Digging out the proof that is stuck in the pudding May 24, 2012
Posted by mareserinitatis in education, gifted, homeschooling, math, older son, teaching.Tags: CLEP, economics, geometry, math, older son, proofs
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Since the older boy was kicked out of school, I’d say he’s been doing more academically than before when he was in school. After he passed his GED in March, I asked him what he wanted to do until summer. He had the choice of getting a job or studying for a CLEP exam. He usually spends a good chunk of the summer with relatives, so he decided to wait on looking for a job and instead aimed to finish another CLEP. He chose to study macroeconomics. To do this, he got up nearly every morning and spent 3 hours at the university library (where he has no internet access), read through the entire textbook, and worked through the study guide. He passed the test on Monday, and we’re all very proud of him for his hard work. (He, however, was disappointed that he didn’t get a higher score and now wants to spend some time going through the text again to figure out the parts he got wrong.)
In addition, we began talking about college things, and I told him that he should take the PSAT in the fall because doing so would automatically enter him into the National Merit Scholarship Program. This is a scary topic because it requires that he go back and do something he hates: math. However, he keeps telling me he really wants to go to college, so he was willing to go back and do some. Of course, saying it and doing it are two different things.
He’d finished algebra 1 two years ago and last year, he’d made an attempt to jump into college algebra. He made it a good chunk of the way and then started having some real difficulties. Therefore, I decided to take a step back and see if he could get geometry done before summer. It turns out that he was better off than I thought because he did the initial evaluation and tested out of about 2/3 of the topics. In the past month, he finished off all the rest except for a handful, all of which had to do with proofs. (Apparently, he is serious about the PSAT.)
I have to admit that this is different than when I took geometry. My geometry class was entirely proofs. It was one of my favorite classes because, to me, doing a proof is a completely different animal than solving an open-ended problem. You know where you’re starting and finishing. All you have to do is find the path between here and there. Usually it was extremely obvious, so I was able to write out my proofs for class and often have time left over to read. I remember being very confused why other people thought the class was hard. Later on, when I took physics in high school, it felt like the same thing. You’re trying to find out a quantity using a bunch of other quantities and formulas. Easy peasy…
I sat down to help the older boy yesterday, and I have to admit I got frustrated pretty quickly. I read the problem, saw what was supposed to happen, and knew immediately the steps in the proof.
Problem was the older boy didn’t.
This really threw me for a loop. I mean, the kid’s obviously smarter than me (and just as obviously less wise and experienced). It really stunned me that there were a couple points where he was struggling to figure out what to do next. He was getting frustrated, though, so I walked him through a few of them, explained the reasoning, and tried to talk to him about how I viewed the problem (which is hard to do when you think in terms of vague notions of going places on diagrams).
It got me wondering, though, if this is why he doesn’t like math. Is it that hard for him to see the end goal? Is the process of finding logical steps difficult? And why is it so easy for me to formulate these things and difficult to him? Do our brains work differently? The whole thing left me with a lot of questions, and I’m still very perplexed.
By the end of the session, he seemed to have it down and was making good progress. I was able to back off and just let him work, and he even found some of his errors when he got things wrong. The best part was, however, at the end when he turned to look at me, grinned, and said that it was actually kind of fun. Mission accomplished.
Second grade logic and rulers February 23, 2012
Posted by mareserinitatis in education, gifted, math, teaching.Tags: angles, education, geometry, kids, rulers, sir cumference
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Today, I went back to work with the second graders. We’ve been spending a lot of time talking about circles and degrees and Babylonian units and π.
My plan has changed from the original one of trying to teach the kids a bunch of applied stuff. I’ve pretty much given in to teaching them historical discoveries in math simply because there’s a lot of stuff you can do that doesn’t require multiplication and division. It’s been a lot of fun, but I decided to try something different but related: I wanted to teach them out to make a formal mathematical proof. Okay, not terribly formal. What I want them to learn is how to use logic to make a proof. I suspect some of them know already (based on some of the arguments I’ve had with my son who has rock solid 7-year-old logic). However, I’d like them to use their brains for good instead of getting out of (or into) trouble.
