Sunday, March 11, 2012

Letter from Huck Finn: Thinking about Inverting, Multiplying, Understanding, and Hanging

Out in Left Field proudly presents the fourth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names (with the exception of Miss Katharine's) have been changed to protect privacy.

Thinking about Inverting, Multiplying, Understanding, and Hanging

Miss Katharine gave me an earful after my last letter. She was bent out of shape over my statement that I am of “average intelligence”. I tried to tell her that enough people believe I'm an idiot or of otherwise low intelligence because of my opinions on education to allow me to get away with saying that. She mumbled something I couldn't hear about who the real idiots are and then told me to "clarify matters". So here goes:

I may be above average in intelligence, but I am neither a math genius, nor am I brilliant. I make this distinction because I do not believe that traditional math serves only those people who are brilliant and/or are pre-destined to study mathematics. I believe it provided many people with sufficient preparation to take calculus in high school or college.

Some think that the old ways of teaching math were just rote memorization, with no understanding. To be blunt, I happen to think that’s a pile of crap. To be more refined, I happen to believe that procedural fluency leads to understanding; once you’re able to do certain procedures, it’s easier to understand why they work. Obviously, there is a lot of back and forth on this. Tina, my supervising teacher, believes that students shouldn’t just be told how to do procedures; they should understand the “why.” She is, however, a firm believer in having students do many problems to gain procedural fluency.

“Do you think you’d like to try teaching a class?” Tina asked me one day. I said I was ready, and we decided on a day that I would do it, which turned out to be the day that fractional division came up in the pre-algebra class. “That one’s a doozy,” she said. “Are you sure you want to take that on?”

“It’s my favorite topic,” I said.

She was intrigued with this and told me about people not being able to explain how the “invert and multiply” rule works. She is correct. In discussions, arguments, and brawls over “understanding”, the invert and multiply rule for dividing fractions is the poster child. For those who view understanding as paramount, the fact that many people are unable to explain why the rule works is considered as another piece of evidence that traditional methods have failed.

“The students already learned how to do this, in fifth grade,” she said. “But for pre-algebra I really like to make sure they understand why it works.” And I agree. Explaining the derivation of the invert and multiply rule is an appropriate topic for a pre-algebra class.

I suggested that since we had just finished the chapter on solving one step equations, we could build on that knowledge. Since a/b divided by c/d equals some number x, then we know that x times (c/d) = a/b. And since they know that c/d times d/c equals 1, we can then isolate x by multiplying each side by d/c, resulting in x = a/b x d/c.

She thought a moment and said she likes to use the “complex fraction” method. So if the problem is 7/8 divided by 3/32 , it’s written as a complex fraction: (7/8) / (3/32).

To simplify, we want to get rid of the 3/32 in the denominator. The easiest way is to multiply by the reciprocal, 32/3 which makes the denominator equal to 1. But if we multiply the denominator by 32/3 we have to do the same for the numerator. This sequence of steps ends up looking like: (7/8 x 32/3) / (3/32 x 32/2) = (7/8 x 32/3)/1 = 7/8 x 32/3.

And there it is; invert and multiply in all its glory. To ensure they really see it, she has students use this method in their homework problems rather than simply inverting and multiplying. This appealed to me because of my belief that procedure leads to understanding, and that the repetition of doing it “the long way” at the very least would make them grateful for just inverting and multiplying. Every time they did it the short way, they would remember with thankfulness not having to go through the rigmarole—which just happened to be the derivation.

Tina taught the lesson to the first period class, and then had me teach the same lesson for the third period class. Unlike the discovery-based algebra class, the pre-algebra classes were taught in a traditional manner. Nevertheless, when my time came to teach, I was a bit nervous, a bit rushed, and probably stood in front of the whiteboard too many times, blocking my own writing.

To make sure they connected with what they already knew, I started off with a question: “How many of you remember how to divide a fraction by a fraction?” Many hands went up. I called on Emilio. I waited. He said nothing.

“Well, Emilio, how do you do it?”

“Oh, I thought you were just asking if we remembered how to do it so I raised my hand.” He will either go into math or law, I remember thinking.

