There's a girl I occasionally tutor for math. She's grown up with people who tend to give her answers rather than teach her how to reach answers. But she's starting to get into math that the people who "help" her aren't able to do. Nothing complicated. The girl is like 10 years old. We're talking basic fractions and things. Two and two thirds divided by one third, for example. I got called in because nobody knew how to do that.
Now...I'm not claiming to be any math genius, but I am a "Person Who Understands Math." When I looked at two and two thirds divided by one third, what I "thought" in my head was "six seven eight. The answer is eight."
Now, immediately I'm guessing there are people who read that and think, "ok. Yeah, I see how you might do it that way. Maybe that's how I'd internalize it, maybe it isn't, but it makes sense, it's reasonable, it works, and however you internalize it doesn't really matter." And then there are people who read it and think "What are you talking about? Six seven eight? Where are you getting that?"
So, as a Person Who Understands Math, but doesn't really use it much and is years removed from any sort of formal setting...when asked to teach a 10 year old how to divide two and two thirds by a third...I admit that I had to sit down on my own for a few minutes and self-examine my thought process and formulate it into something I could explain in a way that didn't generate a "What are you talking about?" reaction.
It ended up becoming a two hour lesson trying to convey some fairly basic ideas.
I think that in public schools, math is generally taught by People Who Do Not Understand Math. People who have memorized processes, and simply teach those processes as The One And True Way. I kind of remember that from my own schooling. Regularly being asked to show my work, and having a difficult time doing so, because the work that they wanted me to show was stuff that I wasn't doing. If somebody asks you how many arms you have, you don't have to stop and count them one at a time. If somebody asks you to divide two by one half, it's a lot faster and easier to simply multiply two by two than it is to convert two into a fraction, give separate treatment to numerator and denominator, then convert the result back into an integer. There's no reason to go to all that effort.
One of the basic ideas I attempted to convey was that division and multiplication are the same thing done in reverse. Two times two is four, and four divided by two is two. A*B=C, therefore C/B=A and C/A=B. Some of us in this thread have the benefit of looking at this and perceiving it as completely obvious, because yeah...that's what those things are. That is the relationship between multiplication and division. You can write out 2 2 4 on a piece of paper, and tap each number with a pencil, correctly saying either "2 times 2 is four" or "four divided by 2 is 2" depending on which direction you tap. You don't have to rearrange the numbers or write it two different ways. But this is not necessarily obvious to a 10 year old being taught math by People Who Do Not Understand Math. And apparently this is not only non-obvious, but a rather difficult concept.
Similarly, a great deal of time was spent trying to convey the idea that division and fractions are the same thing. Like...literally the same thing. "Three divided by four" and "three fourths" is the same thing. And even drawing it out on the paper, I was having to explain that literally the only difference is the angle of the line.
The was a foreign concept. This was something that not only had not been taught...it was apparently such a strange and new way of looking at it that it took a good 5 minutes of providing examples and demonstrating it...and then reinforcing it later on because it was apparently so unintuitive that she forgot that it worked this way. Now...this girl is not stupid. She is not handicapped. She's not a drooling idiot. She's simply been subjected to a school system and a method of learning math that is so...whatever it is...that this was a strange and new concept for her.
There are schoolteachers out there trying to teach math who are apparently teaching via such a rote method, with so little understanding of what's actually going on and why...that apparently these things are difficult. And these students learning via these methods...are doing the math. There are student dividing by fractions who don't understand that fractions and division are the same thing because the teachers don't understand this and would never be able to explain the idea. There's an awful foundation of memorized methods built upon memorized methods with no comprehension of how or why those methods work.
Over the course of my lesson with this girl, I continually used larger numbers and things that weren't even numbers to demonstrate that what the numbers are really doesn't matter so long as you understand the relationships. 2 divided by one third is the same as two times three. 2 divided by one fourth is the same as two times four. Two divided by one billionth is the same as two times a billion. Two divided by one over a smiley face is the same as two times a smiley face. 2 / 1/x = 2*x, and it doesn't matter what x is. This is not complicated...but I invite any of you to try to teach this to a ten year old learning math in a public school.
So continuing on, we were working on a particular problem where I asked the girl to multiply something like one and eight fourths by four. I was pointing the pencil at various numbers on the page and talking her through it. When doing that...I stopped...paused, and explained that there were a couple different ways to do it. When multiplying 1 and 8/4 by 4, it doesn't matter if you convert 1+8/4 to 12/4 then multiply, or if you multiply 1*4 and multiply 8/4*4 then add the two results, or if you look at the fact that "eight fourths" is the same as "eight divided by four" and therefore x divided by y times y is x and there's no need to actually do the math because you can simply cancel the fours out, or you can look at 8/4 and convert it to 2 and then simply multiply three by four...etc. There are a bunch of ways you can internalize the math...it doesn't matter. However...it appears that in these schools students are being taught one particular method, and to apply that one particular method in all cases, always, no matter what. You could hand them (1 + 1/1) * 2 and they would convert it to 2/1 * 2/1, write out a formal three step process of multiplying tops and bottoms then simplifying. Just because that's how they've been taught to do it.
I think that's a very unfortunate way to teach math.
So, that became a bit of a tangent, showing her different ways of doing the same problem, and that it doesn't matter how you do it so long as you understand what's going on. And during this process, during one of those methods, I asked her to multiply 8 by four. And it happens that 8*4 is not something she had memorized. Ok, yes...we were all supposed to memorize at least up to 12*12 and I had one teacher who insisted we memorize up to 25*25, but not all of us did. I'll raise my hand...I never did. There are a few I never memorized. It wasn't totally necessary. For example, I never memorized 8*9. But being a Person Who Understands Math who did memorize 9*9, I was generally able to convert 8*9 into 81-9 and do that in my head fast enough that the examiners couldn't tell the difference.
So, I asked the girl to multiply 8 by 4, and it wasn't something she had memorized. No big deal. So what's eight times two? "Sixteen" she says. No problem. Ok, what's 16 times two? Well, she had some trouble with that one, but eventually she did come to 32. So..."8 times 2 is 16 and 16 times 2 is 32, so what's 8 times 4?"
Silence.
"Ok, what's 8 plus 8?" 16. "What's 16 plus 16?" *thinks* ...32. "Ok, what's 8 times 4?"
Silence.
This became a 5 or 10 minute thing, trying to get her to understand that 8*4 = 8*2*2 = 16+16 = 8+8+8+8, etc. Apparently the concept that "multiplying x by y is the same as adding x, y times" was a strange and unfamiliar concept. x * 4 = x+x+x+x was not an idea that she understood. It was not intuitive, it required explaining, and even after demonstrating it several times, she was hesitant to apply it because in her mind...just because 2*2 = 2+2 and 2*3 = 2+2+2 and 2*4 = 2+2+2+2...didn't in her mind mean that 2*5 would equal 2+2+2+2+2.
These kids are learning fractions, learning algebra, learning math in general...without even a basic understand of basic arithmetic. And I would totally guess that of her classmates who did memorize 8*4=32, probably a lot of them would also not understand that 8*4 and 8+8+8+8 are the same thing, and would see that they both equal 32 as a complete coincidence of no importance and with no application in any other case.
So...this lesson did end up having a happy ending. After about two hours she'd gone from literally crying to doing her homework problems as fast as she could write, with a confused look on her face because she didn't understand why it was so easy. "Is that really all I have to do? Are these answers right?" Yep. That's totally correct. And yes, that's all you have to do.
But some people Do Not Understand Math. And some of those people are teaching math.
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Topic: "People Who Understand Math" and "How Math is Taught" (Read 8285 times)