Thursday, February 11, 2010

Project Wrapup

I want to share my thoughts on how my recent in-class project went. Basically I still think it's a reasonable project. It certainly needs some tweaking though. I'm kinda excited to try it again.


Class began and I started as close to on time as possible. I had already posted the assignment, and some students had seen it. I did a quick run-through of the setup, explaining the two parts (write your own problems, with solutions, and solve a different groups problems). As students divided themselves into groups I distributed printed copies of the assignment. Students got in and settled and started working pretty quickly, which was nice. Throughout, it seemed like students were doing a reasonable job staying focus on the task. I think the time pressure helps with that.

I let students work without any interruption for a few minutes. One student came to me and asked if the problem he had written was ok. It looked quite similar to textbook problems, which I told him, and he went away a bit disappointed. After a few minutes I started wandering around to each group, asking how things were going. Many of the groups were perfectly happy, and didn't have any questions for me. A few groups were having a hard time coming up with things. I tried to indicate how one might make up a new problem. For example, make up crazy rules a bank might use for some sort of account. Or: look at a textbook problem. They solve an equation for one of the 4 variables in our formula. Change the problem around so that a different variable is the one to be found. Sometimes this might not make a particularly new problem, but I think sometimes it could. The students I gave these suggestions to seemed to think they could make progress, and got back to work.

A few other students showed me problems that looked like textbook problems. One told me that it was different, and I challenged him to explain how. He noted that his problem tested the interpretation of the answers from some textbook questions. I can't really argue. He had 3 problems, 2 of which looked like textbook problems, and the third asked for the interpretation. Fair enough I guess.

At the half-hour mark, when students were supposed to be done writing problems, nobody was. I talked to them about it, and it was pointed out that probably solving problems that are already written would be quicker than writing your own. So the second half should go quicker. Makes sense. I told them all to take 15 more minutes on writing their own questions and solutions.

Just after the first group finished, it occurred to me that I should emphasize that they are all getting the same grade. So they should make sure to double check each other's work. I think many groups were divvying up work, so that each person wrote a solution to one problem. This is fine, but I think it is important that all of the group members double-check this work too. I made an announcement about this.

We were running out of time. With 25 minutes, groups were mostly starting to finish, and I re-distributed problems to groups that were done the first part. Since the time wasn't working out as I had anticipated, I told them to solve 2 of the 4 problems they were given.

With about 15 minutes to go, all but 2 groups had begun the second part of the assignment, solving another groups problems. One of the two remaining groups finished (group A), and the other was done 3 of 4 problems (group B). I took the 3 problems over to the group A to start the second part, but let group B continue working on their own problems. Group A eventually finished, about 5 minutes over time, and group B didn't have any time to work on group A's problems when they had finished writing their own. I'm still trying to decide what to do about that. [I assigned them some of the more challenging problems I had written]

While students were working on problems other groups had written, I decided it might be interesting to have them rate the creativity of the problems they were given. I told them to rate each problem they were given with either a 0 (this is a textbook problem), 1 (kinda new), or 2 (terribly interesting). I'm not sure how seriously they took this task, I have not yet (as I write this), looked at the work that was turned in. [Mostly I think this turned out ok, though some 2's were pretty questionable]

As students were finishing, I told them (a) I wanted feedback on the project, what they liked/disliked, how it could have been better, and (b) that I'd probably like to try this again, so to keep track of questions they think of outside of class.

After Class

I've been trying to write daily (we only meet twice a week) blog posts for my class, on whatever we talked about that day. I don't think any of the students are reading them, but I could be wrong. Today I posted questions I had dreamt up, to give students some sort of idea what I had in mind. Some of the questions I had though up before class, others were inspired by discussion in class.

As I was writing up my questions, it occurred to me that several of them weren't quite as original as I had originally (bam!) thought. I could see how to translate my problems into textbook problems. I do still feel like there would be a translation step though, and I guess that's part of the game. Or perhaps I'm fooling myself.

I emailed the class, specifically encouraging them to read my post, and also to provide feedback. Hopefully the reminder generates some feedback.

Initial Reaction

I mis-judged timing. I had originally given more time for doing other groups problems than writing one's own. Definitely backwards. Probably we could do 45 minutes writing problems, and 30 answering others'. Also, I think I need to be firm on the deadline if I try this again. At 45 minutes, you will have to give me your paper, and will lose points for not having written enough problems (well, you just don't earn the points you would have). Of course, that means the group that gets your paper isn't being graded out of as many problems, which must be accounted for. Hmmm.

I think trying this again, students may have a better sense about the project. Writing problems might come easier. Hopefully, then, timing would work out a little better. Perhaps showing students a list of questions before-hand would have been a good idea.

I didn't plan enough about organizing the work that got turned in. I'm sort of dreading looking at the pile. Here's how I think I'd organize it next time: Each group is assigned a number. There are then 4 things they turn in to me by the end of class:
  1. The list of problems they wrote. This should have "Written by group N" on it. It should also have "Solved by group M", the group that gets these problems.
  2. The list of solutions to their own problems. This should have "Written by group N" on it.
  3. The list of solutions to the problems they were given. This should have "Solutions written by group N, for group M's problems" on it. These solutions should not be on the same paper with the questions, since it mucks up re-distributing papers (more on this below).
  4. A paper with the group number and names of all of the group members. I suppose this information could just go on, say, the list of solutions a group writes for their own problems
I need to bring a stapler to class.

