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Redesign from a Research Perspective Recent international come parative studies indicate that some of our basic assumptions about the structure and goals of schooling, about children's abilities, and about the structure of curricula are determined more by tracli- tion than by fundamental educational principles (Cross- white et al,, 19861. For exam- ple: Mothers of American chilciren are far more likely than are mothers of Japanese and Taiwanese children to believe that innate ability underlies chilciren's success in mathe- matics (Stevenson et al., 1 9861. . The curricula of other countries reflect very different beliefs about what children are capable of learning. American -textbooks tend to develop ideas very slowly by progressing through a hierarchy of small, straightforward learning tasks. Texts from Asian countries and from the Soviet Union immerse students in much more demanding problem situations from the beginning (Fuson et al., 1988~. · Mathematics classrooms in Japan use instructional time in quite different ways than American schools. For example, group work and cooperative problem solving are stressed throughout the earliest grades (Easiey and Easley, 1982; Enloe and Lewin, 19871.

28 Reshaping School Mathematics These comparisons underscore that schooling is a reflection of societal values, that we must think of "curriculum" in societal as well as in classroom terms. The emphasis on mathematics in Chinese and Japanese classrooms is a reflection of the impor- tance that the Chinese and Japanese parents ascribe to mathematics learning. The findings of international comparisons are confirmed by National Assessments of Eclucational Progress (Dossey et al., 1988~. Ail studies consistently show disappointing levels of sTu- dent performance in areas of mathematical power, uncler- stancling, and relevant applications. Performance is most dis- appointing in understanding and problem solving, the very aspects of mathematics most important for working and living in a technology-intensive society. Learning Mathematics These difficulties can be overcome only through new modes of teaching (Resnick, 1987~. By synthesizing a broad range of observational research, Romberg and Carpenter (1986) docu- ment a curricular tradition in mathematics built on a massive record of knowlecige divorced from science and other disci- plines. Mathematics education, unlike mathematics itself, is separated into subjects and subdivided into topics, stuclies, lessons, facts, and skills. "This fragmentation of mathematics has divorced the subject from reality and from inquiry. Such essential characteristics of mathematics as abstracting, invent- ing, proving, and applying are open lost." Research on teaching for higher-orcler thinking lends sup- port to the notion that instruction needs to change from the traditional mode where the teacher presents material to a less structured, more indirect style of teaching (Dessart and Suy- clam, 1983; Grouws et al., 1988; Peterson, 1988; Peterson and Carpenter, 1989), Because the development of higher-level thinking in mathematics depends on autonomous, indepen- clent learning behavior, teachers (and parents) must learn how to encourage more self-reliance in students who are learning mathematics. Explicit teaching of cognitive and metacognitive strategies can enhance students' learning (Schoenfeld, 1987), as can small group cooperative learning (Shavelson et al., 1988; Peterson and Carpenter, 1989), Cognitive research in other content areas (Campione et al., 1988) shows the effectiveness of reciprocal teaching in which children take turns playing teacher, posing questions, summarizing, clarifying, and pre- dicting. Reciprocal teaching is based on the premise that the

29 A Philosophy and Framework opportunity to construct meanings communally internalizes the process and gives these constructions permanence (Resnick, 1988), Constructed Knowledge There is now wide agreement among researchers (Resnick, 1983; Linn, 1986) of the need to pay careful attention to stu- clent-constructed knowledge (Piaget, 1954~. For example, Resnick (1976), Carpenter et al. (1982), and Steffe et al. (1983) have shown that when aciding two numbers, say 3 + 6, stu- dents generally invent the method of counting on (that is, "7, 8, 9"~. Children come to school with a rich body of knowlecige about the world around them, including well-developecl infor- mal systems of mathematics (Ginsburg, 19771. Education fails when children are treated as "blank slates" or "empty jugs," ignoring the fact that they have a great deal of mathematical knowledge, some of which surpasses-and some of which may contradict-what they are being taught in school (Erl- wanger, 1974; Clement, 1977, Gelman and Gallistal, 1978; Ginsburg, 1983~. Children are active interpreters of the world around them, including the mathematical aspects of that world. In the words of Piaget (1948), "To understand is to invent." Topics in school should be arranged to exploit intuitions and infor- mal numerical notions that students bring with them to school. Moreover, teaching methocis must adapt to the notion of the child as interpreter and constructor of (possibly wrong) theories as opposed to the child as absorber. Thinking Visually Procedural Knowledge The recommendation to reduce emphasis on procedures used for paper-and-pencil calcu Here is a picture of a roller-coaster track: Sketch a graph to show the speed of the roller coaster versus its position on the track. A _ ~ an, . XL~ . l X~< 1~< , '< >< . ~ X >< X X ~ X X ~ X X X ~

