3
Mathematics

Debates about mathematics instruction have long focused on the relative importance of developing fluency with mathematical procedures and developing the ability to reason mathematically. Few on either side of the debate would disagree that both are necessary for competence in mathematics. There is disagreement, however, on the relative weight and share of instructional time to be given to each and on the approach to instruction that best supports mathematical competence.

Investment in recent decades by federal agencies and private foundations has produced a wealth of knowledge on the development of mathematical understanding and numerous curricula that incorporate that knowledge. As a result, elementary mathematics is ripe for investment in rigorous, independent evaluation to compare the outcomes of alternative approaches to teaching mathematics across a range of students, teachers, and contexts and making that knowledge usable and used widely by schools.

ELEMENTARY MATHEMATICS

STUDENT LEARNING

The Destination: What Do We Want Children to Know or Be Able to Do?

U.S. students fare poorly in international comparisons of mathematics achievement. They show weak understanding of basic mathematical concepts, and although they can perform straightforward computational procedures, they are notably



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Learning and Instruction: A SERP Research Agenda 3 Mathematics Debates about mathematics instruction have long focused on the relative importance of developing fluency with mathematical procedures and developing the ability to reason mathematically. Few on either side of the debate would disagree that both are necessary for competence in mathematics. There is disagreement, however, on the relative weight and share of instructional time to be given to each and on the approach to instruction that best supports mathematical competence. Investment in recent decades by federal agencies and private foundations has produced a wealth of knowledge on the development of mathematical understanding and numerous curricula that incorporate that knowledge. As a result, elementary mathematics is ripe for investment in rigorous, independent evaluation to compare the outcomes of alternative approaches to teaching mathematics across a range of students, teachers, and contexts and making that knowledge usable and used widely by schools. ELEMENTARY MATHEMATICS STUDENT LEARNING The Destination: What Do We Want Children to Know or Be Able to Do? U.S. students fare poorly in international comparisons of mathematics achievement. They show weak understanding of basic mathematical concepts, and although they can perform straightforward computational procedures, they are notably

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Learning and Instruction: A SERP Research Agenda weak in applying mathematical skills to solve even simple problems (National Research Council, 2001c). These results have generally been attributed to the shallow and diffuse treatment of topics in elementary mathematics relative to other countries (National Research Council, 2001c). The panel had the benefit of drawing on a recent synthesis of research on elementary mathematics (National Research Council, 2001c) and on the work of a RAND study group that produced a mathematics research agenda (RAND, 2002b). The National Research Council report presents a view of what elementary schoolchildren should know and be able to do in mathematics that draws on a solid research base in cognitive psychology and mathematics education. It includes mastery of procedures as a critical element of mathematics competence, but it places far more emphasis on conditional knowledge: understanding when and how to apply those procedures than is common in mathematics classrooms today. Conditional knowledge is rooted in a deeper understanding of mathematical concepts and a facility with mathematical reasoning. The NRC committee summarized its view in five intertwining strands that constitute mathematical proficiency (National Research Council, 2001c:5): Conceptual understanding: comprehension of mathematical concepts, operations, and relations; Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; Strategic competence: ability to formulate, represent, and solve mathematical problems; Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification; Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. A well-articulated portrait of mathematical proficiency is an important first step; it provides a well-defined goal for mathematics instruction. But important questions remain regarding the allocation of time and attention to the separate strands, as well as the approach to instruction that best supports the proficiency goal.

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Learning and Instruction: A SERP Research Agenda The Route: Progression of Understanding Research has uncovered an awareness of number in infants shortly after birth. The ability to represent number and the development of informal strategies to solve number problems develop in children over time. Many studies have explored how preschoolers and children in the early elementary grades understand basic number concepts and begin operating with number informally well before formal instruction begins (Carey, 2001; Gelman, 1990; Gelman and Gallistel, 1978). Children’s understanding progresses from a global notion of a little or a lot to the ability to perform mental calculations with specific quantities (Griffin and Case, 1997; Gelman, 1967). Initially the quantities children can work with are small, and their methods are intuitive and concrete. In the early elementary grades, they proceed to methods that are more general (less problem dependent) and more abstract. Children display this progression from concrete to abstract in operations first with single-digit numbers, then with multidigit numbers. Importantly, the extent and the pace of development depend on experiences that support and extend the emerging abilities. Researchers have identified two issues in early mathematics learning that pose considerable challenges for instruction: Differences in children’s experiences result in some children—primarily those from disadvantaged backgrounds—entering kindergarten as much as two years behind their peers in the development of number concepts (Griffin and Case, 1997). Children’s informal mathematical reasoning and emergent strategy development can serve as a powerful foundation for mathematics instruction. However, instruction that does not explore, build on, or connect with children’s informal reasoning processes and approaches can have undesirable consequences. Children can learn to use more formal algorithms, but may apply them rigidly and sometimes inappropriately (see Boxes 3.1 and 3.2). Mathematical proficiency is lost because procedural fluency is divorced from the mastery of concepts and mathematical reasoning that give the procedures power.

