WEBVTT

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I am Hung-Hsi Wu.

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I am Professor of Mathematics at the University of California at Berkeley,

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and I was on the National Mathematics Panel.

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On that panel, I was in two task groups.

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One is the task group on conceptual understanding and skills, and the other one is on teachers.

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Now, the former, on conceptual understanding and skills,

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has to do with explaining what algebra is and also, more or less,

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the content aspect of what students need to know in order to achieve algebra.

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Let me give a brief overview of the major topics of school algebra,

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which is what is in the National Math Panel Report.

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The first one is the use of symbols.

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Now, this is not something usually found

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in school algebra unfortunately, but it's really truly basic.

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Use of symbols is what distinguishes algebra from the previous kind of mathematics

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that students learn, mainly arithmetic and simple things.

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So, that is a foundation skill that has not been sufficiently emphasized,

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but the Math Panel decided that it should come upfront, first thing.

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And the second topic is, you might say, the simplest topic you can imagine in algebra,

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which is linear equations involving, well, sort of the first degree.

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After that, quadratic equations-meaning things of second degree-one degree more.

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Then we get into something that's also truly basic about algebra, the concept of a function.

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And then we get into the basic functions, such as of course linear functions,

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quadratic functions, polynomial functions, rational functions, exponential functions,

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logarithmic functions, periodic functions and other things.

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After that, I think you might say that algebra shifts into a higher gear of abstraction.

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Now, the next topic as listed in the National Math Panel Report is what you might call the

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abstract concept of a polynomial.

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So, that would include things like the binomial theorem,

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things like fundamental theorem of algebra, theory about roots of an equation, things like that.

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That is necessary because the course of school algebra is not just

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to teach students certain basic skills, but it's also to prepare them for higher mathematics.

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It has to serve a dual function.

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And this is the stepping up of the level of abstraction because they will need the ability

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to reason abstractly in all advanced courses.

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When all that is done, then the last topic in the school,

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in the major topic of school algebra is finite probability,

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including permutations and combinations.

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Why is it there?

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Well, it's because of two reasons.

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One is that, as we said, in teaching the abstract theory or concept of polynomial,

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we will have to teach very naturally binomial theorem.

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And when you teach binomial theorem, one way to approach the binomial theorem is

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to consider permutations and combinations of two variables: X and Y. When you do that,

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you enter into what we call the binomial coefficients.

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And the other part is that all high school students must learn something

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about finite probability.

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There is no way you can get around that.

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I would like to address the issue of the importance of the concept of functions in the learning

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of algebra; in fact, in the learning of mathematics as a whole.

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So, I give you a simple example.

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Imagine you are ordering coffee so then suppose you have a cup of hot boiling coffee.

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So, suppose, I ask you, "How hot is your coffee?"

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So then you would say, "Too hot."

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If I ask you five minutes later, you say, "It's cooler."

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Right? So, "How cool, is it drinkable?"

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Maybe not.

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Maybe I asked three minutes later, "How hot is the coffee?"

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then you have third answer again, right.

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Now, what I am trying to get at is the simple question, how hot is the coffee?

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In order for you to answer my question adequately, you don't give me a single number, do you?

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You give me a different number, a different time.

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What you have is not one single number but rather, depending on the time,

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you have associated to that time a particular number, namely,

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the temperature of your coffee at that particular moment.

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That's what a function is.

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It is not a measurement of one single number but it's a measurement

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at each instant that's a different number.

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Anything in real life, in the natural world,

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or even in the financial world, has to do with functions.

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In order for a person to function in a high-tech age like right now,

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if a person has no concept of a function, you can pretty much forget it.

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The topics in the major topics of school algebra should not be taught

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as isolated-disjointed isolated topics but should be taught in a coherent manner.

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This does not apply to algebra alone, but rather this is a statement that applies

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to the teaching of all of mathematics in K-12.

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Mathematics cannot be taught as a bag of tricks, and the failure of mathematics education

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in K-12 is rooted, really, in this one phenomenon that many teachers

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in the classroom teach mathematics: One topic today, you learn it, memorize it;

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another topic tomorrow, learn it, memorize it, and it goes on.

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Now, what's wrong with that?

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Learning actually is a misnomer.

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What do you mean by learning?

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What you mean is, after you have learned something, you can retrieve your knowledge

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and use it when the occasion demands it.

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That's the whole purpose of learning.

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So, learning is not the ultimate goal; rather, it's a means to an end.

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You will learn something in order to be able to use it.

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To be able to use any kind of knowledge, you would have to have a good retrieval system.

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If mathematics is taught to students as a collection of isolated tricks-bags

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of tricks-and then they have no framework to receive that information,

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and then as a consequence, they have no way to retrieve the information, and you learn something

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but you have no way to retrieve what you've just learned,

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then that means you have not learned it.

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So, that's one way to answer this question, but I want to answer it slightly differently, too:

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Why having established connections among topics makes it easier for students to learn,

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to retain, and also makes a deeper impression on them.

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We realize that not all of the topics can be squeezed into two years of algebra.

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We are very upfront about it.

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In some schools or in some school districts, they may never get to the binomial theorem.

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So, then binomial theorem becomes possibly a topic in pre-calculus maybe.

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It should be taught.

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It has to be taught, but where to do it?

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Well, that...there's a whole wide open a world of possibilities,

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and we don't-by no means do we say that you must do it in algebra.

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But we are just saying that among algebraic topics, this is important,

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and we signal to you that this is important by its inclusion here.

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And beyond that, you will have to be a little flexible,

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be a little bit inventive in getting it done.

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The knowledge we want students to have is that mathematics is a whole fabric;

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it's not a collection of isolated facts.

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And what they should learn about the whole fabric is it's one single piece,

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because if they understand mathematics to be one single piece, first of all,

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their impression of mathematics is much better because now you have a whole story,

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rather than little tid-bits, and secondly one single thing is much easier to learn

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that 500 small things separately.

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So, to establish connections among topics, you should not look at that as an explicit skill

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to learn, and rather think of it as, if you want to teach mathematics properly,

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this is what you want students to know: that they are learning one single piece

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and the little bits, and they all belong in its proper place.