If there was ever a time in the United States when no one cared aboutmathematics education, it certainly has not been the past couple years. Mathematics education has been written about in the local, regional, andmost important national newspapers and magazines. Reports have also appeared on radio and national television. The focus of attention has been the so-called "Math Wars" that center on reform in the school mathematics curriculum and its teaching. In particular, a "backlash" against the California Mathematics Framework and the National Council of Teachers of Mathematics' (NCTM) Standards in California has been prominent in the news.
California is the largest among the 50 states. By virtue of its size, it has a very significant influence on school textbook and test publishers. Controversies in education - philosophical, political or social - that come up in California can presage similar controversies and trends in other states. Thus, there is great interest throughout the United States in what transpires in California. As a consequence, and because the 1992 California Mathematics Framework is consistent with or fits closely with the NCTM's Standards, the NCTM has a great interest in these developments in California; in fact, its Standards (currently undergoing revision) have been the focus of much expression of unrest.
The so-called battlefields for the California math war extend beyond concerned parents meeting with teachers and school boards. Involved are state education agencies, their advisory panels, and ultimately the state legislature. As one might expect, newspaper accounts capture only a small part of a rather complex story, so we shall try to provide a few more details here.
Like other states, California has a State Board of Education (SBE). Its members are appointed by the Governor of the state, subject to the approval of the state legislature. There is also a Superintendent of Public Instruction (SPI). This person is elected by popular vote and is the head of the California Department of Education (CDE). [Presently the Governor and the SPI are from different political parties.] The function of the CDE is to provide administrative support for various groups and agencies, implement the policies set by the SBE, prepare and disseminate documents, and provide information to citizens, school districts, and the public.
Due to recent changes in California, however, the curriculum standards, frameworks, and adoptions of instructional materials for the schools now fall under the purview of the SBE, not the SPI. The SBE appoints a Curriculum Commission (CC) whose job is to supervise the drafting of frameworks and instructional materials criteria, recommend instructional materials for state-wide adoption, and make policy recommendations to the SBE. Its members are mainly K - 12 teachers, with a few administrators or representatives fromhigher education. The SBE also appoints (using CC recommendations) members for framework committees and instructional resources evaluation panels (IREP). The mathematics framework committee spends about a year drafting the curricular framework and materials criteria, which is revised by the CC and then submitted to the the SBE for approval. The IREP spends four months reviewing instructional materials submitted for adoption, comparing them against SBE adopted criteria. For details about the materials adoption process and further background, see Jacob (1998).
Standards are new to California. Prior to the SBE approval of the new mathematics standards last December, the state only had curriculum frameworks. Standards that are established at the state level for the schools are voluntary. However, California has established a new testing program for the entire state that must be aligned with the new state standards in 1999. So in reality, the tests will induce districts to adhere to and follow the standards. The standards are drafted by the Standards Commission (SC, which is unrelated to the SBE, CDE, or the CC), whose members are appointed directly by the Governor, the SPI, and the legislature, and what they develop must be approved by SBE.
The California SBE adopted a new Mathematics Framework in 1985. Many ideas in the new framework were new to teachers, as well as to publishers of school textbooks, for in the next year the CC rejected all commercially published materials that were submitted for consideration for adoption in California's schools as none of them met new criteria established by the SBE. In 1987, the CDE published the Mathematics Model Curriculum Guide (CDE, 1987) which included 88 pages devoted to "teaching for understanding" with classroom examples. This document clarified many themes from the Framework and historically proved to be quite influencial, both for teachers interested in change and textbook developers. Also, "state replacement units" were made available to teachers so they could try out some of the new approaches and so textbook companies would have models to consider. However, until 1995, because of the failed adoption, the curricular materials used by nearly all California K - 8 teachers remained almost identical to those sold during the early 1980's. In 1992 a new Mathematics Framework and criteria were approved by the SBE. This new policy document is still in place and is a central ingredient in the the raging controversy that exists at this time.
