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Archive for September, 2016

Vladislav Zorov, a Quora user, brought this to my attention. It is a worthy follow-up to Jerry King’s “The Art of Mathematics,” called Lockhart’s Lament. Lockhart really fleshes out what King was talking about. Both are worth your time. Lockhart’s lament reminds me a lot of a post I wrote, called The challenge of trying to get a real science of computing in our schools. I talked about an episode of South Park to illustrate this challenge, where a wrestling teacher is confronted with a pop culture in wrestling that’s “not real wrestling” (ie. WWE), as an illustration of the challenge that computer scientists have in trying to get “the real thing” into school curricula. The wrestling teacher is continually frustrated that no one understands what real wrestling is, from the kids who are taking his class, to the people in the community, to the administrators in his school. There is a “the inmates have taken over the asylum” feeling to all of this, where “the real thing” has been pushed to the margins, and the pop culture has taken over. The people who see “the real thing,” and value it are on the outside, looking in. Hardly anybody on the inside can understand what they’re complaining about, but some of them are most worried that nothing they’re doing seems to be enough to make a big dent in improving the lot of their students. Quite the conundrum. It looks idiotic, but it’s best not to dismiss it as such, because the health and welfare of our society is at stake.

Lockhart’s Lament — The Sequel is also worth a look, as it talks about critics of Lockhart’s argument.

Two issues that come to mind from Lockhart’s lament (and the “sequel”) is it seems like since we don’t have a better term for what’s called “math” in school, it’s difficult for a lot of people to disambiguate mathematics from the benefits that a relative few students ultimately derive from “math.” I think that’s what the critics hang their hat on: Even though they’ll acknowledge that what’s taught in school is not what Lockhart wishes it was, it does have some benefits for “students” (though I’d argue it’s relatively few of them), and this can be demonstrated, because we see every day that some number of students who take “math” go on to productive careers that use that skill. So, they will say, it can’t be said that what’s taught in school is worthless. Something of value is being transmitted. Though, I would encourage people to take a look at the backlash against requiring “math” in high school and college as a counterpoint to that notion.

Secondly, I think Lockhart’s critics have a good point in saying that it is physically impossible for the current school system, with the scale it has to accommodate, to do what he’s talking about. Maybe a handful of schools would be capable of doing it, by finding knowledgeable staff, and offering attractive salaries. I think Lockhart understands that. His point, that doesn’t seem to get through to his critics, is, “Look. What’s being taught in schools is not math, anyway! So, it’s not as if anyone would be missing out more than they are already.” I think that’s the sticking point between him and his critics. They think that if “math” is eliminated, and only real math is taught in a handful of schools (the capacity of the available talent), that a lot of otherwise talented students would be missing out on promising careers, which they could benefit from using “math.”

An implicit point that Lockhart is probably making is that real math has a hard time getting a foothold in the education system, because “math” has such a comprehensive lock on it. If someone offered to teach real math in our school system, they would be rejected, because their method of teaching would be so outside the established curriculum. That’s something his critics should think about. Something is seriously wrong with a system when “the real thing” doesn’t fit into it well, and is barred because of that.

I’ve talked along similar lines with others, and a persistent critic on this topic, who is a parent of children who are going through school, has told me something similar to a criticism Lockhart anticipated. It goes something like, “Not everyone is capable of doing what you’re talking about. They need to obtain certain basic skills, or else they will not be served well by the education they receive. Schools should just focus on that.” I understand this concern. What people who think about this worry about is that in our very competitive knowledge economy, people want assurances that their children will be able to make it. They don’t feel comfortable with an “airy-fairy, let’s be creative!” account of what they see as an essential skill. That’s leaving too much to chance. However, a persistent complaint I used to hear from employers (I assume this is still the case) is that they want people who can think creatively out of the box, and they don’t see enough of that in candidates. This is precisely what Lockhart is talking about (he doesn’t mention employers, though it’s the same concern, coming from a different angle). The only way we know of to cultivate creative thinkers is to get people in the practice of doing what’s necessary to be creative, and no one can give them a step-by-step guide on how to do that. Students have to go through the experience themselves, though of course adult educators will have a role in that.

A couple parts of Lockhart’s account that really resonated with me was where he showed how one can arrive at a proof for the area of a certain type of triangle, and where he talked about students figuring out imaginary problems for themselves, getting frustrated, trying and failing, collaborating, and finally working it out. What he described sounds so similar to what my experience was when I was first learning to program computers, when I was 12 years old, and beyond. I didn’t have a programming course to help me when I was first learning to do it. I did it on my own, and with the help of others who happened to be around me. That’s how I got comfortable with it. And none of it was about, “To solve this problem, you do it this way.” I can agree that would’ve been helpful in alleviating my frustration in some instances, but I think it would’ve risked denying me the opportunity to understand something about what was really going on while I was using a programming language. You see, what we end up learning through exploration is that we often learn more than we bargained for, and that’s all to the good. That’s something we need to understand as students in order to get some value out of an educational experience.

