Presentation to Algebra students

Today, I presented to Ms. Colwell's algebra students - in two classes, actually. Normally I'm in her fourth hour, today I was in both her third and fourth hours. The presentation was on optimization, which ties into what they've done recently with linear programming. I showed what linear programming versus non-linear programming were by building a model with foamcore, hot-melt glue, and floral foam. The model is shown at the left - the blue and gold model is the linear programming model, and the blue and green model is the nonlinear one. (Unfortunately, they don't make floral foam in U of M colors!) I showed them that if you try to find the minimum, in the linear case, there's only one point - drop a marble in, and it always rolls to the same spot. If you drop a marble into the nonlinear one, it can go to different places, either internal or on the boundary.

The students had a worksheet that I'd made up, and the answers to the questions were all in the presentation that I gave. I explained how to set up an optimization problem - choose the system of interest, get an objective function, select variables, formulate constraints - and a little bit about how to solve them. Next, I gave them an example of a real optimization problem. Most problems I've done have been pretty technical, and a bit dry, so I got information from a lab-mate of mine. Tahira Reid and two other students in ME 555 last winter did a project that involved optimizing the hair of African-American women, involving length, styling, and damage from various sources. Tahira was generous enough to give me a copy of their slides and her permission to show them to the students, so I selected some of her slides and went over the goals of the project and some of the issues involved. Obviously, a lot of the math was too advanced for them, so I didn't dwell on it in any depth, just showed them how the problem was set up.

Next, they turned over their worksheet and there was an exercise we did together: make up a new optimization problem. I wasn't sure at first whether it would be better in small groups or as a class, but we ended up doing it as a class. In the third hour algebra class, they came up with several ideas - optimizing a guitar, for length of time between tuning; optimize life in general (that would be a rather complex objective function!); optimize a textbook, for minimum weight. We set up the optimization of the textbook. They selected quite a few possible design variables (I had asked for two) - thickness of the cover, thickness of paper, size of type used, size of margins. When we considered the constraints, they decided that making the textbook cover resistant to ripping was one constraint, and another big one was readability. In terms of margins, one of the constraints had to be that the margin couldn't be negative. Once the problem was formulated, I asked them whether they thought it would be hard or easy to solve. They decided - and I agree - that this would probably be a fairly easy problem, since a lot of things are linear.

In fourth hour, they decided to optimize popcorn quality. This was a little more difficult, since we had to talk about what makes quality popcorn, and what could be included. Some of the factors that could go into the "popcorn quality metric" were that it doesn't burn, it should all pop, big kernels, and taste. Variables could include the bag's dimensions, temperature, amount of popcorn in the bag, and amount of butter used. Some of the constraints were minimum and maximum temperature, that the bag should be pretty, and that it had to fit into the microwave. When the bag's attractiveness was brought up, I told them about some of the work being done on optimization and aesthetics, and how that could be expressed mathematically in terms of proportions, among other things. When I asked if the problem would be easy or hard to solve, they decided - and again, I agree - that it would probably be a hard problem. The size of the problem was one thing they mentioned. Another was that it was probably non-linear. One person also said that everything is kind of subjective, so how do you know you have the right function? Someone else might make a different function for the perfect popcorn. That brought up a really good point, which I hope they remember - that it's important to start with the right problem before you try to solve it.

Overall, they seemed to be interested. Some of the students really got into the exercise, but they all liked the change - and the fact that, aside from turning in the worksheet I gave them, Ms. Colwell didn't give any homework. The third hour said that if a presentation by me means no homework, could I come back again sometime?

Slippery roads today...

There were quite a few students either absent or late today, due to weather - there isn't much snow, but the freezing rain made the roads a challenge. Many of the buses were late, so attendance in first hour physics was rather low. Today's physics work was reviewing for their test tomorrow. The test is going to cover circular motion and universal gravitation. I looked over the test that Mr. Ambrose is planning to give, after class was over, and I think that it's fair; the students who have been working hard and doing the homework should do just fine on it. After the test, they're going to move on to energy.

