Blog Home : April 2004
Promised Link
One of the best M&M project teams gave me an electronic copy of their report. You can find it here.
Today was a little weird, about half of the 9AM class was missing, but all of the 10AM class was present.
Where is everybody?
M&M Reports
The M&M reports were handed in on Wed. I read and graded them yesterday. Overall, I was pleased. As soon as I
get an electronic copy of some of the best ones, I'll post a link. All groups managed to reasonably address the question
of calculating the number of M&M's in the jar by using a model for a single M&M and extracting the packing density
information from the Science articles. Some groups went much further with this, devising two or three other methods
of computing the number of M&M's, and comparing the methods. Several groups took the problem to the next level
and realized that the Science articles contained enough information to answer questions about jelly-beans, pills, etc.
Most groups did the Viviani-like intersection problem correctly, althought two or three groups had trouble computing the
arc length. A few notes on doing this problem in the future:
M&M Draft One
My classes handed in the first draft of their M&M projects on Wed. Originally, I intended to give
them back to them yesterday and ask for a revised copy by today. After reading them I decided
to meet with each group today in class to discuss their paper and allow them to hand in a revised
copy next Wed. Overall, the content of the projects was good, in some cases even excellent. The
writing was generally terrible. Flaws ranged from poor grammar "we got measurements" to over-flowery attempts to bamboozle the reader. Today in class I gave them a copy of an abstract and introduction I'd written to use as a model. I had told them to use the articles they were reading as models for the language and style. I guess that message got lost in the noise. Other common errors: no figure captions, no table captions, figures and tables not centered, equations not numbered, poor citations, and wordy writing. Next time I should prepare a tip sheet for them to be handed out with the assignment. Our second in-class exam is next Monday...
How to solve it
At the start of the week we started a new in-class team exercise. You can find it here.
Watching the students work through this problem (or struggle through this problem) has been very illuminating.
I didn't think that this problem was very difficult, certainly no harder than any of the others we have done before,
yet for some reason no group was even really on the right track. As I talked to the students I heard comments that led me to believe they were trying to ram this problem into a form of one from the book and the sections we had recently covered. As much as I tried to steer them away from that narrow view, they came back to it, and were locked into to this mode of thinking. Gradually, I began to realize that it is their problem solving skills that are at fault. They are masters at solving problems that look just like ones they have solved before (as long as by before you mean in the last few days), but they were completly adrift when trying to solve a new type of problem. Even though they had all the tools to do so! In thinking more about this I was reminded of a quote that I can't place something about "calculus being the only subject we teach by having the students do a sequence of problems each lasting five minutes or less." I remember mentioning this to a collegue who found no problem with this. Well, here is the problem - under this system students fail to develop problem solving skills. They get good at solving these five minute problems (as long as they are like ones they've just seen), but have no real problem solving skills. Today in class we had a class discussion about this in the context of the current team exercises. I had them tell me why the problem seemed so hard. The answers included "Because we don't have actual numbers," and "Because I can't find one like this in the book",
and "The algebra is too messy for me to do", and "It's Maple's fault." I then asked them what they had to do to solve this problem. They immediatley jumped into writing down equations (often meaningless ones). I then asked "What is the first thing you need to do to solve this problem." Finally I got the answer "Read the problem" and then "Understand the problem." We spent a half-hour on that. No group had thought to really understand the problem before! We then took a page from Polya and asked what was the question. (Where do I put the couch?) What is the data? (The location of the DVD, the fridge, and the bathroom) What are the constraints? (Put the couch equidistant from all three. Put the couch such that the sum of the distances is minimal.) Finally we were getting somewhere! What next? Let's come up with a plan of action. Now the students were into it (a bit). What do we need to develop a plan? We need ideas, a sketch, a coordinate system, etc. I had volunteers go up and sketch a coordinate system to use. Then we criticized these as a class. Gradually, we refined our coordinate system. The students were amazed to see the number of parameters decrease from 6 to 3 and the number of unknowns decrease from 2 to 1 just by formulating a good coordinate system for the problem. And, at this point we hadn't written down a single equation! Finally, we put some equations on the problem. All the algebraic difficulties were gone, all because we spent some time up front thinking through the problem, envisioning the setup, coming up with a good coordinate system, exploiting symmetry, etc. Now, I face a problem - how do I develop these problem solving skills that are so clearly lacking in these students? How do I do this while covering the syllabus. This goes beyond PBL a bit. If the students are going to do problem based learning, they need to know how to think about the problems a bit! They really need some problem solving skills that they don't have. Perhaps that is part of what they get out of the PBL approach, but I'm really concerned that the PBL approach isn't enough. I think they need some more training in problem solving up front. How to give this to them?
Steiner's problem for the masses
Today I continued my policy of trying to introduce new ideas through PBL based exercises before introducing them in lecture.
A classic problem in mimimizing a function of two variables was posed by Steiner: Consider a triangle, in the plane of the triangle find
the location of the point the sum of whose distance to each vertex is minimized. The version I presented to the class may be found
here. In this exercise, the Steiner question arises in a conversation between Diane and Jack (of M&M fame), about where to locate the
couch. Jack wants to place the couch in such a way that the length of his trips to the bathroom, the kitchen, and the DVD player are
as short as possible. The response was very good today. Students hit on an interesting variant of the problem right away. If Jack makes
more trips to the bathroom than the DVD player (on average), how should one pose the problem? The are discovering the notion of
a cost function! I let them work on the problem in class today, with two breaks to discuss as a class. Tommorrow, I'm going to present
a short lecture on max/min topics and then let them return to this exercise. I was really pleased by a student complaint this morning:
"You're making us think too early in the morning."
Back from spring break and M&M's
Spring break is over, but it has taken a few days to get the students mentally back to work. Monday and Tuesday were very quiet classes. Yesterday (Wed), we began the M&M project. The initial response has been favorable.
Students seemed to get a kick out of the project and there was lots of group discussion going on. I'm still amused by the questions I get "Can we assume that the neck of the container isn't important?" "Can we assume all the M&M's are identical?" There is a real timidity among students when it comes to formulating and answering their own questions as opposed to rote questions I might give them. But, they are getting better!
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