Some background. My public statement was intended to raise concerns about math reform programs that are in place in several schools throughout Minnesota. Over the past several years, these programs have been aggressively pushed by organizations like the Dept. of Child and Family Learning and SciMath Minnesota, and also through a multi-million dollar grant from the National Science Foundation. In my view, this campaign has been quite one-sided. I am personally aware of several instances where misleading information has been given regarding the views of my colleagues in the School of Mathematics and the Institute of Technology at the University of Minnesota. I have also witnessed presentations and read literature in which exaggerated claims have been made about the scientific evidence that has been cited to support the effectiveness of math reform texts like Core Plus.
Because of my position at the University and my professional background, I feel qualified and obligated to try to restore some balance by letting people know that there is another side that should not be ignored. I believe my public statement makes this objective quite clear. When I first presented this statement to a state legislative committee, I was informed by several members of the committee that I was the first person representing anything resembling my point of view who had been allowed to testify during the many hearings that had been devoted to related educational issues. I was given less than 3 minutes time (which is why I had to present my statement in written form), compared to the approximately 90 minutes that were given to four different advocates of math reform on that same day. Thus, I find it quite ironic that in his reply, Prof. Fey feels that "it's probably useful to have a reminder of the rest of the story." Is it really necessary to attack my brief 2-page statement after the incredible amount of time and money that has already been spent on promoting math reform programs?
In my remarks here, I will often refer to Core Plus materials. I do this to make it clear that my concerns are with materials that are actually being used in the public schools, not with some idealized product of the math reform movement that may some day appear. The Core Plus books are considered to be among the best available examples of math reform texts, I am well acquainted with them, and Prof. Fey has been heavily involved in their development. So I think it is fair to use them as my main example. In case it is not already clear, let me make my overall opinion of Core Plus quite clear: I consider Core Plus to be quite inadequate for teaching high school mathematics to students who want to enter engineering or scientific majors when they go on to college. Furthermore, I strongly believe that a properly taught traditional math curriculum is much superior to Core Plus for such students. And I am particularly opposed to the course of action that has been taken by several high schools, which is to make Core Plus (or something like it) the only option available to their students. To me, it is clearly a program that is still in the experimental stage (the fourth year Core Plus materials are only just now appearing on the market).
My remaining remarks are organized according to various themes that appear in Prof. Fey's criticisms.
Concerning comparisons with other countries. It is common for math reform advocates to support their arguments by talking about how our high school math students stack up against those of other countries. The implication is that we need to reject traditional math instruction if we want to be competitive. For example, Prof. Fey says: "Traditional American mathematics instruction follows a well-known lesson pattern of explanation, demonstration of procedures for working specific exercises, and student practice of what they have been shown. This routine of low-level tasks is in stark contrast to instruction in many other countries with more effective mathematics programs." I work constantly with math and engineering graduate students from many other countries and have talked extensively with them about their math instruction. I have also taken a careful look at one series of textbooks from one of those countries with "effective mathematics programs" (the Singapore Math texts). What I find is that in those countries, the "well-known lesson pattern" described by Prof. Fey is definitely followed, but without letting the class degenerate into an uninterrupted "routine of low-level tasks". But the real contrast is between the currently available US reform curricula and foreign programs. The only significant similarity between something like Singapore Math and Core Plus is in the integration of algebra, geometry, and statistics. In other matters, such as the logical development of mathematical topics, the importance of precise definitions, the dominant role of lecturing, the use of worked examples, the important role of drill exercises and focussed word problems, the emphasis on facility with symbolic algebraic manipulations, the limited use of calculators, and the style of testing, the differences between Core Plus and Singapore Math are dramatic.
Concerning anecdotal evidence. In reference to my anecdotal evidence about concerned parents, Prof. Fey takes the following cheap shot: "Just a little bit of statistical education would suggest that conclusions from such a sample are unlikely to represent attitudes in the population as a whole." This comment not only distorts my intended use of this evidence, it is also an attack upon me as a professional. My research area is probability theory, and I was made a Fellow of the Institute of Mathematical Statistics in 1989. Since I wrote my original public statement, the number of parents who have contacted me directly has grown from "dozens" to "hundreds", and it is still the case that the overwhelming majority of them have serious doubts about the reform-style classes that their children are taking. Many of them are able to speak quite knowledgeably about the details of their children's experiences. Anecdotal evidence like this is a statistically legitimate cause for alarm, and it would be foolish to ignore it.
Prof. Fey similarly dismisses my statement about the concerns of scientists, research mathematicians, and engineers by saying that: "Again, the sample of scientists, mathematicians, and engineers who have expressed concerns about changes in mathematics education is suspect as a representation of all such specialists." He attempts to counter by citing his own anecdotal evidence, when he says that "There are many scientists, mathematicians, engineers, and professionals in mathematics-intensive fields who believe that the new directions proposed for school curricula are appropriate preparation for where mathematics is going in the future." It is true that one can find some well-qualified mathematicians who generally support the math reform movement. (The current president of the American Mathematical Society is one of them.) But it surprises me that Prof. Fey feels that he knows more about the prevailing views of scientists, research mathematicians, and engineers than I do, or that he feels qualified to judge whether people know "where mathematics is going in the future". I'm sure that Prof. Frey knows that there are literally hundreds of research mathematicians and scientists who have publicly expressed serious doubts about many of the trends in math reform. Whether or not they form a statistically valid sample is irrelevant. We cannot afford to ignore them. For more about the views of scientists and engineers, please see my remarks below about "client disciplines".
