Today was a fairly quiet day... Mr. Ambrose gave his physics student a quiz. There was, as usual, some grumbling about that - no one really likes quizzes that much. They seem to be almost done with the wave material they're covering, so they'll be moving on to a new unit soon.
After going to a teacher's event at U of M, Mr. Ambrose has decided that he'd like to get some design experiences into the class, so we talked about that today. We're going to start with designing and building toothpick bridges - they're easy and simple - and then move on to do some other things. Fridays and Mondays, when I'm there, will be the designated day for those projects. I think the students will like it - we'll see how long they take before deciding how many there will be. This should be a really good way for me to have more of an impact, beyond helping with labs and things, and really let them see for themselves what engineers do.
Friday, February 29, 2008
Friday, February 22, 2008
Doing the Wave
Mr. Ambrose has started a new topic in physics - wave motion. Today, after a brief lecture, he had the students experimenting with long extension coil springs - and by long, I mean about 6 or 7 feet in their relaxed state. They worked in pairs and created both transverse and longitudinal waves, generated standing waves, and saw that waves would bounce off of the other end and return, and that two waves would pass through each other. There was no formal lab report, but they were required to participate, and most of them seemed to be having fun with it.
In Ms. Colwell's algebra class, they're getting ready for a quiz on Monday. They really don't want it then, but the snow day earlier this semester did disrupt the schedule a bit. I can sympathize with them - a day off at the time is nice, but it does have repercussions later. I talked with one of the young women in the class who had trouble grasping the whole purpose of imaginary numbers, and she's happier with some of what they've done more recently - their warm-up today included a problem on compound interest. She could not only understand that problem, but also see how it impacts her life. Imaginary numbers seemed very abstract to some of them. They're also working with powers, and most of them seem to understand that fairly well. Ms. Colwell always tries to tell them about applications for the math they're learning, so she mentioned that a lot of equations in science have a variable to some power.
In Ms. Colwell's algebra class, they're getting ready for a quiz on Monday. They really don't want it then, but the snow day earlier this semester did disrupt the schedule a bit. I can sympathize with them - a day off at the time is nice, but it does have repercussions later. I talked with one of the young women in the class who had trouble grasping the whole purpose of imaginary numbers, and she's happier with some of what they've done more recently - their warm-up today included a problem on compound interest. She could not only understand that problem, but also see how it impacts her life. Imaginary numbers seemed very abstract to some of them. They're also working with powers, and most of them seem to understand that fairly well. Ms. Colwell always tries to tell them about applications for the math they're learning, so she mentioned that a lot of equations in science have a variable to some power.
Monday, February 11, 2008
So, what is all this stuff good for, anyway?
There was a special event going on today, and only five students were in Mr. Ambrose’s physics class. Instead of covering something that the majority would have to make up, he had a class activity with an air track – demonstrating elastic and inelastic collisions. The velocities of the cars on the track were measured with photogates, and we checked to see if momentum was conserved. We expected to see that a little bit of momentum would be lost, but that it would be close. Unfortunately, that wasn’t the case – the cars gained momentum. So, either we broke a law of physics, or else there was a large source of experimental error. We did find one big source – the air track wasn’t level. The cars were being accelerated by gravity, which messed up the results. Despite not being level, one set of results was within 25%, which isn’t bad for an ad-hoc setup.
In Ms. Colwell’s class, I gave a short presentation on the uses for imaginary numbers. They’re really not that hard to work with, given a little bit of practice, but conceptually they seem a bit remote from ordinary experience. Some of the students seem to be very “concrete” thinkers, and couldn’t quite grasp the meaning of imaginary numbers last week when they were learning about them. This is a common problem, I’d expect. I don’t remember what I thought when I first encountered them – I graduated from high school back in 1989, so it has been a little while – but several very prominent mathematicians had trouble with them. One girl’s comments last week echoed a quote from Leibniz - “I did not understand how… a quantity could be real, when imaginary or impossible numbers were used to express it.” She was having trouble with the idea that you could multiply a complex number by its complex conjugate and end up with something purely real. On the worksheet I gave them, I included a few quotes like this, and asked them for their views on complex numbers. There are no right or wrong answers – but I wanted to see them think about what the numbers mean.
In some contexts, they do have a real physical meaning, and that’s what the presentation was meant to show them. I chose two examples of places where imaginaries have real uses – electronic circuits, and control systems. In electrical engineering, use of imaginary numbers allows inductors and capacitors to be treated like resistors, and circuits to be described by algebraic equations rather than differential equations, which is a major simplification. In control systems, a particular system has a characteristic equation, and the real and imaginary parts of the roots tell different things about the system; the imaginary part tells how oscillatory it is – or as I described it to them, how “bouncy” it is. Obviously, in just a short time, I couldn’t go into any depth, but the goal was to let them know that these things do have uses and even physical meaning.
Next, they move on to powers. They’ve long been familiar with the very basics – squaring, cubing, etc. – but now they get to move on to negative and fractional powers. It’ll be interesting to see what kinds of conceptual difficulties this presents; again, it takes a simple concept that they can picture and extends it into a realm where it isn’t as easy to grasp.
I won’t be in on either Friday or next Monday, since they have a break – the students get a five-day weekend, and the teachers get four days, with a professional development day next Tuesday. So, my next post will most likely be a week from this Friday, when I’m back.
