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All the information in the world
February 24, 2005

I mentioned a few weeks ago that in addition to teaching mathematical modeling in Math 518, I'm also teaching a graduate special topics class in nanoscience, Math 824. The goals of Math 824 are two-fold. First, I'm trying to give students a firm foundation in nanoscience. We're doing this by reading and discussing the primary literature in the field. At the end of the course, I hope they'll feel confident enough to be able to explore nanoscience further as they wish. The other goal of Math 824 is to improve their skills in reading scientific papers. So far this semester we've read 5 papers. Two were transcripts of talks by Richard P. Feynman. One was an article from Scientific American. The other two were journal articles. During our discussion of the fifth article, which was from an IEEE journal, it became clear that comprehension had dropped significantly compared to students reading of the other four articles. Yesterday, we spent the first 1/2 hour of class discussing reading again. The discussion was lively and interesting. It was clear that students were feeling a lot of anxiety over their failure to read this one article well. I had them identify precisley why they found it difficult. Most of the answers I expected. A lot of jargon, the article assumes a lot of background, explanations were terse, the context of the work was not well explained, etc. A few of the answers were unexpected. One student said he was fooled by the age of the article. (The paper was from 1967.) Since it was an older paper, he expected it to be easy! I'm not sure what to make of that comment, but I was amused. The other unexpected response had to do with how students used the internet. When I asked them if they had tried to find other sources of information about the topic of the article, many of them responed that they had. All of those who had had simply Googled keywords like "resonant gate transitor," or "tuning problem." They were then faced with the problem of finding relevant information from among millions of hits, or ending up with no hits at all. Since this topic is not like say "nanotweezers," where you can easily find relevant information, they met with little or no success. They did not take the next obvious step of going to the science citation index and tracking the papers that cited the paper we were reading. They didn't try any library databases. The certainly did not physically go to the library. I asked them what fraction of the information in the world they thought was available online. The answer was "most of it." My guess would be that that answer is wildly wrong. I would estimate less than 1% of all the information in the world in online. I would also contend that the information that is available is not really well organized. This is a calculation I need to do! Maybe I can give the problem to my Math 518 students!

How many bricks?
February 23, 2005

Last Thursday I gave my class a homework assignment. This was a follow up to the in-class assignment we did involving making basic estimates and understanding large numbers. One of the problems asked them to estimate the number of bricks on the UD campus. Last night in looking at their answers, I was very very pleased. These guys did excellent work. I think they really get this idea now. I think the last week and a half have taught me some important lessons as a teacher. First, its important to put aside my own emotional response to students work. In particular, I need to be sure to put aside any dissapointment I might feel when they don't measure up to my expectations. Instead, I need to focus on why they haven't measured up and what I can do to communicate what I haven't communicated. The other lesson is the importance of flexibility. Essentially one class period was devoted to this estimation topic. This was completley unplanned, but has turned out to be a wonderful entry point into so many topics in mathematical modeling. It's the type of thing I wish I had thought of doing when teaching this type of class before! If I had tried to adhere to my timeline, I would have missed a wonderful opportunity.

Last night we also resumed our discussion of the Great Lakes project. The discussion was richer than our first discussion. We've moved forward a bit. The next issue I must deal with is in getting them to see how we can focus our efforts on parts of a complex problem. That is, how it is useful to take small bites. We don't need to deal with both pollution and invasive species at once. We don't need to consider every aspect of pollution remediation. I need to get across the notion of higher order effects.

The line between despair and opportunity
February 18, 2005

If you read my last entry you'll recall that after Tuesday's class, I was very concerned about the answers I received to the question "If the U.S. Govt. plans to spend $5 billion to clean up the Great Lakes, what does that cost me?" I worried about this for days. Were these students functionally innumerate? They're almost ready to teach in our high schools! How could this be? I passed through a range of emotions. First I was bewildered, then angry and annoyed, then puzzled and dissapointed. I'm glad that I made no decisions based on those emotions. Eventually, when my head cleared I realized that I had learned something and that I had an opportunity to teach something important. What I had I learned? I'd learned that the first skill needed to engage in mathematical modeling is a feel for the relationship between numbers and reality. I think I've always taken it for granted that everyone grasps that relationship. Now, I think not. What was the opportunity? The opportunity was to highlight this issue, to break down the skills they need to learn to be able to engage in mathematical modeling, to teach guessing, estimation, basic numeracy, at least a little. I started with a lecture last night. We started at a high level examining the question "What is mathematical modeling?" They'd all read the first chapter of their text and so had some idea of one author's answer. We talked about how this is a natural question, no different than "What is philosophy?" or "What is calculus?" Questions I hope they'd asked themselves before. We talked about how if they hadn't thought on that level, they might have thought of various disciplines from within, asking questions like "What is a function?" or "What is justice?" We then began to answer these questions by starting with the notion of definition. We talked about Aristotle and the components of a definition (Proximate genus & specific difference). We illustrated this notion by looking at the definition of a function, of a group, of philosophy. We then built a definition of mathematical modeling and discussed how this is a topic of debate, not something set in stone like the definition of function. We discussed how the way in which we define things frames the subsequent debate. Once we had a working definition, we had more questions. If mathematical modeling is a process, what does that process look like? If it's purposeful, what is the purpose? If it's the business of building models, how do I build them? This led to the question: "What skills do I need to engage in the art of mathematical modeling?" which in turn led us to a discussion of developing a feel for numbers and their relationship to reality. We talked about estimates, about back of the envelope calculations. They then worked in groups and tried to answer the following questions:

