Project NExT Workshop Mathfest, 2010 Pittsburgh, PA What is the Definition of Definition? What is the Definition of Definition? And other mathematical cultural conundrums Carol S. Schumacher Kenyon College

My teaching improved a lot when I stopped thinking so much about teaching and started thinking more about learning.

Teacher as Amateur Cognitive Scientist How do we get our students to think and behave like mathematicians?

Teacher as Amateur Cognitive Scientist Getting into our students’ heads. – How do they learn? – And (thinking cognitively) what do they need to learn?

Teacher as Amateur Cognitive Scientist Getting into our own heads: How do we operate as mathematicians?

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. These are different skills that have to be learned. A great deal of versatility is required....

And we aren’t even talking about proving theorems yet! A great deal of versatility is required....

Let’s try an experiment... Q: Who is non-orientable and lives in the ocean? A: Möbius Dick Q: Why is the contour integral around Western Europe zero? A: Because all the Poles are in Eastern Europe! Q: When did Bourbaki stop writing books? A: When they found out Serge Lang was just one person. Mathematics is a Culture

??? Chasm? What Chasm ? Culture is, by its very nature, completely unconscious

Cultural Elements We hold presuppositions and assumptions that are unlikely to be shared by a student who is new to mathematical culture. We have skills and practices that make it easier to function in our mathematical culture. We know where to focus of our attention and what can be safely ignored.

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 bijection 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= . ???

“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, they will readily agree that it is “obvious” that the sequence 1, 0, 1, 0, 1, 0,... fails to converge. We must be sensitive to some students’ (natural) reaction that it is a waste of time to put any work into proving such a thing.

Our students (and most of the rest of the world!) think that the sole purpose of proof is to establish the truth of something. The Purpose of Proof

First: people don’t begin by proving deep theorems. They have to start by proving straightforward facts. Second: this is a sort of ‘test’ of the definition. It is so fundamental, that if the definition did not allow us to prove it, we would have to change the definition. I Stipulate Two Things

That’s Obvious, too! If I give my students the field axioms, and ask them to prove that they are very likely to wonder why I am asking them to prove this, since it is “obvious.”

The Purpose of Proof Sometimes proofs help us understand connections between mathematical ideas. If our students see this they have taken a cultural step toward becoming mathematicians.

What?... Where?

There is a lot going on. Most of our students are completely overwhelmed. Us Our students!

Karen came to my office one day…. She was stuck on a proof that required only a simple application of a definition. I asked Karen to read the definition aloud. Then I asked if she saw any connections. She immediately saw how to prove the theorem. What’s the problem?

Charlie came by later... His problem was similar to Karen’s. But just looking at the definition didn’t help Charlie as it did Karen. He didn’t understand what the definition was saying, and he had no strategies for improving the situation. What to do?

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   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. 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... Sorting out the Issues Vector Subspaces

Sorting out the Issues Equivalence Relations Equivalence Relations Partitions 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. Are we directing our students’ attention in the wrong direction? ??? Chasm? What Chasm ?

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.

Exploratory Exercises Suppose that f : A  B and g: B  C are functions. For each of the following statements decide whether the statement is true (if so, give a proof) or false (if so, give a counterexample). In the cases where the statement is false, decide what additional hypothesis will make the conclusion hold: 1.If g  f is one-to-one, then f is one-to-one. 2.If g  f is one-to-one, then g is one-to-one. 3.If g  f is onto, then f is onto. 4.If g  f is onto, then g is onto.

Discovering Trees Consider what happens when you remove edges from a connected graph (making sure it stays connected). Your group’s task is to look at example graphs and remove edges until you have a graph. Group A: “with no circuits.” Group B: “ that is minimal in the sense that if you remove any more edges you disconnect the graph.” Group C: “in which there is a unique simple chain connecting every pair of vertices.” Can it always be done? What happens if you take the same graph and remove edges in a different order?

Impasse! What happens when a student gets stuck? What happens when everyone gets stuck? How do we avoid THE IMPERMISSIBLE SHORTCUT ?

Definition: a n  L means that for every  > 0, there exists N   such that for all n > N, d(a n, L) < . In beginning real analysis, we typically begin with sequence convergence: Breaking the Impasse

a n  L means that for all  > 0 there exists n  ℕ such that d(a n, L) < . a n  L means that for all  > 0 there exists N  ℕ such that for some n > N, d(a n, L) < . a n  L means that for all N  ℕ, there exists  > 0 such that for all n > N, d(a n, L) < . a n  L means that for all N  ℕ and for all  > 0, there exists n > N such that d(a n, L) < . Don’t just stand there! Do something. Students are asked to think of these as “alternatives” to the definition. 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.

Pre-empting the Impasse Teach them to construct examples. If necessary throw the right example(s) in their way. Look at an enlightening special case before considering a more general situation. When you introduce a tricky new concept, give them easy problems to solve, so they develop intuition for the definition/new concept. Separate the elements. Even if they are not particularly significant!

But all this begs an important question. Do we want to pre-empt the Impasse?

Precipitating the Impasse Impasse as tool Why precipitate the impasse? The impasse generates questions! Students care about the answers to their own questions much more than they care about the answers to your questions! When the answers come, they are answers to questions the student has actually asked.

Precipitating the Impasse Impasse as tool Why precipitate the impasse? The impasse generates questions! The intellectual apparatus for understanding important issues is built in struggling with them. At least as importantly, when students generate their own questions, they understand the import of the questions.

... the theory of 10,000 hours: The idea is that it takes 10,000 hours to get really good at anything, whether it is playing tennis or playing the violin or writing journalism. I’m actually a big believer in that idea, because it underlines the way I think we learn, by subconsciously absorbing situations in our heads and melding them, again, below the level of awareness, into templates of reality. At about 4 p.m. yesterday, I was working on an entirely different column when it struck me somehow that it was a total embarrassment. So I switched gears and wrote the one I published. I have no idea why I thought the first one was so bad — I was too close to it to have an objective view. But I reread it today and I was right. It was garbage. I’m not sure I would have had that instinctive sense yesterday if I hadn’t been struggling at this line of work for a while. Written by David Brooks In one of his NYTimes “conversations with Gail Collins

Morale: “Healthy frustration” vs. “cancerous frustration” Give frequent encouragement. Firmly convey the impression that you know they can do it. Students need the habit and expectation of success--- “productive challenges.” Encouragement must be reality based: (e.g. looking back at past successes and accomplishments) Know your students as individuals. Build trust between yourself and the students and between the students.