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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 30
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 39
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 40
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 41
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 42
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 43
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 44
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 45
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 46
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 47
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 48
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 49
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 50
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 51
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 52
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 53
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 54
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 55
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 56
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 57
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 58
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
×
Page 59
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 60
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 61
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 62
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 63
Suggested Citation:"THE VISIONS AND THE CHALLENGES." National Research Council. 1987. Teacher of Mathematics: Issues for Today and Tomorrow : Proceedings of a Conference. Washington, DC: The National Academies Press. doi: 10.17226/18770.
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Page 64

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

THE ISSUE OF REFORM: WHAT HAVE WE LEARNED FROM YESTERYEAR? Thomas J. Cooney University of Georgia As I understand the central issue before us, we are concerned with the question of how the National Council of Teachers of Mathematics' (NCTM's) Curriculum and Evaluation Standards for School Mathematics*, currently being developed, could or should impact on mathematics teacher education programs. I think this question and the issues in which the question is embedded are not only highly relevant to our professional interests, but are, ultimately, primary to our professional activity. My experiences, both as a teacher and as a teacher educator, suggest that the task ahead is significant and awesome. A Brief History of Reform We should keep in mind that there is a long history of mathematics educators expressing concern about the teaching and learning of school mathematics. At the turn of the century, E. H. Moore urged that schools abolish the separation of algebra, geometry, and physics. During the same period, John Perry emphasized the importance of applications and laboratory-teaching techniques in the teaching of mathematics. Following World War II, the Commission on Post-War Plans pointed to serious shortcomings in Americans' mathematical knowledge (NCTM, 1970). Kline (1958) questioned the underlying philosophy of much of the modern mathematics movement. Euphoria reigned during the pinnacle of that movement, with some claiming considerable success in promoting curricular reform (see, for example, Adler, 1972). Later reflections, however, were less sympathetic, suggesting that the teaching of modern mathematics was *References to the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics are based on the Working Draft dated October 1987. -17-

-18- sterile and notation-bound (see, for example, Rising, 1977). Good and Biddle (1987) have argued that observational studies that detail carefully what happened in classrooms during the modern mathematics movement are virtually nonexistent, thereby making it difficult to determine just what the nature or effectiveness of the reform movement really was. These points provide some historical perspective for our current situation and some sense of the problems associated with previous reform movements. I recall well Harold Brody's admonition that "Ignorance is the mother of educational innovation." It is important for us to understand that our current effort at reform through the Standards is but one of many efforts to reform mathematics education. I hope we have learned from the lessons of yesteryear. I think it is important to consider the times in which recent reform movements have occurred. In the 1960's, there was considerable belief in the "reasoned" man and in the notion that intellectualism could lead society toward the resolution of its problems--educational, economic, and societal. As a young practicing teacher, I remember the advice that permeated most professional meetings: "Teach the structure of mathematics and all else will fall into place." But something went awry. Politically, the country became less confident that the science that put man on the moon could solve its pressing social, economic, and educational problems as well. In society, the notion of accountability emerged. In education, the buzzword was "relevant." And so, in the 1970's, we saw the emergence and proliferation of competency-based educational programs that were mired in the framework of behavioral objectives and couched in the seemingly infallible slogan: "Practice makes perfect." As Alan Osborne predicted at a 1970 NCTM meeting, we left our decade of renaissance and our flirtation with Camelot and began an era of anti-intellectualism manifested eventually in the "back-to-basics" movement. We were never really sure what "back to basics" meant, but it was certainly something different from the intellectualism that characterized the 1960's. And so, in the late 1970's, we had the lunacy of children--my child--doing a page or two of long division problems and then checking them on a $9.95 calculator. We began to see what Romberg (1987) has described as antiprofessionalism, in which teachers were denied the opportunity of becoming true professionals because their tasks were becoming increasingly "deskilled." Teachers became less the decision-makers and more the implementers, maybe even the technicians, of mandates from above. While once we could have been accused of having an educational system that could be described as "each age in his cage," we now began to see an educational system that featured "each age on the same page."

-19- But the pendulum had swung too far, and people from many walks of life began to sense an impending disaster if something were not done. Witness A Nation at Risk (1983) and the volumes of com- ments that it spawned. Specific to mathematics education, witness studies such as the Second International Mathematics Study and the resulting reports, and the recommendations given in School Mathematics: Options for the 1990s (1984). The middle school curriculum represents a particular concern because of its "flatness" and the fact that it fails to include many new topics (Flanders 1987). This is particularly obvious when the curriculum is compared with middle school curricula in other countries, particularly France and Japan. The effects of the back-to-basics movement with respect to middle school mathematics are revealed when one compares the results from the First International Mathematics Study with those of the Second International Mathematics Study, particularly the achievement in arithmetic, which dropped 6 percentage points (see McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987). A study by Miwa (1987) on the achievement of middle school Japanese students underscores further the need for reflection on our middle school mathematics programs. Miwa presented the following results: Table 1 Achievement Items for Japanese Fifth- and Sixth-Grade Students Fifth Grade Find the value of X which satisfies each X x 4 - 2 - 6 (85.8% correct responses) 5/6 + 3/8 - X (80.8% correct responses) Sixth Grade When we substitute a positive number into [ ] of the following expressions, the greatest is a.[ ] x 1 1/2 b.[ ] x 1/2 c.[ ] - 1 1/2 d.[ ] - 1/2 (63.0% correct responses) We buy apples for A yen and oranges for B yen, and hand a 1000 yen note. How much change do we have? (61.2% correct responses) Various studies by Stevenson and Stigler (Stevenson, Lee, and Stigler, 1986; Stigler, Lee, and Stevenson, in press; and Stigler and Perry, 1987) leave us with the impression that elementary schools in

-20- Taiwan and Japan are more academically oriented, that their students are more motivated, that their teachers have greater support for the time they need to prepare lessons, and that their students learn because of effort, whereas our students learn because of ability. In short, the American teachers' task of teaching elementary school mathematics is much more difficult than that of their Japanese or Taiwanese counterparts. In light of the various reports on the status of American education and studies such as the Second International Mathematics Study, we can hardly reach any conclusion other than that reform of some sort is sorely needed. The need for reform is considerably deeper than altruism on the part of the educational community or society more generally. The long-overdue recognition of the connection between our nation's economic well-being and the state of our educational system, particularly as far as mathematics education is concerned, suggests that the fiber of our nation's economy is very much related to the success of our educational system. Jennings (1987) refers to this connection as the "Sputnik of the 80's," in terms of its impetus for reform. No longer can society view education and competitiveness in the international marketplace as disconnected happenings. Thus, we need to consider the issue of reform, the form it might take, and the means by which the Standards can influence or drive that reform. The Notions of Standards and Reform Uninterpreted, the term "standards" should give us some pause. It suggests universal applicability, regardless of the context or circumstances in which the standards are to be applied. Such application can have the potential to constrain and confine mathematical activity that cries for flexibility and creativity. In my view, any attempt to consider standards for mathematics education must be tempered with a good dose of wisdom and humility when such standards are applied to increasingly complex educational settings. Used as a heuristic for reform, a rebellion against the status quo, the Standards can provide guidance for significant improvements in mathematics education. Used literally with insensitivity to local conditions, I fear that the result could be another instance in the annals of mathematics education of reform foiled from the top down. Allow me to address the question of what shape reform can take. The Cambridge Conference on School Mathematics (1963) focused on curricular reform by addressing the sequencing of topics, but dismissed the practicality of classroom teaching and related teacher education problems as issues that could be resolved when the time came to do so. It was assumed that the "top professional experts" were in the best position to design curricula for both the practitioner and the consumer because they could judge best what mathematics was needed for the "next course" in mathematics. It was

-21- a classic instance of reform suggested from the top down. In contrast, the membership of the NCTM Working Groups that developed the Standards represented a cross-section of classroom teachers, supervisors, teacher educators, mathematicians, and researchers. The issue of reform from the top down, however, still lurks on the horizon. Do we see reform as "paper reform," as was the case with the School Mathematics Study Group and the "modernization" of school mathematics? I would hope not. To me, reform should be considered a humanistic enterprise rather than a matter of paper reform. Reform needs to be based on human innovation and activity. Freudenthal, in his plenary address at the 1980 International Congress of Mathematical Education, concluded that: Curriculum development viewed as a strategy for change is a wrong perspective. My own view, now shared by many people, is educational development. (Freudenthal, 1983, p. 6.) Freudenthal emphasized a broad perspective of educational develop- ment, one that includes research and teacher education. For him, reform is not a matter of paper, but a matter of people. A humanistic orientation suggests quite different strategies for teacher education than does reform conceived of as paper reform. A humanistic orientation emphasizes the teacher as a decision-maker who determines what mathematics students are capable of learning and what strategies are appropriate, given the mathematical maturity of the students. Can you imagine any greater task for teacher education than educating teachers to make such decisions? If reform is viewed from such a perspective, how can the Stan- dards influence and drive reform? Romberg's use of the metaphor of "vision" in the introduction to the Standards seems quite appro- priate: a vision not just of curriculum, but of instruction as well. Despite the fact that Romberg states correctly that the Standards were not "written as criteria to be used in observing whether teachers' actions during instruction are appropriate," there is much embedded in the Standards that has significant implica- tions for instruction. Romberg's use of the verbs "examine," "repre- sent," "transform," "apply," "solve problems," and "communicate" gives an undeniable tone of a certain philosophical orientation toward instruction. In fact, this philosophy is manifested explic- itly in the following statement in the introduction: In summary, in our view, instruction should be based on problem situations....They (instructional situations) should be amenable to individual, small-group, or large-group instruction, involve a variety of conceptual domains, and be open as to the methods to be used. (p. 10.)