  1. How much human blood is there in the world?
  2. If you were to count $1 bills, one at a time, at a rate of one per second, how long would it take you to count $5 billion?
  3. In 2004, the U.S. Govt. spent about $2,292 billion dollars. How much is this per person in the U.S.?
  4. If you folded a piece of paper in half 50 times, how thick would the resulting stack of paper be?
I let them work as long as they wanted. Most groups stayed till about 6:30 (class ends at 6:15). Some stayed longer. Last night I looked at their answers. Each group handed in one team answer. They can do this! They're not totally innumerate - they simply don't have the habit of thinking in this way! Now we can move on, now we can work to make this a habitual part of their thinking. I think we've already made some progress.

The crux of the matter
February 16, 2005

What should a student coming out the other side of Math 518 look like? What do they look like coming in? Last night I began to get a picture of their current state. We began by discussing the spaghetti project. I need to reflect a bit more on the conversation and the whole exercise before I can decide what worked and what didn't. One thing for sure that did not seem to get across is the point that the process of math modeling is the process of problem solving. We then began our discussion of pollution in the Great Lakes. They read two newspaper articles which talked about the problems faced by the lakes, potential solutions, and the cost. I then asked them to write their answers to the following questions:

  1. What problems do Flesher and Paulson claim the Great Lakes are facing?
  2. What are the probable consequences of leaving these problems unchecked?
  3. Is cleaning up the Great Lakes "important"? If so, why? In what sense?
  4. At then end of (Paulson 2004) the claim is made that $4-$5 billion federal dollars are needed to clean up the Great Lakes. How much will that cost you as an individual? Is this amount reasonable? Suppose the actual cost is twice as high? Ten times as high?
They then discussed the answers with their groups and we began to discuss them as a class. At the end of class they handed in their written answers. The answers to the first two questions were reasonable. The first question asked them to extract information from the articles. The second the extract some claims from the articles and to think of other possible consequences of the facts they found in the articles. The answers to the last two questions were mostly nonsense. Let me dwell on the fourth question for a moment. Only 7 out of the 31 people in the class last night even attempted to translate a $5 billion dollar government expenditure into a cost per individual taxpayer. Of those, 4 answers were more guessing than computing. Three answers were reasonable. The remaining 24 students either ignored the question or stated outright that they had no idea how to compute the cost to an individual. (I won't keep you in suspense any more, Total Cost/# of people = cost to individual.) The answers to the third question were also mostly nonsense. If you read the question again, you'll note that it is vague. The question behind the question is one of definition. If you are going to claim that cleaning up the Great Lakes is "important" (or not), you had better make the notion of "important" precise. You had better make a cogent argument. "We should clean up the Great Lakes because..." What follows the "because" is key. So, back to the picture of my students at this point in time:
  1. They have a limited feel for numbers. They don't instinctually "think like a mathematician" and attempt to translate numbers into something that can be understood.
  2. They are not used to or unable to make questions precise. To define terms and give rational arguments.
So, what do I do on Thursday? Press ahead as planned? Or try and bring out some of these issue a little more carefully?

Great books
February 14, 2005

Tommorrow we'll discuss the results of the spaghetti tower exercise. We'll also begin our Great Lakes project. This is a multi-week, multi-part project focused on mathematical modeling of pollution remediation in the Great Lakes. We'll start by reading newspaper articles, then read a piece of legislation related to the Great Lakes, and finally read an article from Science. The first handout is here. The first two readings are here and here. In developing this course and in developing Math 824, I spent a good deal of time thinking about the skill of reading. I was very heavily influenced by Mortimer Adler's "How to Read a Book." If you haven't read it, read it. It's one of those rare books that has the power to create an ontological discontinuity in your life. Well worth it. On the subject of great books, I started reading "A Tour of the Calculus" by David Berlinski. What a wonderfully written book! If you're not a mathematician and you would like to have some sense of mathematics, this is the place to start. If you are a mathematician and you'd like to see what we do expressed as eloquently as is possibly, read Berlinski. Math 518 is about mathematical modeling. I've spent hours thinking about and discussing the relationship of mathematics to the real world. Berlinski sums up my feelings more eloquently than I could ever imagine:

