{"id":1392,"date":"2017-01-31T04:57:02","date_gmt":"2017-01-31T04:57:02","guid":{"rendered":"http:\/\/mathwise.net\/?p=1392"},"modified":"2017-01-31T04:57:02","modified_gmt":"2017-01-31T04:57:02","slug":"r-james-milgram","status":"publish","type":"post","link":"http:\/\/mathwise.net\/?p=1392","title":{"rendered":"R James Milgram"},"content":{"rendered":"\n\n\n
<\/p>\n\n\n\n\n
1999 Conference on Standards-Based K-12 Education<\/span><\/p>\n

California State University Northridge<\/p>\n

<\/span><\/p>\n

\n


\nTranscript of R. James Milgram
\n(edited by the speaker)
\n\"biography<\/a><\/span> Biography<\/a><\/p>\n

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<\/center> <\/td>\n<\/tr>\n

.<\/span><\/td>\n<\/td>\n\n

\u00a0\u00a0Return to conference page<\/a><\/p>\n

Return to transcript of Norman Herr<\/a><\/p>\n

\u00a0<\/span><\/p>\n


\n<\/a>Mr. Milgram: I would like to start by again thanking David Klein and Cal State Northridge for arranging and organizing this wonderful opportunity to get together and compare ideas on the incredibly challenging times ahead of us.\u00a0 Professor Wu brought up a number of critical points in his discussion and one of them that he mentioned — that this is a long term challenge — is particularly important.<\/span><\/p>\n

I’d like to fill in somewhat what the problem is here. First of all, “long term” has generally been understood to be in the order of perhaps three years, and there seem to be real expectations of being able to meet the standards in that time frame.\u00a0 But this is very unrealistic!<\/span><\/p>\n

A realistic long term is maybe 15 years. If we are lucky, in 15 years the average student may get near the standards if everything goes just right. In a shorter time than that, it is almost inconceivable to believe that this will happen. California today ranks just about at the bottom in the United States, in terms of the level of mathematical achievements of students in K-12. The United States ranks near the bottom among all the developed countries in the world in terms of math achievements of students. We have an incredibly long way to go because you have to remember that the new California Mathematics Standards were written to match the levels of the standards of the top achieving countries in the world. Meeting these standards is a daunting challenge and we had better take it seriously.<\/span><\/p>\n

We now look at the reasons we clearly needed new standards in mathematics.\u00a0 They can be subsumed in three main areas.<\/span>
\n<\/span><\/p>\n\n\n\n
R<\/span>EASONS FOR <\/span>N<\/span>EW<\/span>
\nS<\/span>TANDARDS<\/span>
\n<\/span><\/p>\n
\n

<\/center><\/p>\n

    \n
  • The increasing failure of the present system to produce enough technically skilled graduates to meet national needs<\/span> <\/li>\n
  • Curricular problems which leave more and more students without the prerequisites needed for their majors, particularly in technical areas<\/span> <\/li>\n
  • Lack of a clear understanding – on the part of teachers and math educators – of the major goals of the mathematics component of K-12 education<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


    \n<\/span><\/p>\n

     <\/p>\n

    The next three slides explain a little bit about how we see some of this so we cannot escape from these issues.\u00a0 The facts quoted in these slides come from recent newspaper articles for the most part.<\/span><\/p>\n

    <\/span><\/p>\n\n\n\n
    I<\/span>NDICATIONS OF <\/span>F<\/span>AILURES<\/span>
    \n<\/span><\/p>\n
    \n

    <\/center><\/p>\n

      \n
    • From 1990 to 1996 there has been a 5% decline in high-tech degrees — engineering, math, physics, computer science — in this country and the trend is continuing.<\/span> <\/li>\n
    • Of the decreasing number of high-tech degrees awarded a significant and growing proportion go to foreign nationals.<\/span> <\/li>\n
    • At the doctorate level 45% of high-tech degrees were granted to non-U.S. Citizens<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

