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The Math Wars California Battles It Out over Mathematics Education Reform (Part II)

Note: The first part of this article appeared in the June/July issue of the Notices, pages 695–702.

The Numbers Battle examinations but also group assignments, long- “Show us the data, show us that this program term projects, and portfolios of work done over works,” demands Michael McKeown, the head of a period of time. Some also favor the idea of judg- Mathematically Correct, the San Diego-based ing the efficacy of curricular programs by criteria group critical of mathematics education reform. other than standardized tests, such as the num- McKeown makes his living in the data-driven ber and type of mathematics courses students field of biology, as a researcher at the Salk In- subsequently take. stitute and an adjunct faculty member at UC Such are the views in that portion of the math San Diego. One of the things he finds most vex- wars devoted to testing. The scramble to find ing about the reform is that it has been imple- numbers to support certain viewpoints has led mented without having been subject to large- to some questionable uses of data. For example, scale, well-designed studies to show that student the anti-reform group HOLD (Honest Open Log- achievement rises when the reform materials ical Debate on mathematics reform) recently cir- are used. He points out that what convinced culated an e-mail message containing data on the everyone that there is a problem in mathemat- Elementary Level Mathematics Examination, ics education was low scores on tests like the Na- which is administered in the California State tional Assessment of Educational Progress. To University system to place students in appro- show that the reform materials are an improve- priate mathematics courses. The data were stun- ment, he asserts, one had better show that stu- ning: in 1989, 28.2% of students failed; in 1992, dent performance on those tests improves with 45.0% failed; and in 1994, 54.0% failed. The e-mail the new materials. message blamed mathematics education reform For their part most reformers acknowledge in California for this poor showing. What the the importance of showing that their students message neglected to say was that the exami- can perform well by traditional measures. How- nation was changed in 1992 to include not just ever, they are wary of overemphasis on perfor- two but three years of high school mathemat- mance on standardized tests because, they say, ics. According to Scott Farrand of California they are aiming for a far deeper understanding State University, Sacramento, who serves on a of mathematics than can be gauged by such Cal State committee working on remediation, tests. Indeed, some argue that it was standard- the change in the test was the “primary cause” ized testing that drove the traditional curricu- of the rise in failures. There are other causes, lum to sacrifice understanding in favor of the but he believes it is simplistic to blame reform. narrow skills of arithmetic computation and As Farrand puts it, “The greater issue is that symbol manipulation. In this spirit, most reform the high school curriculum serves too many programs use a variety of methods of assessing masters, aims to cover too much, none of it with student achievement, including not only timed enough depth.”