My other use of anecdotal evidence (concerning calculator-dependence) was admittedly not so effective. But even in this case, I think the information I gave is not to be dismissed out of hand, even if many of our successful students used calculators "extensively and wisely" in high school. Please see my comments below about calculator usage.
Concerning our "client disciplines". In addition to his statement about scientists, mathematicians, etc., that I quoted above, Prof. Fey also says the following: "Many client disciplines are far more interested in students who can reason well with data and use the methods of discrete mathematics to solve problems in important fields like computer science and biotechnology. So, even if traditional mathematics is good preparation for traditional calculus, it is not at all clear that it is the right preparation for new approaches to calculus or for the mathematization that is reshaping many disciplines outside of the traditional physical sciences." He also says: "In fact, on most college campuses, the number of computer science majors far exceeds the majors in physics, chemistry, and mathematics combined. Should high school curricula be determined by looking at preparation for traditional science and mathematics careers?" As Director of Undergraduate Studies in Mathematics, I am in regular contact with representatives from departments all over the University, including all of the engineering departments, computer science, statistics, most of the departments in the business school and the college of biological sciences, and even some of the social sciences. They tell me what is needed in the math courses their students take. Furthermore, the School of Mathematics at the University of Minnesota was recently ranked fourth in the nation in applied mathematics. Our industrial math program sends graduate students for internships at companies throughout the Twin Cities. My own research has applications to several applied fields, including computer science, business, and most recently, traffic jams. I flatly reject the implied suggestion by Prof. Fey that the School of Mathematics and I don't know what our client disciplines want.
Concerning the use of calculators in the classroom. Since Prof. Fey speaks about using calculators "wisely", he presumably feels that it is possible to use calculators unwisely in math instruction. Indeed, he says "We have to learn more about the right way to utilize technology...." To this extent, he and I agree. Indeed, my public statement mentions the fact that I helped design several of the computer labs that are used in our sophomore level classes, so it should be clear that I favor the appropriate use of technology in the classroom. My concern is that the heavy dependence on calculators in math reform programs like Core Plus is not only unwise, but downright disastrous. Prof. Fey seems to imply that there is no need to "require mastery of the kinds of skills that calculators now perform routinely". I strongly disagree. In this respect, I wish we would emulate the instructional practices of many other countries, where calculator usage is quite limited in the classroom, and where the skills mastery of students is often tested under conditions that do not allow the aid of a calculator.
Concerning theoretical evidence. It is quite common for advocates of math reform to claim that there is an abundance of evidence in favor of materials like Core Plus. Consider the following statement by Prof. Fey: "However, the new materials have been designed as they are for very sound reasons, and there is consistent evidence of their effectiveness." I dispute this assertion. I don't have the space here to go into details, but I have looked carefully at several oft-cited studies that supposedly give evidence of the effectiveness of actual materials like Core Plus, and in every case, I have seen that these studies are highly suspect, and in some cases, downright misleading.
Concerning valid reasoning. While I recognize that Prof. Fey's critique was not necessarily intended as a carefully crafted document, I think it is important to draw attention to a few of the logical lapses therein, because of the fact that these same lapses seem to appear frequently in statements supporting math reform.
In general, I object to the way in which math reform advocates try to support the effectiveness of currently available math reform curricula by using arguments that, instead, only support the conclusion that changes are needed in math instruction in the US. While Prof. Fey does not explicitly state this argument, I think we can safely assume that when he talks about the need for change, he wants us to conclude that (at least) the Core Plus materials effectively provide such change. In my public statement, I agreed with the need for changes. But just because something like Core Plus represents a dramatic departure from traditional textbooks, one cannot conclude Core Plus represents a step in the right direction.
Prof. Fey states: "The new standards-based school mathematics programs are based on the premise that students will develop deep understanding of important ideas and ability to deploy that understanding in problem-solving if they are consistently challenged to work on rich mathematical problem tasks. Text materials that reduce student tasks to studying worked examples and practicing similar procedures in repetitive exercises do not produce powerful learning. So there is very sound reasoning behind the decision to write different kinds of mathematical text." This statement uses a common rhetorical dirty trick, which is to compare the two sides of an issue by using an idealized description for one side and a pessimistic description of the other. I know for a fact that "powerful learning" can and does take place in traditional mathematics classrooms. When I advocate adhering to a traditional approach in the classroom, I do not envision reducing the entire instructional experience to a series of repetitive exercises.
Here is another one of Prof. Fey's statements: "In weighing the advice from professionals in the sciences, technology, and mathematics, one also has to keep in mind the fact that a fairly small fraction of all high school students choose to major in traditional sciences, engineering, and mathematics. The fraction of any age group that will go on to a Ph.D. in mathematics is less than one one-hundredth of one percent. Thus even the most mathematically able students are unlikely to use mathematics in traditional ways." Note how he improperly uses statistics taken from one population to justify a statement and a conclusion about a very different population. It is not unusual for advocates of math reform to try to explain away the opposition of research mathematicians by making it sound like we are only interested in training Ph.D. in mathematics and some of the physical sciences. When they do this, don't believe them.