       <\/p>\n

      From 1990-96, there’s been a 5% decline in high-tech degrees overall in this country. And the trend is continuing — in fact, the trend is accelerating.\u00a0 Even though the number of high-tech degrees is decreasing, it is vital to note that an ever increasing portion go to foreign nationals. At the doctorate level, for example, 45% of high-tech degrees are granted to non-U.S. Citizens and at Stanford, in the mathematics department, two thirds\u00a0 of our graduate students are foreign-born. Even 10 years ago, less than half were.<\/span><\/p>\n

      As a result of this situation it has been impossible to fill all our technical jobs with United States citizens.\u00a0 This is particularly true in Silicon Valley.\u00a0 To find qualified people to fill these positions Congress was intensely lobbied by Silicon Valley, and Congress was forced, much against their will, to provide 142,500 more visas for foreign nationals to fill jobs in Silicon Valley.<\/span><\/p>\n

      Currently it is estimated that the number of foreign-born residents of Silicon Valley is about 25% of the population.<\/span><\/p>\n

      Among all the states as I said in the beginning, California colleges showed the greatest decline in high tech degrees.<\/span><\/p>\n

      <\/span><\/p>\n\n\n\n
      I<\/span>NDICATIONS OF <\/span>F<\/span>AILURES – <\/span>II<\/span>
      \n<\/span><\/p>\n
      \n

      <\/center><\/p>\n

        \n
      • Last year Congress was forced to provide 142,500 more visas for foreign nationals with high-tech skills<\/span> <\/li>\n
      • Currently it is estimated that the number of foreign born residents of Silicon Valley is about 25% of the population<\/span> <\/li>\n
      • Among all states, California’s colleges showed the greatest decline in high-tech degrees awarded.<\/span><\/li>\n<\/ul>\n
        <\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

         <\/p>\n

        So the first point is that the system today is simply failing to produce enough technically qualified graduates to meet national needs. The foremost problems and most dramatic declines are here in California.<\/span><\/p>\n

        Curricular problems are overwhelming here and leave more and more students without prerequisites needed for developing and learning technical skills in college. When they come to us, even at Stanford, more and more of them are just not able to become engineers and scientists, even though this is their original intent. They just don’t have the background any more. It is a dramatic change.<\/span><\/p>\n

        Finally, and sadly, because I have the utmost respect, and I think we all do, for the practicing teachers, the level of understanding on the part of teachers and above all of math educators — that is members of the educational schools throughout the country — that is required for teaching mathematics in K-12 is just not there any more.<\/span><\/p>\n

        Look at the effect of this lack of understanding on our students.<\/span><\/p>\n

        <\/span><\/p>\n\n\n\n
        I<\/span>NDICATIONS OF <\/span>F<\/span>AILURES – <\/span>III<\/span>
        \n<\/span><\/p>\n
        \n

        <\/center><\/p>\n

          \n
        • The percentage of entering students in the California State University System who are place into remedial mathematics courses after taking the ELM placement exam is about 88%<\/span> <\/li>\n
        • Overall, well over 50% of entering students are placed into remedial mathematics courses.<\/span> <\/li>\n
        • The average level of the questions on the recent version of the ELM is about grade level 6.9 according to the new California Standards.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

           <\/p>\n

          This 88% is a statistic that astounded me. And it is correct, differing from the failure rates commonly reported (which are bad enough). The percentage of entering students in the California State system who are placed into remedial mathematics courses after taking the ELM placement exam is 88%.\u00a0 Let me emphasize this: 88% of those students taking <\/b>the exam fail it. Some of you may know a statistic of about 55% for the failure rate.\u00a0\u00a0 Unfortunately, this is calculated by counting the 40% of the entering students who are not required to take the exam as having passed it.<\/span><\/p>\n