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This example shows how seemingly clear-cut Mathematics Program (IMP), the data that exist statistics can mask a much more complicated pic- may not be sufficiently conclusive to change the ture. Indeed, one of the reasons there is a dearth minds of the critics of these programs. When of data on the reform is that it is very difficult MathLand was put on the textbook adoption list to design studies of educational programs well in California, there were no reliable data show- enough to produce meaningful statistics. Asked ing how it influenced student achievement. Ac- what kind of evidence would satisfy him that the cording to MathLand’s national mathematics reform was working, McKeown reels off a list of consultant, Dennis Estrada, for three years teach- variables that would need to be controlled in a ers extensively field-tested the program with well-designed study. For example, one must con- students and helped to shape and fine-tune all trol for the fact that experimental programs are of the lessons. Estrada explains that in classroom sometimes tested only on students who are al- field testing, authors and editorial team mem- ready doing well rather than on students who are bers work closely with teachers and students. still struggling with the old program. Pretests and Achievement data collected in such an environ- posttests have to take into account the prior ment would not be reliable, he says. achievement of the students. Often it is the bet- Now that MathLand has been out for a cou- ter and more enthusiastic teachers who volun- ple of years, a study has been done using stan- teer to test new programs, and they may also dardized test scores for 30,000 students in make greater efforts if they know that their per- grades one through six in eight school districts formance is being monitored. The sample of in California.2 The districts supplied scores for students and teachers must be large enough to 1995—the year before MathLand was imple- test whether the program can be used in differ- mented in those districts—and for 1996. The ent settings, and a variety of programs should study is based on “normal curve equivalent” be included so that results can be compared. And (NCE) scores, which are a standard way of pre- then there is the “Hawthorne effect”, which sug- senting educational data. The actual NCE scores gests that a change in any variable will cause a are not presented. Instead, the study calculates temporary rise in achievement. the differences in NCEs between the two testing McKeown has one example of an educational points, noting that if the NCE is the same from study that aimed for scientific rigor: Project Fol- one year to the next, then “one year’s growth” low-Through. Sponsored by the U.S. Department has occurred. The study formed 23 cohort of Education, Project Follow-Through was begun groups of students from the same district and in 1967 as part of Lyndon Johnson’s “War on at the same grade level, representing all 30,000 Poverty” and received funding through 1995, students. On tests of applications of math- for a total of about $1 billion. During its main ematics, 13 groups achieved the normal one phase from 1968 to 1976, 10,000 children in 120 year’s growth, 5 were above normal, and 5 were communities were studied. The main objective below normal. On tests of mathematical com- of Project Follow-Through was to come up with putation, data were available for only 8 cohort effective ways to teach disadvantaged children. groups, and all of these achieved the normal Twelve different approaches were included in the one year’s worth of progress. The reason that so study, and many of them utilized ideas in use few cohort groups had computation scores was in reform classrooms today, such as construc- that half of the districts in the study do not test tivism and group learning. One of the approaches for computation. This is exactly the area in which was called “direct instruction” and was based on MathLand has been most strenuously attacked, the ideas of Siegfried Englemann, who devel- so these data are unlikely to quell MathLand’s oped finely honed materials to be used by teach- critics. ers in more or less the traditional mode of in- MathLand has had its share of controversies struction. Mathematics test scores pointed to over test scores. After the program was intro- “direct instruction” as being far and away the duced in San Francisco in the fall of 1995, Math- most effective of the approaches in the study. Land took some heat when the school district’s Before the evaluation of Project Follow-Through mathematics test results showed a decline the was released, questions were raised about the following spring. An article in the San Francisco methodology and some of the findings, and the Examiner reported that schools superintendent project’s results were buried under the ensuing controversy. Thus the largest educational study 1A group at the University of Oregon has assembled a ever undertaken in the U.S. was never used to great deal of information on Project Follow-Through at shape educational policy.1 the Web site http://darkwing.uoregon.edu/ Needless to say, none of the reform programs ~adiep/ft/151toc.htm. has been part of a study as extensive as Project 2A press release about the study is available on the Math- Follow-Through. For the two programs discussed Land Web site at http://192.216.191.114/ in this article, MathLand and the Interactive assessinmath.html.

