Mr. Stewart's MathEd Musings

Posts Not Posted

2008-05-18T21:46:00.003-06:00

As advertised, here is a post about Conversations from Algebra 1 and Research and Teaching.

There are several reasons this post has come a little later than anticipated. One of them is that I had a job change, some would even call it a career change, shortly after my last post. I am now working as an Instructional Designer for a company in Salt Lake City, Utah. Maybe someday I will post about why this change occurred, but not today.

CONVERSATIONS FROM ALGEBRA 1
During the two quarters I spent teaching this school year I worked with my Algebra 1 students on some problem sets obtained from the Philips Exeter Academy, a private school in New Jersey. The problem sets are designed to engage students in problem solving, not simply computing answers using methods shown by the teacher at the start of the class period. I was not sure how the students (or parents) would respond to this strategic shift. I did feel that it would be possible to get the students engaged and interested, and expected that I would have to encourage and prod to get some of them to work. All in all things turned out well.

Let me explain a little about the daily classroom procedures we followed. I would pass out a set of problems at the beginning of class and students would work on them. Problem sets varied in length from 7 to 12 problems. Almost all of the problems were what are generally referred to as 'story problems' among students and teachers. Students would work in groups or by themselves, whichever they preferred (some were initially opposed to talking to other humans, so they chose to work alone, though once they saw that everyone was struggling with the problems they quickly joined their peers). As the students worked on the problems I would stop by each group and ask more questions, get them to share their thoughts and ideas, and help foster discussion amongst themselves. I would then leave them alone to figure out a solution to the problem. Every fifteen or twenty minutes I would have students go to the board and share what they were thinking about a particular problem that was stumping the class. If the class had successfully solved a problem, I would ask several students to go to the board and show their solution. I would usually have two or three different students present solutions, and each would be unique. We would then talk as a class about why the presenters did the things they did. If a student made a comment that was inappropriate I would tell them that it was an inappropriate comment for the setting we were in. They quickly learned how to ask questions and challenge each other in a respectful manner. Students rarely got through more than five problems during our class time (89 minute class periods every other day).

One day we got into a discussion about positive numbers, negative numbers, and the number zero. The discussion may have been about opposites, it may have been about positive and negative numbers, I don't remember exactly what, but the students began asking each other questions and explaining why they felt zero was positive or negative or not a number at all. It was amazing to hear them carrying on a respectful, mathematics based conversation.

Several of the students in the class were students that did not get along well with other teachers. I taught in a high school, and many of the students were not academic over-achievers. They had learned to respect each other and to share their thoughts. I guess I was so impressed because it gave me a glimpse of what I think a mathematics classroom should be: a place where students are engaging with each other and with mathematics.

RESEARCH AND TEACHING
I think the teaching community has a long way to go before we can be called professionals. This is a strong statement, but I think it is an accurate one. I have heard recently the comments of professional athletes regarding their preparation for games. They talk about watching film, about studying other teams' players and plays. They use the latest research on training and health to keep their bodies in top physical condition. They are dedicated to improving themselves, and the better they get the more entertained the crowds that watch them, and ultimately pay them, are.

What about in teaching? Do we ever watch film of ourselves? Do we ever watch film of others teaching the lessons we teach? Do we know if the lessons we teach are effectively teaching the topics we think they are teaching? Do we know how the students see us, and more importantly, how they see the math?

There are some good things happening in math education. There are teachers and leaders who are working to create a quality environment so that their students can learn mathematics. But there is room for improvement.

I once sat in a meeting with those considered to be the leaders among math teachers in the school district where I taught. We were discussing ways to improve math instruction. An idea was presented that was in line with recent research on math education, and one of the veteran teachers in the group, remember this is a group of the leaders in the math teaching community of one of the largest school districts in the state of Utah, said something along the lines of "New teachers have too many other things to worry about." What other things was she talking about? Things like taking roll. Things like learning how to keep a classroom full of students busy so they didn't cause trouble. These are important things to learn as a teacher, but why not learn how to do them by becoming better at helping the students interact with the mathematics? Why not make mathematics education our primary concern, and figure out ways to use the mathematics to accomplish the things that we have to worry about as teachers?

It seems to me that there is not enough focus on, or knowledge of, recent research in mathematics education among secondary math teachers. Math teachers should be recording themselves teach, and then watching those recordings to see how students reacted to what was done. We should take what we learn from these videos and apply it in our teaching. We should have time to do these things, but currently we don't. Teachers, and those who ultimately make the decisions on spending for education, should be working on ways to make time.

CANS OF WORMS
I have probably opened several cans of worms above, more than I could eat in one sitting. I think many are aware of the current problems in education. I also believe that we should all do our part to come up with solutions and to solve these problems. I will write more on one proposed solution in days to come.

