"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.
As a member of the last slide rule generation, I agree. The trusty slipstick was a calculation and thinking aid only. By not automatically handling units, for example, it forced you to think through the steps of your calculation. Back then students learned to approximate the result they were expecting.
I wonder how my college students today have already estimated the total of their purchases when they arrive at the cashier. Such thinking doesn't seem to be part of the culture anymore.
I agree with this. I also think curricula designed with the assumption that a calculator is available can be much more challenging. After all, the student is given a more powerful tool.
The goal is definitely to avoid turning people into calculators. I found that for myself and my peer group, having a calculator was the best panacea to doing that. It meant that we'd use those extra free moments wondering why things worked and what it meant instead of "carrying the ones".
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.