-22- In the K-4 standards, we find the following statements: The major classroom emphasis should be on establishing a climate that places critical thinking at the heart of instruction. (p. 26.) Teachers should focus on the development of inductive reasoning, such as that required in extending patterns, by providing situations which require children to make generalizations. (p. 26.) A spirit of investigation and exploration should permeate instruction. (p. 44.) In the 5-8 standards, it is stated that: Teachers need to provide a caring environment in which students can feel free to explore mathematical ideas, to ask questions, to discuss their ideas, and to make mistakes. (p. 51.) In a separate section on "Instruction" (p. 53), the following prescriptions are given: Students should be actively involved in the learning process, investigating and exploring individually and in groups. Relevant situational contexts should motivate instruction. Students should experience ideas in context--real world and/or mathematical. Teachers should be facilitators of learning, not merely dispensers of knowledge. The 9-12 standards devote a section to the topic "Patterns of Instruction" (p. 90). The following statements appear in that section: The role of the teacher should shift from one of dispensing information to one of facilitating learning. In order for students to internalize the view of mathematics as a process, a body of knowledge, and a human creation, they need many opportunities to experiment with Ideas, develop strategies, formulate and communicate conclusions, apply fundamental skills, and interact in groups. The Evaluation Standards support the same instructional theme, in that assessment is viewed as an integral part of instruction and should be based on multiple and varied kinds of evidence and on continual dialogues between teacher and students. The position is taken that learning is not a matter of collecting, but a matter of constructing. Indeed, the evaluation task force proclaimed that:

-23- Because assessment is such an integral part of instruction and, hence, is basic to the attainment of the standards, teachers need to know as much about assessment as they do about the content they teach. (p. 139.) The message across both the curriculum and evaluation standards is unmistakably clear: Instruction is a fundamental aspect of the Standards. The implication for teacher education in educating teachers to teach as suggested in the Standards is, in my mind, the ultimate challenge for mathematics teacher educators. To accomplish this, two significant obstacles must be addressed: teachers' conceptions and students' conceptions of mathematics and the teaching of mathematics. The Influence of Teachers' Conceptions Whether or not teachers implement the full intent of the Standards and not just the mathematics identified in them depends on how the intended curriculum is filtered through the teachers' beliefs and conceptions of mathematics. Research over the past several years on teachers' beliefs provides strong testimony that teachers' conceptions make a difference in how mathematics is taught (see, for example, Thompson 1982, 1984; McGalliard, 1983; Kesler, 1985; Brown, 1985; and Brown and Cooney, 1986). I will not review the research on teachers' beliefs, although I believe such a review would reveal many tantalizing metaphors for teacher education. Rather, I choose to discuss the work of the Dutch mathematics educator Fred Goffree as a way of conceptualizing how teachers interact with curricula. Goffree (1985) has identified four different perspectives that seem to characterize primary school textbooks (although they probably characterize texts at other levels as well): the mechanistic view, the structuralist view, the empirical view, and the realistic or application view. Their names are suggestive of the philosophical intent of the books. Related to these perspectives, Goffree identified three types of teacher use of textbooks: (1) Instrumental use. The textbook is followed to the letter; learning should occur along the sequence of the learning tasks presented. (2) Subjective use. The teacher first makes a constructive analysis of the material and then elaborates on the material based on personal beliefs and knowledge. (3) Fundamental use. The curriculum material is analyzed constructively, but now the underlying philosophical view of mathematics education is taken into account as well. (p. 26.)

-24- As Goffree suggests, these concepts of teacher use, combined with the four different types of mathematics textbooks, can provide a basis for describing how teachers interpret different types of curricula. Consider, for example, the dramatic change, and probably related trauma, associated with asking an instrumental user of a mechanistic textbook to teach an application (realistic)-oriented curriculum in a fundamental way. A study by Stephens and Romberg (1985) that involved the innovative curriculum materials called RIME (Reality in Mathematics Education) illustrates the problem. A recurring theme throughout their analysis was the difficulty teachers faced when they were asked to teach atypical content in atypical ways. The authors provided the following teacher and student quotations: In RIME, there is a fair chance that someone will put forward an idea, or give you an answer, which you haven't anticipated. That puts teachers on edge, when they first teach RIME. (Remark by one of the experimental teachers.) Why are we doing this? Is this part of a games period? How are you going to assess what we are going to do in this activity? (Student remark.) They (students) expect a certain kind of approach to education in general, and that includes mathematics. Students expect the teacher to be in charge and to give clear directions, to give lots of work for students to do. There is little expectation that they will have to motivate themselves or to show initiative in what they do. (Teacher remark.) [A difficulty is] setting the story and getting the students in the right frame of mind to do the lesson; that is, getting the students to accept a different form of teaching. (Teacher remark.) In terms of Goffree's analysis, teachers were asked to move toward a realistic curriculum with a fundamental perspective in preparing lessons. As was clear from the comments, students and teachers were not accustomed to such an instructional orientation. Similarly, preservice teachers have certain conceptions about how they ought to teach mathematics. Secondary preservice teachers provided the following diary entries in reflecting on their own teaching experiences: Activities must be done in a class in which the teacher has total control because we all know that, when unruly children are given a chance to move around and talk in the classroom, they will go crazy and no learning will take place.

-25- Right now, I'm teaching in an authoritative way, meaning I am not allowing the students to come up with alternative ways of working problems. I always have the fear that someone will ask a question that I don't know the answer to. When you're up there teaching, you're nervous and you just want the kids to hear what you have to say, so it's difficult to listen to what they have to say. It's hard to keep from doing everything yourself when you're up front. A common perspective among preservice teachers seems to be that teaching mathematics is primarily a matter of providing a broadcast. The most prevalent word used by our secondary mathematics methods students to describe their teaching is "present." "If only I can find the right way of presenting mathematics," says the intern, "life in the classroom will be o.k." When the presentation goes awry, as it typically does for the neophyte, it is easy for the preservice teacher to reach the conclusion that students lack sufficient internal motivation to receive the broadcast; their antennas are not up. The notion of presentation embeds the notion of authority. It was revealing that the Middle School Task Force thought it appropriate to quote Steen (1986) who said, "above all else, it [the mathematics curriculum] must not give the impression that mathe- matical and quantitative ideas are the product of authority or wizardry" (p. 64, Working Draft). The issue of authority is central. Those familiar with Perry's stages of intellectual development (Perry, 1970) and their adaptation to mathematics education will recognize the importance of moving teachers through the sequence of a dualistic to a multiplistic to a relativistic perspective. Unfortunately, there is evidence that algebra and geometry teachers communicate a dualistic/multiplistic perspective about mathematics to their students (McGalliard, 1983; Kesler, 1985). Owens (1986) found that preservice secondary mathematics teachers' orientations were generally a mixture of dualistic and multiplistic conceptions, with occasional traces of relativism. Meyerson (1977) demonstrated some success in moving secondary mathematics methods students from a dualistic perspective to a multiplistic or a relativistic one. The relevance of these findings to the Standards is that dualistic teachers emphasize the importance of authority--be it theirs, the textbook's, or famous mathematicians'--when teaching mathematics. Teachers who adopt a broadcast metaphor as a means of teaching mathematics, a metaphor based on the importance of authority, cannot possibly subscribe to a fundamental use of texts, nor can they "present" mathematics as a problematic subject to be created and explored. The realization that

-26- preservice and inservice teachers hold such conceptions should, at the very least, suggest that teachers' philosophies and orientations toward mathematics require attention beyond what mathematics they "know." In light of Romberg's observation that the culture of schooling impedes the professional development of teachers, this concern about teachers' conceptions is of no small significance. This is particularly true with respect to the use of technology. The Standards advocate extensive use of technology as a means of exploring mathematics. The stated assumption is that all students will have ready access to calculators and that teachers will have a computer available for daily classroom use. But if we think that the issue of using technology is solely a matter of availability, we are sadly mistaken. Schofield and Verban's (1987) insightful analysis on teachers' use of computers, which were readily available, paints quite a different picture. Their analysis suggests that teachers' attitudinal considerations, for example, the perceived challenge to their competence, the challenge to traditional means of evaluation, and a general lack of ability to integrate computers and mathematics in a fundamental wav (to use Goffree's term) are significant barriers to computer usage. I see the problem as one of addressing teachers' conceptions of not only what mathematics is, but also of how they envision their role as teachers of mathematics, a problem clearly within the domain of mathematics teacher education programs. The Influence of Students' Conceptions A second factor to be considered with respect to realizing the Standards is the students' conceptions about what constitutes appropriate teaching of school mathematics. In a case study I conducted with a beginning mathematics teacher (Cooney, 1985), I found that the teacher experienced considerable difficulty when students found that his teaching and their expectations, both of what mathematics should be and of how it should be taught, did not coincide. His introduction to insurance in a general mathematics class in which dice and probability were used was viewed by the students as the act of an uncaring teacher because, if he really cared, he would be teaching them to add, subtract, multiply, and divide. My own experiences as a classroom teacher, especially recently as a teacher of secondary school geometry, suggest that the class moves less smoothly when lessons deviate from students' conceptions of what constitutes conventional classroom practice; witness Romberg and Stephens' study with the RIME materials. Sizer (1984) noted that a successful class is one in which students and teacher agree on what they are doing and on how to do it. It is not always easy to achieve such agreement, particularly when reform is the primary objective. Students and teachers bring very different expectations, perspectives, and agendas to the

-27- classroom. Consider Holt's (1964) lucid description of the defensive strategies students use to protect their own self-images at the expense of understanding the mathematics being taught. Krummheuer (1983) observed that students may be involved in the same activity as the teacher, but think about it in quite different ways. Hoyles (1982) reported that students want security and structure in their work; they want to "get it right." When our children ask for help in studying mathematics, there is often the qualifier, "Dad (Mom), could you just tell me the answer?" When students' concerns, likes, and dislikes are not congruent with those of the teacher, some negotiating must be done in order to maintain a successful classroom environment. I remember well on my return to classroom teaching the episode in which we needed the quadratic formula to solve a geometry problem. When asked if they could recall the formula, the students manifested the usual disassociation with what had been taught the previous year in algebra class. As one young lady stated particularly well, "Well, we had it, but you can't expect us to remember it--we weren't tested on it." Given these pressures, it should be no surprise that many teachers try to create successful classroom environments by compromising whatever reforms may be intended in order to accommodate student expectations. 1 suspect that students gravitate toward a mechanistic curriculum and appreciate teachers whose interpretations of the text are quite predictable. If you believe the contrary, listen carefully to the negotiations that take place between students and teacher when test time arrives. Conclusion When I think of the question, "What education should a mathematics teacher have?" I often recall an article by Trevor Fletcher, Her Majesty's Inspector of mathematics teaching in England, entitled, "Is the teacher of mathematics a mathematician or not?" (Fletcher, 1979). I find it an intriguing article, especially appropriate for our discussion of the Standards. In essence, Fletcher answers the question in the affirmative, using the following logic. A mathematician has a broad knowledge of mathematics and a specific area of expertise in it. So, argues Fletcher, is it the case with teachers of school mathematics. They, too, must have a general knowledge of mathematics. But they also must have expertise in curriculum and instruction that allows them to convey the notion that mathematics is a subject to be explored and created. It is important to realize that this special knowledge is rooted in a philosophy of mathematics for, as the mathematician Rene Thorn stated in his address at the 1972 International Congress of Mathematical Education in Exeter:

