WEBVTT 00:00:10.136 --> 00:00:12.016 My name is Wilfried Schmid. 00:00:12.426 --> 00:00:15.236 I am Professor of Mathematics at Harvard University. 00:00:15.506 --> 00:00:19.356 I was a member of the National Mathematics Advisory Panel. 00:00:20.306 --> 00:00:27.226 The panel did some of its work as a whole and some in five task groups and three committees. 00:00:27.776 --> 00:00:33.096 I was a member of two of the task groups: on critical knowledge and skills 00:00:33.566 --> 00:00:35.036 and the task group on assessment. 00:00:35.706 --> 00:00:42.026 Algebra is really the gateway to mathematics, to engineering, to science in college. 00:00:42.606 --> 00:00:47.496 And not surprisingly, there are a number of studies that show a very high correlation 00:00:47.496 --> 00:00:52.566 between success in algebra in high school and graduation from four-year colleges. 00:00:53.316 --> 00:00:57.996 The subject matter of Algebra I is important for technical careers, 00:00:58.466 --> 00:01:01.656 even careers that don't require a college education. 00:01:02.036 --> 00:01:09.656 For students who expect to go to college to major in science, including biological science 00:01:09.656 --> 00:01:16.256 and social science, or in engineering or who expect to go on to a professional school, 00:01:16.576 --> 00:01:19.156 well they certainly also need Algebra II. 00:01:19.316 --> 00:01:23.886 The panel was asked to make recommendations, based on the best available evidence, 00:01:24.426 --> 00:01:28.826 on the skills and skill progressions necessary for success in algebra. 00:01:29.786 --> 00:01:35.076 To do that, one, of course, has to have clarity on what algebra actually is. 00:01:35.506 --> 00:01:41.396 Three of the members of the panel, including myself, are university mathematicians, so we used, 00:01:41.556 --> 00:01:43.646 first of all, our professional judgment. 00:01:44.706 --> 00:01:51.556 We also examined the state standards of all 50 states-the technical term for these documents is 00:01:51.556 --> 00:01:56.396 "curriculum frameworks," and we looked at the curriculum guidelines 00:01:56.396 --> 00:02:02.346 of the high performing countries, such as Japan, South Korea, Singapore. 00:02:03.376 --> 00:02:09.616 Now, there are some controversies in mathematics education, but the nature of algebra is not. 00:02:10.276 --> 00:02:15.096 There seems to be a high degree of consensus of the major topics of algebra. 00:02:15.636 --> 00:02:22.486 The panel gives a list of six major topics, which cover Algebra I, Algebra II, 00:02:23.216 --> 00:02:28.366 and depending on how the curriculum is organized also part of Pre-Calculus. 00:02:28.856 --> 00:02:37.976 The major topics are: symbols and expressions, linear equations, quadratic equations, functions, 00:02:38.846 --> 00:02:43.956 the algebra of polynomials, and combinatorics and finite probability. 00:02:44.846 --> 00:02:49.036 The first five topics really constitute the essence of algebra, 00:02:49.526 --> 00:02:51.306 and they are all equally important. 00:02:51.946 --> 00:02:55.786 The last topic, combinatorics and finite probability, 00:02:56.306 --> 00:02:58.966 is really an application of the binomial theorem. 00:02:59.906 --> 00:03:05.296 A very important application to be sure, but not really basic algebra. 00:03:06.366 --> 00:03:11.216 In all cases, it is not just the skills and concepts that should be taught, 00:03:11.756 --> 00:03:15.246 but also the way these are used in solving problems. 00:03:16.286 --> 00:03:23.146 Algebra involves three main circles of ideas; symbolic computation, the notion of function, 00:03:23.976 --> 00:03:29.306 and the process of translating problems into equations that then can be solved. 00:03:29.756 --> 00:03:36.036 Our list of major topics of algebra is an elaboration of these circles of ideas. 