**************************************** BILL JACOB and JERRY BECKER had an article on the politics of California school mathematics in the March, 2000, Phi Delta Kappan: Becker, J.P. and Jacob, B. (2000 / March) The politics of California school mathematics: The anti-reform of 1997-99, Phi Delta Kappan, Vol. 81, No. 7, pp. 529-537. The article can be viewed at http://www.pdkintl.org/kappan/kbec0003.htm . Here is an ABSTRACT -- The authors tell the story of a powerful group of parents and mathematicians in California who manipulated information and played off of the public's perception of our "failing schools" to acquire political clout. Through this telling, they hope that other states will be able to adopt a more rational course as they reconsider their policies. -------------------------------- A response to the article, written by Professors Deborah Tepper Haimo and R. James Milgram (mathematicians) appears in the October, 2000 Phi Delta Kappan, along with our reply to them. Both their response and our reply are given below with the complete citations -- JPB and BJ *******************************************
RESPONSE BY HAIMO AND MILGRAM
CITATION: Haimo, D.T. and Milgram, R.J. (2000 / October). Professional Mathematicians Comment on School Mathematics in California, Phi Delta Kappan, Volume 82, Number 2, pp. 145-46.
Professional Mathematicians Comment on School Mathematics in California
At Odds/Mathematics Standards
The authors respond to the March article on California school mathematics, by Jerry Becker and Bill Jacob - pointing out only a few of their most serious concerns.
By Deborah Tepper Haimo and R. James Milgram
WE WISH to respond to the article by Jerry Becker and Bill Jacob on the recent changes in school mathematics in California ("The Politics of California School Mathematics: The Anti-Reform of 1997-99," March 2000). Although we would have no trouble providing a more complete treatment and going through the article line by line pointing out difficulties, we will confine our comments to a few of our most serious concerns.
Right at the start is a most unfortunate assertion that cannot be ignored. The subhead of the article, chosen by the editors from the authors' second paragraph, reads: "The authors tell the story of a powerful group of parents in California who manipulated information and played off of the public's perception of our 'failing schools' to acquire political clout."
There is no evidence whatsoever for this statement. Do the authors really want to claim that professional mathematicians in California (Jacob included) who have devoted a substantial portion of their time to K-12 mathematics education over the past five or more years have done so because they wanted "to acquire political clout"? All our observations indicate that these mathematicians have engaged in this activity because of their strong commitment to excellence in mathematics education for California's students. They are deeply concerned about the sharp drop in the performance in mathematics of California students compared to students in almost any other state and, above all, compared to students in other countries.
However, we have even more serious difficulties with the body of the article. We find, for example, that the point of view presented is excessively biased in a direction that tends to widen the gap between the vast majority of professional mathematicians and K-12 teachers.
In the current reality, upwards of 75% of high school graduates in this country attempt higher education. Thus, regarding matters of content, K-12 teachers must seriously heed the concerns of professional mathematicians who teach college mathematics courses, just as elementary school teachers must take account of the concerns of high school teachers. It is no longer acceptable to argue, as Becker and Jacob do, that the reason for dismissing the concerns of professional mathematicians is that, to the authors' knowledge, "none of these mathematicians [who were involved in the development of California's new school mathematics policies] ever taught in K-12 schools."
The authors' persistence in characterizing the current California mathematics standards as a return to the past or a curriculum of the past is strange. Why not call the curriculum internationally benchmarked mathematics? Or, as we prefer to call it, real mathematics? It is amusing that they refer to the mathematics of the previous California standards - standards that have been around for over a decade - as "reform math."
We conclude with two additional items that must be mentioned. Becker and Jacob constantly pretend that suggestions given to teachers in the California framework are meant instead for students. They do this even though the foreword to the framework clearly states that it "provides instructional guidance to teachers to enable them to raise their benchmarks for achievement and mastery in realistic ways."
As an example of this type of distortion, Becker and Jacob consider one of the key examples in the framework.1 They criticize the discussion as set too high when the authors of the framework attempt to inform teachers of the simple way in which the problem can be transformed to be mathematical. Instead of clarifying the difficulties, Becker and Jacob seem to believe that these authors merely confuse. Their own solution is to evade mathematics being introduced. The example in question is the following: "The students are given a picture that shows in succession a rectangle, triangle, square, rectangle, triangle, square, blank, triangle, square. The students are asked to fill in the blank."
This problem is not well posed, as is pointed out by the authors of the framework. At this point, however, there is little explanation about the mathematical way of stating similar examples correctly. Becker and Jacob pounce on this, stating that "we must remember that this is a discussion about teaching mathematics in kindergarten!" (Assuredly, everyone is keenly aware of this.) Furthermore, they assert that these authors wish to introduce formal mathematical language and rules for reasoning in kindergarten.