-28- Whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics. (Thorn, 1973, p. 204.) It is not difficult to identify the philosophy of mathematics behind the Standards. The Standards reflect a remarkably consistent tendency to conceive of mathematics as a subject that is not only replete with problems, but also is itself a problematic subject. One cannot conjecture, explore, transform, and perform all of the other action verbs used throughout the Standards and still maintain the notion that mathematics is a subject handed down from on high. It is difficult to imagine that any other issue could speak more directly to mathematics teacher education than that of how the philosophy set forth in the Standards can be engendered in teachers. The Standards call into question how much we "lie" to students when we teach mathematics, particularly in the face of the finding from the Second International Mathematics Study that 50 percent of the eighth-graders surveyed believed that learning mathematics is basically a matter of memorizing rules. I do not mean the fact that we teach students that 2+3-6, but, rather, the impression they receive that there is no mathematics to be discovered by anyone except only the world's most brilliant and talented mathematicians. I regard this as a serious matter, for teachers are strongly influenced by their mathematical experience at both the college and precollege levels. Both mathematicians and mathematics educators cannot escape the responsibility they have for shaping their students' philosophies of mathematics, no matter how implicitly or subtly those philosophies may be communicated by their instructional methods, the means by which they encourage students to learn mathematics, and the means by which they assess their students' learning of mathematics. George Polya had a long history of teaching mathematics through exploration, conveying concomitantly the belief that mathematics is not just a collection of symbols devoid of meaning save for their own cyclical, symbolic referents. It is in the Polya tradition that the mathematical education of teachers should occur. No longer is it acceptable, if it ever was, for the "right mathematics" to be taught from a basically broadcast perspective. Too often, students confuse the message with the means by which the message is delivered. When students admire teachers for their depth of knowledge, they also tend to adopt their teaching styles. Bush (1983) found that much of what preservice teachers believe about teaching stems from their experiences prior to formal entry into a teacher education program. This is disturbing if one considers that research suggests that true problem-solving episodes are a rarity in the teaching of school mathematics. We run the risk of a cyclic process of mathematics being studied and then taught in a manner antithetical to the intent of the Standards.

-29- How can we educate mathematics teachers to acquire the kind of professionalized knowledge to which Fletcher refers and which is central to the realization of the Standards? The question is immensely complex, as anyone who has spent even a moment listening to teachers or students talk about their mathematical experiences realizes. What evidence we have does not point toward additional study of higher mathematics as a sufficient condition for resolving the problem. Owens (1986) found that preservice secondary mathematics teachers' study of higher mathematics, or what Freudenthal (1973) describes as "sterilized courses of further training in abstract mathematics" (p. 73), only contributes to preservice teachers' conceptions that mathematics is essentially an exercise in manipulating symbols. The preservice teacher sees school mathematics as basically equation-solving in various forms. As Owens observed, the college curriculum attempts to "enlighten" the teacher mathematically not by expanding his or her conceptions of mathematics, but by the wholesale introduction of new ideas. This has led Freudenthal to observe that: Educational programmes and methods are influenced by a belief which is natural for every mathematician, that mathematical education is education to become a mathematician. (Freudenthal, 1973, p. 73.) Owens concluded that the preservice mathematics teacher has neither the constructs nor the incentive to provide the kind of transformation needed to see school mathematics as something other than what Whitehead (1929) calls "exercises in intellectual minuets." Let me be clear. The value of advanced study of set theory, analysis, or abstract algebra by teachers who will implement the Standards is not in question any more than one can question the centrality of deoxyribonucleic acid in human physiology. What is in question is the practice in mathematics teacher education of dichotomizing the study of advanced mathematics on the one hand and of general pedagogy on the other so as to limit the teachers' mathematical experiences to only those involving the study of wholesale new mathematical ideas. Teachers need experiences constructing the same mathematics that they will be teaching. Permit me a personal illustration. When I was teaching high school geometry a few years ago, 1 taught the theorem: If two chords intersect in a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the second chord. My planning was thorough, and I thought the students were reasonably responsive. The theorem was couched in the context of an axiomatic system, with numerical examples to be completed by the students. For some reason, I then posed the following question:

-30- Which chord through point P in circle 0 will give the largest product of the lengths of the two segments? The reactions were quite interesting. For the most part, the students' facial expressions expressed the view that it was, indeed, a strange question. After all, it did not seem to fit the model of what they knew would be assigned as homework. Second, and most important, they had no idea of the answer. Those brave enough to venture a response suggested that it must be the diameter since that would be the longest chord. After several minutes of discussion, one student volunteered that it did not matter what chord is drawn--the product of the segments would all be the same. The students seemed satisfied with this, if only to permit them more time for doing their homework. The next fall, I presented the same question in a methods course for preservice secondary teachers in which we were concerned with various instructional strategies. These students had no idea of the answer either. They, too, suggested that it might be the diameter, although some conjectured that it might be the chord through P that is perpendicular to the diameter. We decided to examine various special cases in order to determine the chord in question. First, we assumed that the circle had a radius of 2 and that P was the midpoint of a radius. We then considered three chords: the diameter, the chord perpendicular to the diameter, and the chord that formed a 45° angle with the diameter. In each case, we determined the length of the two segments and found that their product was 3, to the students' amazement. A conjecture was formed and then a proof was sought. I had the distinct impression that the methods students would not soon forget their theorem and the means by which it was discovered. Knowledge about teaching mathematics in similar ways constitutes one aspect of Fletcher's "professionalized knowledge." It provides the teacher with the background knowledge necessary to teach mathematics in a nonauthoritarian manner, using a construction metaphor rather than a broadcast one. It provides the basis for teaching mathematics in a way consistent with Goffree's fundamental use of realistic textual material. Yes, teacher education should consist of substantial study of higher mathematics and of general pedagogical techniques that ^ emphasize such things as maintaining an appropriate learning environment, preventing or dealing with discipline problems, and being aware of what students are doing, that is, Kounin's (1970) "withitness" (or, in the vernacular of the practitioner, using peripheral vision). But teacher education also must provide contexts in which teachers study the content that they will be required to teach in a way that embodies the philosophy that permeates the Standards. Education on the content of the Standards without

-31- the accompanying experiences to reinforce the Standards' philosophy will doom us to failure as surely as we have seen the failure of reform movements in yesteryear. It requires teacher education not only to persuade teachers to teach students according to the philosophy that "A child's mind is a fire to be ignited, not a pot to be filled" (source unknown), but also to engage in the symmetric issue of teacher educators teaching teachers with the ffflni? philosophy. Such a conception of teacher education requires considerable introspection, determination, and a good dose of patience and fortitude on the part of us all. The stakes are high, but the rewards are many. What we must realize is that the task is formidable but feasible, costly but affordable, and, most importantly, controversial but necessary. A dualistic perspective is rooted in the acceptance of authority as the arbiter of truth. Such a view of mathematics leads to the conclusion that mathematics is nonproblematic and consists of a collection of true propositions apart from the context in which they were developed. A multiplistic perspective recognizes the plurality of "answers," but as a collection of discrete entities without structure. Perry sees relativism as the recognition of a plurality of perspectives, but with accompanying interpretations, each being sensitive to the frames of references within which knowledge is developed. Multiple perspectives are legitimized, appreciated, and evaluated taking into account these frames of references.

-32- References A Nation at Risk: The Imperative for Educational Reform. The National Commission on Excellence in Education. Washington, D.C.: U.S. Government Printing Office, 1983. Adler, I. "Criteria of Success in the Seventies." The Mathematics Teacher, 65, 1972, pp. 33-41. Brown, C. A. "A Study of the Socialization to Teaching of a Beginning Secondary Mathematics Teacher." Doctoral dissertation, University of Georgia, 1985. Brown, S. I. and Cooney, T. J. "Stalking the Dualism Between Theory and Practice." Paper presented at the conference on "Systematic Cooperation Between Theory and Practice." Lochera, Holland: November 1986. Bush, W. S. "Preservice Secondary Mathematics Teachers' Knowledge about Teaching Mathematics and Decision-making During Teacher Training." Doctoral dissertation, University of Georgia, 1982. Dissertation Abstracts International, 43, 1983, p. 2264A. Cooney, T. J. "A Beginning Teacher's View of Problem Solving." Journal for Research in Mathematics Education, 16, 1985, pp. 324-336. Flanders, J. R. "How Much of the Content in Mathematics Textbooks is New?" Arithmetic Teacher, 35, 1987, pp. 18-23. Fletcher, T. J. "Is the Teacher of Mathematics a Mathematician or Not?" In: H. G. Steiner, ed., The Education of Mathematics Teachers. Bielefeld: Institut fuer Didaktik der Mathematik der Universitaet Bielefeld, 1979, pp. 185-199. Freudenthal, H. Mathematics as an Educational Task. Dordrecht, Holland: Reidel, 1973. Freudenthal, H. "Major Problems of Mathematics Education." In: M. Zweng, T. Green, J. Kilpatrick, H. Pollak, and M. Suydam, eds., Proceedings of the Fourth International Congress on Mathematical Education. Boston: BIrkhauser Boston, Inc., 1983, pp. 1-7. Educational Services Incorporated. Goals for School Mathematics: A Report of the Cambridge Conference on School Mathematics. Boston: Houghton Mifflin Company, 1963. Goffree, F. "The Teacher and Curriculum Development." For the Learning of Mathematics, 5, 1985, pp. 26-27.