"It is sometimes said and said by mathematicians that the usefulness of the calculus resides in its applications. This is an incoherent, if innocent, view of things. However much the mathematician may figure in myth, absently applying stray symbols to an alien physical world, mathematical theories apply only to mathematical facts, and mathematics can no more be applied to facts that are not mathematical than shapes may be applied to liquids. If the calculus comes to vibrant life in celestial mechanics, as it surely does, then this is evidence that the stars in the sheltering sky have a secret mathematical identity, an aspect of themselves that like some tremulous night flower they reveal only when the mathematician whispers. It is in the world of things and places, times and troubles and dense turbid processes, that mathematics is not so much applied as illustrated."

Ah, allometry!
February 11, 2005

Math 518 students are shocked at spaghetti tower construction!

Interesting second class. We began with a discussion of the spaghetti bridge exercise from last class. I was very happy with the level of participation in the discussion. I think I've learned a bit about managing such discussions. I think it is effective to push the students when what they say is unclear, or vague, to force them to be precise. This revealed some interesting observations that I think would have remained hidden otherwise. Overall, a few key points came from the discussion.

  • Resource constraints need to be dealt with when solving a problem.
  • It's important to understand exactly what the problem is asking before proceeding.
  • It's important to understand the meaning of words. (What does "bridge" mean?)
  • It's important to understand how available resources are related to problem constraints and goals.
  • Measuring team effectiveness requires some thought.

The last point was especially interesting. Every team said they were "very effective." Then, when I pushed them to define how they measured their effectiveness, it came out what they were really measuring was how well they got along, not how well they solved the problem presented to them. When asked to rate their team effectiveness using how well they solved the problem presented as a measure, the answers changed quite dramatically.

After the discussion, we started the spaghetti tower exercise. The environment in the room was dramatically different than last class. There was a sense of urgency, less idle chitchat, more focus. Overall the constructions were better. Here's a few pics:

A good spaghetti tower. Another good one. A unique idea.

Next class, we'll revisit the discussion of these exercises. We'll see how their thinking has changed. One idea that occured to me this morning - a discussion of allometry (the science of how properties of a system scale with the size of the system) would have been a great follow up to the spaghetti exercises. They all have a hands-on feel for the strength of spaghetti and how the strength of a spaghetti bundle changes as you add more pieces to the bundle. It would be great to measure and plot the strength as a function of the number of pieces in the bundle. This is an allometric measurement. We could then move into an examination of allometry in nature, along the lines of Thompson and "On Growth and Form." Something to keep in mind for next time!

More spaghetti
February 10, 2005

Tonight in class we'll do part two of the spaghetti exercise. We'll start with a group discussion of the following questions about the bridge exercise:

  1. Did your group develop a plan prior to starting construction?
  2. Was there a leader of the group? Do all team members agree on who was or should be the leader?
  3. Was everyone involved in building your bridge?
  4. Did your team members communicate well? Did your team function well? By what measure?
  5. What did you learn about using spaghetti to build a structure?
  6. What made this exercise difficult? What made it easy?
  7. If you were confronted with another spaghetti construction problem, what would you do differently? What would you do better?
Then, we'll do a second construction exercise where the following instructions are given:

Again your team will be given 1 kg of spaghetti and one spool of thread. This time, your task is to build a tower. You will have 35 minutes to complete this task. The tower must be at least 40cm in height and will be required to carry a load. The load will be placed on the tower in a basket. The strength of the tower is defined as the maximum load carried divided by the mass of materials used in the construction. Unlike the last exercise, your team will not be given construction materials until the team presents the instructor with a plan for construction. The team with the strongest tower wins.

We'll follow up with more discussion. I'll post some pictures tommorrow. You might be wondering about the history of this spaghetti construction exercise. I first learned of it from Lise Kofoed and Torben Rosenorn of Aalborg University in Denmark. My implementation is an adaptation of their plan. The focus of their exercise is on teaching the Kolb learning cycle and in getting people to think about their own learning process. As you can tell, I've changed the goals a bit to suit my class. I'm using it as an introduction to group work, an easy way to start to think about mathematical modeling, and as a way to get students to evaluate how they learn. When I was returning from class the other night, I ran into Mary Ann Huntley, a math education specialist here at UD, who told me she'd seen the spaghetti exercise used by her thesis advisor. A student of mine then told me that she'd had to build a spaghetti tower from spaghetti and glue for a class in high school. A quick Google on "spaghetti construction" reveals that variations have been used in many places and for many different goals. Spaghetti is versatile!

Vomit and Spaghetti
February 9, 2005

Math 518 students build a spaghetti bridge.

