By R. James Milgram
April 28, 2001
Dear Ms. Zernike,
I am delighted to see that the NY Times has finally noticed the problems with mathematics education in New York and the nation generally. Your article seemed informed and balanced, but, as I am sure you are aware, it barely scratches the surface.
I, H.-H. Wu at Berkeley, and a number of my colleagues at Stanford are responsible for many of the recent changes in mathematics education in California.
Bas Braams' letter to you articulates many of the concerns of people like myself who are professional mathematicians and who have been fighting the worst excesses of the current system. But I would like to mention a number of other issues that might be of interest in case you or one of your colleagues does a follow-up.
One of the biggest failings of the current system is the disastrously low expectations it has for our students. Let me describe the NORMAL expectations in mathematics for high school graduates in the high performing foreign countries. In Russia and all the former iron curtain countries I've been able to check, Poland, Bulgaria, and Romania, calculus is a high school graduation requirement. This holds even for graduates of the vocational high schools, and, at least in the cities, an extremely high percentage of the populace graduates from high school. Indeed, about the only possible negative in this requirement is that there appear to be three distinct calculus courses aimed at different levels. However, I have checked with people who have had the middle course, and the content of that course is comparable with the content of a typical calculus course at one of our better universities.
The situation in Japan is more interesting. Over the past thirty or so years they have entirely redeveloped mathematics education in that country. 10 - 15 years ago, 45% - 55% of their high school graduates had taken calculus in high school, and 98% of their students graduated high school. Today, according to several discussions with a professor at Tokyo University, 98% of their population still graduates from high school, but fully 90% of these high school graduates have had calculus!
In China, currently, 10 of the 30 Cantons - comprising about 490 million people - have calculus as a regular high school course, and 70% of the high school graduates in these areas have taken calculus - typically in the eleventh grade. (Of course, it could be argued that only about 40% of the population in these cantons graduate from high school in China, but this is said to be largely because, while K - 9 education is free there, the high school education is not free, and not everyone can afford it.)
In the United States, by contrast, the only firm data we have indicates that 6.4% of our high school graduates have taken calculus. Even the most optimistic of math educators will not put their estimate at more than 10%. Moreover, I am not sure one can legitimately count the constructivist "calculus" course, Harvard Calculus, that a large number of our students take, as an honest calculus course.
If you are interested in pursuing the details of the assertions above, I can give you the names of experts that will be able to assist you.
Bas remarks - almost in passing - that the "people in charge" of the constructivist math courses and the authors of these curricula "appear to have in many cases barely an eighth-grade mathematical competence" and I fully agree that this is my experience as well.
Recently, a collaborator of mine at one of the national laboratories pointed out that his daughter's fifth grade math teacher had told the class how to subtract 2 2/3 from 8 1/6 as follows: "since 2/3 is greater than 1/6 you exchange, subtract 1/6 from 2/3, and then subtract the 2 from the 8." Thus, the teacher claimed the correct answer was 6 1/2. When my collaborator's daughter handed in a paper with the correct answer, 5 1/2, the teacher made her redo the assignment.
I interpret this as a direct result of the constructivist insistence on "self constructed algorithms" as the means for students - and, ultimately, teachers - to learn the basic operations of arithmetic.
The teacher was well meaning and, after a number of parents pointed out the difficulty, re-did the lesson. I do not blame the teacher, but I do blame the education system and the hair-brained methods it pushes for teaching mathematics.
The other consequence of minimal knowledge on the part of these math educators is that they have no concept of how much mathematics students really must know if they are to work in technical areas. In most of the high paying fields, this knowledge is at least as much as a full fledged mathematics major at a top level university had to know 15 to 20 years ago. It is precisely this ignorance on the part of these math educators that allows them to say that the mathematics programs they are putting into the schools will give the students what they need to know.
Typically, nationwide, even if our high school graduates want to go into technical areas, they can't, since somewhat more than 40% of them have to take remedial mathematics when they enter college. This puts them too far behind.
I think you can find some detailed statistics on the economic costs of our failure to produce enough mathematically competent high school graduates in my congressional testimony of February, 2000, and the references there should give you further sources. In places like Silicon Valley it is not unusual to see entire engineering departments in major companies that are almost entirely composed of foreign born engineers.
Finally, let me emphasize that even these comments only scratch the surface of the problems.
R. James Milgram
Professor of Mathematics
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