-33- Good, T. L. and Biddle, B. J. "Research and the Improvement of Mathematics Instruction: The Need for Observational Resources." Paper presented at the conference on "Effective Mathematics Teaching," University of Missouri, Columbia, Missouri, March 1987. Holt, J. How Children Fail. New York: Dell Publishing Co., Inc., 1964. Hoyles, C. "The Pupil's View of Mathematics Learning." Educational Studies in Mathematics, 13, 1982, pp. 349-372. Jennings, J. J. "The Sputnik of the Eighties." Phi Delta Kappan, 69, 1987, pp. 104-109. Kesler, R. "Teachers' Instructional Behavior Related to Their Conceptions of Teaching and Mathematics and Their Level of Dogmatism: Four Case Studies." Doctoral dissertation, University of Georgia, 1985. Kline, M. "The Ancients Versus the Moderns, a New Battle of the Books." The Mathematics Teacher, 51, 1958, pp. 418-427. Kounin, J. S. Discipline and Group Management in Classrooms. New York: Holt, Rinehart, and Winston, 1970. Krummheuer, G. "Das Arbeitsinterim im Mathematikunterricht." In: H. Bauersfeld, ed., Lernen und Lehren von Mathematik. Cologne: Aulis Verlag, 1983, pp. 57-106. McGalliard, W. A., Jr. "Selected Factors in the Conceptual Systems of Geometry Teachers: Four Case Studies." Doctoral dissertation, University of Georgia, 1982. Dissertation Abstracts International, 44, 1983, p. 1364A. McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. 0., Travers, K. J., and Cooney, T. J. The Underachieving Curriculum. Champaign, IL.: Stipes Publishing Company, 1987. Meyerson, L. M. "Conception of Knowledge in Mathematics: Interaction with and Applications for a Teaching Methods Course." Doctoral dissertation, State University of New York at Buffalo. Dissertation Abstracts International, 39, 1977, p. 02A. Miwa, T. "Algebra Teaching in Japanese School Mathematics." Paper presented at the conference on "The Teaching and Learning of Algebra," University of Georgia, Athens, Georgia, March 1987.

-34- Natlonal Council of Teachers of Mathematics. A History of Mathematics Education. Washington, D.C.: National Council of Teachers of Mathematics, 1970. Owens, J. E. "A Study of Four Preservice Secondary Mathematics Teachers' Constructs of Mathematics and Mathematics Teaching." Doctoral dissertation, University of Georgia, 1982. 1986. Perry, W. G., Jr. Forms of Intellectual and Ethical Development in the College Years: A Scheme. New York: Holt, Rinehart, and Winston, 1970. Rising, G. R. "Which Way Mathematics Education?" New York State Mathematics Teachers Journal, 28, 1977, pp. 5-17. Romberg, T. A. School Mathematics: Options for the 1990s, Chairman's Report of a Conference. Washington, D.C.: U.S. Government Printing Office, 1984. Romberg, T. A. "Can Teachers be Professionals?" Paper presented at the conference on "Effective Mathematics Teaching," University of Missouri, Columbia, Missouri, March 1987. Schofield, J. W. and Verban, D. "Computer Usage in the Teaching of Mathematics: Issues that Need Answers." Paper presented at the conference on "Effective Mathematics Teaching," University of Missouri, Columbia, Missouri, March 1987. Sizer, T. Horace's Compromise: The Dilemma of the American High School. Boston: Houghton Mifflin Company, 1984. Steen, L. A. "A Time of Transition: Mathematics for the Middle Grades." In: R. Lodholz, ed., A Change in Emphasis. Parkway, MO: Parkway School District, 1986, pp. 1-9. Stephens, M. and Romberg, T. "Reconceptualizing the Role of the Mathematics Teacher." Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 1985. Stevenson, H. W., Lee, S. Y., and Stigler, J. W. "Mathematics Achievement of Chinese, Japanese, and American Children." Science, 231, 1986, pp. 693-699. Stigler, J. W., Lee, S. Y., and Stevenson, H. W. "Mathematics Classrooms in Japan, Taiwan, and the United States." Child Development. In press. Stigler, J. W., and Perry, M. "Cross Cultural Studies of Mathematics Teaching and Learning: Recent Findings and New Directions." Paper presented at the conference on "Effective Mathematics Teaching," University of Missouri, Columbia, Missouri, March 1987.

-35- Thom, R. "Modern Mathematics: Does It Exist?" In: A. G. Howson, ed., Developments in Mathematical Education. Cambridge: Cambridge University Press, 1973, pp. 194-212. Thompson, A. G. "Teachers' Conception of Mathematics and Mathematics Teaching: Three Case Studies." Doctoral dissertation, University of Georgia. Dissertation Abstracts International, 1982, pp. DEP82-28729. Thompson, A. G. "The Relationship of Teachers' Conceptions of Mathematics and Mathematics Teaching to Instructional Practice." Educational Studies in Mathematics, 15, 1984, pp. 105-127. Whitehead, A. N. The Aims of Education. New York: The MacMillan Co., 1929.

MATHEMATICS TEACHING IN SCHOOLS: IMAGINING AN IDEAL THAT IS POSSIBLE Magdalene Lampert Spartan Village School and Michigan State University The current reform documents propose an ideal for mathematics teaching in our schools that is based on what we know about mathe- matics, about learning, and about environments conducive to learning or doing mathematics. This paper is based on what I know about mathematics teaching. I was asked to discuss indicators of ideal mathematics teaching. Later, I would like to comment on the relationships among the ideal, the possible, and the realistic. But, to address ideals for a moment, if I had to pick one indicator of an ideal mathematics teacher, it would be a measure of whether that teacher could give students at the grade level he or she is teaching a mathematically legitimate and comprehensible explanation for why the procedures students are using are appropriate or not, or why the answers they are giving are correct or not. In other words, my ideal teacher would not need to use "the answer book," nor would that teacher think it acceptable to refer students to the answer book. This is an indication of good mathematics teaching because of the nature of mathematical knowledge and the procedures used in the discipline to verify whether knowledge is true. In order to teach mathematics, a teacher needs to be confident that what he or she is teaching and what the students are learning make mathematical sense. This is different from being confident that you "know how to do it," although it probably assumes some degree of procedural knowledge. When the teacher asks a mathematical question in a classroom setting, and a student gives an answer, the student ought to be called upon to explain how the answer was reached. If the answer is wrong, the teacher ought to be able to provide a counter to the student's explanation that will cause the student to recognize the error. For example, if the teacher asks, "Which is the larger number, four-sixths or three-fourths?" and the student an- swers, "Four-sixths, because it means four pieces, and three-fourths only has three pieces," the teacher might come back with, "What about one-half and two-tenths?" This second problem gives the student a most accessible fraction (one-half) to think about, and gives more data from which to develop a general principle. I do not mean to say that the teacher should leave the conclusion to either of these -37-

-38- problems to the student, but that counterexamples are a more appro- priate vehicle for discussing mathematical verity than reference to answer books. Repeating the algorithm: "Get a common denominator and compare the numerators" has no meaning for the puzzled student. It might result in a correct answer, but it will not result in learning how to compare fractions. What does it take for mathematics teachers to wean themselves and their students away from answer books and algorithms, and replace them with public mathematical conjecturing and arguing among students about plausibility? Based on my attempts to do this every day over the past six years in a fifth-grade classroom and my observations and conversations with other teachers making similar attempts from kindergarten through high school, I would say that it is immensely more difficult than I ever imagined it would be, but that it is possible. Establishing whether ideals are possible is a matter of experimentation, but turning ideals into realities, in actual classroom settings, is a political matter. Schools in our society are about accomplishing many more--and sometimes conflicting--goals than engaging children in learning mathematics. Different groups have different ideas about what constitutes good teaching, and different ideas about the sorts of goals that schools should be trying to accomplish. These conflicting goals, and the environment, the conditions, and the distractions of everyday life in classrooms make it difficult for teachers to keep their focus on mathematically respectable ideals, even when they believe in them. Because of the sorts of institutions that schools are, we also must consider what is realistic at a very mundane level. If there are 200 students gathered together in one building (which can be a small number, even for an elementary school), for example, someone needs to worry about how they are all going to manage to eat lunch. The teacher is faced with the job of organizing students' inquiry and their engagement with mathematical ideas in such a way that it does not interfere with their getting to the lunch room on time, because someone is waiting there to serve the food. There are many such logistics that must be arranged when a large number of people are living and working together in a small space, and they almost always interfere with making engagement in the subject matter a top priority. Engagement in subject matter in a school setting is not impossible, except perhaps in some of our most shameful urban schools, but it is difficult to manage along with everything else. It does not simply "happen" because students are presented with interesting problems to work on. Another practical problem which any teacher will tell you about is that even the most enticing computer software or video program is not helpful when the machinery either

-39- does not work or is not available at the appropriate time. When it works, it is wonderful, and good teachers welcome technological tools that can make them better. But when there is a problem with the machinery, and no one to help, one becomes acutely aware of the difference between possibilities and realities. If we espouse ideals which do not take account of these realities, we do teachers a disservice, for it is they who will be blamed if our ideals are not put into practice. At the other end of the spectrum from such mundane issues are questions about what teachers need to know to engage children productively in mathematical activity (and the related, perhaps even more difficult question of how to get the people who know these things to work in schools. What would a teacher need to know in order to do on a regular basis the kind of ideal teaching I have described? There are a number of things, such as knowing: • how to get a large group of students in a small space interested and engaged in doing intellectually challenging work; • how to manage a rather complex set of interactions, both between teacher and students as a whole class and among students as they work together in problem-solving groups; • how students think about mathematical phenomena and knowing how to respond to that thinking in ways that are both supportive and challenging; • how to listen to students and how to organize the classroom so that students can express their thinking and listen to one another with respect; • where the mathematics teaching and learning processes are headed, not in the linear sense of one topic following another, but in the global sense of a network of big ideas and the relationships among those ideas and between ideas, facts, and procedures; • a variety of ways in which to represent big ideas to students, drawing on concrete, pictorial, verbal, and contextual as well as abstract modalities; • how to assess student understanding and being able to represent that assessment in terms that students, parents, and administrators can understand and accept. More probably could be added to this list, but the above are difficult requirements already. It seems reasonable to question whether it is realistic to expect that every mathematics teacher, from kindergarten through teacher preparation courses in college,

