4 ms·

Well, suggested solution of this problem seems kinda backwards to me. I'd much prefer something along the lines of what Stephen Wolfram suggests - offload compu

Well, suggested solution of this problem seems kinda backwards to me. I'd much prefer something along the lines of what Stephen Wolfram suggests - offload computation to computers in the first pre-rigorous stage, which may enable building of the foundation in a more rigorous manner right from the start. Even at rigorous stage, offloading computation increases dramatically exposure to solutions and builds intuition faster by orders of magnitude.

josephagoss 14y ago
What does this mean? For example I self-teach myself math at the moment, easy stuff like calculus. I'm still trying to get my head around it all. How can I offload computation to build up intuition faster? Any tips? (I love programming if that helps)
- dsirijus 14y ago
  Particulary in starting calculus, after laying out the foundation of real numbers, exercises pretty soon start with determing limits, inflexion points, trends, and all. Sure, play with those at first by hand, but try to plot and play with those functions in, say, Mathematica. Pretty soon you'll be able to intuitively "know" how any of the functions you could be thrown at exams should look like, which in itself is pretty usable skill. Unrelated, I'd find it even more usable in physics, where solution guesstimate developed by exposure to numerous examples computationally solved can reliably indicate the correct solution as soon as you read the exercise or a problem.
  - josephagoss 14y ago
    Thanks for replying, I got to get back into Mathematica, its so different from "normal" languages. we use Matlab at uni a lot, but its not really the same at all.
ionfish 14y ago
Having written the following, I now wonder whether you meant something more specific by computation than I did, so I'm not certain whether my point is really a response. Could you spell out the details of your comment a bit more, and perhaps touch on the approach which Wolfram argues for? * * * Offloading computation only works when you understand what the computations are and why we do them. That's something that must be learned, it's not knowledge that springs fully formed into our minds as soon as we step into a classroom. Carrying out computations thus gives us explicit and implicit knowledge of how the things we may eventually automate actually work. But it's also valuable because it trains us to compute in a precise and effective manner—a capability that remains useful later on. For instance, in logic it's often important to be able to carry out syntactic manipulations (e.g. into normal forms) in one's head, or even tacitly. I'm sure there are plenty of examples from other areas of mathematics where computation is important, it's just that we do it so automatically that we don't think about it. Often I've found that students have trouble following proofs that take logically and computationally innocent steps without saying what's going on. Here I don't mean things like applying AC in the background, but just simple tricks like de Morgan's laws or taking the contrapositive. They have difficulty because they haven't taken those steps often enough themselves to have internalised them.
- Avshalom 14y ago
  I just recall the Wolfram article and I seem to think it was pretty handwavey as to what/how things get offloaded (to Mathematica specifically of course). But I will say that at least half the homework of my Calc 1-3 courses was spent well past the "understanding" stage and more into "getting fast enough to do it on an artificial, time limited test situation" and basically memorizing pages of identities that I quickly forgot because they so rarely came up in my physics courses. This was pretty much the case with almost every math class since about algebra 1 in middle school. And in particular I would like to hold up Electricity and Magnetism 2. Calculating the momentum of a magnetic field, in all but the most trivial case, takes a full sheet of paper: being rows and rows of 8 inch long equations as you carry out the tedious work of canceling terms; moving things in and out of square roots; and multiplying large polynomials together. It's all basic algebra stuff you learn in high school but it's a slog to work through and so time consuming that you actually lose track of the big picture and end up with very little better understanding at the end. As far as I know that's why things like tensor and bra-ket notation had to be invented in the first place. Without a compressed notation the ability to get a correct answer to any interesting problem became less a question of knowledge and more a question of probability of transcription/sign flip errors. not that anybody teaches sophmores tensor notation.

josephagoss 14y ago

What does this mean? For example I self-teach myself math at the moment, easy stuff like calculus. I'm still trying to get my head around it all. How can I offload computation to build up intuition faster? Any tips? (I love programming if that helps)

dsirijus 14y ago
Particulary in starting calculus, after laying out the foundation of real numbers, exercises pretty soon start with determing limits, inflexion points, trends, and all. Sure, play with those at first by hand, but try to plot and play with those functions in, say, Mathematica. Pretty soon you'll be able to intuitively "know" how any of the functions you could be thrown at exams should look like, which in itself is pretty usable skill. Unrelated, I'd find it even more usable in physics, where solution guesstimate developed by exposure to numerous examples computationally solved can reliably indicate the correct solution as soon as you read the exercise or a problem.
- josephagoss 14y ago
  Thanks for replying, I got to get back into Mathematica, its so different from "normal" languages. we use Matlab at uni a lot, but its not really the same at all.

ionfish 14y ago

Having written the following, I now wonder whether you meant something more specific by computation than I did, so I'm not certain whether my point is really a response. Could you spell out the details of your comment a bit more, and perhaps touch on the approach which Wolfram argues for? * * * Offloading computation only works when you understand what the computations are and why we do them. That's something that must be learned, it's not knowledge that springs fully formed into our minds as soon as we step into a classroom. Carrying out computations thus gives us explicit and implicit knowledge of how the things we may eventually automate actually work. But it's also valuable because it trains us to compute in a precise and effective manner—a capability that remains useful later on. For instance, in logic it's often important to be able to carry out syntactic manipulations (e.g. into normal forms) in one's head, or even tacitly. I'm sure there are plenty of examples from other areas of mathematics where computation is important, it's just that we do it so automatically that we don't think about it. Often I've found that students have trouble following proofs that take logically and computationally innocent steps without saying what's going on. Here I don't mean things like applying AC in the background, but just simple tricks like de Morgan's laws or taking the contrapositive. They have difficulty because they haven't taken those steps often enough themselves to have internalised them.

Avshalom 14y ago
I just recall the Wolfram article and I seem to think it was pretty handwavey as to what/how things get offloaded (to Mathematica specifically of course). But I will say that at least half the homework of my Calc 1-3 courses was spent well past the "understanding" stage and more into "getting fast enough to do it on an artificial, time limited test situation" and basically memorizing pages of identities that I quickly forgot because they so rarely came up in my physics courses. This was pretty much the case with almost every math class since about algebra 1 in middle school. And in particular I would like to hold up Electricity and Magnetism 2. Calculating the momentum of a magnetic field, in all but the most trivial case, takes a full sheet of paper: being rows and rows of 8 inch long equations as you carry out the tedious work of canceling terms; moving things in and out of square roots; and multiplying large polynomials together. It's all basic algebra stuff you learn in high school but it's a slog to work through and so time consuming that you actually lose track of the big picture and end up with very little better understanding at the end. As far as I know that's why things like tensor and bra-ket notation had to be invented in the first place. Without a compressed notation the ability to get a correct answer to any interesting problem became less a question of knowledge and more a question of probability of transcription/sign flip errors. not that anybody teaches sophmores tensor notation.