In fact, Becker and Jacob have taken this completely out of context. In the previous paragraph, the problem that was discussed in detail was the following: "A picture of three objects, a basketball, a bus, and a tennis ball, is shown to the students, and they are asked to tell which one does not belong." This is similarly not well posed. Here, however, there is substantial discussion of ways of stating the problem correctly and at the appropriate level. For example, with this picture in hand, students might be asked the following: "We want to collect balls. Which of these objects should we select?" A number of other examples of correct and grade-appropriate variations are suggested before the problem criticized by Becker and Jacob is introduced in the next paragraph.
There is some question as to why the authors of the framework did not show, in equal detail, how to replace the second problem by one that was both mathematically correct and grade appropriate. The reason, of course, was that it was assumed that the audience of teachers would be able to do so for themselves, using the example of the first problem as a model. Was this unreasonable?
The message conveyed by the framework is precisely that elementary school teachers must become aware of the fact that there are ways of presenting mathematics both accurately and at the appropriate grade level. We find it ironic that Becker and Jacob attempt to show the complete opposite.
In exactly the same way and with the intent of again seeking to reverse the actual content, Becker and Jacob criticize the introduction to the discussion of the grades 8-12 standards,2 saying, "Students are expected to provide a 16-step, two-column proof of such algebra facts as. . . ." First, it is clearly and repeatedly stated in the surrounding discussion in the framework that this section is for teachers. Furthermore, the objective is to make sure that teachers are aware that problem solving and proof are essentially the same thing. To emphasize this point even further, immediately after the 16-step proof that Becker and Jacob object to, we find these clarifying remarks:
In practice it would be impractical to demand such detail each time a linear equation is solved. Nevertheless, without the realization that a mathematical proof is lurking behind the well-known formalism of solving linear equations, a teacher would most likely emphasize the wrong points in the presentation of beginning algebra.
This is one of the points where the California framework most explicitly rejects the approach of the "new math" of the 1960s. Thus we find it interesting that Becker and Jacob - by omitting the surrounding text - attempt to portray this as a "new math" prescription of deadly logical formalism for children.
In another vein, Becker and Jacob are critical of the use of the term "hidden agenda," which appears on page 110 of the California framework (body of discussion and sidebar). Perhaps this phrase might better have been replaced by "mysterious element" or, as Becker and Jacob suggest, a more formal term, such as "missing assumptions."
Jacob took full advantage of the formal procedures for public input on the final draft, giving substantial public testimony. Also, he never commented on this phrase at the time nor during the long period afterward when public input as well as corrections were sought. Apparently, he chose instead to save that particular objection for a more favorable occasion. The phrase could have been changed easily in proof, and it would have been, if it had been pointed out. That Jacob ignored this phrase until he could use it later for purposes of attack appears to have been more than an oversight.
1. Mathematics Framework for California Public Schools, Kindergarten Through Grade 12 (Sacramento: California Department of Education, 1999), pp. 109-11. 2. Ibid., p. 155.
REPLY BY BECKER AND JACOB
CITATION: Becker, J.P. and Jacob, B. (2000 / October). Look at the Details: A Reply to Deborah Haimo and James Milgram, Phi Delta Kappan, Volume 82, Number 2, pp. 147-48.
Look at the Details: A Reply to Deborah Haimo And James Milgram
At Odds/Mathematics Standards
We need to be vigilant and careful and not be fooled by the seemingly impressive credentials and writing of critics, Mr. Becker and Mr. Jacob respond. Content knowledge is no substitute for knowledge of how students' understanding develops and can be nurtured.
By Jerry P. Becker and Bill Jacob
DEBORAH Haimo and James Milgram question our use of the phrase "political clout" and assert there is no evidence to support it. We ask Kappan readers to consider the following events and judge for themselves.
In 1997, four mathematicians (Milgram included) substantially revised the draft California mathematics standards. Their changes were accepted by the state board of education without seeking public input or the involvement of K-12 teachers. (The fact that four people, acting as advisors to the board, met in private to discuss the revision of a public document appears to violate the state's public meeting act.) In 1998, three mathematicians wrote sample problems for the state mathematics framework, and two mathematicians and a cognitive psychologist wrote significant portions of it. None of them discussed this work in public as part of an open process. Some of the work was presented to the Curriculum Commission, which had little or no time to work with it.
But most of the sections cited in our March 2000 article were not contained in the last draft made available for public comment (on 8 October 1998); they were inserted in November just prior to the state board's vote. During 1999, the state board adopted a new policy requiring panelists to have a Ph.D. in mathematics as a prerequisite to serving on a Content Review Panel (CRP) for the California K-8 mathematics adoption - a doctorate in education was not allowed. Although members of the Instructional Materials Advisory Panel included teachers and did review materials, in the end it was the CRP members who determined the state board's decisions. And in the case of Everyday Mathematics, a CRP report was rewritten by a mathematician two months after the panels had disbanded. Also during 1999, two mathematics professors reviewed and rewrote AB 1331 professional development materials, and the state board accepted their judgments without allowing any further review (and subsequently appropriated $43 million for AB 1331 programs during the 2000-01 school year).
