Postby **SimonMoon5** » Wed May 02, 2012 5:48 pm UTC

As someone who teaches math at a college level but often has to teach remedial classes to students (meaning teaching college students math classes that they were expected to learn in high school or earlier), I often am distressed by the students that think that they don't need to do arithmetic in their head because they can use their calculator. And when they use their calculator to try to work quick and easy problems, it takes them ten minutes to do a problem that might take someone else ten seconds. (I saw someone spending ten minutes trying to answer a question reading "What is one fifth of 30?") Using a calculator is a crutch. Or to put it another way:

Imagine a perfectly healthy able-bodied person who doesn't like walking. Instead, he decides that he will ride in a wheelchair. And, after all, there's no NEED to walk if you have a wheelchair, right? And sometimes you can even do things faster (going down hill, for example). And so he considers himself to have mastered walking as much as he needs to do so; it's not like actual walking is ever needed in real life.

And because he never walks anywhere, his leg muscles atrophy. He quickly finds that he couldn't walk to save his life. But that's okay because he can use a wheelchair. He doesn't understand why his coach sighs so deeply every time he passes by the guy in the wheelchair.

And then he finds that some places are not wheelchair accessible. But that's okay, he doesn't need to go to those places anyway (those grapes were probably sour anyway). You can hire a professional if you need help going to those places.

And then, he decides to get a better (physical) education. And so he signs up for a class in Running. After all, he's mastered "walking" by being in a wheelchair. And he quickly discovers that he can't run as fast as the other students; his wheelchair doesn't go as fast as someone running at top speed. In fact, he's so slow that he's going to fail Running. His coach suggests that he try not using the wheelchair. But it's too late. His leg muscles have atrophied to the point that he can never walk without months of therapy. There's no way he's going to pass Running since he can't walk. And that's a problem because Running is a prereq for all the later classes he wants to take (like Baseball or Basketball). He can't progress at all. And it's all because he chose to never learn to walk... or because his coach gave him a passing grade in Walking when really he could only do it with the help of a mechanical device; he himself was unable to perform these tasks.

That's how I feel about a lot of my students. I've had a student who couldn't multiply 7 by 10 in her head. She needed a calculator. I've had a student who couldn't subtract 2 minus 0. I've had a student who didn't know what 7 times 3 was, without the use of a calculator. And these *college* students want to learn Running when they can't Walk. Being able to do simple arithmetic without a calculator is crucial to being successful in later math classes. It's this ability to do simple math that builds up one's mathematical intuition (oh, of course, 346 is divisible by 2, because the last digit is even) because you realize how math problems work after being able to do math problems. And, yes, there's unfortunate drudgery involved in learning your times tables, but gosh, that's pretty early in your mathematical career. By the time you get to college, it should be no harder than remembering your alphabet. And surely, people wouldn't complain that learning the alphabet is too much to expect of someone, too boring and soulless?

So, what I'm saying is:

(1) There's a huge problem that students are allowed to use calculators instead of their own brains. It's as impossible to teach someone to do algebra when they can't do arithmetic as it is impossible to teach someone to read Shakespeare (much less appreciate it) if they can't read and don't know the alphabet.

(2) But that problem arises before I get the students. By the time I see them, it's too late. They've already crippled themselves by using calculators to do their thinking for them. And so, if they are asked to find seven times five, and they push the wrong buttons, they will have no idea that they've gotten the wrong answer. If they accidentally push 8 times 5 to get 40, they won't realize that that can't be the answer to 7 times 5 because 40 is even. They don't have any mathematical intuition.

(3) That Lament is nice and idealistic, but problematic as well. It's all well and good to say, "Let's not teach the boring part of math, only the fun part," but really, someone who can't do the boring part will have no chance to be truly successful at the fun part. There's a certain sense of logic that's built through formalism that really can't be learned without formal definitions, as unfortunate as they may be. And, yes, giving students only "perform these steps" problems to solve is unfortunate. But a student who can't even perform the prescribed steps, someone who can't think inside the box, they have no chance of being able to do fun stuff and think outside the box. Which isn't to say, you can't show the reasons "why" things are they way they are. I often spend a lot of time explaining why we're doing what we're doing. But the attitude I get from a lot of students is "Just show us what we're supposed to do," which I assume they inherited from previous math teachers.

(4) Learning math is often more about the journey than the destination. The goal (for people who won't use these *specific* skills of solving quadratic equations later) is often simply to show that you can use logical processes to solve problems (after someone has specifically shown you exactly how to solve this problem). Ideally, we'd do this without the qualifier, letting people solve problems that they haven't seen before. But if they can't do the former, they'll never be able to do the latter. You don't want to learn math? Fine, you don't want to be a person who can solve problems. Good thing that you never have problems in your life. Like all the rest of us, your life is perfect.