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I think the problem is a tension between what’s valuable to us in learning and what's valuable to us after learning. So when I was tutoring, I noticed that my

I think the problem is a tension between what’s valuable to us in learning and what's valuable to us after learning. So when I was tutoring, I noticed that my “standard approach” failed: a kid would have a homework problem, I would say, “Here's how you want to think about that problem, it’s (say) a conservation of energy problem, calculate the energy before and after...” and it fostered no real learning! I thought the problem was that I was subsidizing laziness—stepped back a bit but gave lessons after some struggling—and that didn’t hurt much but it didn’t help much either. They were motivated, just the didactic structure was not working for them. I started to ask questions, and that was a bit more helpful, but found that I myself was somewhat unguided about how to accomplish “Socratic teaching” so that they could benefit from it—if you can believe it, my default approach was to try to introduce the same theory ideas as questions from the start, “would conservation of energy apply here?”—so no wonder I only got mixed results: it was basically the same thing. But questions gave me a way to get off that map, because I started to ask tutees to give me expectations and examples, and somehow those worked well. “What can we expect that this system does, within a few orders of magnitude? Can you give me some examples of similar systems and how they work?” I think that in order to learn abstract things, we need to first have a bunch of examples in our heads, things that all kind of fit together but do not make sense. The abstraction unifies the examples, helps us remember them and fit them together. We can use each example to test the abstraction for fit, to massage our cognitive ability from one context into another. When we build a building, we need frameworks and scaffolds to build it. When the building is freestanding, all of that effort of building the scaffolds can be undone, the scaffolding comes down, it's no longer necessary. We go back to a textbook and we want a picture-book: show me the building, and just the building, and nothing but the building. Let me feel lushly indulged by its architecture, by photographs that go into detail in the really rich areas—I want indulgent artistic experience, because that's what the building is! We have a constant tension in textbooks between the how-to-manual—spartan, practical, containing all of these scaffolding steps and nitty-gritty–and the coffeetable book—rich, whimsical, pop-simplified. Griffiths’ Introduction to Quantum Mechanics begins with a prologue to this effect, though I thought it was just a quantum quirk at the time I think it's actually much more general now. He says [these will not be exact quotes] that, “This book will teach you to do quantum mechanics because I strongly believe that this has to come before any sort of strange discussions about what quantum mechanics is.” It’s one step between Feynman’s “don’t even try to understand quantum mechanics, just shut up and calculate” and Searle’s response “I’m sorry, I am a philosopher and literally the only thing anyone pays me for is to try and understand things like quantum mechanics.” And I think that is where I want to focus my teaching efforts, that every textbook should kind of have the appendix coffee-table section, I’d even like to carve out a new name for it and call it an “abridgement” or so, at the end. The book Mazes for Programmers has an appendix like this of “let me summarize all of these maze generation algorithms and their essential properties and their essential approaches” that I really loved, it makes it an absolute joy to keep this thing on the bookshelf and come back to it time and time again. The main text has the “Here’s how to do it” information, the appendix has the polite overview of what was just built that is great for returning to.

roystonvassey 8y ago
This. It’s a daily trade-off, as a working professional, between how deep I should learn the fundamentals and how quickly can I get to solving problem at hand. As an example, I want to learn how to deploy my models on the cloud but the mechanics of how computing on the cloud works is a depth trade-off. I wish I could learn all about distributed computing concepts but I find that I have neither the time or energy for it, at times.
throwawaymath 8y ago
Yes, I think you've captured the problem pretty nicely. This is the problem I have with doing something like defining vector spaces on arbitrary fields F (not even R or C!) before you even talk about linear equations. The theory of linear algebra is rich and useful, but it's hard to motivate it that way. I don't mean "motivate" in the sense of, "Why should I care about this, and when will I use it?", because you don't always need that kind of motivation to do math. I mean motivation in the sense of, "Why did people come up with this theory? What led them to these definitions, and why are these definitions the right ones?" I have seen the hierarchy of math presented like this: 1. In high school, you're taught how to compute arc length. You need to do nothing but calculation, and it doesn't matter why it works. 2. In undergrad, you're taught to prove the formula for computing arc length. Given the theorem statement and the requisite axioms, you can show that the formula works. 3. In grad school, you're taught to derive the formula for arc length from first principles. Given a set of axioms, you come up with the theorem and prove it from scratch. 4. In research, you're not taught anything. Instead you solve the questions, "How should I define arc length? Why does my definition of arc length matter, and where is it useful? What can I prove with it?" Contrary to (quite a bit of) popular opinion, I would hold that a truly rich understanding of the theory can only happen in the context of the trigger for the theory. You don't need to understand the context of systems of linear equations, or why linearity is a useful concept, to understand vector spaces. But it's a lot easier if you have that context. Likewise you don't need to use Gaussian elimination to solve a lot of questions which ask you to prove something about a vector space or a linear map. But if you have that context, you can use it in your proofs instead of miring in towering heights of complexity.
mncharity 8y ago
Incoming tech seems about to change the texts-and-tutoring constraint and opportunity space. I wonder if it's time to start thinking and exploring ahead? Consider an art project, where you sit in a booth, and watch an interesting video drama. But why is it interesting? The video is a graph of segments, like an old "choose your own adventure" story. And there's an eye tracker watching you. So if you're interested in characters A and B, the story mutates to emphasize them. Consider a one-on-one tutor on a good day. Noticing a student engaged and enthused, they might reorder content on the fly to leverage that. Might emphasize different aspects, and alter presentation, based on observed interests and weaknesses. Or consider working with a young child, watching them read word by word, noticing their where they hesitate and frown, probing for their thoughts, adjusting difficulty, providing a path of development. What if that was math content rather than a drama or children's book? What if we could do this at scale? Eye tracking is just one tech coming in on the coattails of VR/AR. A setting for personalization AI is another. What if saying "the best way to organize and present this topic in a textbook", becomes like saying "the Capital mandates, that every teacher of this grade, will today all teach the following lesson, by saying the following words, regardless of local context"? While not on a national level, that is a real thing. What might it take to start encoding an adventure graph for linear algebra? The "oh, if you like this perspective on this topic, you might like this similar perspective on this other topic"? Or if we don't have the tooling for that yet, can we start thinking about the tooling? Or fruitfully do something else now, in preparation for opportunity? Perhaps Kahn academy problems in more flavors, in a richer graph? ML-based textbook aggregation, synthesis and retheming? Perhaps it's all not ripe yet. But something else, maybe?

