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The points brought up here, especially by Thoma but less so by the quoted passage by Woolley, are very valid in my experience. Often times the shortest way surp

by tel 15y ago

The points brought up here, especially by Thoma but less so by the quoted passage by Woolley, are very valid in my experience. Often times the shortest way surpass a difficult abstract challenge is to find a practical instance of it and then slough through the mechanics. You abuse the natural pattern searching of the human brain very productively that way. Simultaneously, these arguments tend to be reactionary and ignore the benefits of calculators. A calculator, use effectively, allows for a far larger number of examples to be considered together than the brain is often able to achieve. This is was most apparent for me in pre-calculus classes where students should be gaining abstract intuition about the behaviors of functions. Here, graphing large numbers of functions varying their parameters allows one to quickly get a sense of a parametric family. There's an implicit statement here that the "gap" is one such that the lower generation is worse off than their elders --- which is a pretty common human narrative, really. I think instead that this difference is less well-ordered than assumed. Technology is definitely capable of improving human cognition and learning by providing new capacity, and curricula need to explicitly study and take advantage of these capacities.

run4yourlives 15y ago
The argument though is that it is simply impossible to use a calculator effectively without first understanding the basics. If you don't understand that 50*80 should give you something starting with 40, you don't even understand that getting 3745 as an answer on your calculator because you mistyped is horribly incorrect. You can chalk it up to "kids these days" all you want, but if you do some basic math problems with a 60 year old the odds are they will simply leave you in the dust while you go looking for a calculator.
- ColinWright 15y ago
  > The argument though is that it is simply impossible to > use a calculator effectively without first understanding > the basics. I don't think that's the case. I think the author is perfectly happy that people know how to use the calculator. I think the point is that without working through the basics underneath, mind-numbingly repeatedly, you don't gain any real insights about what's going on. It's certainly true for me. Many's the time, when I've had monster power under my fingertips, that I've brute-forced solutions and failed to gain any insight. Then I've worked a few small, single instances by hand, and realised that there was structure I hadn't seen before. The pattern-matching, pattern-finding parts of the brain are phenomenal. Sometimes they are best exercised by getting machines to produce loads of examples, and loads of visualisations very quickly. Sometimes they are better exercised by working tedious examples by hand. Balance.
  - run4yourlives 15y ago
    I think we're essentially in agreement; you've just defined "know how to use a calculator" a little more specifically than I did. In my mind, if you don't know the concepts behind what you are doing, you don't know what you are doing. It's like copy and pasting code from the interwebs and then saying you are a python programmer. But it isn't about balance, it's about using a tool to extend your abilities, not to replace them. If you are doing 36+19 on a calculator, I would posit that you simply aren't every good at math.
    - tel 15y ago
      This is precisely the example repeatedly cited by the author of the post and the author he quoted: people doing very basic addition using a calculator. If you focus your vision on these sorts of example, the rise of calculators is reasonably disastrous-looking! My comment meant to destroy the artificial dichotomy that calculators are "good" or "bad". They are clearly capable of generating some appalling lazy mathematical habits, but they're also clearly able to extend your abilities, improve pedagogy, and garner insights impossible to consider unaided. Science and mathematics have wholeheartedly embraced computers as a tool for analysis and exploration. Education cannot ignore that. Simultaneously, many abstract concepts that people must know to make use of computers are best learned by replicating the exact work the computer can save you later. The benefits of trading off toward laborious computation are going to be especially pronounced while learning things, of course.
  - kenjackson 15y ago
    I think the point is that without working through the basics underneath, mind-numbingly repeatedly, you don't gain any real insights about what's going on. I don't think that's true. And I'll say something controversial. Programming is taught more effectively than math and its because you don't spend a lot of time mind-numbingly repeating the basics. Half the people on HN recommend teaching with Python for just this reason. And think about it, when you teach programming do you make your student walk through what every instruction does? When I learned recursion, I walked through fib(n) by hand -- once. That's the sum total of how many times I've done a full hand expansion of a recursion in my life. How many times have I run a full program on paper with the substitution model? Never. How many times have I iterated even a small loop by hand? Never. These are concepts that I understand as well as anything that I know, yet I've never "mind-numingly" worked through them by hand. A lot of traditional math teachers want us to sit down with an instruction pointer, stacks, heaps, physical/virtual pages, laid out and have us repeatedly simulate program. That's really the mathematical equivalent of doing this rote computation. I say teach them the concepts and give them interesting and challenging problems that make use of the concepts. The concepts will stick better, they'll learn faster, and they'll be more engaged.