California developed its own state-wide testing program during the early 1990's (the California Learning Assessment System/CLAS) and this performance-based exam was field tested in 1993 and administered state-wide in 1994. But some of the free response questions on CLAS were considered controversial and the Governor vetoed funding for the program, resulting in a three-year gap in the state's testing program. California's test results in 1993 and 1994 were poor and they were confirmed by the results of the National Assessment of Educational Progress, in which California ranked 41st out of 43 states. This led to an attack on the 1992 California Framework. Well-organized groups claimed that emphasis on cooperative learning, problem solving and applications of mathematics diminished the importance of individual accountability and the importance of mastering basic computational skills (NCTM, 1998). The concern about the "failed reform" led the state legislature to enact a new law (A.B. 170) in 1995 requiring the SBE to adopt instructional materials that are "based on the fundamental skills, ... including basic computation skills." By the summer of 1995 the verdict was clear, the 1992 MathematicsFramework had failed. The timing was amazing. In September 1995 [1] for the first time in over a decade, new instructional materials in alignment with the Framework became available for state-wide use in grades K - 8.
During 1995 and 1996, the SBE became quite active in mathematics, producing a "Program Advisory" in July 1996 that called for a "balance of basic skills, conceptual understanding, and problem solving" in mathematics education. And in November of 1996 the SBE appointed a new framework committee - it rejected the majority of the CC's nominees to that committee and added fourteen others recommended by a recent appointee to the SBE, a person who holds views of mathematics teaching and learning very much at odds with the recommendations in the 1992 Framework. In 1997 the framework committee met, developed and sent a draft document to the CC with all eight CC-recommended framework members voting against it.
Central features of the draft were a listing of topics required at each grade level and an elimination of all discussion of pedagogy.In a supplemental adoption (September 1997), the SBE rejected two programs highly recommended by the CC (both were NSF-funded curriculum development projects) citing mathematical errors and other problems. Examples of mathematical errors noted by the SBE included writing "ratios instead of fractions" and a number theory mistake that "30 divides the product 36x45" which the SBE explains in their written report is an error because "30 is not a factor of either 36 or 45". A middle school program that included a "pizza pirate" in a story problem was cited as violating the states patriotism and morality code. (For details see Nicholas (1997) or Jacob (1998)).
The Standards Commission (SC) approved the school mathematics (K - 12) standards in September, 1997, after a year of deliberation and considering public reactions to the document. In fact, the Commission's standards were substantially revised during July and August as a result of public comment, and the rapidity of these last-minute changes resulted in some glitches that both sides of the debate critcized and worked to correct during the Fall. The SBE adopted a substantially revised document in December, relying almost solely on the work of four Stanford University mathematics professors. They claimed the revisions were necessary to increase the "mathematical precision" in the statements of the standards and to remove the "Clarifications and Examples" whose purpose was to illustrate what the standards mean in the classroom, but were interpreted as "prescribing pedagogy" by the SBE. The process by which this document was approved, and the content of it, are major factors in the controversy that presently rages.
While the media has tried to make sense of the debate that surrounded the controversial developments, it seems that they would not or could not get to the heart of each side's position. Typical news reports stated that the controversy was over "the best way to teach math" and that the arguments over teaching were about such issues as "uses of real-world problems vs. skills" or "integrated vs. traditional curriculum". While on the surface such statements are not incorrect, they miss the central issues. The Framework committee agreed not to "prescribe pedagogy" in its document, but could not come to agreement, or even agree on a format to discuss, how to balance skills with problem solving. Even with pedagogy off the table, the committee could not agree on content. In their written "homework" (and therefore, public documents) for the committee on "How to balance K- 6 Mathematics," two northern California mathematics professors offered contrasting views: "the curriculum should include extended projects or capstone problems that require the student to synthesize and integrate concepts and calculational techniques", and "I suggest that our goals and expectations of elementary school children should be pretty much limited to arithmetic" (CFIR, 1997). How mathematical questions could be posed in the document was particularly problematic. For example, describing an area problem on a geoboard was rejected by the committee majority for two reasons: it "prescribed manipulative pedagogy" and "the appropriate tools for geometry are the straightedge and compass." Disagreement arose over appropriate use of calculators and technology. The media has by-and-large bypassed these central controversies.