-40- could know how to do all of these things. And another reality we must acknowledge is that many teachers teach five different groups of children five different subjects for five hours a day five days a week. It requires an enormous amount of intellectual and physical energy to have them all be engaged in exciting mathematical dis- course, even once the students have been educated to take some respon- sibility for their own learning, which in itself is no simple job. I do this with fifth graders for one hour every day, and it is exhaust- ing. Those of you who teach graduate students--presumably the most talented and motivated learners--know how difficult it is to keep a good, focused, problem-oriented discussion going for a few hours a week. My list of what an ideal school mathematics teacher would need to know how to do is based in part on my own teaching and in part on a research project that I have been doing at the Educational Technology Center at Harvard University. In that project, I have been observing and interviewing a group of secondary school mathematics teachers in a wide variety of high school settings who have chosen to experiment with technology designed at the Educational Development Center at Newton, Massachusetts (a member of the Educational Technology Center consortium) to support the process of "guided discovery" in classrooms. The software used is "The Geometric Supposer," which enables students to explore geometrical relationships on computers inductively. Generally, the students work in pairs, meeting once or twice a week in a computer laboratory, and their findings are processed during regular class sessions. This setting certainly has aspects of the ideal in it. But the teachers who experimented with the Supposer were faced with a serious conflict. The culture of the schools in which they were working had not prepared either the teachers or their students to feel secure that, if the students followed their own intellectual road maps, they would learn what they are supposed to know. Yet the new technology was seductive--to students and teachers alike--and drew them along mathematically interesting paths that did not coin- cide with the routes through the subject defined by the textbook. Students went off on mathematically productive tangents that no one could track. As they made conjectures that they wanted to prove, teachers were barraged with questions. Even teachers went off on tangents as the Supposer captured them in interesting mathematical puzzles. This was both exhilarating and frustrating; the partici- pants enjoyed what was happening, but they were not sure what con- nection it might have with what they had come to know as learning high school geometry. They were faced, at a very practical level, with how to "guide" the inquiry process once students were engaged in it. The situation the teachers experienced was something like that of a tour operator in Paris whose bus had been replaced with a collec- tion of glitzy motor scooters in the middle of the Place de la Con- corde. Many of the "tourists" could see places they wanted to ex-

-41- plore, but did not know how to get to them. Others did not even know where to go or how to begin to make choices. All of them were com- plaining because their expectations for the tour were not being met, while at the same time they were anxious to jump on a scooter and take off. To make matters worse, the cameras that were to have recorded where everyone had been were left on the bus. The theme that dominated the teachers' thinking over the course of the year was how to make a situation like this into one that would fit into the culture of schooling without losing the excitement and engagement that the Supposer engendered. What an evaluator might want to look for in a classroom character- ized by such excitement and engagement, in order to judge the quality of the mathematics teaching and learning, would be quite different from what one would look for in a more conventional setting. Three basic questions--or clusters of questions--suggest themselves: • First, how does the teacher treat the "knowing" of mathematics? Is it all to be listened to, practiced, and remembered, or are there elements of what comes to be known that are arrived at through a process of reasoning about mathematical entities? Are students expected to make conjectures and then try to prove to their teacher and their classmates that what they have said makes sense? Are they given the linguistic, representational, and technological tools to enable them to construct such arguments? • Second, how does the teacher move around in the mathematical terrain that is appropriate to the grade level? Is the teacher able to respond to an individual student's ideas in a way that values and challenges them, and also to maintain the interest of the rest of the class and not stray too far from the teaching goal? Is the teacher able to build on the connections students invariably will make among the intricate web of ideas that constitute mathematics, or are learning opportunities missed by adhering too closely to a linear list of goals? Is the teacher able to present mathematics in a way that is coherent, but also responsive to the diversity of interests and skills that present themselves in any group of students? • Third, is the teacher able to make use of the tools available for teaching and to make those tools accessible to the students? Are curriculum materials being used to their fullest advantage? Are the calculators and computers that are available being used in ways that engage students in mathematical activity? Are connections being made among different concrete representations of a mathematical idea and between the manipulation of concrete representations and symbolic strategies for performing mathematical operations?

-42- Even If all of these questions are answered affirmatively, one will have encountered a mathematics teacher who has a substantial set of relatively novel problems to be considered every day because we are far from having ascertained how to make good mathematics teaching happen routinely. Therefore, rather than trying to decide which results of this kind of teaching we want to measure, I would like to propose an intermediate indicator. Given the state of our practical knowledge about mathematics teaching, it seems that the most mean- ingful indicators of ideal mathematics teaching among people who are trying to achieve it on a regular basis in the schools should be an assessment of whether teachers are grappling with the appropriate problems. Perhaps, in the future, we will have solved these problems and determined how to educate novices in order to put those solutions into practice. At that time, we will be able to develop a more conventional set of indicators. But, at the moment, an appropriate indicator of whether or not a mathematics teacher is moving in the direction of supporting the kind of mathematics learning that all of the reform documents embrace might be whether the following are among the problems he or she is facing: • managing the social structure of the classroom, including large group, teacher-led lessons, small group activities, and individual work on problems, in a way that supports inquiry and coherence; • keeping track of what content is taught and determining how to move through the appropriate mathematical terrain at a compatible pace; • assessing student learning in a way that takes into account both individual understanding and issues of equity, and negotiating the terms of assessment with students and other concerned parties. The mathematics teachers with whom I have worked who are better educated and more experienced than their colleagues and who would be regarded by all of us as good, if not ideal, mathematics teachers at all grade levels feel almost overwhelmed by these problems contin- ually. If they are lucky, they have someone nearby with whom they can commiserate. If they are not, they have a reformer who has never tried to do what they are doing who asks why they are experiencing difficulties.

TOWARD A NEW VISION OF LEARNING AND TEACHING* Albert Shanker American Federation of Teachers I would like to share some thoughts that are not yet the topics of national discussion and debate among teachers and other educators across the country. We have gone through a number of years of educational reforms; most of you have experienced them in your own states. My perception of these earlier reforms is that, if you look back to the period from 1976 to 1980, the teaching profession and public education were in considerable difficulties. Annual ratings of the public schools on the Gallup poll were going down quickly. Our support constituency of parents was becoming an ever smaller percentage of the voting popula- tion, dropping from 55 percent to 21 percent. The nation was facing a host of other problems. We went into a period of stagnant economy, so the public was not so generous as it once was, and the nation had a number of other pressing agendas, such as reindustrialization and rebuilding the infrastructure. Also, both Presidents Carter and Reagan agreed that America had disarmed unilaterally and that it had to have a substantial increase in military spending. Education, therefore, lost a good deal of political influence because of these many other agendas. There were also fights on tuition tax credits, where we lost in the House and won in the Senate. Even today, with people feeling a little better about education, the latest Gallup poll shows 41 percent of the public in favor of vouchers or tax credits for private and religious schools and 41 percent opposed. We have a public divided over public educa- tion and it may be giving public schools one last chance. Now, a new wrinkle on tuition tax deductions and credits has been added through state, rather than federal, action. Minnesota permits them and Iowa now has them, through legislation that was tacked on to a teacher salary bill to ensure more teacher support. *This text was taken from the transcript of a luncheon address that Mr. Shanker delivered during the conference reported in these proceedings. -43-

-44- Another new approach is reflected in the Thatcher educational reform bill in England. You may be sure it will be talked about here and will become a part of our political discussion. There are a lot of pieces to the Thatcher plan and one piece is this: if the parents of any of the children in any public school decide that they want to vote on whether or not they like the way the school is being run, the government has a procedure for it, very much like a collective bar- gaining action. If the majority of the parents say they do not like the way that a school is being run, that school can be removed from the authority of the local education agency, and the parents will be able to elect their own board of education for that school alone. The parents also will have the right under the national law to fire and hire any staff members they want. I make these preliminary remarks as a backdrop because, when the wave of education reforms came along, I did not like most of the things that the reformers said. They were quite mechanical; they were very regulatory; and they were all "top-down." I shall point out later some of the other reasons I did not like them. But I also did not like the extreme softness of America's schools in the late 1960's and 1970's, when many schools permitted students to decide whether or not they wanted to learn particular subjects. But I came very close to embracing these reform proposals and, publicly, I was very positive toward them, mainly because they gave us some political support which we desperately needed. We needed the support of the public, government, and industry. The way to get that support is not necessarily to agree with everything that is being said, but at least to listen and respond constructively. Representatives of government and industry were coming to us as friends of public education, saying that they wanted to help it. None of the reports issued at that time advocated tax credits or vouchers; they were calling for greater resources. The reports were not very perceptive and did not make clear-cut propos- als. How could they? They were written by people who did not work in the schools. It was a failure on our part that we did not go to the authors. If we had done so, and had had some good ideas, we most likely would have had a voice. But that is what happens when you do not take charge of your own profession. Someone else does--and they did. So I made some friendly statements, but not because the substance of the reform proposals was particularly good. Then what happened? We got fat legislative books promoting more rules and regulations, teacher testing, student testing, no automatic promotions, minimum competency tests at graduation, definition of the number of hours per subject--and many more things. These laws were based on the rather

-45- simple notion that, if you put together a few ingredients, you will make major differences and breakthroughs. One element is to have good teachers; I am certainly in favor of that. It also was dis- covered that many teachers were leaving the profession and that those coming into it might not be of the quality we wanted, unless the incentives were changed. Essentially, it was a package that mandated high standards, abolished automatic promotions, defined curricula, and called for a lot of homework and good and harder-working teach- ers --as if a shopping list could be put together and reform would take place. None of us would be against those things in principle, but we have to ask whether they are likely to be successful. First, we need to ascertain where we are. If we were educating most of the children in this country successfully--if, say, 80 to 85 percent were doing quite well and we had to find ways of improving the other 15 to 20 percent, then something like the above list might be a reasonable strategy. That is, we would be saying that the system is basically good; most of the products are fine; we have a little problem with the quality of 15 to 20 percent--a minor quality control problem--but, overall, the system seems to be working well. In fact, quality control was the obsession of the day. I can illustrate it best by a story about the Frenchman, the American, and the Japanese who were captured by a terrible enemy and told they have only an hour to live. They were given one last wish. The Frenchman said he would like to sing his national anthem, and he was told that he could have his wish. The Japanese said he would like to give his lecture on quality control for the last time, and he was granted his wish. Then the American said he wanted only one thing, and that was to be killed before the Japanese gave his lecture on quality control. But where are we really? Let us look for a few minutes at some of the results of the National Assessment of Educational Progress. Take the literacy assessment of 17-1/2-year-old youths who are pre- sumed successful. They have not dropped out of school; they are soon to graduate high school. We are not talking about the 20 percent who did drop out. The good news is that they can all read exit signs; in case of fire, they will know where to go. If they open up a package, they can read simple instructions; they can read simple books and some newspapers. But, when it comes to reading an editorial or a news analysis in The Washington Post or The New York Times, less than 40 percent of those "successful" 17-1/2-year-olds who are still in school could handle it. The hardest task of all in this category on the National Assessment tests was to manipulate about six pieces of information on a bus timetable. One might ask why that is necessary when a telephone call to the busline will provide such information. But we must think of this as the ability to do such things as open up the world almanac and understand a population or other chart or to interpret a spread sheet. I do not have to con- vince this audience that this is a worthwhile skill. But only 4.9 percent of the prospective graduates can read a simple table of that sort.

