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.

70%How Addicted to Blogging Are You?

Utah Population by County

I found the 2000 U.S. Census population data for each county in Utah last night (I guess it was early this morning). I used GEGraph to create a 3D bar chart of population by county. I did it several times to explore the options, and came up with two that I really liked (one I adjusted after importing it into Google Earth).

The one that I will say was naturally occurring was one where both the height and diameter of the cylinder are scaled relative to the distribution of the population across counties. This one really gives you a good sense of how small the populations of most of the counties are when compared with those counties on the Wasatch Front (Utah, Salt Lake, Davis, and Weber).

The second one had only the height of the cylinder changed with respect to population distribution. The diameters of all of them were the same. Once I got this imported into GE I selected each of the 29 individually and edited their points so that the shape was no longer cylindrical, but instead matched the general shape of the county itself. This view is nice when trying to read the data because the polyhedrons provide a solid background for viewing the text.

The .kml files for both of these are available, but you must email me for them, as I have not yet figured out where I can post them online. I have a couple of options, but they are down at the moment. When I find something I will update this post.

My next project is to figure out how to get all my iTunes library into Fathom. I can get the music in easy enough, but am looking at a way to include the podcasts and the movies as well.

Google Earth and Math

This morning I shared with my co-attendees here at PCMI some information about GoogleEarth (hereafter GE) and how I have used it in the classroom. I told them how my classes had looked for circular objects in and around Salt Lake City. We used these objects as our examples when we looked at calculating the area of a sector and the length of an arc. A couple of the circles we found allowed us to see some practical applications of these calculations.

As the day wore on I was playing with the polygon tool on GE and I noticed that a path drawn between two points did not line up with the side of a triangle that shares endpoints with the previously drawn path. As our discussion continued I learned a little more about what constitutes a polygon in spherical geometry (all sides are portions of great circles). We came to the conclusion that the polygons were not true polygons in the spherical mathematics sense. It does look like a path between two points will be the shortest distance spherically, but for some reason the paths and the polygons don't seem to use the same construction when dealing with segments between the two points.

I started browsing the web to find out what I could about GE and how it constructs its paths and polygons. I still have not found the answer to this, but I did run across some interesting software. GEPath (see GE Blog article here or go to download page here) allows you to export a set of placemarks from GE as a KML file and then open this file in GEPath and create a polygon. GEPath will also calculate the area and perimeter of the polygon. This turns out to work nicely.

Using this method, I created a spherical triangle. I then zoomed in to each of the corners and copy/pasted the pictures into Geometer's Sketchpad. I calculated the angles of the triangles and they sum to greater than 180 degrees (181.69 degrees). This type of activity can be used to show students the importance of understanding the importance of the assumptions that are made in problem solving, particularly in geometry.

The other program is GEGraph, which allows you to plot data in a 3D view in GE. I have not tried it yet, but hope to try it soon. I am particularly interested in how GEGraph data can be transferred to and from Fathom.

iTunes Data in Fathom

This afternoon in our PCMI SSTP (Secondary School Teachers Program) data working group we had the chance to mess around with Fathom. I have not been overly impressed to this point with the friendliness of the Fathom interface, but am getting more comfortable as I play with it. I am not convinced that I will be letting my students touch it anytime soon, but as the start of the school year is still two months away, that is ok.

I was wondering how to spend the two hours of Fathom-time and I decided to see if I could import the data from my iTunes library. I had to fiddle with things a bit, but after

1. choosing export from the iTunes File menu (not Export Library...), and then
   
2. importing this .txt file into an Excel spreadsheet,
   
3. cleaning up the data a bit,
   
4. exporting it as tab delineated .txt file, and then
   
5. importing this file into Fathom I had my data in the form of a collection.

I created a box plot that had Genre as the y-axis and Play Count as the x-axis. From this I determined that my country music songs are played more often than the other genres that I had. A histogram showed that the number of country songs in my iTunes is larger than the number of songs from any other genre. A dot plot with file size on the y-axis and time of song (which shows in seconds) on the x-axis showed a linear pattern with very few files outside the lines. The files with larger file sizes relative to the length of the song were files of the MPEG audio format. This makes some good sense, since the default in iTunes is to import songs in the more compressed AAC format.

It wouldn't be too much trouble to design a lesson where we compiled students' iTunes library information and looked for trends among classes or found the equation of the line that fits the dot plot of song length vs. file size. I am sure I will get time to look at this over the course of the next few weeks.

Governor Huntsman Welcomes PCMI Participants

This morning we heard from Governor Huntsman (Utah). He welcomed participants of PCMI (Park City Math Institute) to Utah, and asked us to consider several things as we participate in the three week institute. Here are a few of his thoughts and questions (paraphrased) and some of my responses.

• After the first three years of teaching, 50% of the math teachers leave the classroom. There are many pursuits professionally that lure them away, but we need to get something figured out to keep them. How do we do this?

I think that a major step in keeping teachers will be to pay them competitive salaries. I know there are a lot of people out there who say that teachers only work 9 months of the year, so why should we pay them more. I agree. I propose that new teachers be required to work 12 months of the year, and that starting salaries for this 12 month work schedule be double the current starting salary (in my district this would mean approximately $56,000). Teachers would spend their summers working on curriculum, technology implementation, and/or improving their teaching methods.

• There should be a concern for the new underclass of mathematically and scientifically illiterate. How is it that we begin to close the gap as kids work through their early years of schooling so that we don’t develop the underclass that could exist. Teachers are a key to inspiring our kids to pursue math and science.

One of the reasons that students don't pursue mathematics and sciences is that they don't have the opportunity to experience math and science in the secondary school setting. Allowing students to get into the mathematics on their own is necessary to getting them interested in pursuing math long enough to gain numeracy.

• We are entering a STAR (Science, Technology, and Research) world. How best can we reach back into our high school and middle schools to help them meet the bench marks of literacy and numeracy that will prepare them for this STAR world?
  

This relates to the first two problems I have mentioned. We need to pay quality teachers, as they are the key to inspiring our students to pursue math, science, and technology. We need to require our new teachers to learn to teach better. As one of the presenters today pointed out, teaching is a profession that needs to be learned. We all benefit from practice. As we require teachers to spend their summers improving their teaching, their teaching will improve.

And finally he gave us a homework assignment, which is essentially to answer the questions and problems posed above. Here are the questions as I wrote them this morning.

How do we shift the focus to numeracy? As we move toward heightening literacy standards, what do we do for numeracy? What benchmarks should we have? What do we do for the teachers? What nationally do we look at? What do we do to keep math teachers in place (differential pay, something else)?