“Well, as long as we’re here, why don’t you tell us?”

“You flip the second fraction upside-down and you multiply,” he said.

“Correct,” I said. “But now we’re going to find out why it works.” I went through the explanation, asking questions as I did so to make sure they were following. I gave them some fraction division problems to do, instructing them to do it as I had done on the board.

In the end, the students knew what problems are solved by fractional division as well as the procedure. But I’d also say, that in the days that followed, once they were again allowed to use the short way (invert and multiply), the derivation did not matter much and probably only a few could reproduce it if asked. That doesn’t bother me.  I'd rather they know how to solve problems than be able to reproduce an explanation they don’t fully understand for a procedure they cannot perform.  A teacher friend of mine told me not to say that too loud or they’ll hang me. Well then, I guess when judgment day comes in the education world you’ll find old Huck hanging from a tree

Lsquared said...

I rather like that derivation--not one I've thought about recently. In an ideal world, not only would the students know how do divide fractions, and would know how to do it, but they would also know a way to simplify complex fractions, and they would be closer to really internalizing how to find equivalent fractions (an extremely useful thing!). Keep in mind that it could be worse--you could be in a school district with a curriculum where this is the first time they've seen division of fractions. The least satisfying thing about texts that call themselves standards based is how they sometimes avoid the hard topics altogether (can't think of a way to teach division of fractions in a way that children will understand what they're doing? OK--lets not teach it at all--we'll leave that for the algebra teacher).

Geozel said...

I wonder how you explain to the students that(a/b)*(b/a)=1. In other words, how you explain that dividing by a number x is equivalent to multiplication by the fraction 1/x. This cannot be _proved_ because fractions do not exist as numbers before you define their properties in such a way that the useful rules established for whole numbers remain intact. Of course, to a mathematician this is a known formalism. However, to a young student some motivation based on some non-formal examples or reasoning should be presented. Pumping the difficulty over from the formal rule "inverse and multiply" to a formal rule (a/b)*(b/a)=1 is hardly helpful, especially to a thinking student. To the one who is not thinking both are equally unclear, so why introduce an extra layer?

bky said...

If students don't know how to multiply fractions yet, there is no point in teaching them to divide fractions. So we can assume they get (a/b)*(b/a) = (ab/ab) = 1.

I like to show it this way. Start with understanding that solving
AX = B (finding the unknown factor X) is what division is, so X=B divide by A, for any kind of numbers. I guess that is formalism.
(I am avoiding using the fraction bar for division.) Then if you have

(17/23)*X = 31/11

you want to get at the X; get rid of what you don't want in two stages. Multiply by 23 to get rid of the unit fraction 1/23, thus

17X = (31/11)*23

now divide by 17, thus

X = (31/11)*23*(1/17)
= (31/11)*(23/17)

But also X = (31/11) divided by (17/23)

and comparing these two you get the well-known and somehow derisively named "invert and multiply". The point is simply that multiplying bot sides of
(17/23)*X = 31/11 by the reciprocal of 17/23 solves the equation for X, and that is division. But students have to know about multiplying.

There is something I don't like about the complex fractions method, which is that students have to know and understand that a complex fraction is a division of fractions. To me that seems a harder sell then working with the idea that dividing is solving for an unknown factor.

KimS said...

Huck - quit rocking the boat! Don't you know what happens when you rock a boat too hard?

Katharine Beals said...

Huck responds: I agree with BKY regarding multiplication of fractions. In fact, we had just finished the section on fraction multiplication and showed that for non-zero integers a and b, a/b * b/a = 1 following the rules of fraction multiplication. I also agree with BKY's preferred method of teaching fractional division. I stated in my letter that I wanted to do it that way in the paragraph prior to the description of the complex fraction method. With respect to the complex fraction being a division of fractions, I did in fact introduce the fact that fractions themselves represent division (i.e., a/b where a and b are integers), and from there showed how you could represent division of two fractions by a complex fraction a/b where a and b are fractions.

Rivka said...

Wow! That's such a clear algebraic explanation for something which is often presented as hopelessly confusing. Thanks.

