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I love attempts to shake up maths education, and I reckon the first passage is a pretty neat explanation, but it seems to slide quickly downhill from there. Th

I love attempts to shake up maths education, and I reckon the first passage is a pretty neat explanation, but it seems to slide quickly downhill from there. The first three chapters are a bit airy-fairy - I get that that's its schtick, but they're also pretty light on content. The fourth reads like it's been taken straight out of a maths textbook by a "cool" teacher - engaging but not fundamentally simpler than what one learns in class. The fifth teaches how to use Wolfram Alpha to perform calculus. I could teach a dog to use Wolfram Alpha. It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to simplify the first few lessons on a subject, but you pretty soon hit a wall when you reach the parts that actually require a bit of mental legwork. Calculus is not a simple subject. By teaching it better, you can probably improve the situation by, like, 20%, but you can't magically make a complicated subject trivial. It just doesn't seem like the way forward for education. We need to make the difficulty more rewarding, not pretend that it doesn't exist.

jasode 11y ago
>It all seems to suffer a bit from the "Dummies guide" syndrome I think your criticism is misguided. His tutorial is obviously meant to make the introduction to the topic more accessible. He does that by using pictures and diagrams to build intuitions. He can do that because he extensively uses drawing tools to help the learner "visualize" what Calculus is about. Compare that with typical calculus teacher with a whiteboard. Because the teacher's drawing skills are limited, he/she dives right into the delta-epsilon limit definition: lim h --> 0 for f(x+h)-f(h)/h Yeah there's some hand drawing of tangent lines to converge on a "instantaneous change" but that's not enough for a lot of students to "get" Calculus. The symbolic notation is demonstrated more often than pictures. It therefore devolves into mindless "plug & chug" for the rest of the semester. As for the Wolfram Alpha comment, the idea is about "immersion". There are other tools besides pencil-&-paper that also "do" Calculus and they can be used to verify self-study Q&A. Likewise, if I teach Differential Equations to programmers, it's helpful if I "immerse" them into other tools to numerically solve them such as MS Excel, MATLAB, Wolfram Alpha, C/C++ code, etc. His BetterExplained doesn't replace a rigorous Calculus textbook but it can help with the "aha!" moment to understand what the dry Calc textbook is trying to teach.
- wyattpeak 11y ago
  >His tutorial is obviously meant to make the introduction to the topic more accessible. And I'm saying he doesn't much succeed at that. After the first few chapters, the content is either trivial or very similar to what you'd find in a textbook. >As for the Wolfram Alpha comment, the idea is about "immersion". You could be right, but it seems like a generous appraisal. It reads to me more like an attempt to brush over the fact that the student doesn't actually know any calculus yet. If your intent is to teach that calculus can be performed intuitively, teach that - don't teach how you can phrase your question to get a computer to solve them.
  - jasode 11y ago
    > It reads to me more like an attempt to brush over the fact that the student doesn't actually know any calculus yet. ... - don't teach how you can phrase your question to get a computer to solve them. No, he's not showing WolframAlpha as a mental shortcut to cover up deficiencies of not knowing calculus. The point is to utilize it as a diagnostic tool to check the student's understanding of previous Ch 1 - 4. If the student understands the previous chapters' vocabulary and concepts, he "doublechecks" that understanding by seeing if he can "describe" problems to WolframAlpha. Again, the idea is to create a feedback loop for the self-directed learner. Instead of using WolframAlpha, one could be shown how to use the TI-83 graphing calculator for the same purpose. However, since the student is likely reading Betterexplained on the web, he can just conveniently click over to wolframalpha.com and "check his calculus knowledge" there. As a bonus, WolframAlpha is easier to use, draws prettier graphs, and doesn't cost $100.
- bigger_cheese 11y ago
  I was taught calculus during my last two years of high school from a starting point of the conic sections (parabolas, elipses etc). Basically what you said drawing of tangent lines etc. Limits and the fundamental theorem came after this. We used old school TI graphics calculators in the classroom. This was 15+ years ago so students today probably have better tools available (In Australia some schools give laptops to students nowso at least something like excel is available) we certainly had better then pencil and paper in the late 90's. My first year university calculus course covered 2 years of high school math in about 5 weeks and moved on to things like Taylor series Laplace transforms etc. There were people who hadn't studied calculus at high school in the lectures but they were the minority. I'd imagine if this was their first exposure it would have been a struggle.