In Ms. Colwell's algebra class, they learned how to use matrices to solve systems of linear equations; then, while most students took a quiz, I took one of the students aside who had been absent and wasn't taking the quiz, and went over some of the material he had missed. He hadn't been there when they learned how to solve equations by linear combination, so we went through those problems. First I set up and solved one, then I gave him a problem with slightly different numbers for him to do; then I set up a slightly different problem, more complex, and did the same thing. By the end of the class period, he seemed to be catching on pretty well. He'll still need practice, of course, but that's true of everything.

The students in both classes have gotten to know me fairly well by now, and they're more comfortable just talking. I had mentioned to them, several times, the need to do a "sanity check" and gave an example of how it had helped me to catch an error that could have haunted me down the road in my own work. I had to re-do a fairly big bunch of derivations, but at least I knew where the algebra started going wrong. One of the algebra students said that sounded frustrating, and asked how I kept wanting to do it. It was a serious question, so I told him the truth; that I did occasionally have days when I DIDN'T want to. Everyone has days when nothing goes right and it would be easy to just quit trying. Sometimes work doesn't go right, and the world as a whole seems terribly dark and pointless. But if I do my work right, it will make a difference - if I can develop a better method of doing something, that matters. I told him about the pleasure, when I was working in industry, of seeing a machine that was just sketches on paper at one point turn into a real, physical thing - and that in a special way, it was MINE, regardless of who it was actually sold to. I told him about the special feeling of realizing something that quite likely no one had seen before, looking up at the stars in the quiet night sky, and thinking that at that moment, you know some small thing that no one else does. The rewards are more than worth the difficulties, in the end. I don't know if he totally understood what I was trying to describe, but if he gets a little bit of it, maybe he'll gain more of a joy in learning.

Circular motion and Inequalities

On Friday in Mr. Ambrose’s physics class, the students did an experiment. They ran a string through a tube and connected one end to a weight; the other end was connected to a small rubber stopper. One person spun the stopper in a circle, and the other one timed it. The idea was to measure the period and the radius, calculate the speed and centripetal acceleration, and compare it with the force provided by the weight. Most of them had pretty good experimental data and relatively low errors, once they did the calculations properly. In the rush of doing the lab, there were quite a few careless errors, but they’re smart enough to see that if the results are THAT far off, something is wrong. I helped several of them find their mistakes, and many times it was a decimal point in the wrong place, or reading a number out of the wrong column. Running a “sanity check” on results is a good skill to get in any field, so I think making those errors actually had a use. Today, they moved on to doing problems on curves – things like how fast can a car go around a curve without skidding. They haven’t gotten through banked curves yet, but that will be the next topic, and then they’ll move on to universal gravitation.

In Ms. Colwell’s class, they’ve been working with inequalities, and graphing them on number lines. I’ve mentioned that inequalities are useful in linear programming, and it turns out they’re actually going to cover that. Since that falls into the area of optimization, Ms. Colwell is going to give me a class period next Friday to talk about optimization and non-linear programming. Obviously, I can’t teach them in one class what it takes multiple graduate classes to learn, but I can give them an idea of what you can do with this type of math and why it’s important.

Rotating and translating

Today was an easier day for both classes. In Mr. Ambrose's class, they had a test just before the break, so they were starting a new unit. He gave them a brief introduction to the equations involved in uniform circular motion, and a worksheet. Most of the students were having a bit of a hard time settling down - I can sympathize, it was hard to get up this morning after a long weekend. They were also checking with him on what their grades are - the marking period ended right before Thanksgiving, so report cards are coming out soon. There are several students who have pulled their act together and really improved a lot, and they can be proud of that. Also, a few people are starting to get college acceptances, and one young lady just got notified that she's admitted to her first-choice college. Now, she says it's a matter of scholarships, since she can't afford them.

In Ms. Colwell's class, they're working on using matrices for translations. I helped a few students, but most of them seemed to have the idea pretty well. A few of them had some minor confusion, but it was easily resolved, and most of them got a good start on the homework during class. One student didn't feel like doing his homework - he said he'd take it home. I asked if he wouldn't rather get it done and free up that time, but he said no. I think it's a shortsighted decision, but he has to deal with the consequences. By that class period, most people had settled down and were better able to work than during first hour. They're just about done with Chapter 4, and will be taking a test pretty soon.