          These 40% are counted as passing it probably so the statistic will look reasonable.\u00a0 I reiterate that the actual statistic is 88% taking the ELM fail it, and it is not that hard an exam overall.\u00a0 In any case, well over 50% of entering students in the California State University system are placed into remedial math courses.<\/span><\/p>\n

          Those are some of the reasons for our current problems. They stare at us. We can’t avoid or deny them.<\/span><\/p>\n

          <\/span>Now, I would like to give you an idea of the real complexity of the problem and the consequent difficulty with trying to fix it.<\/span><\/p>\n

          On our first slide the second problem with mathematics that I indicated is the lack of understanding of curricular development on the part of math educators.<\/span><\/p>\n

          Curricular development is a very complicated issue.\u00a0 As an illustration, I’m going to look at one topic, long division, now. Long division is something that a lot of professional math educators want to take out of curriculum. So let’s just look at why it is in the curriculum.<\/span><\/p>\n

           <\/p>\n\n\n\n
          C<\/span>URRICULAR <\/span>P<\/span>ROBLEMS<\/span>
          \n<\/span><\/p>\n
          \n

          <\/center><\/p>\n

            \n
          • The recent fashion of not teaching material like long division and factoring polynomials is based on claims that such skills are no longer useful.<\/span> <\/li>\n
          • This reflects a deep lack of understanding of the role of mathematics in fields like science, engineering and economics.<\/span> <\/li>\n
          • In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

             <\/p>\n

            First of all, I claim that taking — even asking to take it out of the curriculum — shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they’ve been introduced.\u00a0 Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.<\/span><\/p>\n

            <\/span><\/p>\n\n\n\n
            S<\/span>OME <\/span>S<\/span>KILLS <\/span>D<\/span>IRECTLY
            \n<\/span>A<\/span>SSOCIATED WITH <\/span>L<\/span>ONG
            \n<\/span>D<\/span>IVISION<\/span>
            \n<\/span><\/p>\n
            \n

            <\/center><\/p>\n

              \n
            • Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.<\/span> <\/li>\n
            • Long division is essential in learning to manipulate and factor polynomials.<\/span> <\/li>\n
            • Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

               <\/p>\n

              So just to start, understanding that decimals represent rational numbers if and only if they are terminating or ultimately repeating — a skill that was requested be put into the standards by math educators — cannot be understood without long division.\u00a0 It is only in understanding of the process of taking the remainder in long division that you see the periodicity or termination happen.<\/span><\/p>\n

              I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it’s minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials.<\/span><\/p>\n

              So what, you ask?\u00a0 Again, this is a question that doesn’t come up until the third year in college.\u00a0 At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory.\u00a0 It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory — the chief underpinning of optimization and design in engineering, economics, and other areas.<\/span><\/p>\n

              <\/span>The previous slide indicated\u00a0 what I call the static applications of long division. The next slide illustrates\u00a0 some of the “dynamic” applications.<\/span>
              \n<\/span><\/p>\n

               <\/p>\n\n\n\n
              D<\/span>YNAMIC <\/span>S<\/span>KILLS <\/span>A<\/span>SSOCIATED
              \nTO <\/span>L<\/span>ONG <\/span>D<\/span>IVISION<\/span>
              \n<\/span><\/p>\n
              \n

              <\/center><\/p>\n

                \n
              • The process of long division is one of successive approximation, with the accuracy of the answer increasing by an order of magnitude at each step.<\/span> <\/li>\n
              • The skills associated with this process become more and more fundamental as students advance.<\/span> \n
                  \n
                • They include all infinite convergence processes, hence all of calculus, as well as much of statistics and probability, to say nothing of differential equations.<\/span> <\/li>\n<\/ul>\n<\/li>\n
                • Long division is the main application of the previously learned skills of approximation.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                   <\/p>\n

