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Welcome to "An I in the Air?" Talk about Tuesday, where I talk about something related to
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Sudbury School. So this week I'm doing a math theme, so I thought I'd talk about something related to math.
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In particular, there is a notion kind of floating around that one of the problems with math education is
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a focus on precise right answers. And, you know, I thought I'd comment on that.
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I am a mathematician, and one would expect, perhaps, a mathematician would like the idea of right answers in math.
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And certainly there are situations where there's a clear right answer, and, you know, that's just that.
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But most of the use of math by people, it's not really about a precise right answer.
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It's, you know, it's about getting it generally in the right ballpark.
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Certainly my Millions Monday podcasts, those are all kind of guesstimation kind of stuff,
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which is very much just try not to get an absurdly wrong answer, right?
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And, you know, that can propagate even, you know, like when you're doing a recipe.
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Some recipes require very precise amounts.
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You know, baking is supposedly famous for this. I don't really bake, so I don't actually know.
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But I've heard that, you know, if you're slightly off on one thing or another, it can be quite dramatic.
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So that, you know, requires presumably a lot of precision.
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So if you were measuring things, you need to do that precisely.
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And if you're changing the amounts of the recipe, you need to do those computations correctly too.
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But there's other recipes, you know, where it's fine to, you know, just kind of roughly eyeball things
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and just be like, "Oh, yeah, just do this and that."
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And, you know, they can easily just come out fine. They're not very sensitive.
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And so, you know, in math education, ideally one would want to, you know,
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have that as part of what one's talking about, this kind of sensitivity to just how precise it needs to be.
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And, you know, one particular example is the notion of quadratic equations, which is, you know, like you've got some unknown thing
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that you're squaring and you're adding it, you know, multiplying by something else, some constant,
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and maybe adding something times that unknown plus something else, that's kind of a quadratic.
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And if you want to know when that's zero for what inputs, you can change a lot of problems to just being setting equal to zero.
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And there's a little formula called the quadratic formula. It involves a square root.
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And, you know, people who have been through that kind of math stuff can often rattle it off.
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Sometimes they don't know what it means, but they can say the formula, and many do know what it means.
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But that is an example of an infinitely precise formula with the square root in there.
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Now, if you have something like, you know, square root of 17, that's precise, but it's not really all that useful.
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You know, like if you want to know, like, you need to, I don't know, have square root of 17 inches from where you are for some reason.
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And, you know, it's like, I can't use that, right? I need to know something more.
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Ideally, one would say, well, 17 is close to 16, and 4 squared is 16, so it's a little over 4, maybe 4.1.
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And, you know, 25 is the next square, so it's kind of a thing of a 9, so maybe, you know, yeah, 9th or 10th, yeah, 4.1, 4.13 maybe, I don't know, whatever it is.
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And so that's just in the mind, and then, you know, you can check it by hand by squaring it out and seeing if it comes close to 17.
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Or you can just use a calculator and it just gives you the answer, right?
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So, that is the, you know, kind of precise thing, you can even see with the precise thing, it's kind of messy.
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Now, with quadratic equations, and many other equations, there's sort of a more guess-and-check kind of method,
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where you kind of take, you know, two guesses, and you see, okay, how close they are to, you know, whatever it is you're trying to get to.
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And then you kind of figure out, you know, what direction you need to go in terms of your next guess, and how much, and boom.
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You get closer and closer, and if you're doing it right, it can get close really quickly.
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And so that gives, you know, a lot of insight.
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And, you know, it naturally leads to questions of, well, how close do I need to get, and all of that kind of thing.
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And, I mean, that is really valuable. That is math as practical and as useful as you can imagine.
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The quadratic formula has its uses, but it's actually pretty limited for, you know, certain special cases compared to this other technique, which is quite general.
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But yet, it's the quadratic formula that gets taught and not this other technique, even in calculus, which is where this technique derives from.
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You know, it doesn't, it's not even taught that way. I mean, it's kind of mentioned, but most teachers don't teach it.
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They teach some complicated thing that it's like, well, I don't know what's going on. And it's not the method, it's the teaching.
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And that kind of saddens me. But, you know, one of the big things is really kind of, you know, trying to think about how you would teach this, right?
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And when you have people who are ready and interested and have a use for it, it can happen.
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We have people who are going to take it and figure it out and explore with it.
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But when you don't, when you have people who are just there because they're told to do, to be there, it goes very poorly.
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And you end up needing these kind of like precise rules. And this is the precise answer, even though that's not what one wants.
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And I'm having a real hard time seeing how to teach math appropriately to people who don't want to learn it.
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Math is a tool of exploration and creativity. That's just what it is. It's a tool for understanding the world around us.
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And if the people trying to learn it have no appreciation for, you know, exploring the world with it, understanding the world with it, playing around with it, then it has really no purpose.
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So that's sort of my thoughts. And of course, as it relates to Sudbury School is like, in many ways, the core of mathematics is exploring and understanding the world.
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And that is exactly what students here are doing all the time. They are trying to figure things out.
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And then, you know, there will be a point in time, very likely, that they want to figure something out that really does require some math.
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Obviously, there's arithmetic, like going to a store and needing to figure out how much something, these things cost when you add them up and all that kind of thing.
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Or trying to figure out how much you can spend from a budget or, you know, all those things.
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But, you know, there's a lot of other things like, you know, if you need to know how tall something is, you know, all of a sudden you need, you know, some trigonometry might be helpful along with the right tool set, of course.
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And so, you know, that's when that can arise. Of course, many of our kids, if they need to know how tall something is, they probably just climb up the side of it and measure it from there.
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But anyway, so, you know, I guess that's a point that this kind of education, Sudbury education, is laying the groundwork for what really, you know, is the foundation of mathematics, science, history, literature.
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All these things, all these experiences, all of it is making the world a place of interest and curiosity, experience, and so that when one needs it, when one eventually comes upon it, it all slots in, it all figures out.
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And contrast that with someone in a conventional school, say in a math class, where the stuff being talked about has really no clear application in their lives, they're not interested in it, they're not really allowed to play around with it and be curious with it,
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and they just have to go through these steps that don't make any sense.
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I mean, they might make some logical sense, but it's like, what am I doing? Why am I doing this? It doesn't make any sense. And it's even just further insulting in this day and age of computers and calculators, because the very routine steps that are often taught, you know, that's what they're based, you know, that's what the calculators do, right?
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It's the next step beyond that, understanding what's being done, how it works. That is, you know, what one really requires. And of course, that's extremely hard to deal with in a grading situation.
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Some people attempt it to varying degrees of success, but I think it's extremely difficult. And I've encountered far too many adults who seem traumatized from their math education experiences.
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I've had to undo a lot of stuff that they've experienced, learned, what they think of themselves. It's a real shame, because, you know, math is something that every single human being should feel empowered and entitled to.
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It is an amazing tool to really bring things to clarity. And it saddens me that in service of an attempt to make it accessible to everybody, they essentially drive everyone away from it.
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Not everyone, of course. I survived. But many, many, many people are like, I can't do math. Yes, they can. They just need to believe it. And so, you know, that's my fervent hope for all the students here, that they realize they can do it, if they want to.
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They can, you know, put in the work to figure it all out. It does require work. It's not easy. But it's also not impossible or truly difficult. It just requires a playful attitude, which I hope they all have.
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Alright, well, that's enough of me droning on for now. I will see you when I see you.