Research Basis of EM - page 2

Download PDF

Page: 1 | 2 | 3 | 4

Interviews conducted by Bell and Bell (1988) confirmed the relationships between developing mathematical capabilities and social situations. Questions that seemed beyond primary children’s capabilities were often solved during the interview. Children would say, “I didn’t know I could do that!” The availability of tools further extended what a child could explain, and a simple rephrasing of a question often made it quite understandable. For example, children who could not make sense of the symbol “1/2” or of the expression “12 divided by 3” could easily respond correctly to the request, “Please give me half of these blocks” or “Share these blocks among you, me, and my friend [a doll].” While learning and understanding are sometimes individual activities, they are often social activities, greatly influenced by the situation, the language used, and the materials available.

The Johnson brothers also began to publish their pioneering work on cooperative learning during the 1970s (Johnson & Johnson, 1974, 1978).

The Underachieving Curriculum

In the early 1980s, the UCSMP Resource Development Component began studying mathematics education in the Soviet Union, Japan, China, and other high-achieving countries (Wirszup & Streit, 1987, 1990, 1992). Wirszup found that other nations were much more ambitious in the scope and sequence of mathematics covered. Even in arithmetic, textbooks in other countries presented topics earlier, had a consistent pattern of spaced practice with mixed operations, and included both more types of word problems and more challenging problems than U.S. textbooks (Stigler, Fuson, Ham, & Kim, 1986; Fuson, Stigler, & Bartsch, 1988). For example, although kindergarten and first grade children had notions of doubles and other multiples, a sure grasp of the demands of equal sharing, and a clear understanding of “half of,” multiplication and division were not in the U.S. curriculum until late in second or third grade, and then primarily as rote memorization of the simplest facts. Children also had substantial capabilities from their everyday experience with decimals (money), numbers less than zero (winter temperatures), measurement, and geometry. In teaching experiments by UCSMP researchers, children showed readiness for algebra, functions, and data analysis, but all these topics were deferred to later grades or given scant attention in U.S. textbooks of the 1980s. Not surprisingly, in international studies, U.S. students consistently ranked near the bottom in comparisons with their peers in other industrialized nations. (Stevenson, Lee, & Stigler, 1986; McKnight et al., 1987).

Classroom observers found that teaching practices in the higher-achieving nations differ greatly from those in the U.S. For example, researchers found that Japanese elementary teachers employ more child-centered, problem-solving approaches to instruction in mathematics (Stevenson & Stigler, 1992; Stigler & Perry, 1988). Problems are posed in realistic contexts, and students find their own solution methods. To support these explorations, each Japanese student has a tool kit of manipulatives. Following an exploratory lesson segment, the Japanese teacher asks students to explain their reasoning and multiple solutions. This pattern– problem posing, exploration with manipulatives, and discussion of multiple solutions– fits very well with what we now know about how children learn.

The use of mathematical modeling, aided by increasingly powerful computers, had transformed research and practice in many areas. Important decisions in work and daily life required greater knowledge of mathematics, as well as greater problem–solving and reasoning skills–but results from the second NAEP (Carpenter et al., 1981) showed that most U.S. students completed school without basic problem-solving and reasoning skills and with little appreciation of the utility of mathematics. Educators, leaders of industry, and governmental agencies realized that the U.S. was failing to produce citizens competent in the mathematics that would be needed to compete in the twenty–first century (Education Commission of the States Task Force on Education for Economic Growth, 1983; National Commission on Excellence in Education, 1983).

Mathematical Modeling and Problem Solving

Investigations in problem solving (Polya, 1948, 1962; Lesh, Post, & Behr, 1987; Schoenfeld, 1987; Janvier, 1987) showed that an important step in solving a problem is choosing a model or representation for the problem situation. Researchers and theorists stressed the importance of natural language, concrete models, physical or mental visual images (including pictures, graphs, and diagrams), and symbols in representing mathematical ideas (Bruner, 1964a, 1964b; Lesh, Post, & Behr, 1987; Silver, 1987; Hiebert, 1988). Facility with multiple representations, especially the ability to translate among representations, was found to be important in problem solving.