                  Long division is the only process in the K – 12 mathematics curriculum in which approximation is really essential. The process of long division is a process of repeatedly approximating and improving your estimates by an order of magnitude at each step. There is no other point in K – 12 mathematics where estimation comes in as clearly and precisely as this. But notice that long division is also a continuous process of approximation, the answer keeps getting more and more accurate and when the students learn how to do long division with decimals they learn to carry the process to many decimal places.\u00a0 This leads naturally — in a well conceived curriculum — to students understanding continuous processes, and ultimately even continuous functions and power series. The development of these skills are all contingent on a reasonable development of long division.\u00a0 I don’t know of any other or any better preparation for them.<\/span><\/p>\n

                  What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I’m referring to here is the experience of my students in a differential equations class in the fall of 1998.\u00a0 The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations.\u00a0\u00a0 Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.<\/span><\/p>\n

                  But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster.\u00a0 Moreover, it was very difficult for them to fill in the gaps in their knowledge.\u00a0 It seems to take a considerable amount of time for the requisite skills to develop.<\/span><\/p>\n

                   <\/p>\n\n\n\n
                  A<\/span>PPLICATION OF THE <\/span>S<\/span>KILLS
                  \n<\/span>A<\/span>SSOCIATED TO <\/span>L<\/span>ONG <\/span>D<\/span>IVISION<\/span>
                  \n<\/span><\/p>\n
                  \n

                  <\/center><\/p>\n

                    \n
                  • The combination of these skills is used critically in economics, engineering and the basic sciences via Laplace transforms and Fourier Series.<\/span> <\/li>\n
                  • Without a thorough grounding in these topics it is impossible to do more than routine work in most areas of engineering, the most active current areas of economics and generally, any area involving optimization.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                     <\/p>\n

                    So you see the problem. The problem is that the scope of things in mathematics is so long that an ordinary second, third, fourth grade teacher is not equipped to make a judgment about whether a subject is needed or not needed.<\/span><\/p>\n

                     <\/p>\n\n\n\n
                     <\/p>\n\n\n\n
                    <\/td>\n\n

                    S<\/span>COPE IN THE <\/span>M<\/span>ATHEMATICS<\/span><\/p>\n

                    C<\/span>URRICULUM<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


                    \n<\/span><\/p>\n

                    \n

                    <\/center><\/p>\n

                      \n
                    • The long division story illustrates one of the chief problems with curricular development in mathematics. The period needed before a learned skill can be fully utilized can be as long as eight to ten years.<\/span> <\/li>\n
                    • It takes real knowledge of mathematics as well as how it is applied to make judgements regarding curricular content.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                       <\/p>\n

                      I think the long division problem illustrates the problem described on the slide above very well. And I put that dragon up there advisedly.<\/span>
                      \n<\/span><\/p>\n

                       <\/p>\n\n\n\n
                      E<\/span>DUCATORS TELL US OF THE
                      \nNEED FOR CONCEPTUAL
                      \nUNDERSTANDING AND MATH
                      \nREASONING SKILLS IN OUR
                      \nSTUDENTS<\/span>
                      \n<\/span><\/p>\n
                      \n

                      <\/center><\/p>\n

                        \n
                      • These skills ARE critical in todays technological society.<\/span> <\/li>\n
                      • What many math educators tell us represent examples and exercises for developing these skills are NOT relevant and\/or NOT correct.<\/span><\/li>\n<\/ul>\n
                        <\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                         <\/p>\n

                        The first slide mentioned a third aspect of the problem, which was the lack of knowledge of the subject on the part of math educators. To make it clear, I’m talking about math educators and not teachers. Teachers learn what they are told in the education schools and just hope that this background prepares them sufficiently.\u00a0 They do the best they can and have the most demanding job that I know of.\u00a0 As a group I believe they are the most dedicated people I know of. But if you do not provide teachers with the proper tools, they can’t do a proper job.<\/span>
                        \n<\/span><\/p>\n