by tel 15y ago

run4yourlives 15y ago

The argument though is that it is simply impossible to use a calculator effectively without first understanding the basics. If you don't understand that 50*80 should give you something starting with 40, you don't even understand that getting 3745 as an answer on your calculator because you mistyped is horribly incorrect. You can chalk it up to "kids these days" all you want, but if you do some basic math problems with a 60 year old the odds are they will simply leave you in the dust while you go looking for a calculator.

ColinWright 15y ago
> The argument though is that it is simply impossible to > use a calculator effectively without first understanding > the basics. I don't think that's the case. I think the author is perfectly happy that people know how to use the calculator. I think the point is that without working through the basics underneath, mind-numbingly repeatedly, you don't gain any real insights about what's going on. It's certainly true for me. Many's the time, when I've had monster power under my fingertips, that I've brute-forced solutions and failed to gain any insight. Then I've worked a few small, single instances by hand, and realised that there was structure I hadn't seen before. The pattern-matching, pattern-finding parts of the brain are phenomenal. Sometimes they are best exercised by getting machines to produce loads of examples, and loads of visualisations very quickly. Sometimes they are better exercised by working tedious examples by hand. Balance.
- run4yourlives 15y ago
  I think we're essentially in agreement; you've just defined "know how to use a calculator" a little more specifically than I did. In my mind, if you don't know the concepts behind what you are doing, you don't know what you are doing. It's like copy and pasting code from the interwebs and then saying you are a python programmer. But it isn't about balance, it's about using a tool to extend your abilities, not to replace them. If you are doing 36+19 on a calculator, I would posit that you simply aren't every good at math.
  - tel 15y ago
    This is precisely the example repeatedly cited by the author of the post and the author he quoted: people doing very basic addition using a calculator. If you focus your vision on these sorts of example, the rise of calculators is reasonably disastrous-looking! My comment meant to destroy the artificial dichotomy that calculators are "good" or "bad". They are clearly capable of generating some appalling lazy mathematical habits, but they're also clearly able to extend your abilities, improve pedagogy, and garner insights impossible to consider unaided. Science and mathematics have wholeheartedly embraced computers as a tool for analysis and exploration. Education cannot ignore that. Simultaneously, many abstract concepts that people must know to make use of computers are best learned by replicating the exact work the computer can save you later. The benefits of trading off toward laborious computation are going to be especially pronounced while learning things, of course.
- kenjackson 15y ago
  I think the point is that without working through the basics underneath, mind-numbingly repeatedly, you don't gain any real insights about what's going on. I don't think that's true. And I'll say something controversial. Programming is taught more effectively than math and its because you don't spend a lot of time mind-numbingly repeating the basics. Half the people on HN recommend teaching with Python for just this reason. And think about it, when you teach programming do you make your student walk through what every instruction does? When I learned recursion, I walked through fib(n) by hand -- once. That's the sum total of how many times I've done a full hand expansion of a recursion in my life. How many times have I run a full program on paper with the substitution model? Never. How many times have I iterated even a small loop by hand? Never. These are concepts that I understand as well as anything that I know, yet I've never "mind-numingly" worked through them by hand. A lot of traditional math teachers want us to sit down with an instruction pointer, stacks, heaps, physical/virtual pages, laid out and have us repeatedly simulate program. That's really the mathematical equivalent of doing this rote computation. I say teach them the concepts and give them interesting and challenging problems that make use of the concepts. The concepts will stick better, they'll learn faster, and they'll be more engaged.