1. What we say / what they hear Culture shock in the classroom

2. Mathematical Culture We hold presuppositions and assumptions that may not be shared by someone new to mathematical culture.

3. 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 our students, it is a phrase or sentence that is used to help understand what a word means.

4. What does it mean to say that two partially ordered sets are order isomorphic? Most students’ first instinct is not going to be to say that there exists an order- preserving bijection between them!

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

6. 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= . ???

7. Mathematical Culture We know where to focus our attention for maximum benefit and we know what can be safely ignored.

8. What?... Where?

9. Helping our students focus Example: Equivalence Relations Equivalence Relations Partitions We want our students to understand the duality between partitions and equivalence relations. We want them to prove that every equivalence relation naturally leads to a partitioning of the set, and vice versa.

10. There is a lot going on in this theorem. Many of our students are completely overwhelmed. Us Our students!

11. Equivalence Relation on A Partition of A Every partition of a set A generates an equivalence relation on A. & Every equivalence on A relation generates a partition of A. Sorting out the Issues

12. Relation on A Collection of subsets of A. Every collection of subsets of A generates a relation on A. & Every relation on A generates a collection of subsets of A. Sorting out the Issues

13. 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? And...It’s not just about logical connections Motivation for defining equivalence relation?

14. Mathematical Culture We have skills and practices that make it easier to function in our mathematical culture.

15. 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. A great deal of versatility is required.... And none of these are even talking about proving theorems!

16. First Day Out Ed Burger Mike Starbird Carol Schumacher--- “the thought experiment” What are the goals of each instructor? Are there common elements/goals ? The impermissible shortcut

17. Logical Structures and Proof Proving a statement that is written in the form “If A, then B.” Disproving a statement that is written in the form “If A, then B.” Existence and Uniqueness theorems Other useful ideas: e.g. “If A, then B or C.” Beyond counterexamples: Negating implications!

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

19. Breaking the Impasse Definition: a n  L means that for every  > 0, there exists N  ℕ such that for all n > N, d(a n, L) < . Given any tolerance there is some fixed position beyond which a n is within that tolerance of L In beginning real analysis, we define the convergence of a sequence:

20. a n  L means that   > 0  n  ℕ  d(a n, L) < . a n  L means that   > 0  N  ℕ  for some n > N, d(a n, L) < . a n  L means that  N  ℕ,   > 0   n > N, d(a n, L) < . a n  L means that  N  ℕ and   > 0,  n > N  d(a n, L) < . Don’t just stand there! Do something. Make it “real”

21. 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 theorems to prove, so they develop intuition for the definition/new concept. Separate the elements. even if they are not particularly significant!

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

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

25. What they say/what we hear Listening to our students (Sometimes) hearing what they mean instead of what they say

26. 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. A great deal of versatility is required....

27. 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?

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

29. “That’s obvious” To a mathematician this 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.”

30. I find that it is helpful to stipulate two things: First: students don’t begin by proving the deep theorems. They have to start by proving straightforward facts. Second: 0  x = 0 is a sort of ‘test’ for the axioms. It is so fundamental, that if the axioms did not allows us to prove it, we would have to add it to our list of assumptions. Then we make an amazing observation: The field axioms discuss only additive properties of 0, but because addition and multiplication are assumed to interact in a certain way (distributive property), this multiplicative property of 0 is obtained “for free.” 0  x = 0 holds because nothing else is possible.

31. If our students see this they have taken a cultural step toward becoming mathematicians. Our students (along with the rest of the world) think that the sole purpose of proof is to establish something as true. And while this is the case, sometimes proofs can help us understand deep connections between mathematical ideas.

32. In what sense is that teaching? Susie is a pretty good student. But “work on these problems and we will talk about them next time” is a little nebulous for her. She thinks student presentations are a waste of everybody’s time. (She may secretly believe that I don’t lecture because I’m lazy or unprepared.) She is conditioned to respond to what I say and she doesn’t believe that her work starts (or even can start) before she understands the material. She is wondering when I will get around to actually teaching her something! Basically, Susie doesn’t get it.

33. 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.

34. 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?

35. 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 ℝ 2 and ℝ 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.

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

37. 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?

38. 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?

39. 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.