finite dimensional vector space".)I think it would be helpful to discuss particular issues involving definitions that students have trouble with. I can think of these: equality, existence and uniqueness, isomorphism. What others should be discussed?

I don't know whether it is better to discuss logic before getting to those specifics.