-46- Let us move on to writing. This is very important because it is not just mechanical; it involves persuasion, expression, and the organization of ideas. The writing task was not Shakespearian; it was to write a letter to a prospective employer, convincing him or her that you should get the job. It was designed to see if students could use critical thinking and apply rules of evidence to bolster arguments. If an employer is looking for someone reliable, it will impress him or her to be told that the applicant has held three jobs and has never missed a day's work, even coming in when sick. If the employer wants someone who can handle money, the student needs to be alert to point to experience such as working in a pharmacy and being the treasurer of the Boy Scouts troop. A letter that shows a little persuasion, a little ability to muster evidence, is required. Of these successful students who were tested, only 20 percent could write such a letter at the adequate level. Not to be able to understand a serious newspaper, not to be able to take a chart that contains a few numbers and words and make sense of it, not to be able to write a persuasive letter adequately is very alarming. Does anyone really believe that the system works? Remem- ber that these are not the 20 percent who drop out of school; they are the 20 percent of the 80 percent who are successful. Essential- ly, the conclusion that has to be drawn is that the system by and large is not working, that it is not a simple matter of improving it only for those who are either deviant, or who did not have good teach- ers , or who went to a school that was particularly ineffective. The failures are not the result of an abnormality; the failures are the ordinary. In terms of reasonable expectations of what we need in our society, of what we want to produce, success is the extraordinary. I would like to ask the authors of these reports if they really believe that if you put a teacher in the classroom who is a little better than the one there now, provide a textbook that is a little better, and do a little more homework, those changes will result in an improvement from 4.9 percent to 75 percent. Maybe, by tightening up a bit, an increase from 4.9 percent to 6 to 10 percent might be achieved, but, after that, it is unrealistic to project a jump from 4.9 percent to 75 percent because a better teacher is being put in the classroom or a better book is being used. Let me use another piece of evidence to show why the "tightening up" approach will not work. It will not work because we have already conducted the experiment. Did we ever have schools that had excel- lent teachers, a well-defined curriculum, no automatic promotions, family support, and a lot of homework? Of course, we did. I went to a school like that in New York City. That is what the schools were like in the 1930's and 1940's. We had wonderful teachers who could not work anywhere else. They earned four or five degrees during the Depression because there was nothing else to do. They took their examination in New York, Chicago, or elsewhere, and then waited eight years to be appointed to work as substitute teachers for $2 per day. They were brilliant people and the selection was such that, if you

-47- did not work out, you were not retained. There were no unions then, not in the sense that we have them today. What was the result? Was it good education? It was great; I am an example! But looking back to 1940 and 1941, for whom was it great? Twenty percent of the youth in this country graduated from high school in 1940. 1953-54 was the first year in which a majority of young people graduated from high school in this country. Isn't it interesting? In 1940, we were educating 20 percent of the children; now, we are still educating 20 percent, but we are keeping the rest a lot longer. Those who are staying longer are learning more; they are better off than those who left. But, that is where we are. Essentially, what this tells us is that we now have reached a point where we can develop a hypothesis. If you look at England and read that country's current material about education, you find that about 20 percent of British youth are succeeding. It is hard to get such figures from France. We have affiliates who meet with French officials several times a year. The French used to say to me, "There are absolutely no problems in our schools; no child leaves illit- erate." Then, under Mitterand a few years ago, an attempt was made to remove subsidies from parochial schools. About one million people came to a rally about the number of illiterates graduating from public schools. Then we started having honest discussions. It seems to me that there are only two hypotheses which will explain the failure to reach so many children over such long periods of time. One is that God or Darwin only makes 20 percent of us intelligent. Maybe there are another few percent we could reach if we tried a lot harder, but that would be all. I am not ready for this hypothesis and I doubt if you are. Then there is a second hypothesis. The results we are getting are not reflections of the students, nor are they reflections of a lack of effort by the people in the system. The results are the reflection of a very faulty process of production--in other words, of the way the schools are organized. The situation is comparable to going to an automobile manufacturer 15 years ago and asking why one million cars per year are produced, but 350,00 of them have to be recalled and rebuilt. We thought that was the result of mass pro- duction until the Japanese came along and showed us that, if you do your homework first and use a different system that involves the employees, and so forth, a system with radically different results in terms of successful productivity can be developed. Another way of thinking about this is that, for about two thou- sand years, people went to medical practitioners hoping to be cured; often, they were killed--killed by the fact that the medical profes- sion only recently discovered the practitioner should wash his hands and sterilize his instruments. That was not a result of viciousness or evil. It was Just ignorance and the standard operating procedure of the profession.

-48- If we do not accept the theory that only 20 percent are intel- ligent, then we have to ask ourselves: are there other things similar to the nonsterilization of instruments or the nonwashing of hands that we are still doing in the schools? Are there factors in the ways in which we handle children in school that, instead of helping them, hurt them? Are there things we do that lose the interest of a large number of students? Are there things that, instead of being functional, are radically dysfunctional? I want to spend a few minutes talking about something that is almost never talked about and which is, to me, at the core of why we need professionalism. I do not view professionalism as the business of creating a bureaucratic hierarchy which results in some people getting more money. Over the years, although we have talked a great deal about what constitutes good teaching, we may not have talked about it enough. We ought to talk more about it--about what consti- tutes good materials, good textbooks, good lessons, and good incen- tives. There is not very much thinking about learning and there is not a lot of very detailed thinking about what the schools do to build or destroy the self-esteem of children. Basically, if the child concludes he or she is no good or not intelligent and, ulti- mately, decides not to become involved, that is the end of it. Nothing can be done if a child decides to give up. So, keeping the flame alive in a student becomes an absolutely necessary condition for any sort of education. Let us look at a few things that happen routinely. Last fall, two million new children went to school and entered the first grade. In essence, we tell these children, "You're all six years old and you're in the first grade," and the children start making comparisons among themselves. In a sense, by the way we sort them and put them in classrooms, we are telling these children that they are supposed to be the same. Then they begin to look at each other, to see who is faster, stronger, smarter, and so forth. Now, are they all six years old? No, they are not, because the way that children enter school is through an arbitrary date. If a child's birthday is before a certain date, he or she goes to school this year. After a certain date, it will be next year. This means that we have taken a whole year's age range of children; the oldest child in the class is likely to be a year older than the youngest. A year makes a tremendous difference at the age of six. The difference is evidenced by the development of major skills such as the use of language and numbers, physical coordi- nation, and so forth. It is not like the difference between being 60 and 61. It is much more like the difference between 25 and 40--it is big! So what do we find? If we look later on to see which children have dropped out or which ones are not doing well, there is fairly substantial evidence to show that they are the ones who were younger at the beginning of school. The oldest person in the class feels stronger, smarter, and faster, even though all sorts of tests might indicate that this person is not so well endowed in many ways as others in the class. Of course, this is not an absolute rule. The

-49- youngest child who overcomes such a handicap is like anyone else who overcomes a handicap--stronger for it. The statistics to support this are quite persuasive. It is akin to putting a heavyweight boxer into a ring with a lightweight boxer and saying, "Go to it; you're both boxers." Now, I have never seen the younger children in a classroom tell the teacher that "The reason I can't answer this is because I'm a year younger than he is, so you shouldn't ask me the same question--or you should use a different marking system." Why do all of the students have to come to school on the same date? It is because that is when the teachers start talking. If some of them were to wait until they were a little older or more mature, until their birthdays caught up to the other birthdays, it would be like arriving in the middle or at the end of the movie, so we cannot do that. It has to be on one particular date. One of the questions we have to ask is: Can we build a learning environment that does not create unfair competition at an age when children usually cannot handle it? It was better in the old days when groups of children were admitted to school twice a year. Of course, that requires more paper work and movement, and that goes against administrative convenience and creates problems with keeping track of large numbers of children. If there were three or four semesters, it would be easier, but there are other ways of dealing with the problem. Obviously, each child cannot start school on his or her birthday if the teacher has started the program already. There would have to be a totally different organization. But think of how it is now! When children come in on the first day, they have the teacher at a disadvantage. They know the teacher's name, but the teacher does not know their names. Unless the teacher is very skilled, children can play a lot of games. Many teachers are driven out by that first day of school; they realize they cannot handle it. It would be quite different if part of the class were there and other children arrived on their birthdays, without knowing anyone. Then, there would be a different kind of atmosphere. Let us move on to the next problem. This problem is where I am the teacher and, after I have talked to the students and they have done some reading and other things, I call on them to answer ques- tions. Some of them always have their hands raised. School is great for these children; they would come to class on Christmas day. They are ahead of their classmates and they shine. Then there are a few who sometimes answer, and sometimes do not. And then there are the children who never raise their hands; they are constantly engaged in what is constitutionally prohibited--prayer in the schools! They are praying that I will not call on them. And, every time I do call on them, they either turn red and green or they guess wildly at the answer and everybody laughs at them. After they have not answered or have answered incorrectly a few times, the other students start making up names for them. What am I doing when I call on such children? I know what I think I am doing--getting pupil par- ticipation; getting their attention. But to call on a child once in