I sorta like the idea of getting students to rate the problems they are given for creativity. I can't quite decide what to do about giving points based on those ratings, as they'd be pretty easily gamed. One option I came up with: Have students rate the problems they were given on the 0-2 scale above. As I'm grading, type up all the problems that earned a 2. Distribute this list, without any identifying marks, back to the students, and have them pick their favorite n, say (outlawing voting for your own (more bookkeeping for me, but doable as long as I have them write their name on the paper)). Any problem that gets more than m "favorites" earns the authoring group p points (1, likely).


As I organized papers, and got to a point I could start grading, I realized that I could extend this project to have the students do the grading. Group A gives their answers to Group B, and when Group B is done, Group A grades the solutions. This could be a valuable exercise for the students, seeing how their questions were interpreted, perhaps seeing other ways to solve questions they had designed. Of course, it leads to problems about what grade gets written down in the grade book... suggestions?

As I was grading, I realized that students can game the grading system by writing easy questions. This guarantees 3 points for solutions to each problem the group writes, even if they lose a point or two for writing uninteresting problems (I was pretty relaxed about taking points off for this). Perhaps this can be corrected for by having other groups rate "originality" first, (maybe "difficulty" too) and then base the score solutions are worth out of that grade?

Maybe take interesting problems in to class, have everybody do them, and talk about answers. Also problems with difficult wording.

On re-distributing papers: Group A did Group B's problems. Giving A's solutions back to A, they probably won't be able to see B's questions, to look back at.

Student Feedback

At this point, I still haven't gotten much. One said they enjoyed the project, even if making up problems was difficult. Another suggested just writing up questions individually and exchanging papers with partners; that working in larger groups was hard. Also the timing issues were pointed out.

Current Thoughts for Next Time

Be more organized about what names I need to see on which papers. I found that the paper-shuffling aspect of grading was easier for groups where group A did group B's problems and vice-versa. I'm not sure that it matters too much though.

Perhaps break this project into two class period. In the first, groups will meet to try to create interesting problems that they could solve (though I won't ask for solutions just yet). The problems will be re-distributed around the room, and groups will rate the originality and difficulty of the problems they were given. I will then gather up all of the problems, and we'll end class. Before the next class, I'll go through all the problems, find all those that earned good originality/difficulty ratings, and compile them into a list. In class the next time, all of the students will work those problems, maybe just working with a neighbor. Then we can spend time in class discussing solutions to the problems, talking about different interpretations of questions, what makes questions well-written, etc. There's issues here about turning in solutions - if students just wait until we talk about solutions in class, they can just copy those.


I'd really like to know what you think about any or all of this. I know it's rather a lot to read through, sorry.

Wednesday, February 10, 2010

Approximating Functions of Shapes

Every now and then, I end up writing about my research at a not-too-technical level. Here's my latest attempt:

The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible. If one flexible shape can be flexed and twisted and not-too-drastically mangled into another flexible shape, topology deems them to be the same. It becomes much more difficult, then, to tell if two shapes are different. The question of topology is, ``Given two flexible shapes, how can we tell if they are the same?''

In geometry, much time is spent studying shapes like triangles and quadrilaterals. If you draw one triangle on one sheet of paper, and another triangle on another sheet of paper, then there is an easy way to tell if those two triangles are the same. Namely, shift the papers around, possibly flip one or the other over, rotate them, and see if you can get the triangles to line up. If you can, then the triangles are ``the same'' (congruent). If you can't line up the triangles (and have put forth enough effort), then the triangles are not the same. Similarly, you could draw a square on one paper, a triangle on the other, and try to line those shapes up. Of course, you will never succeed, because the two shapes will always have a different number of sides.

In topology, we might consider the same scenario of triangles and squares, but instead of using ordinary paper, we would draw our shapes on a sheet of flexible rubber (an inflatable balloon, say). Indeed, topology is often referred to as ``rubber-sheet geometry.'' Now that the surface is flexible, by pulling it in the appropriate directions one can get the square to look like a triangle. Or either shape to look like a circle. To a topologist, this means that all of these shapes are ``the same'' (homeomorphic). The topologist still has some rules in place, though. You are not allowed to cut holes in the balloon, or glue parts of the balloon together.

With all of the additional flexibility, we have a harder time telling two shapes apart. We can no longer rely on the number of sides, or the lengths of sides. Other methods must be found.

