WEBVTT 00:00:10.086 --> 00:00:11.736 I am Hung-Hsi Wu. 00:00:11.736 --> 00:00:15.296 I am Professor of Mathematics at the University of California at Berkeley, 00:00:15.966 --> 00:00:19.596 and I was on the National Mathematics Panel. 00:00:20.396 --> 00:00:23.736 On that panel, I was in two task groups. 00:00:24.376 --> 00:00:31.246 One is the task group on conceptual understanding and skills, and the other one is on teachers. 00:00:32.415 --> 00:00:39.686 I think, if I may say so, at the moment, the main problem with our teachers in algebra, 00:00:40.096 --> 00:00:51.826 what's holding them back, really, is a deep enough understanding of the subject matter 00:00:53.046 --> 00:00:59.526 so that they can tell their students correct information. 00:00:59.796 --> 00:01:00.476 That's important. 00:01:01.346 --> 00:01:07.596 And second, they know what are the most important aspects of algebra 00:01:08.056 --> 00:01:12.206 to focus the students' attention on exactly those things. 00:01:13.106 --> 00:01:21.446 Unfortunately, I think our education system has not gotten to the point 00:01:21.446 --> 00:01:26.576 where we actually teach our teachers what they really need to know in order 00:01:26.576 --> 00:01:28.716 to function effectively in the classroom. 00:01:29.326 --> 00:01:31.426 It's a matter of failure of content knowledge. 00:01:31.976 --> 00:01:37.766 For example, the fluency in the use of symbols being so fundamental and so important, 00:01:38.416 --> 00:01:44.506 I don't think this piece of knowledge is at present known to most teachers. 00:01:46.226 --> 00:01:51.376 We simply have not taught our teachers what they really need to know. 00:01:51.696 --> 00:01:57.576 The second point about, our teachers don't quite know what 00:01:57.576 --> 00:01:59.916 to emphasize in the teaching of algebra. 00:02:00.296 --> 00:02:06.196 For example, it is supposed to be a very big deal to factor trinomials. 00:02:07.156 --> 00:02:15.636 Well, of course standardized tests are full of questions about x squared 00:02:15.636 --> 00:02:25.126 minus x plus 6 = a product of what linear factors, and this is a big deal. 00:02:25.336 --> 00:02:28.826 Now, teachers ought to be taught that it's not a big deal. 00:02:28.936 --> 00:02:31.636 It's a very minor topic. 00:02:32.126 --> 00:02:38.706 It's something that yes, students ought to have some practice with it, but ultimately, 00:02:38.706 --> 00:02:40.436 it's not that important because why? 00:02:41.196 --> 00:02:49.366 Because when students get to learn the quadratic formula-using the quadratic formula-you don't 00:02:49.366 --> 00:02:50.316 need to know anything. 00:02:50.646 --> 00:02:53.476 Plug the numbers in and out comes the factorization. 00:02:54.526 --> 00:02:56.726 So, it's good to spend a little time, 00:02:56.756 --> 00:03:00.096 get students some practice about learning how to factor a trinomial. 00:03:00.746 --> 00:03:04.526 It is important on some occasions. 00:03:05.236 --> 00:03:11.876 Why is it important for teachers to emphasize the connection among topics in mathematics? 00:03:12.846 --> 00:03:19.796 Having established connections among topics makes it easier for students to learn, to retain. 00:03:19.976 --> 00:03:22.206 Also, it makes a deeper impression on them. 00:03:22.206 --> 00:03:29.416 For example, factoring trinomials, this is not an isolated topic. 00:03:30.366 --> 00:03:36.336 Given any polynomial, the fundamental theorem of algebra tells you, 00:03:36.336 --> 00:03:40.096 no matter what the polynomial is-polynomial of degree five, 00:03:40.096 --> 00:03:46.336 for example-you can always write it as a product of five linear polynomials. 00:03:47.666 --> 00:03:51.256 So, it gives you a trinomial, which is degree two, 00:03:51.626 --> 00:03:55.416 you will know automatically it must be factorable into a product 00:03:55.726 --> 00:03:59.286 of two linear polynomial, first of all. 00:03:59.286 --> 00:04:03.856 So, conceptually, you have a framework to receive this skill. 00:04:04.336 --> 00:04:08.856 Moreover, then you will also learn the quadratic formula. 00:04:09.386 --> 00:04:12.906 Now, quadratic formula is a basic skill that people should learn. 00:04:13.466 --> 00:04:19.065 But if you learn it and at the same time once you learn it, you look back and say "Oh, 00:04:19.065 --> 00:04:21.206 the thing I learned way back that made me tremble 00:04:21.206 --> 00:04:25.526 about factorizing trinomials is a consequence of quadratic formula. 00:04:25.706 --> 00:04:27.766 Oh, that's a useful skill for me to learn." 00:04:27.766 --> 00:04:31.366 It gives you more incentive to learn the quadratic formula. 00:04:31.606 --> 00:04:35.196 At the same time, it makes you look back and say "Now, 00:04:35.196 --> 00:04:38.726 I understand this factoring of trinomial, what this is all about." 00:04:39.776 --> 00:04:41.726 So, this is the function of connections. 00:04:41.776 --> 00:04:46.706 It will make you integrate these disparate pieces of knowledge you have gained, 00:04:46.706 --> 00:04:48.956 you have learned before, into one whole. 00:04:49.416 --> 00:04:53.856 And you begin to see the whole thing, you feel much better about it. 00:04:54.066 --> 00:04:55.776 Mathematics is really quite simple. 00:04:56.916 --> 00:04:58.766 You don't need to know 1000 things. 00:04:59.176 --> 00:05:06.046 I, myself, I know three or four important things, and once I know them well enough, 00:05:06.736 --> 00:05:09.646 I know how to make deductions from them. 00:05:10.776 --> 00:05:11.826 That's plenty good enough. 00:05:12.266 --> 00:05:16.746 And I would rather have our teachers know that there are a few key things they must know. 00:05:17.436 --> 00:05:21.636 Once they know that, of course, they have to learn how to teach their students well. 00:05:22.126 --> 00:05:24.296 They themselves must learn how to make deduction, 00:05:24.296 --> 00:05:28.326 make logical deductions they can reason,they can do that. 00:05:28.326 --> 00:05:36.046 Limit themselves to a few key concepts, few key skills, they teach those well, 00:05:37.096 --> 00:05:39.326 and they would be a very successful teacher.