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I grew up learning math from calculators. I still today am terrible at mental math, but having immediate, accurate graphing capabilities from precalc onward was

by tel 15y ago

I grew up learning math from calculators. I still today am terrible at mental math, but having immediate, accurate graphing capabilities from precalc onward was an incredible boon. Even before that, mechanical skills such as polynomial factorization or algebraic reorganization ("fiddly symbol manipulation") were actually, in my opinion, aided by having a calculator which could perform them for me. By releasing the burden of actually performing the computations, I was focused on larger patterns which made the computation meaningful and possible. There are two camps here, of course. As an engineer or mathematician, doing mental arithmetic is a dead skill---all the effort comes from modeling and understanding the structure of math. However, "practical" math such as computing tips or adding up grocery bills---and this was the kind of math application stressed in my public school education---probably benefits much more from rapid mental sums. So, ironically, I'd support the heavy involvement of calculators (even ones as powerful as Mathematica itself) in education if it's to be aimed to produce engineers, scientists, or mathematicians. Otherwise, they're probably damaging.

moped 15y ago
WolframAlpha is a good visual tool also, and I often use it when I tutor math.
lnguyen 15y ago
The issue is whether you learned the actual skill (graphing, factorization, etc.) or just learned how to get the calculator to do it for you. In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.
- gergles 15y ago
  Factorization and graphing aren't actual skills, though. They are mechanical computations that basically scream "please use a tool to do this". We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.
  - gmichnikov 15y ago
    This is not true. Calculating a log by hand would be the same process every time. Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not. Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals. There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.
  - lnguyen 15y ago
    You do realize that the tools have to be created based on some underlying process. If you have an understanding of that process you'll have a better understanding (and appreciation) of that tool. And maybe be able to come up with a better one.
beloch 15y ago
"A computer lets you make more mistakes faster than any invention in human history - with the possible exceptions of handguns and tequila." ~Mitch Ratcliffe As a physicist, I use math environments like Matlab and Mathematica to do calculations all the time. Some are simulations that are too massive to even contemplate without a fairly powerful computer. Sometimes I just use them to speed up things I could do by hand. In the latter case, I usually do some samples by hand just to check if my program is working as expected. Later, when using the program, I always try to have some estimate of what the results should look like, even if that intuition of what is correct comes from a very different way of thinking. This is something a lot of students I've taught simply do not do, and it really bites them in the ass a lot! For example, there's one standard junior lab experiment where students work with a very weak sample of a radioactive isotope. It's just barely strong enough to register a small amount of clicks if you practically stuff the sample inside of a Geiger tube. As an exercise, we have students calculate the equivalent dose of radiation they receive over the course of the lab. If the calculation is done correctly it should come out to being barely above background exposure and nowhere near as much as an intercontinental flight or diagnostic X-Ray. However, without fail, every year at least some students will calculate an exposure that is fatal, either by incorrectly converting units or just simple "calculator error". "Calculator error" is what we call it when a student writes out their calculation and everything is correct except the final answer. They got all the input numbers and units right, but they just failed to punch it into their calculator correctly. Many students probably make this mistake but catch it right away. Others do not catch their mistake and how they deal with it in their lab reports varies quite a bit. Some simply don't notice it. They clearly had no idea what to expect from the calculation and didn't bother to compare their result to what was in the lab materials. Other students are at least smart enough to note that they think their answer is too high, although they don't know why. Occasionally, a student will actually write that they are concerned that the experiment was unsafe, perhaps because a horrible mistake was made and they were given the wrong isotope samples. These last students are easy to laugh at, but at least they are thinking, unlike the first group of students who write down a dose that would have left them dead without any further comment. That first group of students who trusts their calculation blindly (or simply don't care) is much larger than the other groups, and that's my greatest beef with our education system. It turns children into calculators. i.e. Devices that can mechanically turn input into output without understanding the process they are executing or having any intuition about what the answer should look like. Grade school texts rarely teach students to critically examine their results. They present students with a way to do a calculation, ask them to execute it like a mechanical device, and then knowing whether they're right or wrong boils down to checking the back of the book to see if the book's answer matches their own. Grade school students can get perfect marks without having any understanding at all, provided they practice the methods enough. Understanding the method and having the ability to recognize what results should look like are skills that should be at the very core of math curriculum's. Learning to be a calculator is boring. Math is infinitely more interesting when you know the why and not just the mechanical how.