We had our first class yesterday. I've been struggling with my grading policy. As I mentioned in a previous entry, I'm trying to break away from a formulaic approach to assesment. Yesterday morning I was reading the Enchiridion (Epictetus) and came across a passage that captured my view on assesment quite well.

For even sheep do not vomit up their grass and show to the shepards how much they have eaten; but when they have internally digested the pasture, they produce externally wool and milk.

While identifying students with sheep is not necessarily an identification I want to make, I think this passage captures what I really want to assess. I do not want to force feed information to students all semester then watch them vomit it onto the final exam and assess them on the quantity of their vomit. If I do that, there is not only little to be gained, but it is also likely that they leave the vomit (knowledge) behind. Instead I want to focus on what they are able to do as a result of what they learn. What kind of milk and wool do they produce?

We spent most of the class yesterday working on a team exercise. I divided the class into teams of 4 students. This time I divided them by birth month. I figure this is just as random as any other scheme I've used in the past. It also let us discover that their were two pairs of students who share the same birthday. We got to put the question of the probability of this happening out there. We can revisit this later in the semester. After the students were divided into groups, I put up a slide with the following information:

Your team has been given 1 kg of spaghetti and one spool of thread. You have 40 minutes to construct a bridge using only these materials which shall connect two tables that have been placed 50cm apart. During the testing of the strength of your bridge, one person from your group may hold the bridge in place. The strength of your bridge will be measured by placing a weight on the bridge or suspended from the bridge. The strength of your bridge is defined as the ratio of the maximum load it can carry divided by the mass of the material used to build the bridge. The team with the strongest bridge wins. Math 518 students build a spaghetti bridge.

I had several reasons for doing this exercise. I wanted to introduce them to group work and to their team members in a relatively low stakes way. Next class, we'll spend some time discussing how effectively they worked as a group. I'll try to get them to focus not only on whether or not they got along, or had fun, but on how successful they were at solving the problem presented to them. As you can see in the picture of the blackboard below, there was wildly different levels of success. I'm also trying to get the class to think about how they learn. Next class we'll begin with a discussion of this as well. Finally, there are lessons about mathematical modeling to be learned from this exercise. They had very limited resources and very limited time. They had all that they've learned before to use if they could. They needed to understand how success was to be measured and how that measure effected how they solved the problem. They were free to test their own solutions. Etc.

The strength of various bridges.

We'll also do a second spaghetti construction exercise, more on this later. A few final thoughts about the class from yesterday. I asked (and had them write their answers) whether or not they had ever had math modeling in high school. No students had. I also asked what their learning objectives were for this class. It was very clear from their answers that they had given no thought to this whatsoever. Almost every answer was "I want to learn what this class is about and some math modeling." I'm hoping that through discussion of the spaghetti construction exercises, I can get them to think about their own learning objectives at least a little bit.

Gearing Up
February 3, 2005

The new semester is almost upon us. This semester, I'll be teaching two courses, Math 518, which is a mathematical modeling class for mathematics education majors, and Math 824, which is a special topics course on mathematics and nanoscience. In this blog, I'll chronicle my efforts in teaching Math 518, although from time to time I may comment on Math 824. If you are looking for my blog on Math 243, just view this blog by month and go back to the spring semester of 2004.

It looks like I'll have roughly 30 students in Math 518. The majority of these are in their junior year and are majoring in secondary mathematics education. These are our future high school mathematics teachers and Math 518 is probably their one and only real course on mathematical modeling. I've decided to teach the course using a very PBL based approach. I'll be taking this approach much further than Idid with my Math 243 course. I've also made major changes in the way I've prepared the syllabus and in the way I'll grade the students. If you'd like to see the syllabus, you can find it here. You'll notice that I don't give a week by week breakdown of what we will cover. In fact, the orientation is not on the material that we will cover, but rather on what the course promises to the student and what the student must invest in order to benefit from the course. Rather than thinking of this as a syllabus, I think of it as simply a course outline. It tries to address the questions - Why should I take this course? What will I learn if I take this course? What must I do in order to learn what this course promises? You'll probably also notice that there is not a grade "formula." No percentage breakdown, that is, nothing which says "Homework = 15%, Exam 50%, Final Exam 35%." I'm very concerned about the focus on grades and the treatment of a course as a game where the objective is to maximize one's grade with minimal effort. I'd like to break that philosophy. I'd like to try and take the focus off of exams and homeworks as "points scored" and make them into simple tools used to assess student learning. I'm going to try and base grades on a much more individual assessment of each student. Their grades will be based on what they learn. Exam, homework, etc., is simply evidence of learning. I want to approach this like a scientific problem. The students are black boxes, I need to figure out what's going on inside. I'll evaluate the picture I build of their learning in the course. This might fail miserably.

 

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