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.
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that is a good ending! =D
If he wants to do more economics, he needs the math!!! (Though this mommy is hoping her son doesn’t grow up to be an economist )
I think I already recommended my favorite proof-based geometry textbook. http://www.amazon.com/Geometry-Enjoyment-Challenge-Richard-Rhoad/dp/0866099654 There’s two views about what a geometry class should be, and I think this one does a great job teaching proof-based geometry. I taught my sister out of it so she would be able to not have to take geometry again when she transferred to Catholic school (she’d had the “proofs are useless” version at the public school whereas I’d had the “proofs” version and the Catholic school still did proofs).
Being able to do proofs does not have to come automatically. The reason the class is so valuable is because it trains minds into a specific system that is useful for many many fields in the future including computer programming, engineering, and yes, economics. If he can let things go in economics, he’s got what it takes to make all these other things fall into place, but it’s going to take time, effort, and training his mind. If you’re already thinking like a scientist or engineer etc, of course geometric proofs are going to seem easy, but for most people geometric proofs are the *first* time we see this way of thinking. And that’s why they’re so valuable.
The important thing is to be able to get into that growth mindset and not let the frustration over things not coming automatically stymie you. And there’s a lot of value in working through that too.
I think part of the frustration is that I knew how the proof was supposed to work, and it was very automatic to me. Explaining how I figured it out was hard. Something like, “Well, I know that if I have this and this angle, then it says something about that angle, and then we can infer that these are parallel lines.” But I really can’t explain how I knew that. I have suspected it is as a result of learning to program at 9, because programming follows the same logical progression (if done halfway decently) as proofs. (And I’ve been trying to get him to learn to program for a long time, but he’s not interested in doing anything involving a command line…) The other part is that we learn in similar ways, so I was rather surprised the proof thing wasn’t automatic to him as it was to me.
Today he said, “Proofs are easy once you get how they’re done.”
So I think he’s made that breakthrough. He hates math, but he’s really good at it.
I am hoping the older boy takes a long-term interest in economics because it’s the first thing I’ve seen him get fairly interested in. He still says he wants to be a writer, but I’ve told him he’ll have better luck with it if he can find another area to study about which he can write. Also, it’s helpful if that area happens to be one that can earn better money than an English major. (Nothing against English majors except that every one of them I know has had a hard time finding a job that makes them happy.)
There are a *lot* of perks to being an economist. And I frequently get freelance writing offers. Being able to do technical writing, especially if you can explain technical concepts to a wider audience, puts someone way ahead of the game.
If I ever had to leave academia, my options are pretty wide.
There is a downside, however, in that a lot of economists are kind of jerks, especially the male ones. We don’t have the best social skills, we often have high opinions of ourselves, and many of us have taken the rationality and self-interest lessons to heart. (Again, more common male economist stereotype than female.) That’s not to say there aren’t economists interested in the public good, but there are definitely studies that show that economists donate the least of any major in public goods games. Whether that is selection or training is still an open question.
If it helps, I think the reason he got interested in economics is that he saw people using it to argue points that he strongly disagreed with. In some cases, it was being used as a rationalization for being selfish and he wanted to know if their arguments really were being backed up by economic theory or if they were just cherry picking or even completely misinterpreting things.
He’s fairly analytical, but he (at least to me) doesn’t seem like he’s the kind of person who is engrossed in self-interest. He’s clueless in the way that teenage boys are, but he genuinely tries to be a nice person. I can easily see him trying to use it as a tool to develop better social policy. (One of his other interests is politics, so I can see those going together quite nicely…)
When he gets to college he will have to take a Public Finance/Economics class. The first few courses teach the basic economic theory about when the market works, but then after you have the basics you can learn about market failure and why we need government.
He may actually be interested in majoring in Public Policy instead of Economics itself. Michigan’s Ford school is *excellent* if he wants to stay in state. There are some pretty amazing people in the economics department as well.
College is a lot more fun than high school in so many ways, especially if you’re into social science. There’s so much to learn.