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Bill Rojas believed the drop in scores was due One aspect of the WCER evaluation that has to MathLand. “There’s math and math substi- left some leery of its conclusions is the fact that tutes,” he was quoted as saying. “Let’s get real all results must be cleared by the directors of math in there.” In fact, it is unlikely that Math- IMP before they are released. This stipulation is Land was responsible for the drop in test scores, made clear in the introductory pages of the because San Francisco was phasing the program WCER reports. Alfred Manaster is a mathemat- in gradually, and teachers had only taught with ics professor at UC San Diego and has had ex- three MathLand units during 1995–96. Rojas tensive experience with the Mathematics Diag- also suggested that one problem was that classes nostic Testing Project, which is used by many were too large. During the 1996–97 school year colleges and universities in California to place San Francisco made substantial reductions in students in appropriate mathematics courses, class size. The spring 1997 test scores, released and with the videotape portion of the Third In- in May, showed marked gains in mathematics ternational Mathematics and Science Study achievement. A news release from the San Fran- (TIMSS). Manaster is dismayed by the lack of in- cisco Unified School District said Rojas was “ju- dependence of the evaluation of IMP, but he bilant” at the results, which he attributed to says it is typical of NSF-funded programs. After smaller classes. Did MathLand deserve criticism looking at an NSF report on evaluations of the for the drop in test scores the previous year? Did curricular projects it funds, he says, “it seemed it deserve some credit for the rise in test scores clear to me that all these project evaluations this year? These are not simple questions to an- were being controlled by the project directors swer. themselves.” He says the lack of independence MathLand was produced by a commercial of the WCER evaluation “confirms why I’m trou- publisher, whereas IMP was developed through bled” about IMP. “That’s not a good way of doing business, but that seems to be the way the NSF a grant from the National Science Foundation has embraced,” he remarks. “It makes it very (NSF), which typically requires evaluation com- hard to say which programs are good.” ponents in its curriculum development projects. In the background of the controversy over test In 1992 the Wisconsin Center for Education Re- scores and reform, one senses the implicit mes- search (WCER) began a five-year evaluation of sage that teachers cannot be trusted in their IMP. One part of the evaluation has focused on judgments of what works for their students. whether IMP students take more mathematics When a teacher like Margaret DeArmond—who courses than students enrolled in traditional has a degree in mathematics and many years of courses. In a study of three schools offering teaching to her credit—says that IMP works for both IMP and the traditional sequence of high her students, her statement carries weight. But school mathematics, the percentage of IMP stu- most teachers do not have that background: in dents taking three years of mathematics ranged California in 1990–91, 51% of public high school from 77% to 94%, compared to a range of 69% teachers who taught one or more mathematics to 76% for non-IMP students. This effect was classes did not have even a minor in the subject. more pronounced for African-American stu- The calls for data about the reform programs are dents: 79% of those in IMP took three years of to some extent calls for “teacher-proof” pro- mathematics, compared to 51% of those in tra- grams. UC Davis mathematics professor Sherman ditional courses. Very few of the students in ei- Stein wrote in a recent “Letter to the Editor” ther IMP or traditional programs went on to cal- (Notices, March 1997, page 312), “If all teachers culus in high school, and their performance was were mathematically well prepared, I for one about the same, with an average grade of C. IMP would stop worrying about the age-old battle still students also had higher grade-point averages raging between ‘back to basics’ and ‘under- in mathematics and overall; SAT scores were standing’.” And maybe the concerns over test not available for all students, but when they scores would ease as well. were available, they were about the same for the two groups. In one analysis, a group of IMP Propaganda Warfare students with high scores on a seventh-grade Abigail Thompson, a mathematics professor at standardized mathematics test was matched UC Davis, became concerned about mathemat- with a group of non-IMP students with compa- ics education reform when her daughter started rable scores. An IMP-produced newsletter called attending public school in Davis and “I realized Evaluation Update says this analysis found that she wasn’t learning any math.” Thompson has the mean mathematics SAT scores for the IMP since helped to launch Starting With Math, a UC students was 545, compared to 531 for the non- Davis program aimed at strengthening teachers’ IMP students. However, the more detailed WCER knowledge of mathematics, and she has made report says that this difference is not statistically presentations and written about her views on the significant. reform approach. While her main concerns are

