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Return to conference page Return to transcript of Norman Herr
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. But this is very unrealistic! 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. We now look at the reasons we clearly needed new standards in mathematics. They can be subsumed in three main areas.
The next three slides explain a little bit about how we see some of this so we cannot escape from these issues. The facts quoted in these slides come from recent newspaper articles for the most part.
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. 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 of our graduate students are foreign-born. Even 10 years ago, less than half were. As a result of this situation it has been impossible to fill all our technical jobs with United States citizens. This is particularly true in Silicon Valley. 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. Currently it is estimated that the number of foreign-born residents of Silicon Valley is about 25% of the population. Among all the states as I said in the beginning, California colleges showed the greatest decline in high tech degrees.
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. 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. 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. Look at the effect of this lack of understanding on our students.
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%. Let me emphasize this: 88% of those students taking the exam fail it. Some of you may know a statistic of about 55% for the failure rate. Unfortunately, this is calculated by counting the 40% of the entering students who are not required to take the exam as having passed it. These 40% are counted as passing it probably so the statistic will look reasonable. I reiterate that the actual statistic is 88% taking the ELM fail it, and it is not that hard an exam overall. In any case, well over 50% of entering students in the California State University system are placed into remedial math courses. Those are some of the reasons for our current problems. They stare at us. We can’t avoid or deny them. 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. 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. Curricular development is a very complicated issue. 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.
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. 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.
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. It is only in understanding of the process of taking the remainder in long division that you see the periodicity or termination happen. 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. So what, you ask? Again, this is a question that doesn’t come up until the third year in college. 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. 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. The previous slide indicated what I call the static applications of long division. The next slide illustrates some of the “dynamic” applications.
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. 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. I don’t know of any other or any better preparation for them. 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. 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. 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. 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. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop.
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.
I think the long division problem illustrates the problem described on the slide above very well. And I put that dragon up there advisedly.
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. They do the best they can and have the most demanding job that I know of. 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.
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. I will concentrate on problems that these people suggest for testing mathematical knowledge.
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. 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.” But that’s not right. The words need to be read carefully. 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.
Hmm? 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. So what is the moral here? 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. The next slide presents a problem that Wu is very fond of (Laughter). 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.
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. 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.
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. Mathematics says the correct answer to the problem is all 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.
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. 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. One of the first things that mathematics should prepare student for is making the best possible (rational) decisions when faced with real problems.
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. It helps prepare students to defend themselves from tricky claims and fake uses of statistics. Students also have to learn how to survive in the monetary world. So a key part of our request for changes 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.
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. But there is one more example that is worth noting (Laughter).
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. 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.
It is a huge process — of re-education on everyone’s part, it is a process we all have to contribute to and work on with full attention. But I think there are grounds to hope that we can actually do it. And the one thing that has the potential to help with this process is the Framework. The Framework is something that Wu and I worked on with Janet and the Curriculum Commission, and with many of the best people in many aspects of education throughout the country. The Framework has been designed to ease our way into the teaching to the Standards. It’s something that I think we have to focus on a lot more in the next few months as we try to figure out how to reach the levels needed. I would like to just say one word about one of the ways in which the new Framework can help.
The critical thing about this is that the Standards for first grade have about 30 basic topics. Well, those topics are, for the most part, quite difficult at the first grade level and will take a great deal of time and effort to teach properly. Fortunately, it turns out that only 5 or so of them are essential. The Framework identifies the essential standards and makes your jobs as teachers and your jobs as curriculum developers much easier because the textbooks in the next textbook adoption will be focused on the emphasized topics, rather than the entire 30 topics in the Standards. So this will allow us to focus on just a few pieces and make your job of reaching the levels needed a little simpler. I think this is where I’ll stop (Applause).
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1999 Conference on Standards-Based K12 Education Telephone: (Dr. Klein: 818-677-7792) FAX: 818-677-3634 (Attn: David Klein) email: david.klein@csun.edu clipart: http://www.clipartconnection.com/, http://web2.airmail.net/patcote1/partydan.gif |
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