Oh, and the subject of birthdays came up somehow... I don't remember the conversation... but mine is coming up this week. One student asked how old I'd be, and I told her that "it's 25 in hexadecimal." If she's interested enough, she can figure it out. I doubt if she'll go through the math to figure it out, though.

Almost to the break...

Most of the students seem really excited about Thanksgiving break. They don't have class Wednesday, Thursday, or Friday. I'm a little envious - when I was in high school, I only got Thursday/Friday off for Thanksgiving. A five-day weekend is pretty generous!

In Mr. Ambrose's class, they were reviewing for a test tomorrow. As usual, some of the students are getting the material really well, and others still need work. We'll see how they do on the test. There's one young man who's been doing a tremendous job - he's obviously been working hard, since he's pulled up his grade from the "OK/good" range to "great". He's doing all the stuff that he's been told he should to succeed - taking notes, at least trying problems even if he doesn't understand them, paying attention in class - and it's paying off. I'm really happy to see it, since it'll carry through to his future in college as well. What some of these kids need to learn is that putting in an effort is a big part of the battle - if they do that, they're ahead of a lot of people already. Anyway, their test is over Newton's Laws, including the good old familiar F = ma, and they also have friction in that unit. After the break, they'll be getting to the mathematics and force behind circular motion. They've already touched on it in lab - the fact that things won't go in a circle unless there's a force acting on them - and now they'll get the detailed treatment.

In Ms. Colwell's class, they're working on reflection and rotation using matrices. Some of the students still need practice with matrices, but they do have a fair amount of homework so they're getting the practice. On Friday, I gave a short presentation in the class, on how matrices are used in robotics. Those are rotation matrices, so they tied in quite well with what they're doing now. It also gave them a preview that the trigonometry they're going to learn later on will also be useful.

It's amazing to see how fast the year is going - it seems like just a short time ago, I was over at YHS for the first time, and now we're approaching the holidays. Before long, it'll be the end of the year - what an incredible thought.

Recognizing the points and the uses of things....

Today in Mr. Ambrose's physics class, they were reviewing a worksheet from Friday, and started on a new one. Some of the students started out class not really being sure what they needed to do to solve some of the problems, but by the end of class they seemed much more sure of themselves. One point that was brought out in one of the problems, and which we discussed, was the existence of irrelevant information. One of the easiest problems given was something like, "What is the weight of a 50-kg object when it moves with a horizontal acceleration of 5 meters per second squared?". The answer is just mg - 50 times 9.8. The acceleration is irrelevant. Mr. Ambrose talked about how, in labs, you can measure any number of things - the question is which things you need. I pointed out that it's useful in other fields, too - including humanities. Some of the students in the class, while talented in the sciences, don't plan on science-based careers, so tying everything together could be useful to them. We discussed the fact that whether writing an essay or solving a problem, one of the key things to ask yourself is, "What is the point?" In science, what is the problem I'm trying to solve? In English, it might be "What am I trying to say?" Another useful point they learned was that the normal force and the weight are NOT the same, even though they often are the same magnitude. To demonstrate that, Mr. Ambrose showed that you can change your "weight" on a spring scale by pushing down on a nearby object. We also noted that the scale reads in pounds and in kilograms, which isn't strictly correct since the scale does NOT directly measure mass. It really should have a label indicating that it's only calibrated for Earth-based use.

In Ms. Colwell's class, they learned how to multiply matrices. She had mentioned last week that some teachers only taught how to multiply them by hand, others only taught how to do it on calculators, and she taught both. I think she's got the right approach - sometimes, for small matrices, it's quicker to just do it by hand, and it's always a good idea to understand the background instead of just having a "black box" - or gray, in the case of some calculators. But it's also useful to know how to use the tools to make it quicker, once someone understands it. I may know how to multiply a 100 x 100 matrix by a 100 x 80 matrix by hand, but that doesn't mean it's a good use of my time. Also today she told the students about some of the uses of matrices. Her list didn't have any good engineering examples, so I added one - robotics. Robotics is way cooler than business, at least to a lot of people (myself included), so she's asked me if I can put together a quick example of how matrices are used in robotics. I'm going to look in my notes from robotics classes - I took a couple of robotics courses in my masters' degree at UIC back in about 1999/2000 - and put together something they can understand that will show how you can use them to work with really cool stuff.