In Ms. Colwell’s class, I gave a short presentation on the uses for imaginary numbers. They’re really not that hard to work with, given a little bit of practice, but conceptually they seem a bit remote from ordinary experience. Some of the students seem to be very “concrete” thinkers, and couldn’t quite grasp the meaning of imaginary numbers last week when they were learning about them. This is a common problem, I’d expect. I don’t remember what I thought when I first encountered them – I graduated from high school back in 1989, so it has been a little while – but several very prominent mathematicians had trouble with them. One girl’s comments last week echoed a quote from Leibniz - “I did not understand how… a quantity could be real, when imaginary or impossible numbers were used to express it.” She was having trouble with the idea that you could multiply a complex number by its complex conjugate and end up with something purely real. On the worksheet I gave them, I included a few quotes like this, and asked them for their views on complex numbers. There are no right or wrong answers – but I wanted to see them think about what the numbers mean.
In some contexts, they do have a real physical meaning, and that’s what the presentation was meant to show them. I chose two examples of places where imaginaries have real uses – electronic circuits, and control systems. In electrical engineering, use of imaginary numbers allows inductors and capacitors to be treated like resistors, and circuits to be described by algebraic equations rather than differential equations, which is a major simplification. In control systems, a particular system has a characteristic equation, and the real and imaginary parts of the roots tell different things about the system; the imaginary part tells how oscillatory it is – or as I described it to them, how “bouncy” it is. Obviously, in just a short time, I couldn’t go into any depth, but the goal was to let them know that these things do have uses and even physical meaning.
Next, they move on to powers. They’ve long been familiar with the very basics – squaring, cubing, etc. – but now they get to move on to negative and fractional powers. It’ll be interesting to see what kinds of conceptual difficulties this presents; again, it takes a simple concept that they can picture and extends it into a realm where it isn’t as easy to grasp.
I won’t be in on either Friday or next Monday, since they have a break – the students get a five-day weekend, and the teachers get four days, with a professional development day next Tuesday. So, my next post will most likely be a week from this Friday, when I’m back.
Monday, February 4, 2008
Back from a Surprise Three-Day Weekend
Friday morning, I checked the website for the local school closings, and discovered that the Ypsilanti Schools were closed due to snow. So, today when I went in, I hadn't seen the kids since last Monday, and they'd just had a three-day weekend.
In physics, they started off with a quiz on momentum. Some of the students said they'd forgotten too much over the weekend, but Mr. Ambrose lets them use their notes, books, and worksheets - everything but their neighbors - so if they've kept up with the work, then they've got plenty of material to draw on. Most of them seemed to finish up without problems, though two need to come in to finish later. (Makes me wish I could do that with tests - it would make them a lot easier!) After the quiz, they went over elastic and inelastic collisions. They should be finishing up momentum soon, and then they'll be moving on to other topics like electricity.
In Ms. Colwell's advanced algebra class, they're working with complex numbers. One girl couldn't quite understand how a number that "isn't there" could mean anything - she asked, how could you have "i" apples? Her math skills are good, but she seems to need to tie everything to physical objects. Another girl mentioned that her older brother is studying electrical engineering - I'm sure he uses plenty of complex numbers. Since I use them a lot - they're ubiquitous in controls work - I offered to put together a brief presentation for the class. Ms. Colwell and I agreed that next Monday would be a good time for it, since they have a test scheduled for Friday. I can also make up a brief worksheet to go with it that can be used as a warm-up - the assignment that gets them to settle down and work, and which keeps them busy while she takes attendance. Maybe, if they see areas where complex numbers are used, they'll seem a little more "real" to them. I can't blame them for having some conceptual difficulty - after all, for a long time mathematicians had trouble figuring out complex numbers. Descartes, who certainly was a talented mathematician, dismissed them as being meaningless. I've told some of them about things like that, and that it's OK to have trouble visualizing these things. They're easy enough to work with but they can be hard to understand. The fact that we don't even blink at them doesn't mean they're easy - it means they're familiar.
In physics, they started off with a quiz on momentum. Some of the students said they'd forgotten too much over the weekend, but Mr. Ambrose lets them use their notes, books, and worksheets - everything but their neighbors - so if they've kept up with the work, then they've got plenty of material to draw on. Most of them seemed to finish up without problems, though two need to come in to finish later. (Makes me wish I could do that with tests - it would make them a lot easier!) After the quiz, they went over elastic and inelastic collisions. They should be finishing up momentum soon, and then they'll be moving on to other topics like electricity.
In Ms. Colwell's advanced algebra class, they're working with complex numbers. One girl couldn't quite understand how a number that "isn't there" could mean anything - she asked, how could you have "i" apples? Her math skills are good, but she seems to need to tie everything to physical objects. Another girl mentioned that her older brother is studying electrical engineering - I'm sure he uses plenty of complex numbers. Since I use them a lot - they're ubiquitous in controls work - I offered to put together a brief presentation for the class. Ms. Colwell and I agreed that next Monday would be a good time for it, since they have a test scheduled for Friday. I can also make up a brief worksheet to go with it that can be used as a warm-up - the assignment that gets them to settle down and work, and which keeps them busy while she takes attendance. Maybe, if they see areas where complex numbers are used, they'll seem a little more "real" to them. I can't blame them for having some conceptual difficulty - after all, for a long time mathematicians had trouble figuring out complex numbers. Descartes, who certainly was a talented mathematician, dismissed them as being meaningless. I've told some of them about things like that, and that it's OK to have trouble visualizing these things. They're easy enough to work with but they can be hard to understand. The fact that we don't even blink at them doesn't mean they're easy - it means they're familiar.
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