More Thoughts on Problem Solving

We have finished up our second group project focused on using problem solving to teach mathematics. As we wound it up we were presented with a couple of thoughts that, I think, show us where we are today in our understanding of such a task.

The first thought was that, in 1980, NCTM released a document titled 'An Agenda for Action: Recommendations for School Mathematics of the 1980s'. Akihiko Takahashi, our instructor, discussed with us a couple of the phrases from that document on how problem solving should be used in a mathematics classroom. He then noted that a major reform in Japanese mathematics teaching took place as a result of this document. They, the Japanese, took what was said and began implementing it. Today they are seen as leaders in the teaching of mathematics in their k-12 schools. What about those of us in the U.S? Where are we? It seems that we missed out on the general ideas of this document. It was our organization that crafted this document, but somehow, it seems, it did not lead to widespread change in the teaching of mathematics. I may be seeing it wrong, but from what I have read, studied, and experienced, I think I am right in my assessment.

The second thought I had relates to teachers working together to create problems and lessons that will help students learn mathematics through problem solving. As we struggled with each other, or more accurately against each other, while trying to put two such lessons together in two weeks, there were a lot of groups that struggled to work well together. We were told that we needed to spend more time learning each other, rather than trying to teach each other. Apparently we were busy trying to tell each other the way it should be and why other ideas would not work.

The last thought I had came as we discussed why we had been given such a task. Aki reminded us that we are often bombarded with great activities and problems at conferences, but never take the time to figure out what mathematics the students could learn from such an activity. We simply go and use the activity as is, and as a result we often have a less than desired outcome. In short, we need practice turning good problems into tasks where students learn mathematics.

Finally, although at times I was frustrated with these tasks and with the dynamics of the groups I was working with, I did learn a few things that I would like to implement in my teaching. I do think that mathematics as is commonly taught is more boring than it should be. I hope to figure out some of this stuff in time to try a few things in the classroom this year, and then record what happened and use that to improve in the coming years.

Video Recorder In The Classroom?

I am for it. A few schools in the Seattle are participating in what they call a Video Club. Several teachers get together once a month for a couple of hours to discuss a 5-7 minute video clip from one of their classrooms.

The main focus of the meetings is the mathematics that the students are doing. The first topic of discussion is the math of the problem. Once the problem has been discussed they move on to possible misunderstandings the students could have. Once this has been done they watch the video. Then comes discussion of the what evidence the video provided about what the students understood about the math. Finally there is some discussion about questions that are still open.

All of the teachers said that they have gained insight into their own teaching as they have participated in the video club. It makes sense. Coaches at all levels watch video of their athletes in order to understand how best to help them. There are things you see in a video that are much harder to see in real time. If this practice has helped so many coaches to help athletes over the years, it should also help us help our students each year.

Mathematical Eye?

Today we were told about the early morning fish markets in Japan. Those bidding in the tuna auction have to decide what is on the inside of the tuna before it is cut open. All they have to go by is as cross section where the tail is cut off. Those with a good tuna eye are able to pick the good fish and so end up making more money. Those without a good eye for what is on the inside don't get the money.

What's this got to do with math education? It goes back to the idea of using problem solving to teach mathematics. In order to use a problem in this way you must be able to see what math can be learned, not just what math can be used to solve it. Our ability to distinguish between these two ideas will determine our success when using problem solving in such a powerful way. This ability is a talent. For the sake of my students I hope this talent can be learned.

How the World Really Works

2006 Fields Medalist Andrei Okounkov discussed the Law of Large Numbers and a few of its consequences. Essentially if we are looking at a random path of n steps that starts at ordered pair (0,0) and goes to (T,X), where T is some time and X is some location, then as the number of steps gets very large the probability that the path varies from a straight line is next to nothing. In other words by far the most likely thing to happen is a straight line between (0,0) and (T,X), which turns out to not be that random after all.

This result was extended to cover three dimensional walks as well. What this means, at least as far as I understand it, is that the things we see as very precise and not random at all are actually so random that they gain, as Okounkov said, an Optimality and Elegance that makes us think they are not random.

All in all I felt it was one of the most understandable of the Clay Lectures I have attended in my two years at PCMI. The speaker was funny, and to my surprise, a little bit uncomfortable at times. This seemed to be in part because English is not his native language and in part because of the challenges of addressing such a diverse crowd. He took the time to state things in a way that was very accessible, which was nice. Having the chance to interact with such individuals during mealtimes and in such intimate settings is a feature that I have only experienced at PCMI.

And so we see that our world is, in fact, very random.

Problem Solving FOR Learning

Today we finished discussing an activity we created last week. We were to take a paper fold and use it to teach mathematics. The most challenging aspect of this task was to design a problem that would actually teach a mathematical concept. I, and it seemed like many of the others, have traditionally used problem solving activities as a means to test our students' knowledge about a particular topic.

Several times in our discussions we were asked "What new math are they learning as a result of this problem"? This struck me as an important aspect to teaching mathematics that I have overlooked. One portion of what we do as teachers at PCMI is to learn more mathematics. In these lessons I find it much more interesting to explore things that I don't necessarily know the formula for. I think that, with some training, our students will be the same. They will be much more interested in learning the math when the problems they are solving are teaching them this math.

I will let you know when I have it figured out.

What is Problem Solving?

Today in one of our PCMI sessions we discussed problem solving.

According to the Principles and Standards for School Mathematics (NCTM document), 'problem solving means engaging in a task for which the solution method is not known in advance.' It also says that 'problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program.'

I have enjoyed working on problems, but I don't know that I have given my students problems that are true problem solving problems. The problems I give my students are story problems that have a specific method for solving. I am usually hoping the students will see that method and use it to get an answer. I need to do a little more digging and find some problems that can help students get the math they are supposed to get.

I read a little on formative assessment last week, and read that formative assessment is assessment for learning, not assessment of learning. Problem solving would then be problems for learning new mathematical techniques, not problems that practice learned techniques.

Addicted to Blogging

I saw this in a blog I read (Drape's Takes), and so thought I would find out how addicted I am to blogging. I think it is a little high, as it represents me in an environment where blogging is a handy way to record notes and thoughts. When it is no longer handy we will see how it lasts.

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