roystonvassey 8y ago

This. It’s a daily trade-off, as a working professional, between how deep I should learn the fundamentals and how quickly can I get to solving problem at hand. As an example, I want to learn how to deploy my models on the cloud but the mechanics of how computing on the cloud works is a depth trade-off. I wish I could learn all about distributed computing concepts but I find that I have neither the time or energy for it, at times.

throwawaymath 8y ago

Yes, I think you've captured the problem pretty nicely. This is the problem I have with doing something like defining vector spaces on arbitrary fields F (not even R or C!) before you even talk about linear equations. The theory of linear algebra is rich and useful, but it's hard to motivate it that way. I don't mean "motivate" in the sense of, "Why should I care about this, and when will I use it?", because you don't always need that kind of motivation to do math. I mean motivation in the sense of, "Why did people come up with this theory? What led them to these definitions, and why are these definitions the right ones?" I have seen the hierarchy of math presented like this: 1. In high school, you're taught how to compute arc length. You need to do nothing but calculation, and it doesn't matter why it works. 2. In undergrad, you're taught to prove the formula for computing arc length. Given the theorem statement and the requisite axioms, you can show that the formula works. 3. In grad school, you're taught to derive the formula for arc length from first principles. Given a set of axioms, you come up with the theorem and prove it from scratch. 4. In research, you're not taught anything. Instead you solve the questions, "How should I define arc length? Why does my definition of arc length matter, and where is it useful? What can I prove with it?" Contrary to (quite a bit of) popular opinion, I would hold that a truly rich understanding of the theory can only happen in the context of the trigger for the theory. You don't need to understand the context of systems of linear equations, or why linearity is a useful concept, to understand vector spaces. But it's a lot easier if you have that context. Likewise you don't need to use Gaussian elimination to solve a lot of questions which ask you to prove something about a vector space or a linear map. But if you have that context, you can use it in your proofs instead of miring in towering heights of complexity.

mncharity 8y ago

Incoming tech seems about to change the texts-and-tutoring constraint and opportunity space. I wonder if it's time to start thinking and exploring ahead? Consider an art project, where you sit in a booth, and watch an interesting video drama. But why is it interesting? The video is a graph of segments, like an old "choose your own adventure" story. And there's an eye tracker watching you. So if you're interested in characters A and B, the story mutates to emphasize them. Consider a one-on-one tutor on a good day. Noticing a student engaged and enthused, they might reorder content on the fly to leverage that. Might emphasize different aspects, and alter presentation, based on observed interests and weaknesses. Or consider working with a young child, watching them read word by word, noticing their where they hesitate and frown, probing for their thoughts, adjusting difficulty, providing a path of development. What if that was math content rather than a drama or children's book? What if we could do this at scale? Eye tracking is just one tech coming in on the coattails of VR/AR. A setting for personalization AI is another. What if saying "the best way to organize and present this topic in a textbook", becomes like saying "the Capital mandates, that every teacher of this grade, will today all teach the following lesson, by saying the following words, regardless of local context"? While not on a national level, that is a real thing. What might it take to start encoding an adventure graph for linear algebra? The "oh, if you like this perspective on this topic, you might like this similar perspective on this other topic"? Or if we don't have the tooling for that yet, can we start thinking about the tooling? Or fruitfully do something else now, in preparation for opportunity? Perhaps Kahn academy problems in more flavors, in a richer graph? ML-based textbook aggregation, synthesis and retheming? Perhaps it's all not ripe yet. But something else, maybe?