agumonkey 11y ago
What I'd love to see in math education is not dumbing down. Although finding a way to make people enthusiastic at first is important. I like math because I liked math. If people don't have pleasure then education fails. But here's my point I want teachers to state clearly what is to understand. "working hard" doesn't cut it for me. Give me a few 'really hard' exercise that will require my brain to change fundamentally then leave me on it. Affirm patience is key, that one WILL understand after looking at this elusive idea. Because that's why odd concepts are odd, they don't fit in our brain patterns, and thus it takes time to reconnect the dots.
deleted 11y ago
[deleted]
nerdy 11y ago
Learning new material is hard at the beginning. You don't have any context and the terms are foreign. You might not even know Wolfram Alpha exists if you're uninitiated. Despite its ease of use, some people appreciate a guided approach to learning. While the subject can't be trivialized, if someone gains learning momentum (whether by interest or confidence) then this kind of material did a great deal to help them.
forgetsusername 11y ago
>It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to simplify the first few lessons on a subject, but you pretty soon hit a wall when you reach the parts that actually require a bit of mental legwork. And yet Dummies Guides are often the first book I'll look for as an introduction to a subject. I know they'll be light reading, easily consumable, and convey information that is simple and considered "mainstream". The wall you speak of likley means you've leveled up your understanding, and now have a better idea of where to go next.
InclinedPlane 11y ago
Calculus is an elegant subject that is not nearly as difficult to pick up as its reputation leads people to believe. But there are two main problems with learning calculus today. One is that mathematics in general is taught very poorly (well, all subjects really, but mathematics bears the brunt even more so). There's a great concentration on "teaching to the test", rote memorization, "plug and chug", churn through as much "material" as possible and then forget it over summer, that sort of thing. Cargo cult education, basically. This is very damaging to the teaching of calculus because it is very much a subject that is dependent on some very fundamental core concepts. If you don't "get" those core concepts, then calculus will be forever opaque to you. Worse, this pattern of "education" applies all the way down the line to every math subject taught in school, including the pre-requisites for calculus: advanced algebra and trigonometry. And calculus is very dependent on actually knowing that pre-requisite math. Not being dependent on plug-and-chug, guess-and-check, "eliminate 2 wrong answers from the multiple choice question and then guess", or "cram for the test and then forget it". All of those techniques work well at gaming the educational system, but they leave you weak in the actual skills of applying trigonometry and advanced algebra to problems, and using them as tools while learning other, more advanced, mathematics. So you have a typical calculus course trying to lay down enormous concrete slabs of new concepts and "material" but it just sinks into the bog rather than building up a structure sitting on a solid foundation. Additionally, calculus is often used as a filter course. Calculus has a potentially infinite amount of "trivia" associated with it. All manner of rules, substitutions, identities, definitions, theorems, and so forth. As such, calculus effectively has a "difficulty dial" than any teacher can set arbitrarily depending on how much memorization and busy work they want their students to slog through. And many teachers decide to set this at just high enough of a level to make calculus a real pain in the ass for most students. There's always a million justifications for doing this, but most of them are pretty weak. There's a justification for acquiring familiarity with all of these things, and proving you can walk through the steps to do harder problems, but in practice it's mostly useless. People who actually use calculus day to day generally just keep a cheat sheet around of integration and derivation rules and substitutions and whatnot, or they employ the use of a computer to do the heavy lifting. Once you understand the concepts there's not much value in slogging through many mindless examples just for bragging rights. All of this leads to people getting the impression that Calculus requires superhuman skills to learn (no, it just requires being actually caught up with the pre-requisites before hand) and also that the application of calculus tends to be boring and monotonous. If education did a better job of ensuring the pre-calculus fundamentals were thoroughly in place, and concentrated more on concepts and applications of calculus (like physics, engineering, geometry, control systems, computer graphics, and so much more) would, I think, lead to not only a lot more students succeeding at learning calculus well but also enjoying it.