It turns out that a good way to get a sense of how similar two shapes are is to understand how the shapes are related to yet another shape. Suppose we have two flexible shapes, called Shape 1 and Shape 2. One might try to obtain information about Shape 1 in isolation, and compare that to information obtained about Shape 2 in isolation. However, as the shapes are flexible, there's only so much we can tell by looking at each shape independently. Instead, we introduce another flexible shape, Shape 3. Generally Shape 3 is a shape that has been much studied, like a circle. Now we ask, ``How many ways can Shape 3 be put into Shape 1?'' and, ``How many ways can Shape 3 be put into Shape 2?'' We call a single way of putting Shape 3 into Shape 1, if it is reasonably well-behaved, an ``embedding.'' If the embeddings of Shape 3 into either of the two original Shapes are similar, we might decide that the two shapes themselves are fairly similar. We can then replace Shape 3 with Shape 4, repeat the question, and get an even finer appreciation for how similar Shapes 1 and 2 are.

The entire collection of embeddings of Shape 3 into Shape 1 can all be gathered up and reasonably interpreted as making up yet another shape. However, it can be a fairly difficult shape to get a good sense of. Instead, one might ask if there is a way to approximate this shape. Perhaps an initial approximation can be attained, and then that approximation can be refined, and refined further, giving a better and better understanding of the complicated shape we originally sought. Using geometry as an example, one might approximate a circle by a square, and then refine the approximation to an octagon, and carry on the process of adding more sides to the approximating figure. This was the method used, historically, for calculating the area of a circle.

But the idea of approximating the shape of the collection of embeddings can be further generalized. If we have a strong handle on a particular shape, say S, then our first step in understanding any new shape that comes along might be to find all the ways S can be put into the new shape. We have, then, described a function. Given a shape, X, we can ask for the shape of ways that S can be embedded in X. We might denote this function by E(X), and call it the embedding function for S. Now instead of understanding a particular value, E(X), of this function, we might ask about understanding the function E as a whole. While the function itself is likely complicated, we can try to approximate it, as before.

In my research, I study a family of embeddings functions. The functions are based on the ``Euclidean spaces'' of various dimensions: a point, a line, a plane, three-dimensional space, and on to higher dimensions. From a topologist's perspective, these are the easiest shapes there are, but they can be used to build up nearly all of the other shapes worth studying, so are of fundamental importance. Instead of just thinking about one such space as the basis for my function, I might make a collection of them; say, 3 lines and 2 planes. I only ask about E(X) when X is, itself, a single Euclidean space.

It might seem that by restricting my study to particularly nice shapes S and X, and particularly nice ways to put S into X, I have eliminated any hope of doing anything useful. However, this is not the case. An important class of shapes, called manifolds, are built up from Euclidean spaces by gluing them together. By obtaining a particularly nice description of embedding functions using my restricted shapes S, there are known ways to assemble them into the embedding functions for more general manifolds.

Previous work has found nice ways to describe embedding functions for some of the cases I am considering. If the base space, S, is a finite collection of points, then spaces of embeddings are what are known as ``Configuration Spaces,'' and these detect one type of behavior for embeddings. Alternatively, if S has just a single space, but of any dimension, the spaces of embeddings are detecting a different sort of behavior. Nice descriptions for the embedding functions in both of these cases are known. It has been my task to unify the two descriptions, allowing for an understanding of embedding functions when the known space, S, is a collection of any number of Euclidean spaces.

Update 20100225: There's a somewhat different version of this on my other blog.

Challenge Your Neighbor

I think I'm going to try a new exercise in my financial math class tomorrow. We've just finished Chapter 1, which is on simple interest (yes, a whole chapter on simple interest). If you are visualizing how the following project might go, here are some parameters: 45 students (we'll see how many actually show up), 75 minutes. If you've done a project similar to what follows, or even if you haven't, and you have some feedback for me (things to watch out for), I'd love to hear about it (before noon Thursday :)). Below is the assignment I'll be giving my students:


The goal of this project is for you to write your own questions to challenge your classmates, and help review chapter 1 material.

Outline: In groups of 5, you will be given half an hour to create a list of 4 ``interesting'' questions that cover chapter 1 material. The questions must not use formulas or concepts that are not covered in chapter 1. Your questions must not be re-writes of textbook problems, simply obtained by changing numbers or dates. You must be able to solve the problems you create (with the aid of a computer if the algebra is difficult). The 4 questions may all be based around a central scenario.

You will then exchange your list of problems with another group, and, in turn, will be given the problems created by a different group. You will have the remainder of the class period to produce solutions to the problems you have been given.

Specifics: In the first half hour, you must produce 2 papers, both containing the names of all group members. One paper should contain your list of questions. This paper will be given to another group in the second part of this assignment, so should only contain the questions you have written. Questions must be written well enough to understand without further explanation. Your second paper should contain well-written solutions to the problems you have written. To save time, you do not need to copy the text of the questions to this second page.

In the second portion, you must write solutions to the challenge questions you have been given. You may use the paper you have been given. You must write the names of all your group's members on this paper. As it will have two sets of names on it (the problem authors, and the solution authors), insure that the two sets are clearly identified.

Grading: Unless it is clear that a group member is not contributing, all group members will receive the same grade for this assignment. You may earn 2 points for each problem you write, up to 8 points total. Points may be deducted if the problem is not relevant or is poorly written. The write-ups for the problems you author, and the problems you are given, will all be graded out of 3 points.

You will not earn extra points for writing extra problems. You may have points deducted if the problems you author are not distinct from textbook problems.