The Dilemma of Math Skills versus Math Insights

Di-llama

Many mathematicians and some math educators are aware that a lot of the stuff that passes for “math” in schools is simply number manipulation, or rote memory of math “facts.” 

If we “teach to the test” or use some state-sponsored curricula, or bureaucratically sanctioned “standards,” we are just providing some hoops for students to jump through, which may or may not have anything to do with actually engaging their minds and leading them to mathematical thought. 

There are great insights to be gleaned from mathematics, which fall by the wayside as we instead try to inculcate the greatest amount of students with “material that must be covered this semester.”

Here’s a little dilemma:

It’s probably true that in most cases students have to be able to proficiently master basic skills. It’s also probably true that the insights should come during that initial learning phase. 

On the other hand, for some of the reasons mentioned above, as well as others, most schools are not set up to handle fostering actual insights. There is no time and there are no resources to try to do that for most of the students. 

So what can we do? 

I.

I’d say the first thing is to do no harm.

If you just teach the basics, at least make it clear to the students, over and over, that there is more to learn, and that learning does not end when you learn a a certain way to perform a particular operation. 

Let me give you and example of what I mean: When you teach addition of natural numbers, addition will always give you a greater number than any individual addend. 

In other words in a + b = c, c will always be greater than a or b. 

That may lull students into believing that “addition always makes bigger.” 

Exactly that myth is what leads to frustration when you teach them about adding negative numbers.

So even though you may not be teaching the “negative numbers unit” yet, you must take it into account, and mention to the students that they should not get the impression that “addition always makes bigger,” because there is more to learn. 

That way, even though they may not get the deeper insight about negative numbers yet, you are preparing them for it. At the very least you will not be damaging them by teaching that “addition always makes bigger.” That was a common mistake among teachers in past years. 

II.

The second thing we could do would be to make sure that to get each student to master those basics. 

Good enough is not good enough. It is very easy to make sure that the four basic operations are understood and mastered by every non-neurologically damaged student, and many damaged ones as well. 

I know I am setting myself up for some frustrated souls to rant at me about this, but I feel it’s necessary to say it:

There is a much higher percentage of teaching-disabled teachers than learning-disabled students. 

That is not a dis at teachers. If someone is learning disabled, and you recognize it, you are not saying something mean about them. On the contrary, it is mean to know someone has a disability, and not try to get the the help they need to help themselves, and then expect them to perform, although we practically have set them up for failure. 

The same goes for teaching-disabled teachers. If we do not get them the resources they need (because we are to timid to point out that they need some new resources), and then send them out to teach without those resources, well, then we are setting them up for disaster, and of course, then everyone will blame them for the problem. 

That is tragic. What I am pleading for is that we should make sure the teachers have the means to teach every child in their classrooms at least the four basic operations very well.

It should be considered a crime for an elementary school to send children on to middle school unless those children can confidently and accurately add, multiply, subtract and divide positive and negative whole numbers, fractions, and decimals (at least). 

III.

The the third thing would be to find more than one way to teach those basics, and that at least one of them be superior to the one that the district usually teaches. No matter what method a school uses, one can always find a better one. There is no one best way – that means no matter what way you have, somewhere there is a better one. 

It would be reasonable to ask what those ways might be. Of course it’s not possible to list the definitive set. It’s not even possible to list a partial set, because each case is different. 

What I’d like to offer is a short set of tenets that we might take into account when coming up with alternative solutions. 

1) Respect the student’s minds. Assume that there is a way they can “get” the material. Strive to find a way that’s better for them, and try to let them know that you expect them to be open-minded enough to give each way a chance. Of course it won’t always work out, but beats any mind-set I’ve heard about yet. 

2) Remember, it’s about the students, and not the standards. We are there to help students, not promote any particular curriculum. 

3) Don’t patronize students. Don’t teach down to them. Don’t dumb-down the material because you think the child is too slow to grasp it. You can present it in ways that may be more accessible, of course, but non of them should involve the Tele-frikkin’-Tubbies, if you follow my drift. If you’re teaching math, teach math. Don’t make them think that learning isn’t do-able if it isn’t sugar coated. It’s like bribing a tantrum-throwing tyrant with Coke and ice-cream. (Sheesh, I don’t even know how we got to the stage in our society where that isn’t considered child-abuse.)

4) Keep it human. Ask you students what they have learned, and how they might apply it. That might pre-empt any of that “why do we have to learn this stuff, man?” baloney. Put the onus on them, and let them know that you are really curious about what they can learn from what you are teaching them that you didn’t plan on teaching them. 

5) Consider that the dilemma of math skills versus math insights is probably a false dichotomy. They can go hand-in-hand. Keep looking for ways that that combine the two. You don’t have to look in textbooks, rubrics, etc. If you keep your eyes and ears open, and listen to what the students ask about and talk about, you may find some insights yourself. It’s a two-way street. (Yeah, that’s vague, but how the heck should I know what your students are going to talk about?)

6) I dunno. I’m just a lonely blogger. You tell me. What have you got? I’d love to hear from you. I just think these Ideas are a jumping-off point. They may all be crap. They work for me, in general, but I could use all help I can get. I hope some of these thoughts helped you, too. 

Sound off!

Professor Homunculus