jasode 11y ago

>It all seems to suffer a bit from the "Dummies guide" syndrome I think your criticism is misguided. His tutorial is obviously meant to make the introduction to the topic more accessible. He does that by using pictures and diagrams to build intuitions. He can do that because he extensively uses drawing tools to help the learner "visualize" what Calculus is about. Compare that with typical calculus teacher with a whiteboard. Because the teacher's drawing skills are limited, he/she dives right into the delta-epsilon limit definition: lim h --> 0 for f(x+h)-f(h)/h Yeah there's some hand drawing of tangent lines to converge on a "instantaneous change" but that's not enough for a lot of students to "get" Calculus. The symbolic notation is demonstrated more often than pictures. It therefore devolves into mindless "plug & chug" for the rest of the semester. As for the Wolfram Alpha comment, the idea is about "immersion". There are other tools besides pencil-&-paper that also "do" Calculus and they can be used to verify self-study Q&A. Likewise, if I teach Differential Equations to programmers, it's helpful if I "immerse" them into other tools to numerically solve them such as MS Excel, MATLAB, Wolfram Alpha, C/C++ code, etc. His BetterExplained doesn't replace a rigorous Calculus textbook but it can help with the "aha!" moment to understand what the dry Calc textbook is trying to teach.

wyattpeak 11y ago
>His tutorial is obviously meant to make the introduction to the topic more accessible. And I'm saying he doesn't much succeed at that. After the first few chapters, the content is either trivial or very similar to what you'd find in a textbook. >As for the Wolfram Alpha comment, the idea is about "immersion". You could be right, but it seems like a generous appraisal. It reads to me more like an attempt to brush over the fact that the student doesn't actually know any calculus yet. If your intent is to teach that calculus can be performed intuitively, teach that - don't teach how you can phrase your question to get a computer to solve them.
- jasode 11y ago
  > It reads to me more like an attempt to brush over the fact that the student doesn't actually know any calculus yet. ... - don't teach how you can phrase your question to get a computer to solve them. No, he's not showing WolframAlpha as a mental shortcut to cover up deficiencies of not knowing calculus. The point is to utilize it as a diagnostic tool to check the student's understanding of previous Ch 1 - 4. If the student understands the previous chapters' vocabulary and concepts, he "doublechecks" that understanding by seeing if he can "describe" problems to WolframAlpha. Again, the idea is to create a feedback loop for the self-directed learner. Instead of using WolframAlpha, one could be shown how to use the TI-83 graphing calculator for the same purpose. However, since the student is likely reading Betterexplained on the web, he can just conveniently click over to wolframalpha.com and "check his calculus knowledge" there. As a bonus, WolframAlpha is easier to use, draws prettier graphs, and doesn't cost $100.
bigger_cheese 11y ago
I was taught calculus during my last two years of high school from a starting point of the conic sections (parabolas, elipses etc). Basically what you said drawing of tangent lines etc. Limits and the fundamental theorem came after this. We used old school TI graphics calculators in the classroom. This was 15+ years ago so students today probably have better tools available (In Australia some schools give laptops to students nowso at least something like excel is available) we certainly had better then pencil and paper in the late 90's. My first year university calculus course covered 2 years of high school math in about 5 weeks and moved on to things like Taylor series Laplace transforms etc. There were people who hadn't studied calculus at high school in the lectures but they were the minority. I'd imagine if this was their first exposure it would have been a struggle.