So in California, the mathematics standards, framework, instructional materials, and professional development have all been very tightly controlled by a small group of university mathematicians. High-stakes tests and accountability measures that have profound effects on the lives of teachers and students are linked to all of these policies. Yet the voices of the teachers who know their students best have been omitted from the process. In our view, being allowed to circumvent the public process and get a single vision of policy uniformly imposed on a large state like California is ample evidence of "political clout."
The other main criticism leveled by Haimo and Milgram is that we distort the California framework discussions by confusing comments intended for teachers with expectations for students. The discussions in question all come from "Grade Level Considerations," chapter 3 of the Mathematics Framework for California Public Schools, Kindergarten Through Grade 12, not from the "Instructional Strategies" or "Professional Development" chapters, where such information for teachers might have been appropriate. The focus of chapter 3 is on how mathematics is to be presented to students.
In fact, the question of whether the passages are written "for teachers" or "for students" completely misses the point. Of course teachers will read the framework, not students. The point of our examples was to demonstrate that the formal thinking of mathematicians about mathematical content is now driving California policy with respect to how children are first to encounter mathematical ideas. This is true whether discussions focus on how teachers might view the mathematics being presented or on how students should receive the mathematics. Readers who review our March 2000 article after reading the commentary by Haimo and Milgram will see this point clearly. The issues of well-posed problems or formal proof are concerns of mathematicians. They are certainly relevant, but there are many other issues of far greater educational significance. We do not agree with the statement of Haimo and Milgram that "problem solving and proof are essentially the same thing." Proofs are of enormous importance in higher mathematics, but they come at the end of a long developmental process, and they are not the same as children's problem solving. Yet such mathematical formalism drives the approaches in the California framework, and, in our view, this is not appropriate.
Haimo and Milgram comment that "Jacob took full advantage of the formal procedures for public input on the final draft, giving substantial public testimony." The fact is that, although Jacob did write some letters to the Curriculum Commission during 1998, he never once gave public testimony that year (largely because of travel time and expense). So their comment is inaccurate. Regarding the phrase "hidden agenda" and why Jacob "never commented on this phrase at the time," there is a simple response. The entire Preface to Kindergarten Through Grade 7, in which it appears, was not included in the final draft of the framework that was available for public comment on 8 October 1998. So it would have been impossible for any person outside of the state board's select authoring group to read this section prior to state board approval of the framework. The fact that this particular phase was singled out by the authors of the framework prior to publication to be highlighted as the sidebar on page110 suggests that they felt strongly about its message. Now that Haimo and Milgram feel uncomfortable about it, they wish to place the blame on somebody who was not part of the authoring or proofreading team.
After the last paragraph, many readers may be thinking that Californians must be crazy to be arguing over such points. If so, we agree. Such wording will probably have little or no effect on classroom practice. But the existence of these arguments and the tenor of the discussions about them are revealing and point to real problems in California. The state has failed to forge a true collaboration among parents, teachers, mathematics educators, mathematicians, and others, and instead a small collection of professors of mathematics have autocratically set state policy relating to mathematics curriculum and instruction. The widening "gap" between professional mathematicians and K-12 teachers, noted by Haimo and Milgram, began in California when the state's mathematics standards were forced on K-12 teachers - many of whom do not subscribe to them. The continuing practice of the state board in relying almost exclusively on the same small groups of content experts only widens the gap.
On the other hand, we never dismissed the interest and concerns of either K-12 or higher education. Indeed, as professors of mathematics and mathematics education, we both work closely with teachers and bring our concerns to them. In doing so we have developed the highest respect for the understandings of our K-12 colleagues and believe that those who have devoted their professional careers to working with our children should play a leading role in setting policy. Understanding mathematical content as an adult is not the same as understanding how children's thinking develops. In our article we showed how California's politicians and some professors have failed to understand this. In our view, the situation is a travesty. It is time to set a national agenda - paying highest respect to the professionalism and knowledge of our nation's dedicated K-12 teachers.
Finally, as we mentioned in our article, the National Council of Teachers of Mathematics (NCTM) has now released the final version of its Standards 2000 document, Principles and Standards for School Mathematics. It outlines a balanced view of teaching for understanding and also addresses the issues of both skills and problem solving. We hope that in response to the Principles and Standards we do not again hear about lack of mathematical precision, lack of skills (with emphasis on "standard algorithms"), mathematical errors, inappropriate calculator use, low standards, and the "research" that supports the critics' views.
As we mentioned in our article, we need to be vigilant and careful and not be fooled by the seemingly impressive credentials and writing of critics. We need to look carefully at the details. While we recognize that there is always room for improvement in any endeavor, these critics - instead of joining forces with teachers and contributing to the process - may well keep up their efforts to interfere with those who seek to reach more students in a constructive manner. They are likely to continue to work to secure their vision of 13 years of precalculus symbol manipulation. Content knowledge is no substitute for knowledge of how students' understanding develops and can be nurtured, but this point seems lost on these critics. Again, we ask readers to examine the NCTM document. We are convinced that they will see the same merit in it as we do. *************************************************** -- Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 62901-4610 USA Phone: (618) 453-4241 [O] (618) 457-8903 [H] Fax: (618) 453-4244 E-mail: firstname.lastname@example.org