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A Veteran Sums Things Up Weissglass of the Center for Educational Change Richard Askey, a professor of mathematics at the University of in Mathematics and Science at UC Santa Barbara Wisconsin, is an experienced veteran of the math wars, having compares opposition to the reform movement followed changes in mathematics education for many years. He to opposition to the Emancipation Proclamation has also been a vociferous critic of the NCTM Standards—and one and to giving women the vote. The controversy who has actually read the Standards cover to cover. His influence has also sometimes taken on religious over- on mathematics education reform can be seen in a recent article tones. According to Tom Sallee, a UC Davis math- in the NCTM publication The Mathematics Teacher entitled “Who ematics professor and co-developer of the re- Is Dick Askey and Why Is He So Upset about the Standards?”, by form-minded program, College Preparatory Susan Addington and Judith Roitman. The following sums up, in Mathematics (CPM), a Baptist minister in Davis Askey’s words, some of the problems he has seen with math- went to his daughter’s teacher and asked if CPM ematics education reform over the years. taught kids to be independent and think on their “Mathematics is like a stool; it sits on three legs. In the New own. When the teacher replied yes, the minister Math period the only leg used was the structure of mathematics. pulled his daughter from the class, saying that The feeling was that if you understood the structure of math- he did not want his daughter to think she could ematics, then you could compute and solve problems. That turned understand things on her own. According to his out to be false for all except a small group. Then we got ‘Back beliefs, knowledge is only handed down from to Basics’, which was founded on computation. However, the those who are older and wiser. level was too low, and good problems and structure were both The Web sites maintained by the anti-reform ignored. This failed badly. Then NCTM tried Agenda for Action side have done much to keep the controversy and later the Standards. Both of these were built on the idea that smoldering. According to Mathematically Correct if you could solve problems, then you could do mathematics. You founder Michael McKeown, their Web page re- can, but at too low a level. All three are needed—problems, tech- ceives 70–100 hits per day. Because those visit- nique, and structure. I fail to see why this obvious fact is not ap- ing the Web page write in to add their views, there preciated, but it does not seem to be. The New Math failed for are long lists of personal testimonies by parents good reasons, and the New New Math will fail for good but dif- ferent reasons. Maybe then we can start to try to do this right.” and others who have viewed the onslaught of re- form with dismay. While the material ranges from serious analyses to scattershot gripes, the Web sites make clear that there is deep and mathematical—especially the mathematics back- widespread dissatisfaction with the reform. The ground of teachers—Thompson has found that reform side has not used the Internet as a pro- the disagreements over reform have taken on a paganda tool in the same way. There are Web political slant, “which is just ridiculous,” she sites for CPM, IMP, MathLand, and some of the says. After she made a presentation at a meet- other reform programs, but those sites simply ing of the State Board of Education, she was in- give information about the programs; they do not vited to speak at a local Republican convention. attack traditionalist views. A liberal Democrat, Thompson was stunned. What has caused the debate to become so ac- Mathematicians tend to jump into such issues rimonious? According to Phil Daro, who worked with both feet, she says, “and then they find on the California Mathematics Project and now themselves labeled as right-wing conservatives. heads a national project called New Standards, And it’s pretty hilarious. I don’t know any math- part of the reason is that the reformers were un- ematicians who are right-wing conservatives.” prepared for the responsibilities that fell on Thompson found herself labeled according to their shoulders. Originally theirs was the “voice the stereotypes that have arisen in the math in the wilderness,” he says. When they got their wars, in which caricature has become a weapon. way, “they celebrated victories, and for someone For example, conservative syndicated columnist who wasn’t part of them, and who thought they Debra J. Saunders has adopted the fight against were wrong, it must have seemed very grating reform as a personal crusade with columns sport- and hideous,” he remarks. “Of course, it was ing titles like “Creatures from the New-New- probably because they didn’t realize they were Math Lagoon” and “Crashing the Educrat Oli- winning.” Then, within a relatively short period garchy”. Her columns, harshly sarcastic in tone, of time, the reformers suddenly found they were depict the reformers as low-standards pushovers responsible for “all those teachers who were who just want to get kids to “feel good” about doing a poor job in the old system.” These teach- math. For its part, the reform side has tried to ers continue to be ineffective, but now people cast its opponents as privileged elitists who fear are noticing. “The critics are pointing to the that the reform movement will force their kids same poor teaching that has always been there, to mix with “those kids”. Sometimes the re- and now it’s poor because of [reform].” formers describe their struggle in rather Social agendas have entered into the debate grandiose terms. In an essay aimed at teachers as well. For example, some on the reform side who have been attacked over the reform, Julian have emphasized the elimination of “tracking”,