agumonkey 11y ago

What I'd love to see in math education is not dumbing down. Although finding a way to make people enthusiastic at first is important. I like math because I liked math. If people don't have pleasure then education fails. But here's my point I want teachers to state clearly what is to understand. "working hard" doesn't cut it for me. Give me a few 'really hard' exercise that will require my brain to change fundamentally then leave me on it. Affirm patience is key, that one WILL understand after looking at this elusive idea. Because that's why odd concepts are odd, they don't fit in our brain patterns, and thus it takes time to reconnect the dots.

deleted 11y ago

[deleted]

nerdy 11y ago

Learning new material is hard at the beginning. You don't have any context and the terms are foreign. You might not even know Wolfram Alpha exists if you're uninitiated. Despite its ease of use, some people appreciate a guided approach to learning. While the subject can't be trivialized, if someone gains learning momentum (whether by interest or confidence) then this kind of material did a great deal to help them.

forgetsusername 11y ago

>It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to simplify the first few lessons on a subject, but you pretty soon hit a wall when you reach the parts that actually require a bit of mental legwork. And yet Dummies Guides are often the first book I'll look for as an introduction to a subject. I know they'll be light reading, easily consumable, and convey information that is simple and considered "mainstream". The wall you speak of likley means you've leveled up your understanding, and now have a better idea of where to go next.

InclinedPlane 11y ago

Calculus is an elegant subject that is not nearly as difficult to pick up as its reputation leads people to believe. But there are two main problems with learning calculus today. One is that mathematics in general is taught very poorly (well, all subjects really, but mathematics bears the brunt even more so). There's a great concentration on "teaching to the test", rote memorization, "plug and chug", churn through as much "material" as possible and then forget it over summer, that sort of thing. Cargo cult education, basically. This is very damaging to the teaching of calculus because it is very much a subject that is dependent on some very fundamental core concepts. If you don't "get" those core concepts, then calculus will be forever opaque to you. Worse, this pattern of "education" applies all the way down the line to every math subject taught in school, including the pre-requisites for calculus: advanced algebra and trigonometry. And calculus is very dependent on actually knowing that pre-requisite math. Not being dependent on plug-and-chug, guess-and-check, "eliminate 2 wrong answers from the multiple choice question and then guess", or "cram for the test and then forget it". All of those techniques work well at gaming the educational system, but they leave you weak in the actual skills of applying trigonometry and advanced algebra to problems, and using them as tools while learning other, more advanced, mathematics. So you have a typical calculus course trying to lay down enormous concrete slabs of new concepts and "material" but it just sinks into the bog rather than building up a structure sitting on a solid foundation. Additionally, calculus is often used as a filter course. Calculus has a potentially infinite amount of "trivia" associated with it. All manner of rules, substitutions, identities, definitions, theorems, and so forth. As such, calculus effectively has a "difficulty dial" than any teacher can set arbitrarily depending on how much memorization and busy work they want their students to slog through. And many teachers decide to set this at just high enough of a level to make calculus a real pain in the ass for most students. There's always a million justifications for doing this, but most of them are pretty weak. There's a justification for acquiring familiarity with all of these things, and proving you can walk through the steps to do harder problems, but in practice it's mostly useless. People who actually use calculus day to day generally just keep a cheat sheet around of integration and derivation rules and substitutions and whatnot, or they employ the use of a computer to do the heavy lifting. Once you understand the concepts there's not much value in slogging through many mindless examples just for bragging rights. All of this leads to people getting the impression that Calculus requires superhuman skills to learn (no, it just requires being actually caught up with the pre-requisites before hand) and also that the application of calculus tends to be boring and monotonous. If education did a better job of ensuring the pre-calculus fundamentals were thoroughly in place, and concentrated more on concepts and applications of calculus (like physics, engineering, geometry, control systems, computer graphics, and so much more) would, I think, lead to not only a lot more students succeeding at learning calculus well but also enjoying it.