                         <\/p>\n\n\n\n
                        M<\/span>ATH <\/span>E<\/span>DUCATORS <\/span>O<\/span>FTEN<\/span>
                        \nH<\/span>AVE <\/span>L<\/span>IMITED <\/span>K<\/span>NOWLEDGE<\/span>
                        \n<\/span>OF <\/span>M<\/span>ATHEMATICS<\/span>
                        \n<\/span><\/p>\n
                        \n

                        <\/center><\/p>\n

                          \n
                        • For example, three of the 14 problems originally proposed by the presidential commission on the eighth grade national mathematics text and\/or the “solutions” they gave were INCORRECT. This commission included many of the best known math education experts in the country.<\/span> <\/li>\n
                        • The next slides discuss one of these problems.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n

                          I just want to spend a few minutes, now, looking at some of the problems that we have seen in the last few years when we — as professional mathematicians — have looked at some of the things that math educators are trying to tell the world is mathematics.\u00a0 I will concentrate on problems that these people suggest for testing mathematical knowledge.<\/span><\/p>\n

                           <\/p>\n\n\n\n
                          A<\/span> PROBLEM FROM THE<\/span>
                          \nN<\/span>ATIONAL <\/span>E<\/span>IGHTH <\/span>G<\/span>RADE<\/span>
                          \nE<\/span>XAM<\/span>
                          \n<\/span><\/p>\n
                          \n

                           <\/p>\n

                          <\/center>We are given the following pattern of dots:<\/span><\/p>\n

                          <\/p>\n

                          <\/center>
                          \nAt each step more dots are added than were added at the last step.<\/p>\n

                          How many dots are there at the twentieth step?<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n

                          This is a problem from the original proposed 8th grade national exam, produced by a presidential commission including most of the best known math educators in the country.\u00a0 The problem appears to be simple and every person I’ve asked, who I haven’t warned to think hard and carefully about it, has answered immediately, “Oh, it’s of the form n times n plus 1, so you are looking at the 20th stage, therefore the answer is 20 times 21.”<\/span><\/p>\n

                          But that’s not right. The words need to be read carefully.<\/span><\/p>\n

                          The point is, the words tell you the only thing you are actually given — namely, that there are more dots added at each stage than the previous stage. That’s all you are given, and the picture is just a picture.<\/span>
                          \n<\/span><\/p>\n

                           <\/p>\n\n\n\n
                          A<\/span>NALYSIS OF THE <\/span>P<\/span>ROBLEM<\/span>
                          \n<\/span><\/p>\n
                          \n

                          <\/center><\/p>\n

                            \n
                          • The answer given by the Presidential Commission on the National Eighth Grade Exam was<\/span><\/li>\n<\/ul>\n

                            20 X 21 = 420<\/span><\/p>\n

                            <\/center><\/p>\n

                              \n
                            • This is incorrect! The correct answer is that any number of dots is possible as long as there are at least 267.<\/span> <\/li>\n
                            • As was pointed out, the Presidential Commission that proposed this problem included many of the best known math educators in the country.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                               <\/p>\n

                              \u00a0<\/span><\/p>\n

                              <\/center>

                              <\/p>\n

                              <\/span><\/p>\n

                              <\/center> <\/p>\n\n\n\n
                              A<\/span>NALYSIS OF THE <\/span>P<\/span>ROBLEM<\/span> – II<\/span>
                              \n<\/span><\/p>\n
                              \n

                              <\/center><\/p>\n

                                \n
                              • This can be seen by considering that you must add at least seven dots to get to the fourth stage, eight to get to the fifth, nine to get to the sixth, and so on, but, of course, you can always add more.<\/span> <\/li>\n
                              • So the formula for the number of dots at the n<\/span>th<\/sup><\/span> stage with n>2 becomes:<\/span>\n
                                  \n
                                • any number at least as big as<\/span>\n