The press reports on the Standards debate (see Wu, 1998) usually reduce the question to high standards vs. low standards. SPI Delaine Eastin's quote that the SBE "dumbed down" the CS's standards received prominent coverage, as did the SBE's statements that it is "removing pedagogy from the document" and striving for "greater accuracy". But here again, while publicizing each side's favorite "one-liners", the press has failed to dig out the basic differences over content. Central issues included the SBE's consistent removal of such phrases as "estimate" or "explain" and replacing them with "calculate," the removal of the study of patterns from the Algebra and Functions strand in elementary school, or the complete removal of all exemplars that were designed to help K -12 educators (and text designers) understand how topics are to be approached at a given grade level. The press seemingly never examines why both sides claim their views represent "high standards". Does moving mastery of computational skills to a lower grade level raise standards? Or, does adding an expectation that students explain what a number procedure means geometrically raise standards? However one views the situation, these actions by the SBE have seemingly established California as the center of the opposition to the NCTM's Standards-based reform. Due to many factors, the media a prominent one, many parents, policy makers and some teachers and professors came to the view that the 1992 Framework had failed. Following this thinking, therefore the NCTM Standards were also wrong and could be regarded as the culprit (Ross, 1998).
An important factor in the California debate is the requirement that state-adopted instructional materials "incorporate principles of instruction reflective of current and confirmed research" (CA Education Code 60200c-3). The SBE invited Prof. E. D. Hirsch to speak on this issue in April 1997. In the written version of his comments, he described "mainstream educational research" as found in "journals such as the Educational Researcher" explicitly stating "This is a situation that is reminiscent of what happened to biology in the Soviet Union under the domination of Lysenkoism, which is a theory that bears similarities to constructivism." (Hirsch, 1997, p. 3) After some explanation, Hirsch continues: "I shall briefly outline the conflicts between educational Lysenkoism and mainstream science in testing, math, and early education ...," and citing math education experts Anderson, Geary, and Siegler about what research shows math students need, "They would tell you that only through intelligently directed and repeated practice, leading to fast, automatic recall of math facts, and facility in computation and algebraic manipulation can one do well at real-world problem solving" (Hirsch, 1997, p. 6). Hirsch received a standing ovation from the SBE, and indications are that it is proceeding, based on his recommendations.
In spite of the SBE instructions to base the Framework on research, the Framework committee never discussed any research articles. In July 1997 the SBE awarded a contract to Prof. Douglas Carnine to provide a review of high quality mathematics research upon which the Framework's instructional strategies would be based. In this document, presented to the SBE in March (see Dixon, (1998)), we find "From a total of 8,727 published studies of mathematics in elementary and secondary schools ... only 110 passed the multi-level evaluation criteria we developed to identify high quality studies". All studies are experimental, most consider interventions over very short time intervals, many deal with learning disabled students, and many use "instructional booklets" in order to eliminate teacher-pupil or pupil-pupil interaction (which were considered "confounding variables".)
The American Educational Research Association's Special Interest Group for Research in Mathematics Education has written a public letter to the SBE (signed by 73 researchers) protesting the poor design of the Carnine report. But in spite of numerous errors (for example incorrect reporting of grade levels, content, or experimental design), the SBE will prepare new statements on "math instructional strategies" for inclusion in the Framework this summer, based upon the Carnine document. Due to the lack of time, the CC will not be able to participate in this part of the process. Various observers anticipate an endorsement of direct instruction followed by repeated practice, opposition to having students try to develop ideas through problem solving, and quite possibly (based upon Carnine's work) discontinuation of the use of manipulatives in elementary schools (since they interfere with automaticity in fast recall).