by tel 15y ago

moped 15y ago

WolframAlpha is a good visual tool also, and I often use it when I tutor math.

lnguyen 15y ago

The issue is whether you learned the actual skill (graphing, factorization, etc.) or just learned how to get the calculator to do it for you. In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.

gergles 15y ago
Factorization and graphing aren't actual skills, though. They are mechanical computations that basically scream "please use a tool to do this". We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.
- gmichnikov 15y ago
  This is not true. Calculating a log by hand would be the same process every time. Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not. Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals. There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.
- lnguyen 15y ago
  You do realize that the tools have to be created based on some underlying process. If you have an understanding of that process you'll have a better understanding (and appreciation) of that tool. And maybe be able to come up with a better one.

beloch 15y ago

"A computer lets you make more mistakes faster than any invention in human history - with the possible exceptions of handguns and tequila." ~Mitch Ratcliffe As a physicist, I use math environments like Matlab and Mathematica to do calculations all the time. Some are simulations that are too massive to even contemplate without a fairly powerful computer. Sometimes I just use them to speed up things I could do by hand. In the latter case, I usually do some samples by hand just to check if my program is working as expected. Later, when using the program, I always try to have some estimate of what the results should look like, even if that intuition of what is correct comes from a very different way of thinking. This is something a lot of students I've taught simply do not do, and it really bites them in the ass a lot! For example, there's one standard junior lab experiment where students work with a very weak sample of a radioactive isotope. It's just barely strong enough to register a small amount of clicks if you practically stuff the sample inside of a Geiger tube. As an exercise, we have students calculate the equivalent dose of radiation they receive over the course of the lab. If the calculation is done correctly it should come out to being barely above background exposure and nowhere near as much as an intercontinental flight or diagnostic X-Ray. However, without fail, every year at least some students will calculate an exposure that is fatal, either by incorrectly converting units or just simple "calculator error". "Calculator error" is what we call it when a student writes out their calculation and everything is correct except the final answer. They got all the input numbers and units right, but they just failed to punch it into their calculator correctly. Many students probably make this mistake but catch it right away. Others do not catch their mistake and how they deal with it in their lab reports varies quite a bit. Some simply don't notice it. They clearly had no idea what to expect from the calculation and didn't bother to compare their result to what was in the lab materials. Other students are at least smart enough to note that they think their answer is too high, although they don't know why. Occasionally, a student will actually write that they are concerned that the experiment was unsafe, perhaps because a horrible mistake was made and they were given the wrong isotope samples. These last students are easy to laugh at, but at least they are thinking, unlike the first group of students who write down a dose that would have left them dead without any further comment. That first group of students who trusts their calculation blindly (or simply don't care) is much larger than the other groups, and that's my greatest beef with our education system. It turns children into calculators. i.e. Devices that can mechanically turn input into output without understanding the process they are executing or having any intuition about what the answer should look like. Grade school texts rarely teach students to critically examine their results. They present students with a way to do a calculation, ask them to execute it like a mechanical device, and then knowing whether they're right or wrong boils down to checking the back of the book to see if the book's answer matches their own. Grade school students can get perfect marks without having any understanding at all, provided they practice the methods enough. Understanding the method and having the ability to recognize what results should look like are skills that should be at the very core of math curriculum's. Learning to be a calculator is boring. Math is infinitely more interesting when you know the why and not just the mechanical how.