                                  6+(6+7+8+…+(n+3)) which equals<\/span><\/p>\n<\/li>\n

                                • any number at least as big as<\/span>\n

                                  (n+3)(n+4)\/2 – 9 = 267<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  Hmm?\u00a0 Actually that problem was about as complicated as any problem I’ve seen at this level, and it was proposed for the 8th grade national exam! When you read it carefully, it is a problem a 12th grade senior would have trouble solving.<\/span><\/p>\n

                                  So what is the moral here?<\/span><\/p>\n

                                  If you want to learn mathematics, you must learn it precisely. Mathematics is precision and one of the first objectives in teaching K – 12 mathematics is for students to learn precise habits of thought.<\/span><\/p>\n

                                  The next slide presents a problem that Wu is very fond of (Laughter).\u00a0 It can be found in many sources, but in particular it was included as part of the original Mathematics Standards Commission’s proposed California Mathematics Standards.<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  A<\/span> PROBLEM FROM THE
                                  \nORIGINAL <\/span>S<\/span>TANDARDS <\/span>C<\/span>OMMISSION <\/span>
                                  \nS<\/span>TANDARDS<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>You have a friend in another third grade class and want to determine which of your classrooms is bigger. How do you do it?<\/p>\n

                                  <\/span>This problem is often proposed as an example which shows that “there is no single correct answer” since you could use perimeter or volume or area to measure size.<\/b><\/span><\/p>\n

                                  Of course, this is incorrect!<\/b><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  The trouble is that bigger is not precisely defined. And if every term is not precisely defined, your problem is not well posed. So technically this is not a well-posed problem. Of course, we realize that is a little technical.\u00a0 We have an idea that bigger has certain connotations — but unfortunately, a lot of them: perimeter, area, volume, and maybe even combinations of the three such as 3A + 2.4P + 7V.<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  A<\/span>NALYSIS OF THE
                                  \n<\/span>C<\/span>OMMISSION <\/span>P<\/span>ROBLEM<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>The difficulty here is that bigger<\/b> is not precisely defined, and to do mathematics you generally have to know exactly what each term means.<\/p>\n

                                  However, mathematics does provide for the situation where terms can have different meanings. There is still a single “correct” answer. It consists of the set of all answers.<\/p>\n

                                  But since bigger can mean anything, the set of answers is uncountably infinite, and this problem is totally inappropriate for any but the most advanced high school students.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  You see, when you put in a linear combination of the three, you get an uncountable number of possible definitions of bigger. That’s all right. Mathematics allows for this, as long as you can make some sense of the problem.\u00a0 Mathematics says the correct answer<\/b> to the problem is all<\/b> possible answers to the problem (Laughter). If you are going to take that problem at face value, you have to give me an uncountable number of answers.<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  M<\/span>ORE <\/span>D<\/span>ETAIL ON <\/span>S<\/span>OLUTIONS<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>Here is an example which illustrates the point that the “answer” is a collection of “all solutions”.<\/p>\n

                                  Consider the system of two equations in three unknowns:<\/span><\/p>\n

                                  2x + y + z = 1
                                  \nx + 2y + z = 0<\/span><\/p>\n

                                  <\/center>
                                  \nA solution is x = 1, y = 0, z = -1. The answer is<\/span><\/p>\n

                                  x = 1 + y
                                  \nz = -1 – 3y<\/span><\/p>\n

                                  <\/center> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  So, what is the point? One of the most important things, as I indicated, that students should learn in doing mathematics is precise habits of thought. Suppose we start with a “real world problem”, given, as is typical for such problems, very imprecisely.\u00a0 We want students to be able to break the problem apart into smaller problems, make sense of them, and solve them or recognize that it is not possible to solve them with the information given.\u00a0 One of the first things that mathematics should prepare student for is making the best possible (rational) decisions when faced with real problems.<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  S<\/span>UMMARY <\/span>– I<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>One of the most important things that students should learn from studying mathematics is precise thinking.<\/p>\n