The thing about geometrical proofs is that they really aren’t that hard. At least, I never found them to be. I remember sitting in 10th grade geometry and being given T-charts. I would race through them and ace them all. I was horribly surprised to see that my classmates had difficulty with them as well as complaining the teacher was too abstract. I threw the idea of proofs out to Mike, and he said that pretty much the only tool you need is a brain, so it’s probably a good idea. (I’d have to disagree…you need a brain…but you also need a pencil, paper, ruler, and protractor. But otherwise, I think he’s right.)
Today, we started with the concept a line and measuring its angle. I know my former math instructor wouldn’t approve, but I’m teaching them to use degrees (aka Babylonian Units) because that’s what’s on the protractor. Also, I’m not sure how versed they are in fractions, so we’re not going to get into fractional parts of π. (Actually, if anyone has ever seen a protractor with units of π rather than degrees, please let me know as I’d love to buy it.)
Once we had a line, then I told them to draw a point on the line with another line coming out of it, so that it would look like this (without the measurements):
Each of them drew the line coming out an a different angle. They all measured their angles and found that they all summed to 180°. A couple of the kids seemed surprised that they all ended up with the same number. Incidentally, those that didn’t seem surprised were very absorbed with the flexible rulers I had brought to use. (Note to self: second-graders are easily distracted by anything novel.) We then talked about how any two angles, if they formed a straight line, would add up to 180° and how this was known as the supplementary angle theorem.
Once we had that down, we used it to prove the vertical angle theorem. It took them a bit to realize that the line created by adding supplementary angles doesn’t have to be horizontal (like in the picture above).
That’s all we got through today, but I plan on using this to show them that the interior angles of a triangle always add up to 180°. It might take us a couple weeks to get there, especially since next week I’m supposed to read them a couple more of the Sir Cumference books.
Sterred, not shaken March 15, 2011
Posted by mareserinitatis in electromagnetics, engineering, grad school, physics, science.Tags: fields, geometry, radians, spheres, steradian, stiridian
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A couple years ago, I was taking a class, and the professor put up a slide talking about stiridians.
What the heck is that, I thought?
It turned out that it was a misspelling. Apparently one of the references my prof chose happened to use that misspelling, and he had merely copied it. After class, I tried to politely let him know his error.
So what the heck is a steradian, anyway? And how would I know such an obscure word?
In order to understand what a steradian is, we should step back and look at it’s one dimensional analog, the radian. (Because I’m using an analog, I believe that qualifies me as an analog engineer!) If you’ve had trig, you’re familiar with the radian: it’s an angle in a circle that creates an arclength equal to the radius of a circle. For those of you who prefer Babylonian Units, one radian is approximately 57.3°. Graphically, it looks like this:
When you work with antennas, you generally have to work in three dimensions (unless you get lucky and have an axis of symmetry). The reason we need three dimensions is because we’re working with both electric and magnetic fields, both of which are vector quantities and change within a sphere. As an example, this shows the fields for a dipole antenna:
The electric field direction is in blue and the magnetic in red.
It turns out that when we’re describing these patterns, it’s useful to think of the surface of a sphere. We need to describe where the field is strong or weak over that sphere. Unfortunately, using a two dimensional measure of angle is inadequate for a field.
This is where the steradian comes in. If we want to describe an area of strong field, we can describe it’s span in steradians. This is a measurement of ‘unit solid angle’ – although it’s easier to think of in terms of the area of a sphere.
A steradian is the area equal to the square of the radius of the sphere. That’s it! There are 4π steradians on the surface of a sphere (similar to the 2π radians in a circle). And you can imagine that this is what it looks like:
Also, like a radian, the steradian is a unitless measure, but you can annotate it using sr (much like some people indicate radians with rad).
As I mentioned, I used this in antennas, and I actually wrote an explanation in my MS thesis. So, obscure as it may be, I had actually spent a bit of time dealing with the topic. And now that you all know about it, I sure hope that professor corrected his notes lest one of you sees it.
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