Researchers also noted that the symbolic manipulations that students carry out in school are often disconnected from reality and common sense (Hiebert, 1984, 1988; Baroody & Ginsburg, 1986; Van Lehn, 1986; Silver, 1986; Resnick, 1987b; Kaput, 1987a, 1987b; Romberg & Tufte, 1987). As a result, students produce nonsense but don’t realize it, as Van Lehn and others have shown in their analyses of arithmetic errors. Research also showed, however, that if symbolism is closely related to actions and referents that are familiar to young students, then they are able to deal effectively with it (Hiebert, 1984, 1988; Carpenter, Fennema, & Franke, 1992).

Calls for increased tool use in schools were common before 1990. Both research findings (Suydam, 1984, 1986) and theoretical considerations (Bruner, 1964a, 1964b; Hiebert, 1984, 1988; Lesh, Post, & Behr, 1987; Resnick, 1987b) supported increased use of tools (a.k.a. manipulatives) in school. One particular tool coming into use during this period was the hand-held calculator. Bell (1976) recognized that calculators should play a role in curriculum and learning. Initial research (Hembree & Dessart,1986, 1992; Suydam, 1982, 1985, 1987), since confirmed (Smith, 1997), found that calculators can be valuable tools in school mathematics.

Applications and a Broader Curriculum

Interest in using applications in school mathematics increased during the 1970s and 1980s (Sharron, 1979). Bell (1972) made it clear that people’s ordinary lives provide a rich source of brief but interesting problems for school arithmetic. Usiskin and Bell (1983) proposed a scheme for categorizing the uses of numbers and operations with numbers, so that the actual uses of numbers could easily be included in an organized way in school mathematics programs.

Bell (1974) outlined content for a new and ambitious mathematics curriculum. In contrast to traditional K-6 textbook programs, the proposed curriculum framework included investigations in measurement, geometry, algebra, and statistics, as well as in arithmetic. Bell’s ideas were taken up in a series of authoritative reports on the content of school mathematics (NCSM 1977, 1988; Pollak, 1983; NCTM, 1989).

Pacing and Practice

While research in reading showed that students achieved best when topics were presented at a brisk pace (Barr, Dreeben, & Wiratchai, 1983), most mathematics texts of the 1970s and 1980s moved quite slowly. An investigation by UCSMP of U.S. mathematics textbooks found that from first through eighth grade, more than half of each year’s program was typically devoted to a review of topics from previous years (Flanders, 1987). In those textbooks, a topic was typically introduced and practiced for several weeks and then largely ignored until the following year, when it was reviewed, practiced, and perhaps slightly extended. This cycle of annual repetition with little substantive development was severely criticized by researchers who studied U.S. and foreign textbooks (McKnight et al., 1987; Schmidt, McKnight, & Raizen, 1997). Texts that were essentially medleys of disconnected topics arranged in a flat ”spiral” were identified as a prime reason for U.S. students’ poor performance on international tests ¹.

Besides a brisk pace, research findings from before 1990 supported continuous review and distributed practice. Practice has long been recognized as essential if children are to retain what they learn (Brownell, 1935, 1956; Brownell & Chazal, 1935; Rathmell, 1978; Chase & Chi, 1981; Cook & Dossey, 1982; Coburn, 1989). The positive effects of “spaced” rather than “massed” practice were recognized as early as 1885 when the German psychologist Hermann Ebbinghaus published his seminal work on memory. Over the past century, Ebbinghaus’s findings have been repeatedly confirmed and extended (Caple, 1996). Research about the role of distributed, or spaced, practice in the learning of mathematics was summarized in Suydam’s 1985 ERIC digest (ED 260891): “Long-term retention is best served if assignments on a particular skill are spread out in time rather than concentrated within a short interval.” Transfer of a skill or concept is also more likely to occur when it is practiced in a variety of contexts and situations (Anderson, Reder, & Simon, 1996).

Page: 1 | 2 | 3 | 4