Going in Circles

Today in physics, Mr. Ambrose had them doing a simple lab. Each group had a piepan with a section cut out of it, and they had to roll a golf ball around the edge. The important task was to predict what would happen when the ball came to the empty section, and then to see what really happened - would it continue on a curved path in the absence of anything to force it to do so, or would it go straight? Anyone who's had physics can tell what it did - it went straight. Some of the students predicted that, but others thought it would keep going in a curved path. I told them not to kick themselves over being wrong - Galileo had the idea that there was some form of "circular inertia", that the planets would naturally move in circular paths without any applied force. It wasn't until Newton that someone realized the truth, and formulated it as a law of nature - which is why this one is Newton's first law. It's instructive for some students to realize that this stuff which is so obvious once you learn it was once a new, shocking idea.

In Ms. Colwell's algebra class, they were working a practice test - they have the real thing on Wednesday. Many of the students are doing quite well, but there are some who still struggle. One of the challenges for me is to figure out who needs help, and who needs to be left alone for a few minutes to think. Some people catch on quickly; others catch on, but they need to be given a chance to do so. If someone jumps in to help too quickly, it short-circuits the process. One student in particular needed to be given a few minutes. He knew what he was doing, but needed a chance to think about it. The key, I think, is to know when someone is getting frustrated because they're stuck, and when they just need space to work through the idea at their own pace. I wish I knew the answer to that - I try to read it from their expression and the way they're acting. One thing that does seem to work sometimes is to look at how they're holding the pencil. Someone who's frustrated tends to clench it, whereas a person who's thinking holds it a little more loosely. I don't know if it always holds true, but it seems to be a good enough guide to start with. The truth is, everyone's different, so there won't be an answer that's true 100% of the time about 100% of the students. It's not like a problem in math or science that always has the same answer - people are far more complex than that. Of course, that's part of what makes them interesting.

Off-Schedule...

Today isn't normally a day that I go to the high school, but my schedule's a bit off; I was out of town at a conference and re-arranged when I do what for this week. Friday I'll be back to the normal schedule. Today was also kind of an unusual day at Ypsilanti High School. They had MME testing going on, so there were no regular bells, no announcements, and a fair number of students weren't in class. Mr. Ambrose used the class time to review, let students work independently, and for students to complete make-up work like quizzes that they'd missed. Some of them used the time wisely, but others just sat and talked. I'm not sure whether they were caught up, or just didn't feel like concentrating on their work. As much as it's a good idea to use your school time to work, I know sometimes you just can't focus well. I did learn something interesting from one young woman; she mentioned that her mother had once studied engineering, but didn't complete her engineering degree. She said her mom has considered going back to school. I really hope she does give it serious thought - not only would it set a great example for her daughter (who is a very bright and ambitious young woman), but it would be a great thing to do for her own sake. I know a lot of people say you can't go back to school, but that's just not true. I worked for 13 years before quitting to start on my Ph.D., and while being an older student presents special challenges, I bring a different perspective which I think can be helpful. Diversity isn't just racial and ethnic, although that is important - it's everything that makes a person who they are and influences how they think. That includes race, ethnicity, gender, age, and life experiences - and such a long list of things that if I typed them all, this post would be HIDEOUSLY long and no one would ever finish reading it.

In Ms. Colwell's algebra class, they were working on linear regression and on sequences. She's really trying to get them to think - for instance, they had a set of points, figured out the slope and intercept of the equation for the best-fit line, and then she asked them, "Is a line a good fit for these points? Why or why not?" In this case, the correlation was around 0.18, and after they discussed it, most of the students understood that it really isn't a good fit. I explained how, when you work with data, it's important to look at whether the model you're trying to use (in this case, a line) really matches. You can find the "best fit", but it might still be pretty bad if the points are all over the map - or if you try to fit a line to a quadratic relationship. When they started working with sequences, I mentioned that I've had homework (in an optimization class) that involves sequences; often, when you're using an iterative method to find a solution, you start with some value and use a specified rule to find the next value in the sequence, then the next, until you're done. Obviously, the sequences they study are far simpler than what a graduate student has to use - simple ones like 3, 6, 9, 12, ... but I was able to let them know that this, too, actually has some real-world purpose beyond just filling a section of the algebra book.

