Teaching Students to Prove Theorems Session for Project NExT Fellows Presenter: Carol S. Schumacher Kenyon College and The Legacy of R. L. Moore Project Mathfest, 2007 San Jose, CA

Some Things I Learned the Hard Way

“That’s obvious.” To a mathematician it means “this can easily be deduced from previously established facts.” Many of my students will say that something they already “know” is “obvious.” For instance, if I give them the field axioms, and then ask them to prove that they are very likely to wonder why I am asking them to prove this, since it is “obvious.”

What is a definition? To a mathematician, it is the tool that is used to make an intuitive idea subject to rigorous analysis. To anyone else in the world, including most of your students, it is a phrase or sentence that is used to help understand what a word means.

For every  >0, there exists a  >0 such that if... ? ??

What does it mean to say that two partially ordered sets are order isomorphic? The student’s first instinct is not going to be to say that there exists a function between them that preserves order!

As if this were not bad enough we mathematicians sometimes do some very weird things with definitions. Definition: Let  be a collection of non-empty sets. We say that the elements of  are pairwise disjoint if given A, B in , either A  B=  or A = B. WHY NOT.... ??? Definition: Let  be a collection of non-empty sets. We say that the elements of  are pairwise disjoint if given any two distinct elements A, B in , A  B= .

Great Versatility is Required We have to be able to take an intuitive statement and write it in precise mathematical terms. Conversely, we have to be able to take a (sometimes abstruse) mathematical statement and “reconstruct” the intuitive idea that it is trying to capture. We have to be able to take a definition and see how it applies to an example or the hypothesis of a theorem we are trying to prove. We have to be able to take an abstract definition and use it to construct concrete examples. And these are different skills that have to be learned.

Cultural Elements We have skills and practices that make it easier to function in our mathematical culture. We hold presuppositions and assumptions that may not be shared by someone new to mathematical culture. We know where to focus of our attention and what can be safely ignored.

What?... Where?

It is not what I do, but what happens to them that is important. Whenever possible, I substitute something that the students do for something that I do. Total Immersion

Small scale stuff to enhance lectures. Activities that help students become familiar with a with a new definition. Students work on proving theorems in small groups while you circulate around the classroom helping out. Have students present results of their work to each other. Projects and Activities instead of lectures. GAP projects in abstract algebra: conjugating permutations. (Google: Judy Holdener Gap) There are many ways to do this

Sorting out the Issues Equivalence Relations Partitions Equivalence Relations We want our students to understand the duality between partitions and equivalence relations. We may want them to prove, say, that every equivalence relation naturally leads to a partitioning of the set, and vice versa.

Relation on A Collection of subsets of A. The usual practice is to define an equivalence relation first and only then to talk about partitions. Furthermore, there is a lot going on. Many students are overwhelmed. They don’t know how to focus their attention on one piece at a time. Are we directing our students’ attention in the wrong direction?

a n  L means that   > 0  n    d(a n, L) < . a n  L means that   > 0  N    for some n > N, d(a n, L) < . a n  L means that  N  ,   > 0   n > N, d(a n, L) < . a n  L means that  N   and   > 0  n > N  d(a n, L) < . Students are asked to think of these as “alternatives” to the definition of sequence convergence. Then they are challenged to come up with examples of real number sequences and limits that satisfy the “alternate” definitions but for which a n  L is false. I usually have the students work on this exercise in class, perhaps with a partner.

Scenario 1: You are teaching a real analysis class and have just defined continuity. Your students have been assigned the following problem: Problem: K is a fixed real number, x is a fixed element of the metric space X and f: X  R is a continuous function. Prove that if f(x) > K, then there exists an open ball about x such that f maps every element of the open ball to some number greater than K. One of your students comes into your office saying that he has "tried everything" but cannot make any headway on this problem. When you ask him what exactly he has tried, he simply reiterates that he has tried "everything." What do you do?

Scenario 2: You have just defined subspace (of a vector space) in your linear algebra class: The obvious thing to do is to try to see what the definition means in  2 and  3. You could show your students, but you would rather let them play with the definition and discover the ideas themselves. Design a class activity that will help the students classify the linear subspaces of 2 and 3 dimensional Euclidean space. (You might think about "separating out the distinct issues.”) Definition: Let V be a vector space. A subset S of V is called a subspace of V if S is closed under vector addition and scalar multiplication.

... closure under scalar multiplication and closure under vector addition...

Scenario 3: Your students are studying some basic set theory. They have already proved De Morgan's laws for two sets. (And they really didn't have too much trouble with them.) You now want to generalize the proof to an arbitrary collection of sets. That is..... The argument is the same, but your students are really having trouble. What's at the root of the problem? What should you do?

Scenario 4: A very good student walks into your office. She has been asked to prove that the function is one to one on the interval (-1,  ). She says that she has tried, but can't do the problem. This baffles you because you know that just the other day she gave a lovely presentation in class showing that the composition of two one-to-one functions is one-to-one. What is going on? What should you do?

Scenario 5: Your students are studying partially ordered sets. You have just introduced the following definitions: Definitions: Let (A,  ) be a partially ordered set. Let x be an element of A. We say that x is a maximal element of A if there is no y in A such that y  x. We say that x is the greatest element of A if x  y for all y in A. Anecdotal evidence suggests that about 71.8% of students think these definitions say the same thing. (Why do you think this is?) Design a class activity that will help the students differentiate between the two concepts. While you are at it, build in a way for them to see why we use “a” when defining maximal elements and “the” when defining greatest elements.