For younger kids, I think a cooking/baking example really helps them visualize why you invert and multiply. Suppose that you have 6 cups of sugar, and you want to divide it into portions using a 1/2 cup scoop. It is easy to picture this scenario and understand that you will wind up with twice as many portions as you have cups. With a 1/3 cup scoop, three times as many portions as cups. With a 2/3 cup scoop? Half as many as the previous example, or multiply by three and then divide by two.

It just isn't that difficult of a concept, as long as the teacher understands it.

Michael Paul Goldenberg said...

I know I'm spitting into the wind here, but nonetheless, allow me to suggest that the method you promote, the values that inform that method, likely reflect YOUR way of having been taught mathematics, your way of "mastering" that which you believe you've mastered in mathematics, and hence, you presume it all to be "best" and "logical" and "sensible." Further, you clearly imply (repeatedly and smugly) that other viewpoints just don't "add up" (to use one of the most overused cliches connected with mathematics education and the Math Wars over the past couple of decades).

And yet, it moves. Which is to say, some of us out here in the real world (but a different part of it than the one you inhabit) find that rote is a waste of our time (speaking as a learner) when it comes to things that can be UNDERSTOOD. Please, no analogies to physical activities like golf or baseball, where muscle training is an essential part of the process, or to learning foreign language, where unless/until one gets some deep knowledge of the guts of the language - roots and prefixes and derivations - rote is pretty much all there is. I'm talking about the lion's share of mathematics, where I rarely memorize but rather look for structure, patterns, and reasons why. If mathematics doesn't make SENSE, then it's not mathematics, but rather some sort of magic. And if there's one thing I know about math, it's that while it often gives us beautiful, magical experiences, it's as far from magic as one can get.

I first realized that it wasn't weird to approach mathematics the way I increasingly have come to do (but only as an adult: my K-12 teachers never hinted at this possibility) when my calculus II teacher made clear that he only had the derivatives for sine and cosine memorized: the rest he could easily derive from his understanding of trigonometry and some derivative rules (mostly the quotient rule, which, of course, he could also derive, but didn't need to because he USED it a lot). If he used the derivative of secant a lot, he'd likely have it at his fingertips, as I do the quadratic formula (though I can derive that on request, of course).

Michael Paul Goldenberg said...

You have decided that practice MUST precede understanding. That's certainly your right as a learner, but not as a teacher. You're imposing your prejudices and learning style on everyone you try to instruct. But you will (predictably) fail with many of those students. A few months down the road, many of those kids you're so sure "got it" will not recall how to divide fractions. Some will swear no one ever taught it to them. Unless, of course, the derivation, the WHY, has sunk in and made sense to them of nonsense. In that case, they'll very GLADLY return to that derivation and figure out the 'short way' again for themselves. If you've given them nothing to ground the concept, however, they'll be hopelessly lost. And in the inner-city schools where I've worked for two decades, that's been what mathematics has been for most kids most of the time: meaningless rules to be memorized and repeated without sense. And it fails them, time and time again. Do two negatives make a negative (addition) or maybe a positive, maybe a negative (subtraction) or a positive (multiplication and division)? Who knows? Certainly not most of these kids, because there's no conceptual ground upon which they can stand.

You really need to rethink your biases, though I know you won't (you've made that abundantly clear in post after post, anonymous or attributed). You're SURE you're right. Maybe that's what it means to be left-brained, in which case I am thankful that I am whole-brained. I have successfully completed graduate degrees in literature,educational psychology, and mathematics education. And I continue to learn in as many areas as I can, in whatever ways I can, without needing to insist that there's no place for rote, but merely that it isn't the best option for many sorts of learning. Would that you could see past your nose.

Katharine Beals said...

"rote is a waste of our time (speaking as a learner) when it comes to things that can be UNDERSTOOD."

Wasting time is very much relevant to the tradeoff between rote processes and conscious understanding (and not just when it comes to muscle training).

The acoustics of speech can be understood, involving lots of interesting patterns. Why not consciously analyze the speech signal we receive when people speak to us rather than doing it by rote?

