My 14yo is a highly gifted math student. 4 years ago, if I’d asked him how many ways you could choose 3 people out of a group of 8, if order didn’t matter, he’d have reasoned something like this:
Well, you have 8 choices for the first person, 7 choices for the second, and 6 for the third. So that’s 8x7x6 possible ways. But that counts the same sets of 3 people in several different orders, and since order doesn’t matter, you have to divide by the number of different orders you could have chosen the 3 people. That’s 3x2x1 since there would be 3 choices of which you chose first, 2 for which you chose second, and 1 for which you chose third. So that’s (8x7x6)/6 = 56
Now he’s in honors geometry. They are learning combinatorics and probability first. He asked me a question about a homework problem that was bothering him, so I asked him, “well, what’s the formula for combinations look like?” His reply, “I don’t know, I just know how to type it into my calculator.” ARGH!
I can understand that for large numbers, it is efficient to know how to do such a thing on a calculator, and in one easy step even. But I can’t stand that they are being taught to use calculators to mask the fact that they have NO CLUE what they are really computing! This is a bright kid. He was perfectly capable of understanding the ideas behind permutations and combinations many years ago, and worked with them successfully in many contexts (mainly contest math). I assume that every student in honors geometry is perfectly capable of understanding this topic, which is well within the reach of most middle school students. WHY are these students being taught to be trained monkeys and push the right buttons on their calculators without knowing what the button does?
My first clue that calculators were being over-used by high schools was when I was proctoring a computer science exam at a very selective engineering college. I watched an undergraduate pull out a calculator to multiply a natural number by 10. Really. I was dumbfounded.
Fast forward many years to when I began volunteering to help with math teaching at my kids’ K-8 school. I handed out a mixed set of MATHCOUNTS problems for the older (10-13yo or so) kids to work on. I didn’t specify whether or not calculators could be used, and apparently they were used to being allowed to use them all the time. There was a problem which required the kids to read a bar chart and then figure out the percentage of people who fit into a particular category. Conveniently, 100 people had been surveyed. So, they read the correct bar and found that 8 people were in that category. Then they took out their calculators in order to calculate 8 out of 100 as a percent. Several of them got it wrong! coming up with answers like .8% or 80%. They had recently been taught how to compute a percent using their calculators (and yet had clearly not yet mastered the skill) and it never occurred to them to engage their brains first (or even second) and just write down the obvious answer (or sanity check what their calculator had presented them with).
So… I went on a rampage against calculator use in our school. Their regular math teacher argued that she didn’t want to “torture” them with messy calculations. This is fine. There is a time and a place for calculators, and calculating a messy percent is a perfectly good place for them. Once you’ve shown mastery of long division (and all of these students had done that!) I agree that there is no need to make middle school students regularly work math problems that require messy long division. But the solution is not to just throw a calculator at it. A better solution is to, at least most of the time, engineer the problems so that the numbers work out neatly and no messy calculations are required. Once students are used to doing the problems by hand, if at some point you want to introduce messy ones and allow a calculator, at least the students, with a good feel for what the numbers should look like, have half a chance of noticing if the calculator comes up with something absurd. Fortunately, the teacher agreed with my arguments, and they now use calculators much less frequently.
MATHCOUNTS is interesting in this regard. Their contests have sections where calculators are not allowed, and others where they are. On the “no calculators” sections, they fix it so that the numbers are usually (but not always) nice. On the “calculators allowed” sections, they don’t. I suspect that the reason they design any part of the contest to employ calculators is that MATHCOUNTS is run by engineers, rather than mathematicians. In any case, the free problems they sent to schools to use with their students during the year share this split between the ones with “nice” numbers where calculators should not be needed, and those with messy numbers where they are, frankly, pretty useful. I tend to use mostly the “no calculator” problems to get around the whole issue. I really prefer to make them practice using numbers mentally and by hand as much as possible. I really do think it makes a difference. K-8 students should be working on becoming quick and comfortable with numbers, whatever else they are working on.
There is an argument that calculators are important tools, and students need to learn to use calculators properly and powerfully. This is true, but only to a degree. Yes, calculators are tools that students should learn how to use, BUT, not as a substitute for learning how to work with numbers by hand. There is a skill to working with a calculator, knowing how to use the tools it provides effectively, and absolutely knowing how to sanity check your answer. But when calculators are used without understanding, as a substitute for knowing how to do the problem by hand, they’re horrible tools that lead to innumeracy.
If I had my choice, calculators would not be used at all in K-3, and only very minimally in grades 4-6, in separate “how to use a calculator and estimate if it gave you the right answer or not” type lessons. And then, maybe, pull it out occasionally when working on an interesting real-life application that brings in big, messy numbers. But generally, K-6 students should be using their brains, pencil and paper as their “calculators”.
For pre-algebra students, I’m not opposed to a little more calculator use, but again, only in specially selected situations, when messy numbers are being used either because of the nature of a real-life application, or explicitly for calculator practice. The rest of the time, practice working with the numbers by hand is extremely important and valuable.