Zudents and Zune in the Classroom

2007-11-29T21:00:00.000-07:00

So I am a Zune user...sort of. I picked one up the other day, and have been pretty impressed. The improvements in version 2.0 are very nice. The new feature I was most pleased with was the firmware upgrade. Before the update all my podcasts were listed as music files, which meant that when I played them they started from the beginning each time. The Teachers' Podcast has episodes that are long enough that it often takes me two or three sittings to listen to them. I listen here and there as I have time. It was pretty annoying to start from the beginning each time. Zune 2.0 puts podcasts in their own category and allows you to resume playback from where you were last listening.

And now on to the big news. I sent my first file to a Zudent (this is what I will call students with a Zune). I noticed he had a Zune, and so I turned my Zune on and sent him an episode of The Teachers' Podcast. The episode was quite long (about 50 minutes), but it took less than a minute (I would guess it was about 20 seconds) to send him the file. It was quick and easy.

So what would I do with this technology? Am I actually saying I will be encouraging the use of an mp3 player in my classroom? During class? Yes.

I am thinking that I could prepare a few different files and send each of my Zudents a different file. They could get in groups and listen/watch whatever it is I want them to listen/watch, and then discuss it as a group. We can then come together as a class and hold a class discussion on what was heard/learned.

I will hopefully have more Zudents come January (hopefully Zanta is coming to a lot of houses near me).

The race is on

2007-11-28T08:59:00.001-07:00

It looks as if a tennis match of sorts is starting between Google and Microsoft. Microsoft's new OfficeLive product (http://officelive.microsoft.com/), set to be released soon, was seen as a response to Google's online document, spreadsheet, and presentation tools. From the looks of things the OfficeLive would be much nicer, as it would allow you to work with your files in their true format as office documents. According to this article (Google Plans Service to Store Users' Data) in the Wall Street Journal, Google now plans to offer a storage space for users' files.

One thing that will hopefully result from these new tools is that teacher collaboration will increase in quantity and quality. It will be up to the teachers to make it happen, but it seems a logical and easy way to collaborate on lesson plans.

Adapting Texts for Quality Use

2007-09-18T22:18:00.000-06:00

I read a something (in Advanced Web-Based Training Strategies by Driscoll & Carliner) that caught my eye. The authors were discussing the problems associated with using off-the-shelf content as a broad solution to a training problem. They gave an example of a company that said they had to modify 75% of the off-the-shelf e-learning courses because the content would not work for their staff.

What about with teaching math in the classroom? I am currently using textbooks that are several years old. I assigned one of the story problems from the textbook to my students the other day. They were supposed to set up an equation that modeled the situation described. They struggled mightily. I changed the context to something that they were more familiar with, and they were immediately able to set up an equation to model the situation. The only reason they had been struggling with it in the first place was that the context was unfamiliar to them. And now the connection to the dilemma posed by Driscoll & Carliner.

Should I be adjusting about 75% of the textbook to make it work better with my students?

I think so.

Am I doing this?

No.

So what is going on? I know what I should be doing but I am not doing it as much as I should be. Jordan School District does great things with their math teacher trainings. We have the opportunity to collaborate and a very good structure in place to support this collaboration. I think there needs to be more, though. The vast majority of coordinated training efforts are done during the school year. The vast majority of the teachers have the summer off, at least from teaching. I propose allowing teachers the opportunity to spend their summers doing what the rest of the world does during their summer, namely working.

I realize some teachers may not take anyone up on the offer to work more, but I think that a lot of the young teachers will. We (the young) are in a position where we have to take second jobs to be able to afford necessities, such as a home, anyway. The low pay is what keeps a lot of qualified and quality teachers away from the profession. I often hear things like 'you don't get paid, but you get the summers off'. I may get the summer off from teaching, but it only because I have not yet found the opportunity to get a full-time summer job doing what it is that I am trained to do, namely teaching math or improving my ability to do so.

Are Common Assessments Valuable?

2007-08-26T09:14:00.000-06:00

As we prepared for the start of the new year, my colleagues and I got into a discussion about common assessments and their value.

Our math department does quite a good job collaborating and using common assessments, and we were lauded as a good example by our administrators and others during the school-wide training meetings that happen the week before students come back. This was fine, but then we were informed that we did not meet AYP due to our math scores. In fact, we had a 47% pass rate for 2006 and a 34% pass rate for 2007. If I understand correctly, we were supposed to have improved by at least 10%, not dropped by 13%.

This was the news that prompted a few of us to start talking about what to do and how the results of the common assessments should be used. One fellow math teacher pointed out that as long as we are giving common assessments we should also be using the results of these assessments to guide our instruction. As we discussed the different approaches we are taking in our Algebra 1 classes she mentioned that having a common assessment may prove more valuable now that different approaches are being taken, as long as the results are being reviewed. If it turns out that one class does much better on the common assessments then we should consider the approach that teacher and class took, and the rest of us should adjust our teaching accordingly.