In California, as noted above, there were two sets of standards - those developed by the SC and the revision of that set under the supervision of the SBE. To observe that these two standards were the focii of unabated interest, across the land and for many months, is an understatement! For example, the NCTM devoted the front page of one issue of its News Bulletin to unflattering comments about the SBE's revised standards. (NCTM, 1998) Wu (1998) reported that "The reaction to the revision was swift and violent" (p. 1).
To illustrate the changes between these two documents we consider the original third grade Number Sense section where one has "[2.2] build up multiplication table from 0x0 to 10x10 and commit to memory," which has in its corresponding Clarification and Example column: "Students see that understanding properties and relationships within the multiplication table can assist them in memorizing facts (e.g., 4 x 7 = 7 x 4; 7 x 6 is easy if you know 7 x 3 and know that you can double it to get 7 x 6, etc.)" In its revision, the SBE replaces these with "[2.2] Memorize to automaticity the multiplication table for numbers between 1 and 10". In Grade 7, where preparation for grade 8 algebra has became a central Standards and Framework objective, we find the SC's Algebra and Functions standard including "[3.2] generalize numerical and geometric patterns using algebra, and relate the equation, graph and table of values resulting from the generalization." The Board revised this standard to read "[3.2] plot the values from the volumes of a 3-D shape for various values of its edge lengths." As with the Framework, one finds that discussion of students' learning of mathematical relationships (like the use of commutativity to facilitate learning basic facts) was eliminated either because they "prescribed pedagogy" or "lacked precision".
University of California, Berkeley mathematics Professor H. Wu prepared a paper based on a lecture he gave in California describing his assessment of the two standards (Wu, 1998). He does this from both his mathematical and educational perspective. Since he is known for his critiquing of the current reform, his thinking is worthy of examination. He regards the SC's standards as a thoughtful document, into which a lot of care was used in setting forth its goals (p. 2). But overall, Wu focuses on the importance of "getting the mathematics right" in his article. He felt there are many errors that need to be corrected (for details see pp. 3-10), topics are omitted, and there is an ambiguous mixture of pedagogical statements with content statements. For example, he cites the exclusion of the division algorithm in the elementary grades and the Fundamental Theorem of Algebra in the higher grades.
Wu strongly objected to a grade 4 Geometry Standard which reads: "Students understand and use the relationship between the concepts of perimeter and area, and relate these to their respective formulas" mentioning that the trouble "is that there is no relationship whatever between perimeter and area, or between volume and surface area, unless it be the isoperimetric inequality. However, the latter would be quite inappropriate for students at this level" (p. 4). About his perceived errors, Wu has strong language: "I very much regret to say that this kind of mathematics standards would guarantee the deterioration of mathematics education for a very long time" (p. 4). While this standard may constitute an error in the eyes of one research mathematician, a fourth grade teacher explained to us how she interprets it: "We want students to understand at their level that perimeter 'goes around' and area 'covers,' and then to be able to explain (for example) in the case of a rectangle why 2xl + 2xw can be understood as measuring the 'going around' while l x w counts covering (say by square tiles)". We think a teacher can learn more about this from the Clarification and Examples in the SC's Standards that were eliminated by the SBE. So, we find in the standards debate a serious breakdown of communication between elements of the mathematics community that values precise abstract constructions upon which they conduct their work, and members of the K - 12 educational community who have learned to interpret the informal presentations of ideas that children use as they encounter mathematical concepts. Finally, we note that Wu supports the position in the SBE's standards that calculators should not be used in state-wide testing for grades K - 6.