More Projectiles, and Slope-Intercept Lines

Today in Mr. Ambrose's physics class, the students were doing a lab. In this lab, they dealt with projectiles fired at an angle. In the first part of the lab, they fired a marble with the "cannon" at a horizontal position, and used the point where it landed to determine the muzzle velocity. Next, they chose an angle, predicted where it would land based on the muzzle velocity, and fired it to test the prediction. Generally they did pretty well with their predictions, though they were a little off. Most of them see to grasp projectile motion fairly well, although there are still a few who are confused. One thing that I see is that some of them instantly start putting things into equations without ever asking themselves, "Is this equation applicable?" As an example, they have an equation for the range in terms of initial velocity and angle - but it only applies if the projectile is launched from the same height where it lands, which isn't always the case at all, and certainly wasn't in the lab. I tried to explain that you always have to consider what your circumstances are, and when a certain equation is valid, and I hope that came across. Generalizing what they've learned to new problems can be a challenge for some of them, though it's a critical skill that they would find useful in other areas.

In Ms. Colwell's algebra class, they're working with lines in slope-intercept form and graphing them. Most of them seem to be getting it pretty well - I helped a few with questions, but for the most part, they were doing OK.

Fire one! Fire two!

Today, Mr. Ambrose had his physics class doing a projectile motion experiment. They had a small launcher set up on a lab bench ready for them, suitable for marbles. There was a piece of paper on the floor with distance markings from the muzzle of the launcher. Each student fired a marble once to test range, and then to get data, they put a piece of carbon paper down in the proper area. The whole idea was to measure where the marble landed, and using the height of the launcher, landing point, and acceleration due to gravity, figure out what the speed was when it was launched. For extra credit, they were supposed to figure out where to place a hoop of a given height so that the marble would go through it. Unfortunately, time ran out before that could get done, though several teams did very nice calculations of where to put the hoop. Some of them have caught on very well to the concepts, though there are still a few who seem a little shaky on it, and one or two who are really struggling.

In Ms. Colwell's advanced algebra class, they're learning about joint variation, including combined variation - all those multi-variable equations we all know and love. Some students needed help, but many of them really just need practice. Several students who didn't seem to pay attention before are now taking notes and appear to be trying, which is a great sign. One student who was doing badly has improved tremendously - he raised his hand to answer a question (correctly) today, did well on the last quiz, and appeared to be doing fine on the worksheet they were given. He's one of the students who I suggested go to tutoring; if he did in fact go, it certainly helped. Obviously he's doing something differently. That's one of the best parts of this program - seeing that something has made a difference to one of the students.

Conic sections everywhere…

The physics class is working on projectile motion now – one of the most basic applications of two-dimensional kinematics, and of course, that’s our familiar parabolic motion that we all love. Today, Mr. Ambrose was doing examples for them. Some of them seemed to understand, but there are a few who are having trouble with the concept that you can separate out the X and Y motions, and treat them separately. I’m trying to think of something that would help – a lab, or demonstration, or some particular example, or something that would make the concept more concrete to them. This is really a key point and if people don’t get it, they may be able to do problems, but they won’t truly understand the material.

In Ms. Colwell’s algebra class, they’re working with hyperbolas and inverse-square functions. As they were going through them, talking about the domain and range, I pointed out that they really do need to be aware of domain and range of functions. In my optimization class that I’m taking, and in the work I’m doing for my research, that is something that has to be considered. Some of them seem to think that once they’ve finished a particular section of the book, they’re done with it. I’m trying to help them see that it comes back – as I put it today, “like the night of the living dead” – a bit of a weird twist, but it seemed to get a few people’s attention. There are a few students who really don’t seem very motivated, and I’m wondering how to reach them. I don’t know if they’re distracted by outside issues, or whether they simply aren’t interested in the material. Right now, I don’t know what the answer is, except to work with them when they’re having trouble and not give up.