30 Reshaping School Mathematics rations rests in part on extensive evidence that these proce- dures, in themselves, do not aid conceptual understanding (Mestre, 1987), The literature on arithmetic "bugs" (Brown and Burton, 1978, Maurer, 1987) reveals that many mistakes stu- dents make follow predictable patterns. Such consistent but mistaken procedures learning bugs-have a natural origin in the invention of the student: they are intelligent attempts to modify memorized procedures that ore poorly understood. Even when correctly learned, purely procedural knowl- edge-the ability to implement mathematical algorithms with- out underlying conceptual knowledge~can be extremely fragile. Clement et al. (1979) have shown that even a solid procedural knowledge of algebra, such as is held by universi- ty-level engineering students, does not in most cases (over 80%) imply an ability to interpret the meaning of algebraic symbols, Several recent studies show that mathematical learn- ing is more robust when taught in a fashion that stresses under- lying conceptual models (e.g., Carpenter et al., 1982; Davis, 1984; Hiebert, 1986; Romberg and Carpenter, 1986~. Mastery of Subject Matter Mastery of subject matter has for years been the predomi- nant focus of mathematics education research. Yet researchers have only just begun to construct a detailed map of the phases children can go through as they build under- standing of arithmetic (Steffe et al,, 19831. Even at early ages, the picture is quite complex, Contrary to much present prac- tice, it is generally most effective to engage students in mean- ingful, complex activities focusing on conceptual issues rather than to establish all building blocks at one level before going on to the next level (Hatano, 1982; Romberg and Carpenter, 1986; Collins et al., 1989~. In certain cases, the order of presentation appears critical. Wearne and Hiebert (1988) showed that students who learn to calculate too early may find it more difficult to reach an understanding than students who have had no such experi- ence. On the other hand, Steffe et al, (1983) showed that chil- dren must be able to recite the number words in order before they can develop a concept of counting or number. There is some evidence to suggest that paper and pencil calculation involving fractions, decimal long division, and pos- sibly multiplication are introduced far too soon in the present curriculum. Under currently prevalent teaching practice, a very high percentage of high school students worldwide never