                                  They should understand how to recognize when a problem is well-posed.<\/p>\n

                                  They should be able to decompose a possibly ill-posed problem into pieces which can be made well-posed, and solve the individual sub-problems.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  Now, I don’t for a minute want to minimize the fact that students have to learn basic number skills, certainly they have to do that too. And they have to learn things like statistics, I mean, this is critical in our world today, and it is a wonderful thing that it is commonly taught today.\u00a0 It helps prepare students to defend themselves from tricky claims and fake uses of statistics.\u00a0 Students also have to learn how to survive in the monetary world. So a key part of our request for changes\u00a0 when the State Board of Education asked some of us at Stanford to help revise the California Math Standards was that compound interest be put back into the 7th grade standards.<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  S<\/span>UMMARY <\/span>– II<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>They should also learn the basic mathematical skills needed to survive in today’s society.<\/span><\/p>\n

                                  These include basic number-sense<\/p>\n

                                  They also include skills needed to defend themselves from sharp practices, such as being able to determine the real costs of borrowing on credit cards.<\/p>\n

                                  Additionally, they include being able to recognize illegitimate uses of statistics.<\/span><\/p><\/blockquote>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  I think everybody has the idea now. I have many more problems here, all of which are incorrect and all of which are due to some of the top math educators in the country. But I think you all get the idea of what the level is here and what we are trying to deal with, so I think we can skip most of them.\u00a0 But there is one more example that is\u00a0 worth noting (Laughter).<\/span><\/p>\n

                                   <\/p>\n\n\n\n
                                  A<\/span> P<\/span>ROBLEM FROM THE <\/span>N<\/span>EW
                                  \n<\/span>NCTM<\/span> S<\/span>TANDARDS<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>The following is proposed as a Kindergarten problem:<\/span><\/p>\n

                                  How big is 100?<\/span><\/p>\n

                                  <\/center>
                                  \nThis suffers from exactly the same difficulty. I asked one of our best graduating seniors this problem (he has a fellowship to study in Germany for next year and the year afterwards will continue his graduate work at Harvard).<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  <\/center>This is from the current new proposed version of the NCTM standards. “How big is 100?” It suffers from every one of the flaws I mentioned before. But I loved the response from the student above.<\/span><\/p>\n

                                  <\/span><\/p>\n\n\n\n
                                  A<\/span> P<\/span>ROBLEM FROM THE <\/span>N<\/span>EW
                                  \n<\/span>NCTM<\/span> S<\/span>TANDARDS<\/span> II<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>Without even a moment’s hesitation he answered:<\/span><\/p>\n

                                  Oh, about as big as 100!<\/span><\/p>\n

                                  <\/center>
                                  \nIndeed, any other answer would involve elements of perception and psychology, not mathematics.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  Okay. I think probably I’ll finish up now and say again that it’s a long process ahead. It is a serious, serious thing we are trying to do. But I think it is something that we can do. It’s just something we cannot treat lightly and cannot treat in any way as a casual enterprise.\u00a0 For example if you hear someone say something to the effect that “Oh, we’re going to give the teachers the Standards. We are going to say, now teach — and it’s over — no problem,” be very suspicious.<\/span>
                                  \n<\/span><\/p>\n\n\n\n
                                  I<\/span>MPLEMENTING THE <\/span>M<\/span>ATH<\/span>
                                  \nS<\/span>TANDARDS<\/span>
                                  \n<\/span><\/p>\n
                                  \n

                                   <\/p>\n

                                  <\/center>
                                  \n<\/span><\/p>\n\n\n\n
                                  \n
                                    \n
                                  • Problems<\/b><\/span>\n
                                      \n
                                    • California students rank at or near the bottom among all the states in average mathematics competency<\/span><\/li>\n
                                    • Generally teachers in grades K-4 have little competence in mathematics above their grade levels<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/td>\n
                                  		\n