I am excited to see what happens. Last year we gave common assessments and, once they were given, we decided that the majority of students were doing poorly. Those of us teaching Algebra 1 were taking the same approach to teaching and getting the same results. We did nothing to change what we were doing, we just kept going because that was the way it worked. The common assessments didn't do much for us because we didn't do anything to respond to the results they showed. This year we will be looking at the results and seeing what was happening that made the difference between the successful classes and the less successful classes.

What is a 'Teacher'?

2007-08-08T22:10:00.000-06:00

I was reading a little bit in 'E-Learning by Design' by William Horton, and came across a question that made me wonder about my role as a teacher. The question was one of a list meant to help the reader decide whether to skip the first chapter. The question reads, in part, "Do you lack either education and experience in instructional design? Perhaps you are a subject-matter expert or instructor..." So which am I, as a teacher? Am I a subject-matter expert? An instructor? An instructional designer? What does it mean to be any of these? Does an instructor have to be a subject-matter expert and/or an instructional designer?

My initial answers to these questions is that each of these are separate things. To be one is not necessarily to be the other. College professors, for example, may be subject-matter experts in their chosen field, but may not be able to instruct others in similar pursuits, and may not be able to design quality instruction. Is there value in becoming a professional with training in all of these areas? If one did obtain such training, would they be able to more effectively help students learn? Should all teachers receive specific training in each of these areas, or should teachers team up based on their strengths and have specific tasks assigned them? Should there be teachers working behind the scenes to create lessons for those who are actually in the classroom?

Chinese Philosophy and Math

2007-08-01T22:09:00.000-06:00

I have been reading Fung Yu-Lan's book 'A Short History of Chinese Philosophy'. In the introductory chapter the author is talking about the spirit of Chinese philosophy. In the latter portions of the chapter he addresses why it is that Chinese philosophy is often written in brief and disconnected ways when compared with the Western philosophy and its more articulate style. In Chinese philosophy aphorisms, allusions, and illustrations are used to create this brief and disconnected style. Their use leads to suggestiveness, and Fung points out that this suggestiveness is less limiting than a more articulate approach. There are ideas that are suggested and understood that could not be articulated well without limiting the scope of thought.

It seems to me that the traditional methods of teaching math, namely memorization, drill, and practice, are more articulate (there is a certain way to solve this type of problem, and that method should be used to finish some set of exercises). Once a student has completed the exercises they should have a grasp on the mathematics.

A problem that is to be used to teach mathematics cannot be this articulate. There needs to be more suggestiveness, thus allowing the students to use whatever mathematically accurate approaches they see fit. Having a classroom discussion on the ideas that come from the various solutions will help the students define certain mathematical concepts in ways that will be beneficial to themselves.

Fung quotes a passage from the Chuang-tzu, a significant book of philosophy, which may help distinguish between ideas suggested and methods articulated, and how ideas suggested can be the more powerful of the two.

"A basket-trap is for catching fish, but when one has got the fish one need think no more about the basket. A foot-trap is for catching hares; but when one has got the hare, one need think no more about the trap. Words are for holding ideas, but when one has got the idea, one need no longer think about the words." (pg 12)

Later on Fung says that 'words are something that should be forgotten when they have achieved their purpose. Why should we trouble ourselves with them any more than is necessary?' (pg 13) It seems to me that the traditional methods of teaching math are having our students deal with the words. The formulas they memorize and the specifically formatted equations they learn to solve are the words used to describe the ideas of mathematics. Our students, however, don't get to experience the ideas of the mathematics because we are spending so much time focusing on the words.

The last paragraph of the first chapter of Fung's book says the following.

"Kumarajiva, of the fifth century A.D., one of the greatest translators of the Buddhist texts into Chinese, said that the work of translation is just like chewing food that is to be fed to others. If one cannot chew the food oneself, one has to be given food that has already been chewed. After such an operation, however, the food is bound to be poorer in taste and flavor than the original."

Do we assume our students are not capable of chewing the food of mathematical ideas? Are we continually chewing this food for them? I think we are. It is no wonder that our students are not enjoying math classes as much as we would like them to, or even as much as they would if we would let them.

Video Recorder In The Classroom?

2007-07-17T20:17:00.000-06:00

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?

2007-07-17T19:59:00.000-06:00

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

2007-07-16T15:24:00.000-06:00

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

2007-07-16T15:19:00.000-06:00

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?

2007-07-09T18:10:00.000-06:00

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

2007-07-08T13:02:00.000-06:00

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%

Utah Population by County

2007-07-06T18:56:00.000-06:00

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

2007-07-05T22:14:00.000-06:00

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

2007-07-03T19:59:00.000-06:00

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

choosing export from the iTunes File menu (not Export Library...), and then
importing this .txt file into an Excel spreadsheet,
cleaning up the data a bit,
exporting it as tab delineated .txt file, and then
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

2007-07-02T13:19:00.000-06:00

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)?