-50- the morning and once in the afternoon, for five days, in front of all of the other pupils, with the child never getting the right answer, would be called public humiliation by an impartial visitor from Mars. Fortunately, that has not happened too often to most of us; that is why we are here. But it has happened to most of us at some time and, when we think back to such an occasion--when we were called upon in front of people we respected and were unable to deliver the cor- rect answer--it was a very painful experience. For that to happen regularly--once, twice, three, and four times a week--is enough to convince a child that school is not for him or her. Children decide this very early. This probably is what happened at a very early age to a large part of that 60 to 80 percent who are unable to perform when they leave school. We worry about the dropouts who leave school, but we do not worry about the 80 percent who dropped out in their heads when they were very young and are just sitting in class, going through a ritual. That is the second item: Can we create a learning environment in which children are not humiliated publicly, in which learning during the early years is relatively private rather than exposed to the entire group? We will come back to that. The third item is that there is essentially a single method of teaching and learning throughout most of the western industrial societies and most of their schools. That is, the teacher talks and the children listen, or they read a textbook or write in a workbook. Those are the main ways of teaching and learning. There are cer- tainly exceptions, but those are still the main ways. If you cannot learn in those ways, you are unlucky. In the recent past, there were no other techniques, but, today, there are video tapes, audio tapes, and peer tutoring, although the latter is used very infrequently. There are many ways of teaching and learning, but we do not use them. A lot of people could not learn in these ways and could not endure school; they dropped out because they could not sit still for five or six hours a day, lis- tening to someone else talk. Or they are not sponges who absorb words that someone else is saying. Does that mean such people cannot learn? No; it only means that they cannot learn in that way. Basi- cally, what the results of the National Assessment of Educational Progress tell us is not that these children are stupid, and it does not tell us that their teachers are doing a poor job of organizing their material, or that their textbooks are no good. These results show that the number of people who can sit still and absorb words, either from the written page or from someone else's mouth, and respond by turning them into complex and meaningful pictures and patterns which help to organize experience, is very small. That is all that they tell us. They do not tell us that, if we used other methods, we would not reach these children. We have all heard of people who dropped out of school and later became millionaires or were elected to public office; they are very bright; they go to the

-51- theater; they read; and they are people you enjoy being with. You realize that you are in the presence of amazing people, but these were not people who could learn in a way that we typically teach them. A final example. Most children enter school in September, but the final grade is not given until the following June. Well, if this is September, and it is a nice day outside, and the final mark will not be given until June, a lot of us will decide not to do our home- work tonight. What we require of children is that their characters be so well organized that they will realize that every little thing they do not do each day ultimately will have a cumulative effect which will be disastrous. Now, that is something we would like to develop in people. However, suppose I were to give each of you your salaries for the year on the first day of school. How many of you would have any money left in June? It is the same characteristic that we are asking of children in terms of their investment in studying each day. Most teachers, if they were given their salary a year in advance, would go on strike. Even though they could invest it--if they had it, they could even make money on it--they do not want it because they do not trust themselves to have any money left when they need it. So what happens? Toward the end of October, some of these children who thought they had plenty of time discover that they are falling hopelessly behind. Now, every time they are called on, they are embarrassed. What do you do if you are hopelessly behind at the end of October? Do you sit there and become humiliated every day, or do you drop out? If you drop out, when can you drop back in again? Not until next September. And then you drop back in with students who are one year younger, after the school has told you that you should always be with students your own age. Is it possible to organize things differently? Let me share with you the story of my youngest son, Michael, who graduated from high school as an average student in an above-average school, so he felt he was no good and decided not to go to college. He went to work in a French restaurant, washing dishes. Six months later, he was taught how to make salads. Six months after that, he started making soups. About a year later, he came to me and said, "Dad, I've decided what I want to do with my life." I asked, "What's that?" and he replied, "I'd like to go into the CIA." I looked at him and he said, "No, it is not what you think. I want to go to the Culinary Institute of America. If I'm going to be in this business, I might as well be a chef." So I said, "All right, I'll get the application for you." He got in. I was very worried because I knew he thought it was a hands-on vocational school and I knew that one had to learn nutri- tional content, culinary French, contracts with vendors, hotel management, restaurant management, labor relations, and so forth. It is a very demanding program; it is not all cooking. Even the cooking is difficult. After he had been there for a week or so, I thought it was time to go by and have dinner with him, to encourage him to stay the course. So I called him and said, "Michael, I'm going to be driving by Hyde Park. Can we have dinner?" And he replied, "No, we can't, dad; I'm sorry." "Why not?" He answered, "I'm up past

-52- midnight, writing up my notes and studying." I said, "What do you mean, you've only been there a week and a half?" He said, "You don't understand; the semesters here are only three weeks long." Well, a semester three weeks long! He told me that, if students are 15 min- utes late for class, they are suspended for that three-week semester because they have already missed a major part of the course. The three-week semester concentrates the student's mind and it concen- trates the teacher's mind. No teacher will tell a joke that is more than 10 seconds long unless there is a real pedagogical purpose to it. While on this topic, let us consider the young people who are dropouts. A fellow meets a lovely girl. They drop out and run off to Ft. Lauderdale together and they know it is forever, but, a few weeks later, they break up. With a three-week semester, they could drop back in school every three weeks. Also, what happens if one fails? Think of the big debate we have had throughout our history. Do we promote students automatically, or do we hold them back? Both options are undesirable. To move a pupil ahead automatically when he or she is lost is a terrible thing. But research shows that to hold them back does not accomplish anything either. So, essentially, we have two answers, neither of which works. What do you do when you have two answers that do not work? You have to find a different way of thinking about a problem. My son did fail one course, but it was not devastating because it meant taking three weeks of one subject over again, which was not a big problem. He was not with students who were a year younger than he was. They were only three weeks younger. What this shows is that, by rethinking problems that are part of the daily school structure from the point of view of their impact on the students and on their feelings about themselves and their willing- ness to be engaged, solutions can be found for most problems. Often, the solutions are things like washing one's hands or sterilizing one's instruments. There is nothing deep about any of these sugges- tions. Maybe doing some of these things will not have miraculous results; we do not know. But they are new ways of thinking about these issues. The angle of approach is correct; that is, we ought to be thinking about the things we were unable to learn in school, no matter how hard we tried. And when we finally did learn them later, under what circumstances did we learn them, and how? What was the approach? By thinking back to such a point, can we understand why we were unable to learn? Why do we not spend enough time thinking about those experiences that acted as blocks to our own learning? We also have a problem with the organization of curriculum. John Dewey observed that one sees children looking very inquisitively at things that interest them, but that school manages to turn inter- esting things into subject matter. So I will share with you the story of that great educational philosopher, Father Guido Sarducci. He does a comic routine where he stands in front of an audience and says, "I have opened a college. How many of you don't have a

-53- baccalaureate degree?" Half the audience raises its hands. He says, "At my college, you can earn a baccalaureate degree in one day." They all laugh. They think the whole thing is a fraud, but he says, "No, this is real education. You're going to take the courses, and you're going to take the final examinations, and you'll do it all in one day. I want to tell you how I do it. I go out and find people who have been out of college for two or three years and I ask them, 'When did you graduate? Two or three years ago? Did you take Spanish when you were in college? What do you remember from your three years of Spanish?' and the person answers 'iComo esta usted? Muy bien.' 'Is that all you remember?' 'Yes.' All right, that is my curriculum for three years of Spanish. So I teach it to you; I give you the final examination; then we go on to American history." Why is that funny--and sad and true? It demonstrates that a lot of material is covered in school, but that we do not develop under- standing. We concentrate mostly on factual, short, and multiple choice answers. I was reflecting on some educational experiences that I had had outside school, and I wondered, "What would happen if a teacher in school were given the assignment to teach children about birds?" One would teach them either with a number of flashcards illustrated with birds or one would put a bird chart on the wall, showing pictures of many birds, placed in different categories such as mountain, plain, and water birds. Some might be categorized in terms of migration or parts of the country where they are found. When that was finished, the students would be given an examination and two things would happen. One, they would forget about birds a few weeks after that, a la Sarducci, and, two, they would probably end up hating birds. So I thought of a different experience I had had--through the Boy Scouts bird study merit badge. One actually has to go out and see 40 birds and keep a record of them. And what happens when you go to find 40 birds is that you realize you will not be able to see so many birds where there are only a few trees. You find you will have to get up at 5:00 a.m. and, at sunrise, you will have to be at a watering hole. You are afraid to do that alone, so you ask a few friends to go with you. Off you all go and, as soon as you start looking through the binoculars, something else becomes apparent. Seen through binoculars, birds do not look the way they look when they are in a museum. In nature, they have a red crest, a certain shape, a tail bobbing a certain way--something called field marks. Then you start looking through a standard handbook with your friends and you say, "There it is, that red crest." And your friends say, "No, it can't be; the book says they're native to Texas and we're in New York." So you keep looking. It may take several months before you find the 40 birds. But do you know what else happens? After several months of bird-watching, you are able to walk down the street and see a little movement in a tree which nobody else sees and you develop a sense of knowledge, of power, that becomes a part of you. I have not known anyone who learned about birds this way who hated birds, forgot about them, or stopped trying to learn about them.