By learning this way, we own what we learn, and as such, we also learn to respond to criticism of what we think we know. We come to understand that everything that’s been developed has been created by fallible human beings. We learn that we make mistakes in what we own as our ideas. That creates a sense of humility in us, that we don’t know everything, and that there are people who are smarter than us, and that there are people who know what we don’t know, and that there is knowledge that we will never know, because there is just so much of it out there, and ultimately, that there are things that nobody knows yet, not even the smartest among us. That usually doesn’t feel good, but it is only by developing that sense of humility, and responding to criticism well that we improve on what we own as our ideas. As we get comfortable with this way of learning, we learn to become good at exploring, and by doing that, we become really knowledgeable about what we learn. That’s what educators are really after, is it not, to create lifelong learners? Most valuable of all, I think, is learning this way creates independent thinkers, and indeed, people who can think out of the box, because you have a sense of what you know, and what you don’t know, and what you know is what you question, and try to correct, and what you don’t know is what you explore, and try to know. Furthermore, you have a sense of what other people know, and don’t know, because you develop a sense of what it actually means to know something! This is what I see Lockhart’s critics are missing the boat on: What you know is not what you’ve been told. What you know is what you have tried, experienced, analyzed, and criticized (wash, rinse, repeat).

On a related topic, Richard Feynman addressed something that should concern us re. thinking we know something because it’s what we’ve been told, vs. understanding what we know, and don’t know.

What seems to scare a lot of people is even after you’ve tried, experienced, analyzed, and criticized, the only answer you might come up with is, “I don’t know, and neither does anybody else.” That seems unacceptable. What we don’t know could hurt us. Well, yes, but that’s the reality we exist in. Isn’t it better to know that than to presume we know something we don’t?

The fundamental disconnect between what people think is valuable about education and what’s actually valuable in it is they think that to ensure that students understand something, it must be explicitly transmitted by teachers and instructional materials. It can’t just be left to chance in exploratory methods, because they might learn the wrong things, and/or they might not learn enough to come close to really mastering the subject. That notion is not to be dismissed out of hand, because that’s very possible. Some scaffolding is necessary to make it more likely that students will arrive at powerful ideas, since most ideas people come up with are bad. The real question is finding a balance of “just enough” scaffolding to allow as many students as possible to “get it,” and not “too much,” such that it ruins the learning experience. At this point, I think that’s more an art than a science, but I could be wrong.

I’m not suggesting just using a “blank page” approach, where students get no guidance from adults on what they’re doing, as many school systems have done (which they mislabeled “constructivism,” and which doesn’t work). I don’t think Lockhart is talking about that, either. I’m not suggesting autodidactic learning, nor is Lockhart. There is structure to what he is talking about, but it has an open-ended broadness to it. That’s part of what I think scares his critics. There is no sense of having a set of requirements. Lockhart would do well to talk about thresholds that he is aiming for with students. I think that would get across that he’s not willing to entertain lazy thinking. He tries to do that by talking about how students get frustrated in his scheme of imaginative work, and that they work through it, but he needs to be more explicit about it.

He admits in the “sequel” that his first article was a lament, not a proposal of what he wants to see happen. He wanted to point out the problem, not to provide an answer just yet.

The key point I want to make is the perception that drives not taking a risk is in fact taking a big risk. It’s risking creating people, and therefore a society that only knows so much, and doesn’t know how to exceed thresholds that are necessary to come up with big leaps that advance our society, if not have the basic understanding necessary to retain the advances that have already been made. A conservative, incremental approach to existing failure will not do.

Related post: The beauty of mathematics denied

— Mark Miller, https://tekkie.wordpress.com

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It’s been 5 years since I’ve written something on SICP–almost to the day! I recently provided some assistance to a reader of my blog on this exercise. In the process, I of course had to understand something about it. Like with Exercise 1.19, I didn’t think this one was written that clearly. It seemed like the authors assumed that the students were familiar with the Miller-Rabin test, or at least modular mathematics.

The exercise text made it sound like one could implement Miller-Rabin as a modification to Fermat’s Little Theorem (FLT). From providing my assistance, I found out that one could do it as a modification to FLT, though the implementation looked a bit convoluted. When I tried solving it myself, I felt more comfortable just ignoring the prior solutions for FLT in SICP, and completely rewriting the expmod routine. Along with that, I wrote some intermediate functions, which produced results that were ultimately passed to expmod. My own solution was iterative.

I found these two Wikipedia articles, one on modular arithmetic, the other on Miller-Rabin, to be more helpful than the SICP exercise text in understanding how to implement it. Every article I found on Miller-Rabin, doing a Google search, discussed the algorithm for doing the test. It was still a bit of a challenge to translate the algorithm to Scheme code, since they assumed an imperative style, and there is a special case in the logic.

It seemed like when it came right down to it, the main challenge was understanding how to calculate factors of n – 1 (n being the candidate tested for primality), and how to handle the special case, while still remaining within the confines of what the exercise required (to do the exponentiation, use the modulus function (remainder), and test for non-trivial roots inside of expmod). Once that was worked out, the testing for non-trivial roots turned out to be pretty simple.

If you’re looking for a list of primes to test against your solution, here are the first 1,000 known primes.

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