31 A Philosophy and Framework masters these topics just what one would expect in a case where routinized skills are blocking semantic learning (e.g., Benezet, 1935~. The challenge for curriculum development (and research) is to determine when routinized rules should come first and when they should not, as well as to investigate newer whole-language strategies for teaching that may be more effective than traditional methods. This is an area where far more research needs to be done. Problem Solving Problem solving is a central focus of the mathematics cur- riculum. There is by now an extensive body of literature (Krulik, 1980; Mason et al., 1982; Schoenfeld, 1985; Silver, 1985; Charles and Silver, 1988; Nodclings, 1988) indicating that strategies for problem solving can be taught effectively. The main warning from the research literature is that one should be careful not to trivialize problem-solving strategies, teaching a collection of isolated tricks (e.g., "of" means multiply, or cross-cancelling factors). Problem-solving strategies, in the spirit of Polya (1945), are subtle and complex. Important strategies such as "look for a pattern by plugging in values for n = 1, 2, 3, 4,, . ." cannot be taught effectively apart from the situational clues that indicate when such strategies are appropriate. For a nice collection of "starter" problems and a discussion of mathematical thinking, see Mason et al. (19821. For a discussion of problem posing (as opposed to mere problem solving), see Brown and Walter (1 983~. An effective approach to solving problems is provided by metacognition, the self-conscious ability to know when and why to use a procedure. There is ample evidence (Schoen- feld, 1985; Silver, 1985; Campione et al!, 1988; Collins et al., 1989) that students who know more than enough subject mat- ter often fail to solve problems because they do not use their knowlecig0 wisely. They may jump info problems, doggedly pursuing a particular ill-chosen approach to the exclusion of anything else; they may raise profitable alternatives, but fail to pursue them; they may get side-trackecl into focusing on trivia while ignoring the "big picture." Research indicates that such "executive" skills can be learnecO, resulting in significant improvements in problem- solving performance. Effects can be obtained with interven- tions as simple as holding class discussions that focus on exec- utive behaviors, and by explicitly and frequently posing

32 Reshaping School Mathematics questions such as: "What are you doing?" "Why are you doing it?" "How will it help you?" Making Sense of Mathematics The current curriculum fails badly in teaching genuine appli- cations of mathematics (Dossey et al,, 1988~. In a report of the Thircl National Assessment of Educational Progress (Carpenter et al., 1983), nearly 30% of children reported the answer of "31 remainder 12" to the problem: An Army bus holds 36 solcliers. If 1 128 soldiers are being bused to their training site, how many buses are neecled? Fewer than one in four gave the correct answer to the prob- lem. Approximately 70 percent of the students who took the examination performed the right operation (1 128 divided by 36 yields "31 remainder 12"), However, fewer than one-third of those students checked "32 buses" as their answer. Many stu- dents extract from their school experience the general view that the result of an algorithm (e.g., "31 remainder 12") is the correct and complete answer Very little in their experience would suggest the need to interpret the result of a mathemati- cal procedure. For most stuclents, school mathematics is a habit of prob- lem-solving without sense-making: one learns to read the problem, to extract the relevant numbers and the operation to be used, to perform the operation, and to write down the result-without ever thinking about what it all means. Reusser (1986) reports that three school children in four will produce a numerical answer to nonsense problems such as: There are 125 sheep and 5 dogs in a flock. How old is the shepherd? Typical responses, produced by a student solving the prob- lem out loud, exemplify the mindless world of school mathe- matics: 125 + 5 = 130 . . . this is too big, and 125 - 5 = 120 is still too big . . . while 125/5 = 25. That works. I think the shepherd is 25 years old. Students constantly strive to make sense of the rules that govern the world around them, including the world of their mathematics classrooms. If the classroom patterns are per

33 A Philosophy and Framework ceived to be arbitrary and the mathematical operations meaningless-no matter how well "masterecl" as proce- dures-students will emerge from the classroom with a sense of mathematics as being arbitrary, useless, and meaningless. The classroom culture in which students learn mathematics shapes their understanding of the nature of mathematics as well as the ways they will use the mathematics they have learned. Many studies (Fawcett, 1938; Bruner, 1964; Mason et al., 1982; Burton, 1984; Love et al., 1988; Schoenfelcl, 1988b) indicate that it is indeed possible to create classroom environ- ments that are, in essence, cultures of sense-making-cultures from which students emerge with an understanding of mathe- matics as a discipline that helps to make sense of things. The goal of teaching sense-making via mathematics should be a central concern of all curricular reform. To "see" abstract ideas such as limits, it is helpful to use pictures. For example, the sum 1/2+ 1/4 + 1/8 + 1/16 + 1/32+ . . . can tee seen as totaling 1 either by cone-dimentionalline: 1 1 1 1 1 11 1/4 1/8 1/16 1/2 or by a two-dimensional square: 1/8 1/2 . = 32 _ 1/16 1/4 Proofs Without Words