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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

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.
  - jules 15y ago
    This is even true in college, where for example a large part of a physics education is spent computing integrals and solving differential equations in special cases, using approximation methods by hand. It would be more appropriate for an intuition of the physics to use symbolic and numerical methods on a computer a lot more. You realize that doing arithmetic by hand is a largely pointless exercise when you get a calculator, and we should similarly recognize that doing integrals and differential equations by hand is largely futile when you have things like maple and scipy.
    - hugh3 15y ago
      Maybe, but maybe not. Just like driving through a city doesn't give you the same understanding as walking through a city, immediately solving (say) the Schroedinger Wave Equation for a finite square well doesn't give you the same appreciation for the underlying physics as struggling with the terms one by one.

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.
- jules 15y ago
  This is even true in college, where for example a large part of a physics education is spent computing integrals and solving differential equations in special cases, using approximation methods by hand. It would be more appropriate for an intuition of the physics to use symbolic and numerical methods on a computer a lot more. You realize that doing arithmetic by hand is a largely pointless exercise when you get a calculator, and we should similarly recognize that doing integrals and differential equations by hand is largely futile when you have things like maple and scipy.
  - hugh3 15y ago
    Maybe, but maybe not. Just like driving through a city doesn't give you the same understanding as walking through a city, immediately solving (say) the Schroedinger Wave Equation for a finite square well doesn't give you the same appreciation for the underlying physics as struggling with the terms one by one.