Vectors and Variation

The physics class is working on vectors, in preparation for two-dimensional kinematics. Today they were going over some examples and doing a problem in class. One thing that hurts some of them is that they don't take notes. Some do, of course, but there are a few students who never write anything down. I'm not sure if it's because they don't see the point, or if they never learned how to take good notes, but it's definitely something they need to work on, especially since a lot of them intend to go to college. Perhaps the tutors can help make the point that note-taking is an important skill, and maybe a few hints on taking good notes would help.

In algebra, they spent the class on the Fundamental Theorem of Variation. It seems pretty obvious to someone who's been doing math in various contexts for years - if you double "x" in this equation, or multiply it by 3, what happens to "y"? They're all capable of doing the algebra, but it takes some time and practice to get confident with it, since to them it is a new concept. A few of them could use extra help and may go to tutoring.

Update on both classes

In Mr. Ambrose's physics class, they just got back their first test today. Some of them were happy with how they did, but some of them would like to have done better. I've let them know that tutoring will be available from NSBE, and some of them do seem interested in it. A few of them may find it very helpful to get caught up.

So far, they've gone through one-dimensional kinematics. The next thing they're going to be doing is going over vectors, in preparation for two-dimensional kinematics. I was talking with Mr. Ambrose about how to get the concepts illustrated, possibly through some kind of lab or activity.

In Ms. Colwell's algebra class, they're getting ready for a test tomorrow. I was helping some of them work problems in class, and I noticed something interesting. Many of them really do have the right idea, and can do the problems if you ask them what each step is, but they seem to lack confidence. That may be something that practice would remedy. It's a big problem, though, since I can see it holding many of them back. They're given an equation to solve, and they know what to do, but they don't feel confident that they're doing it right.

Recently, they've been working on solving equations, on functions (also on recognizing what is and is not a function), and on word problems - taking a problem and formulating the equation, then solving it. It seems to me that they do need more practice on word problems, as well as solving equations, but most of them understand what a function is pretty well.

Uses for Math

So far, I've mostly been observing in Ms. Colwell's Advanced Algebra class and occasionally helping a student with their individual work in class, but today I had a short presentation put together, which I gave to both her third hour and fourth hour classes. It was titled " What’s math good for, anyway? A true story of an engineer on vacation, a desperate customer, and a literal back-of-envelope calculation." It was about a drive-train calculation that I did on an envelope, in a parking garage in Milwaukee, on my cell phone with a customer. I'm not sure if the students followed all of the math, though it wasn't anything really beyond algebra, but at least they got to see that math has some actual uses in real life. I tried to engage them in the presentation by asking them questions and showing them that they already know at least some of the stuff that I was using as a practicing engineer. Ms. Colwell seemed to think that it was of some use to them - breaks up the usual routine and shows them something new. As an assignment, she asked them to take a few notes on what I had to say and include them with their homework when they turn it in tomorrow, so I know they were at least half listening. Some seemed more interested than others, of course - not everyone has the same level of interest in the same things.

I'm going to e-mail her a copy of the presentation so she can have it for her files - it might come in useful in the future as an example for an advanced math class.

Physics Experiment

The physics class did an experiment on Friday & analyzed the data today. On Friday, we went outside and measured off how far we were from the wall of the school, and clapped two pieces of wood together. The idea was to synchronize the claps with the echo, time how long 20 claps took, and use that data to calculate the speed of sound. The class also calculated what it should be based on the air temperature for the day.

Everyone's value was a little high, but some people were closer than others. One set of lab partners got a value of 335 m/s, where we'd calculated 333 m/s - extremely close! The farthest value was the teacher's - he measured over 400 m/s for the speed of sound. One of the students asked about eliminating that piece of data from the set when calculating the average for the whole class, which started an interesting discussion of when you can or can't eliminate data points - and how do you know?

The whole class seems very bright and interested in the subject - it's a joy to help out with them. They'll be starting on kinematics next, and Friday when I'm there Mr. Ambrose is planning to do another lab.

Starting out the semester

Well, it's the start of the semester and the beginning of a new experience working with the TA partnership program. I'm looking forward to it - I'll be working with one of the math classes (advanced algebra) and one of the physics classes, spending Monday and Friday mornings at the school.

Thus far, I've met with both teachers - Thomas Ambrose and Doriane Colwell - over at the high school, and have gotten a bit of an idea what they intend to cover in their classes.