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which sorts kids according to ability or interest. bound up in it. That’s why I’m not so interested Historically, tracking has had the effect of steer- in entering the debate.” ing certain groups into serious mathematics The main problem Stigler sees is that in the courses and others, especially minorities, into U.S. “we don’t have a tradition of studying how dead-end courses like “consumer math”. So the things get implemented in the classroom. We suggestion that by high school students should make policy by having people argue at an ideo- be given a choice of mathematics courses has logical level.” In Japan, he says, there is “a whole brought charges of elitism from the reform R&D system” for improving mathematics in- camp. In this way, says Daro, the reformers struction, in which teachers try experiments, “have created enemies where there weren’t any.” discuss results, write papers, and evaluate their For its part, the anti-reform side has its share of work. It does not matter how good or bad the people pursuing social agendas, who believe instruction is deemed to be; the goal is contin- “that the conservative, family values are what the ual improvement. The absence of this kind of schools should be perpetuating,” he notes. “They careful tending to instruction could in fact doom want only ‘right’ answers, and they make fun of the reform if teachers fall back on traditional ma- math where there is no right answer. So it be- terials because they are easier to use and less comes a war of ridicule. It’s very corrosive and controversial. destructive.” The reform materials also place greater de- What is missing from the debate, says Daro, mands on teachers’ knowledge of mathematics. is a practical, moderate viewpoint that values a In fact, one reason drill and memorization have balanced approach to mathematics instruction. dominated mathematics teaching may be that In his view, there are three aspects that need to teachers’ knowledge of the subject was too weak be addressed in mathematics curricula: basic for anything deeper. Though he has spent a lot skills, conceptual understanding, and problem of time critiquing reform materials, H. Wu, a solving and applications. “That to me defines mathematics professor at UC Berkeley, actually what a balanced curriculum is,” he says. “My believes that teacher preparation is a far more view is that if any one of those three wins, it’s a serious problem.3 And he points out that the re- disaster, because if any one of those three loses, sponsibility lies squarely with higher education. it’s a disaster.” The reform has swung math- “I’m responsible, all of us are responsible, in ematics teaching away from skill-and-drill to an that we have failed in discharging our obligations activity-based, problem-solving approach. “We’ve as teachers,” he says. He points out that the made things more engaging and interesting, but most conservative estimate would put the per- we haven’t dealt with conceptual development,” centage of mathematics majors going on to do he notes. Therefore the reform as it now stands graduate work in mathematics at no more than will not cure the ills of mathematics education. 20%; he speculates that at Berkeley the figure is “And I think more and more people are realiz- no higher than 25%. “But we teach almost every ing that.” one of our courses as if all our students are going to do graduate courses,” he says. “This is Smoke from the Battle Obscures Deeper totally absurd.…It’s a horrible misapplication Issues of our authority.” Wu is planning an alternative James Stigler is a mathematics education re- sequence of courses at Berkeley for those who searcher at UCLA. He headed the TIMSS video- will not go on to graduate school in mathemat- tape study, which provides the most detailed ics. The courses will try to cut back on some of comparisons ever of how mathematics instruc- the technical details that are traditionally em- tion is carried out in Japan, Germany, and the phasized for later use in graduate school in United States. Japanese students’ scores in math- favor of broad themes and major problems in ematics have been the envy of the U.S. for years mathematics. now, so there has been a great deal of interest Efforts like Wu’s address the future cadre of in seeing what Japanese teachers do in the class- teachers, but what about those already in the room. Ironically, both sides of the reform debate classroom? The reform programs offer training have tried to claim Stigler’s work as proof that workshops for teachers, but often the emphasis their approach is right. He says that Japanese is on implementing new teaching techniques teachers use some techniques that might be la- rather than on developing a deeper under- beled reform and some that might be labeled tra- standing of mathematics. A few programs, such ditional, but the Japanese style is neither— it is 3 simply Japanese. “What is interesting is the po- Wu’s article “The Mathematician and the Mathemat- ics Education Reform” (Notices, December 1996, pages larized views of instruction,” he remarks. “One 1531–1537) sets out many of his views on reform. An- type of teaching is liberal, one type is conserv- other article, “On the Training of Teachers”, is avail- ative, one type is Republican, one type is De- able on his home page, http://math.berkeley. mocrat. People have their personal ideologies edu/~wu/.