-54- So that is another item we must consider because, if all we have is the meager Guido Sarducci curriculum, we might as well ignore it. Somehow, we must put curricula together, organize information, and present it so that learning is much more like studying birds in nature and it becomes a part of the children. Otherwise, we will continue with the present failure rates. I have a few more things to add. What might a school look like if we were to make one where children were not learning mainly from lectures or textbooks, if they were not all starting on the same day, if they did not have to sit still and be quiet, if they were not exposed to public humiliation, especially at the earlier grades, if there were privacy, if their time were not planned a year in ad- vance- -a school incorporating all of the items I have mentioned, as well as others? It probably would look far more like a Boy Scout troop. Think about it! Boy Scouts have hundreds of different pieces of curriculum. There is one scoutmaster and 40 to 60 young people. How does one scoutmaster handle the learning experiences of 40 to 60 children? One thing he cannot do is to give a lecture on how to tie knots. Only one or two people want to do knots that day. He teaches by saying to one child, "See if you can do this by using the hand- book." To a few others, he says, "Why don't you see if you can learn it from him?" Essentially, it is a school where the adults advise the youngsters. Such a learning environment could be replicated in the schools, using adjuncts such as computers, audio tapes, video tapes, computer instructions, and adults who can do some helping; there also would be cooperative learning. A variety of learning techniques could be used, with people working mainly in small groups, looking for answers together. This would not be a memorization or fact absorption process; a process of true discovery would be created, which is what involves children. Children know what is artificial. If such a learning environment were created, certain things would happen to teachers. There would be teams of adults, instead of a self-contained classroom. You could use volunteers to help. Volun- teers, under our present system, can only be a distraction. Children ask, "Who is that in the back of the room?" Or, worse than that, the volunteer might be a witness to something untoward that happens. So, the way things are organized now, teachers have difficulty incorpo- rating volunteers. However, in a Boy Scout troop with 40 to 60 youngsters, a parent or volunteer can be trained easily over a period of time to help at least one or two children. Those volunteers can be helpful and they are welcome, but not in the school system. According to Goodlad , teachers now spend about 80 percent of their time dispensing information. Here is one reason only 20 percent of our children are able to write a decent letter. Because they are not asked to write very much. Why aren't they asked to write very much? Because it takes a long time to mark a set of papers. And it is often impossible to do the next thing that should be done after marking the papers (which, by itself, does not do the child much good). This is to sit down with Johnny and ask, "What's

-55- this essay about?" "Well, it's about so-and-so." "What you just said is wonderful. I think it's better than the first sentence there. Look at it, do you agree with me?" Johnny says, "Yes." "Well, would you change it that way? Now Johnny, look at this sentence down here. If you read that to the class, do you think most of them would agree with you?" "No, they wouldn't." "Then why do you say a thing like that." "Because of these reasons...." "Then, why don't you put them in there?" "All right, I'll do that." The child has to be coached individually for a few minutes. If a writing exercise is done, someone reads it, then reviews it with the student, and the student redoes it over and over again, then eventually there will not be only 20 percent who can write. But how can that be done now? If there are 30 children per class, and 5 periods per day, that comes to 150 children. It takes 25 hours to mark a set of papers and coach the children if you allow even 5 minutes for marking and 5 for coaching. The solution seems to be smaller class size, so do we reduce the class size by half, from 30 to 15 students per class? Instead of America having 2.2 million teachers, there would be 4.4 million. And where do we get that 4.4 million if we are having difficulty replacing the 2.2 million with qualified people? Would 50 percent of our college graduating classes go into teaching, even with higher salaries? Would other sectors permit that or would they just outbid education? Reducing class size alone will not do it. As I have said, teachers need to be taken out of the self-contained class- room to work with colleagues and given the opportunity to work with children who need individual help. This cannot be done as an add-on; it can only be done by restructuring schools and the teaching profession. It cannot be done by making teachers work "harder" or through some additional programs. It is irrational to think that we can go from 4.9 percent to 75 percent by making only minor reforms. What we have is a structural problem. And what we need is a community of educators that gets away from many of the traditional questions and responses and starts to discuss the issues and the points of learning to which children respond, such as what involves them, what engages them, organiza- tional materials, what in the present system impedes learning, and so forth. The stakes are very high. State and local governments, politi- cians, and industry have put a lot of money into education in the past few years. The nation has been told there is a reform movement going on and it expects results. I am not optimistic about the success of most of these so-called reform measures. Nor do I much trust the commercial standardized tests that are being used to measure performance. I am much more optimistic about some of the more structural and fundamental reform experiments that are going on, but in all too few places. But they will not show "results" quickly, nor should they be measured in our silly, conventional ways. Sadly enough, a structural view, a reform of education from the point of view of how children learn, is not even popular with educators. And all but teachers seem caught up by the prevailing tests. Will we

-56- stop making the standard responses and start more real reforms before this reform movement is judged a failure? I leave you with this thought: We must remember that, if we do not work out our own reforms, we will find ourselves in a struggle to fight privatization of the schools. Privatization will not make education any better. Most private schools are like most public schools, only private schools are able to choose and reject their students. But public education would be gone and, with it, the kind of public visibility and, yes, accountability that lead to efforts to do better for all students. It would be very hard, if not impossible then, to reinvent public education. And that would be a tragedy for this democratic nation.

-57- Reference Goodlad, John I. A Place Called School. New York: McGraw- Hill, 1984.

PROFESSIONAL ASSESSMENT FOR PROFESSIONAL TEACHERS Rick Marks Stanford University In the past year, Lee Shulraan's research group at Stanford University developing prototypes of measures to assess teachers has done some preliminary work on new means of assessing teachers, based on a view of teaching as a professional activity. This paper describes that work in breadth and touches lightly on some of its implications. Our work acquires perspective when placed in a larger context. One of the tasks of the Carnegie Task Force on Teaching as a Profession was to establish a National Board for Professional Teaching Standards; among the charges of this newly formed board is one to develop standards and means for assessing teachers and for granting certification. Such an assessment probably will consist of a combination of written tests, assessment center exercises, documen- tation of supervised performance, and direct observation of teaching. At Stanford, we have designed and tested preliminary models of what some of the assessment center exercises, and the center itself, might look like. Our work entails a good deal of basic research and is intended both to increase the knowledge base for teaching and to extend the conception and the range of tools for assessing teachers. The most concrete products of our research will be a library of prototypes for the board to draw on as it continues to build a workable assessment procedure. We have designed 10 to 15 exercises in two domains: high school history (specifically, the U.S. Constitution and the formation of government) and elementary school mathematics (fractions, especially equivalence as taught in the fifth grade). Each exercise takes between 45 minutes and 3 hours and asks the candidate to perform in or talk or write about a situation that teachers would encounter in their work. In a real assessment center, a number of local candidates might spend two days in an unoccupied school, taking four or five exercises each day administered by other (probably board-certified) teachers. In the summer of 1987, many of these exercises were field-tested under similar conditions. In the fall, we are working on scoring techniques and other analyses of the field-test data. In the winter, we will begin to design exercise prototypes in two new areas--high school biology and elementary school literacy (reading and writing); to explore documentation and attestation procedures; and to develop instructional materials to enable candidates to prepare for the -59-

-60- assessment. We also will continue to study the impact of the assessment on minorities and ways in which to increase the number of qualified minority teachers. In the 10 mathematics exercises in the field test, candidates were asked to demonstrate a range of knowledge and skills. In one exercise, the candidate is given a standard textbook, a topic, and 30 minutes to work; he or she plans a day's lessons, then discusses the plan with an examiner, including such items as the main points, the significance of the material, examples that might be used, previous knowledge required by the students, and what they might find hard or easy about the lesson. In another exercise, the candidate is pre- sented with four vignettes of a tutoring situation and discusses ways of responding to each student's questions and difficulties. One exercise describes three algorithms commonly used with fractions and asks the candidate why each one works, what advantages or disadvan- tages it carries, and what alternative procedures might be useful. Another exercise stipulates that the candidate bring a lesson plan to the assessment center, where she or he discusses the plan with an examiner, teaches the lesson to a group of six students who have been coached for the exercise, then reflects on the lesson, and finally discusses what happened with the examiner. Additional exercises include such tasks as critiquing a videotaped portion of a real mathe- matics class; discussing instructional uses of a standard mathematics manipulative, a computer program, and a collection of everyday objects with potential for mathematics applications; describing alternative classroom routines; analyzing and critiquing a textbook; and others relevant to the teaching of elementary mathematics. The design of these exercises is based on a variety of sources. Within the established knowledge base, we have relied on Stanford University's Knowledge Growth in Teaching studies of the past four years, on our own project's Wisdom of Practice studies of elementary teachers expert at teaching mathematics, and on other research in teacher knowledge--in mathematics, Leinhardt's and others' work at the Learning Research and Development Center at the University of Pittsburgh, and Romberg's and Carpenter's work at the University of Wisconsin-Madison, for example. We have commissioned several papers, subcontracted with other institutions, and held two conferences--on assessment technologies and minority impact--to inform ourselves in particular areas. We also formed expert panels in each content domain and a local teacher advisory board, all of which met regu- larly, participated in our research, and gave us continuous guidance. Our vision of the professional teacher for whom this assessment is designed corresponds to the view expressed by the Mathematical Sciences Education Board in its Curriculum Framework. The profes-

-61- sional teacher is able to learn new subject matter and new methods for teaching old subject matter. This teacher is a complex reasoner, problem-solver, and decision-maker, not just an actor or actress; he or she not only performs, but also gives a thoughtful, justifiable performance. The ideal teacher represents and adapts academic content in ways that are understandable and interesting for the students. This vision requires the teacher to possess the usual knowledge--of content, of general pedagogy, of children, of con- texts --plus knowledge of how to synthesize and adapt them under continually changing circumstances, plus the ability to rationalize her or his own actions and judgment. Of course, this vision of the teacher implicit in our assessment exercises holds certain implications for the institutions and programs that prepare candidates for teaching. Two of these impli- cations are very general, but fundamentally important. First, the teacher's special way of knowing content for instruction implies a more intimate connection between subject matter and pedagogy and, by extension, between disciplinary departments and schools of education. Second, this teacher is not to be trained in effective techniques only, but also must be educated for understanding and provided with significant opportunities to synthesize and apply this understanding. There is little doubt that teacher education institu- tions will adapt their programs to help prepare their students for board certification. This shift carries great potential for improv- ing teaching as well as considerable responsibilities. The board must design an assessment in such a way that preparing to pass it will, in fact, make a candidate a better teacher. Teacher education programs must respect the spirit and not just the letter of reform by aiming to educate teachers rather than simply coaching them to pass the test. Educational institutions in general will have to find ways in which to enable increased numbers of minority candidates to meet higher standards in the teaching profession. In summary, the board's assessment and certification processes will be worthwhile only to the extent that they encourage individual teachers, the teaching force as a whole, teacher educators, and their institutions to achieve higher professional standards.

SUPPORTING THE CHANGES

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