Why not conduct etymological deductions rather than rote memory lookups when listening or reading?

Why not do conscious applications of the parabolic trajectory law every time we try to catch a ball?

Why not translate all arithmetic problems into their set theoretic underpinnings?

Alternatively, one could handle these things as the Quadratic Formula used to be handled in traditional math. Have the students derive it a few times on their own (no Reform Math algebra book I've seen has students do this very important activity, preferring formula memorizations and guess & check plug-ins) and then move on.

One might also explore the Strawman-free middle ground between "practice MUST precede understanding" and "understanding MUST precede practice," and make decisions based on feedback from students and student performance. And, doing our best not to impose our "prejudices and learning styles" on "everyone we try to instruct", keeping our minds open to what might be holding them back versus helping them forward.

GC said...

"Please, no analogies to physical activities like golf or baseball, where muscle training is an essential part of the process, or to learning foreign language, where unless/until one gets some deep knowledge of the guts of the language"

So, practice preceding understanding is important in just about every other thing we may learn, such as sports, a musical instrument, a foreign language, etc. but not in school subjects such as math. Did it ever occur to people who hold these kinds of views that maybe rote and practice is an important part of how our brains actually learn things in general? Why would our brains make exceptions for subjects like math?

For myself, I've found that when something is hard to grasp, repeated practice actually aids understanding. I see this with my own 1st grader. After doing math problems over and over, she starts to get why they work. Trying to explain the why is often completely useless. It's with repeated practice that she starts to get the concept behind it.

Kevin said...

I don't understand Michael Paul Goldenberg's rant about rote here. Now I sympathize with much of what he says, in that I, like his calculus teacher, don't bother memorizing much math—one of the reasons I loved math as a student is that it had so little memory work in it.

But the post wasn't extolling rote work—it was presenting a simple explanation of how division of fractions worked that allowed the students to understand "invert and multiply" (and rederive it if needed). For that matter, the explanation for understanding came before the practice, so Goldenberg's rant seems to miss the mark entirely.

I have found for my son that the amount of practice to master a subject (and retain it for years) is often an order of magnitude less than teachers assign. I much prefer the practice approach of Singapore Primary math or the Art of Problem Solving Books, where you practice by using the ideas and methods in subsequent sections of the book, rather than drill-test-forget cycle of most American math texts.

Anonymous said...

To some extent conceptual understanding is overrated. The assumption seems to be that if a student understands something conceptually they have mastered it. I homeschool and I've found that isn't always the case. For example, we do Singapore Math. To make addition of a problem like 8 + 5 easier you add 2 to the 8 and subract 2 from the 5. My 1st grader easily got this and can do it without any problem.

However, she continually forgets to actually apply this rule. If she comes across 8 + 5 she'll get frustrated and tell me she can't do it. When I remind her of what to do, she says oh yeah and quickly answers the problem. I've started giving her multiple practice problems everyday to ensure that she actually remembers to apply a concept she easily understands. There really is no escape from practice if you want to really master a subject.

R. Craigen (WISE Math co-founder) said...

Hi Katherine. I was just pointed to your blog by Barry Garelick, whom I believe is a mutual friend. What a fantastic story! The critical insight is buried right in the middle: "Procedure leads to understanding".

I frame much of my discussion of these subjects around what I call "procedural understanding". I was taken aback to see it so precisely embedded in this story that I could have written it myself (except I'm nowhere near that gifted as a writer of dialogue!).

I probably can't improve on your excellent insights in this comment thread, so I'll just say "you're exactly right".

Allow me to introduce myself: Assistant Professor of Matheamtics, University of Manitoba and cofounder of WISE Math, the Western Initiative for Strengthening Education in Mathematics -- a response to the WNCP Math curriculum, currently in use across all of Western and Northern Canada, and parts of the Maritimes.

What's WNCP like? Anti-algorithms. Anti-skills. "personal strategies". Need I say more?

Katharine Beals said...

Thanks, R. Craigen, for your kind words. Just to clarify, I am not Huck. Huck is someone else, who prefers to remain anonymous, and I am Miss Katharine.