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as Starting With Math at UC Davis, that are un- people simply are not interested in doing that. related to any particular program or pedagogy That’s not what this round of reform is about.” but simply aim to give teachers more math- Isn’t this an area where mathematicians can ematics. According to elementary school teacher help out? “I think for most mathematicians this Elaine Rosenfield, if you want “human calcula- stuff is just too distant,” says Stanley. Some tors” who can correctly perform computations mathematicians will write down lists of topics on cue, then no teacher training is necessary. “But students should know or advocate that certain if you want to help kids to understand and make topics that reformers have proposed to cut connections, the teachers have to be able to see should be in the curriculum. “But making a con- them too,” she says. Rosenfield calls for ten tribution to conceptualizing whole high school days of staff development each year, which courses in a clear and coherent way—they don’t would cost California approximately $200 mil- seem to be able to do that,” he says. lion. To those who say it is too expensive, Rosen- field replies, “How expensive is ignorance?” Fighting to Reach Common Ground Three years ago California plunged into a war Has the Reform Gone Far Enough? over mathematics education, with the reformers Dick Stanley of UC Berkeley has worked over the on one side and their critics on the other. It is past several years on a number of mathematics not clear yet which side is winning, but those op- education projects and has accumulated and posing reform have won at least one victory. studied a wide variety of textbooks and materi- Late last year, amid mounting criticism of the re- als. One of the benefits of the reform move- form, the state appointed a committee to pro- ment, says Stanley, is that it has gotten more stu- duce the next mathematics curriculum frame- dents to like mathematics. “Bad attitudes toward work a year ahead of schedule. Out of fifteen mathematics, which we hear all the time— ‘I was candidates recommended by the Curriculum never good at math, I never liked math,’ and Commission, the State Board accepted five and this coming from educated and intelligent peo- appointed another fourteen of its own choosing. ple—were a failing of math education in the There were immediate complaints that the Board past,” he says. “So when students like math and had stacked the commission with anti-reform- are engaged in it, that’s a plus.” ers, and the California Mathematics Council as- However, he says, a deeper look at some of sembled a 3,000-signature petition protesting the the reform programs shows that the content— Board’s actions as well as the lack of reform- and students’ understanding of it—is often weak. minded appointees to the committee. In De- An observer in a reform classroom, finding a cember the Board added three appointees per- hum of activity and students discussing and en- ceived as in favor of reform. The mathematicians gaged in their work, might conclude that the re- on the commission are Henry Alder of UC Davis, form is working well. But according to Stanley, Ralph Cohen of Stanford University, Deborah “observers largely have not understood the ex- Tepper Haimo of UC San Diego (who chairs the tent to which the surface positive features of [the committee), William Jacob of UC Santa Barbara, reform] classrooms mask an unchanged lack of Calvin Moore of UC Berkeley, and Bob Stein of real understanding of the content.” This round California State University at San Bernardino. of reform has focused on pedagogy and process, The committee also includes members of Math- not content. It has largely adopted the existing ematically Correct and HOLD. content of the traditional mathematics curricu- After the controversy over its appointment, lum and overlaid it with new teaching methods. the committee had to contend with deep inter- And the problem, says Stanley, is that the tra- nal divisions over exactly what it was supposed ditional curriculum is “in a terrible, sorry state.” to do. Some on the committee believed they It is a confusing hodge-podge of topics, he says, should scrap the 1992 framework and construct with little connection among them and little a new one from scratch; others thought they sense of what the really important ideas are. should keep the 1992 framework largely intact Stanley says that he sees whole courses, flaws and only introduce refinements and balance. and all, being taken over by reform. For exam- The differences of opinion about what the com- ple, the reform versions of some algebra courses mittee was to do, combined with radically dif- have retained all “the old terrible things” like the ferent outlooks on mathematics education, led overemphasis on factoring. Rather than chal- to some very tense meetings. One member called lenging that overemphasis at its roots, the reform them “the most heated meetings I’ve ever been materials simply supply new ways of factoring, to.” An article in Education Week even speculated using manipulatives or diagrams or other “cre- that the committee was so divided it might not ative” activities. “You could make a much bet- be able to complete its task. However, by the time ter curriculum by focusing on what content is the committee met in early June it had produced taught and how it’s taught,” he remarks. “But a working draft of the framework. The commit-

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tee is scheduled to deliver the framework to the Curriculum Commission in July. At the same time that California is revising its mathematics framework, the National Council of Teachers of Mathematics (NCTM) is undertaking a wide-ranging effort to revise by the year 2000 the NCTM Standards, which in 1989 sparked this round of reform. The NCTM is soliciting ideas from a wide range of mathematical and ed- ucation groups, as well as from any individuals who wish to comment.4 The AMS will provide input through a committee it has appointed, chaired by Yale University mathematician Roger Howe. Writing teams have begun to meet to dis- cuss the comments and what revisions are needed to the Standards, and the commission overseeing the entire process will meet in the fall. These two parallel efforts—the preparation of a new mathematics curriculum framework for California and the revision of the NCTM Stan- dards—are bringing together the two warring factions to discuss their differences and try to secure a truce in the math wars. Whether they will succeed depends in part on how much good- will each side can bring to the table. “We come from different sets of beliefs about what con- stitutes mathematics and how children learn it,” says Elaine Rosenfield. “But what we have in common is that we all want students to learn more mathematics.” If the two sides fight for this common ground, they may find they are both winning.

—Allyn Jackson

4For further information and to submit comments, visit the NCTM Web site at http://www.nctm.org/.

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