00:03:36.816 --> 00:03:44.216 The idea that one can compute with symbols as if they were numbers-provided one uses the rules 00:03:44.216 --> 00:03:49.846 that apply to computations with numbers-is absolutely crucial to algebra 00:03:50.146 --> 00:03:52.716 and all the mathematics that comes after algebra. 00:03:53.626 --> 00:03:57.576 The idea of function is related to symbolic computation. 00:03:58.476 --> 00:04:01.656 Let's imagine somebody standing on the top of a tall building 00:04:02.126 --> 00:04:04.646 and throwing a stone so it can fall freely. 00:04:04.936 --> 00:04:12.366 From basic physics or from observations that have been made, one knows that after one second, 00:04:13.086 --> 00:04:18.216 the stone has fallen approximately 16 feet; after two seconds, 00:04:18.846 --> 00:04:23.796 64 feet; after three seconds, 144 feet. 00:04:25.086 --> 00:04:30.356 Now, one can make a list of how far the stone falls after various time intervals. 00:04:30.896 --> 00:04:33.366 But the far more efficient way is to say 00:04:33.656 --> 00:04:40.476 that the stone falls 16t squared feet when t is measured in seconds. 00:04:41.106 --> 00:04:46.906 And it's important to realize that t does not have to be an integer, it can be any real number. 00:04:47.436 --> 00:04:50.716 How long will it take the stone to hit the ground? 00:04:50.996 --> 00:04:53.106 Well, that depends of course on the height of the building. 00:04:54.056 --> 00:04:58.876 Let's say the height of the building is h measured in feet. 00:05:00.046 --> 00:05:08.846 Then, the amount of time, t, it takes to hit the ground satisfies the equation 16t squared =h. 00:05:09.586 --> 00:05:14.976 And that can be solved for t. This is a very simple example, of course, 00:05:15.686 --> 00:05:19.206 but the process of translating the problem into an equation 00:05:19.206 --> 00:05:24.186 that can be solved is entirely typical for applications of algebra. 00:05:24.906 --> 00:05:30.266 Is a traditional single subject approach more effective than an integrated approach? 00:05:31.106 --> 00:05:34.346 Well, in the abstract, this question is almost impossible 00:05:34.346 --> 00:05:36.866 to answer based on really solid evidence. 00:05:37.896 --> 00:05:43.916 High quality studies really do not look at the two approaches in general. 00:05:43.986 --> 00:05:48.626 They may compare individual curricula but not the approaches in general. 00:05:49.656 --> 00:05:54.186 What is important is that in either approach, 00:05:54.736 --> 00:06:02.446 the subject be presented coherently in a logical sequence and that enough attention is paid 00:06:02.446 --> 00:06:04.686 to the connections between various subjects. 00:06:05.536 --> 00:06:10.686 Symbols and expressions and linear equations are definitely Algebra I subjects. 00:06:11.306 --> 00:06:15.756 At the other end, complex numbers, the fundamental theorem of algebra, 00:06:16.266 --> 00:06:21.236 the binomial theorem and its applications are sometimes taught in Pre-Calculus. 00:06:21.886 --> 00:06:25.686 Everything in between belongs to Algebra II. 00:06:25.686 --> 00:06:31.786 For an integrated sequence, the grade level at which various topics are taught depends very much 00:06:31.786 --> 00:06:38.076 on how the curriculum is organized, how geometry topics and algebra topics are intertwined. 00:06:38.986 --> 00:06:45.866 But the sequence must be organized logically, and all of these topics are to be covered 00:06:45.866 --> 00:06:48.656 by the end of eleventh grade or the beginning of twelfth grade. 00:06:49.436 --> 00:06:55.456 Serving on the National Mathematics Panel was a really interesting experience and I hope, 00:06:55.456 --> 00:06:58.116 as I am sure all of my colleagues on the panel do, 00:06:58.716 --> 00:07:01.706 that our recommendations can make a real difference 00:07:01.706 --> 00:07:03.906 to how Mathematics is taught in this country.