In California, the governor's staff has announced that $250 million ($60 per student) will be spent in the next year on new math textbooks (Morain, Los Angeles Times, May 13, 1998). The legislature is revising the process by which curricular materials are adopted (proposed AB 2517), so that the new criteria to be written by the SBE this summer will take effect in 90 days. This means that schools may be limited to using materials that follow the "three-phase" approach outlined as "best" in the Executive Summary of the Carnine Report: (1) Direct instruction in skills ("other regulation"), (2) followed by a "help phase", concluding with (3) "self regulated" drill and practice. In recent (public) discussions with the CC, SBE members have stated that the "terrible" manipulative-based elementary math programs must be stopped immediately. Exactly how all this will play out depends upon fast-track legislation that will be considered during the coming months.
Nationally, a Standards forum was held on the program of the Baltimore AMS-MAA mathematics meetings in January, 1998. The panel was comprised of representatives of sixARGs. [An ARG is an "Association Review Group" from which NCTM requested assistance and input in revising its Standards - e.g., the Mathematical Association of America (see for example, Reports 1,2, and 3 of the MAA Task Force on the NCTM Standards at http://www.maa.org/ ); the American Mathematical Society; the Association of Symbolic Logic, etc.]. Many mathematicians shared their views on the NCTM Standards. In particular, NCTM was credited for tackling important issues in mathematics education, for addressing the needs of all students in mathematics and for drawing people inside and outside mathematics to discuss the issues. (Ross, 1998) At the same time, various concerns were expressed; for example, that the Standards need to be made less vague and less subject to misinterpretations, since a multitude of things are being done in the name of the Standards. Also, it needs to be made clear that mathematics is not always fun, not always easy, and that it is a myth that only some people can do mathematics (Ross, 1998, p. 4). Other commentators expressed views consonant with Wu's views; for example, that the Standards need to be made shorter and crisper and more specific, and that more attention needs to be paid to logic and reasoning in mathematics. The over-arching theme of problem solving was supported in the discussions. But the ARGS only provide suggestions for the NCTM to consider. The NCTM revision of its Standards (called Standards 2000) is scheduled for completion in 2000.
As U.S. mathematics educators deal with the "backlash," there are other important issues that are being raised. Among them, the manner in which precise mathematical language and logical arguments (from informal reasoning to proof) are developed - in particular, what are our expectations across grade spans and how do educators help students develop these understandings - and communicating how "real-world" problems can help enhance mathematical understanding and eliminating a possible over-emphasis on them where the distractions of the context obscure the mathematics. Beyond the curricular issues, there still remain those of teacher preparation. There is insufficient support for continuing teacher education and there is a great need to revamp preservice teacher preparation programs. Using the new curricula requires greater teacher understanding of both the mathematics and the approaches different students will take in learning. Finally, there are those who feel a need to further examine under what circumstances cooperative learning is effective and when it may not be, and more generally, the issue of how constructivist thinking is influencing, or should influence, approaches to teaching (cf., Kilpatrick, 1997).
U.S. Secretary of Education Richard W. Riley has concerns about the deep divisiveness of the debate over mathematics education reform. In a major talk at the Baltimore AMS-MAA meeting, he mentioned in a forceful manner that "This leads me back to the need to bring an end to the shortsighted, politicized and harmful bickering over the teaching and learning of mathematics. I will tell you that if we continue down this road of infighting, we will only negate the gains we have already made - and the real losers will be the students of America" (Riley, 1998). Referring to the California "math wars" he continues, "Let me say right now that this is a very disturbing trend, and it is very wrong for anyone addressing education to be attacking another in ways that are neither constructive nor productive. It is perfectly appropriate to disagree on teaching methodologies and curriculum content. But what we need is a civil and constructive discourse". Perhaps we can see that the California "math wars" have, in the final analysis, served a useful purpose. Overall they have served as a lesson on how not to behave in the future, in rethinking and reconstructing school mathematics education for the benefit of ourstudents.
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Jerry P. Becker is Professor of Mathematics Education, Department of Curriculum and Instruction, Southern Illinois University, Carbondale, IL 62901-4610, USA [jbecker@siu.edu]
Bill Jacob is Professor of Mathematics, Department of Mathematics, University of California, Santa Barbara, CA 93106-0001 USA [jacob@math.ucsb.edu]