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Learning and Instruction: A SERP Research Agenda BOX 3.1 Buggy Algorithms When students attempt to apply conventional algorithms without conceptually grasping why and how the algorithm works, “bugs” are sometimes introduced. For example, teachers have long wrestled with the frequent difficulties that second and third graders have with multidigit subtraction in problems such as: A common error is: The subtraction procedure above is a classic case: Children subtract “up” when subtracting “down”—tried first—is not possible. Here, students would try to subtract 4 from 1 and, seeing that they could not do this, would subtract 1 from 4 instead. These “buggy algorithms” are often both resilient and persistent. Consider how reasonable the above procedure is: in addition problems that look similar, children can add up or down and get a correct result either way: Bugs often remain undetected when teachers do not see the highly regular pattern in students’ errors, responding to them more as though they were random miscalculations. BOX 3.2 Rigid Application of Algorithms Many examples can be cited in which students attempt to plug numbers into algorithms without thinking about their meaning, a phenomenon that stretches through all grades of schooling and all mathematical subjects. Even when students are capable of solving a problem correctly informally, they are found to produce incorrect answers when they use formal algorithms. In studies by Lochhead and Mestre (1988), for example, college students who were told that there are 6 times as many students as professors, and there are 10 professors, could correctly give the number of students. But when students are asked to write the formula to represent that situation, the majority write 6S = P. The formula seems correct to students even though the solution would yield 6 times as many professors as students. The occurrence of the word 6 near the word students is sufficient to lead to a formal representation of the problem that is at odds with their informal knowledge.

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Learning and Instruction: A SERP Research Agenda The Vehicle: Curriculum Development Past investments in R&D have produced curricular interventions to address each of the two problems raised above. With respect to the first, several curricula have been developed that introduce children to whole-number mathematics, with particular attention to the needs of young children who have had little preparation outside school. The most extensively researched of these is the Number Worlds curriculum, which has been tested in more than 20 matched, controlled trials. The results suggest that well-planned activities designed to put each step required in mastering the concept of quantity securely in place can allow disadvantaged students to catch up to their more advantaged peers right at the start of formal schooling (see Box 3.3). The curriculum has a companion assessment tool (the Number Knowledge Test) to help the teacher monitor and guide instruction. If results in controlled trials could be attained in schools across the country that serve disadvantaged populations, this would represent a major success with respect to narrowing the achievement gap, a long-standing national goal that has proven difficult to realize. Number Worlds is not the only curriculum designed to achieve this end. Others include Big Math for Little Kids (Ginsburg and Greenes, 2003) and Children’s Math Worlds (Fuson, 2003). While research to compare these curricula on a variety of dimensions is in order, it is clear that the tools to better prepare disadvantaged children for mathematics are now available. With respect to the second concern—building children’s mathematical reasoning ability—controversy persists. While there is evidence that procedural knowledge without conceptual understanding leads to poor mathematical reasoning, it is also well documented that procedural knowledge is a critical element of mathematical competence (National Reasearch Council, 2001a; Haverty, 1999). Without adequate procedural knowledge, not only are children unable to engage in more challenging problem solving, but also, they are unable to engage in basic everyday transactions, like making change. The goal, then, must be one of strengthening mathematical reasoning without sacrificing procedural knowledge. Research done in the 1990s investigated the effects on student achievement of instruction that builds on informal understandings and emphasizes mathematical concepts and reasoning. Cobb et al.’s problem-centered mathematics project (Wood

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Learning and Instruction: A SERP Research Agenda and Sellers, 1997), and cognitively guided instruction in problem solving and conceptual understanding (Carpenter et al., 1996) both reported positive effects. With support from the National Science Foundation (NSF), several full-scale elementary mathematics curricula with embedded assessments have been developed, directed at supporting deeper conceptual understanding of mathematics concepts and building on children’s informal knowledge of mathematics to provide a more flexible foundation for supporting problem solving. Three curricula developed separately take somewhat different approaches to achieving those goals: the Everyday Mathematics curriculum, the Investigations in Number, Data and Space curriculum, and the Math Trailblazers curriculum (Education Development Center, 2001). All three curricula show positive gains in student achievement in implementation studies, in which the developers collect data on program effects. While such findings are encouraging, they must be viewed with a critical eye, both because those providing the assessment have a vested interest in the outcome and because the methodologies employed generally do not allow for direct attribution of the results to the program.1 Third party evaluations using comparison groups have been done in some cases, but none of these has involved random assignment, the condition that maximizes confidence in attributing results to the intervention. Nor do these studies measure either fidelity of implementation of the reform curriculum for the experimental group or the specific program features of the alternative used with the control group (see, for example, Fuson et al., 2000). How students taught with these curricula compare with each other in mathematical proficiency and, perhaps more importantly, how they compare with students taught with curricula that devote more instructional time to strengthening formal procedural knowledge have not been carefully studied. From the perspective of practice, these are important omissions. To make informed curriculum decisions, teachers and school 1   Implementation studies generally do not involve controlled experimentation that allows for comparison of results of one intervention with another. It is also widely understood that the introduction of a new program can have positive effects not because of program content but because something new is being tried.

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Learning and Instruction: A SERP Research Agenda BOX 3.3 Primary School Mathematics From an early age, children begin to develop an informal understanding of quantity and number. Careful research conducted by developmental and cognitive psychologists has mapped the progression of children’s conceptual understanding of number through the preschool years. Just as healthy children who live in language-rich environments will develop the ability to speak according to a fairly typical trajectory (from single sound utterances to grammatically correct explanations of why a parent should not turn out the light and leave at bedtime), children follow a fairly typical trajectory from differentiating more from less, to possessing the facility to add and subtract accurately with small numbers. But just as a child’s environment influences language development, it influences the rate of acquisition of number concepts. For many children whose early years are characterized by disadvantage, there is a substantial lag in the development of the number concepts that are prerequisite to first grade mathematics. Between the ages of 4 and 6, most children develop what Case and Sandieson (1987) refer to as the “central conceptual structure” for whole number mathematics. The concepts are central in the sense that they are vital to successful performance on a broad array of tasks, and their absence constitutes the major barrier to learning. That structure involves four steps (pictured in Figure 3.1) that are developed in sequence: FIGURE 3.1 “Mental counting line” central conceptual structure. The bottom row of Figure 3.1 indicates that children recognize the written numerals. This information is “grafted on” to the conceptual structure above.

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Learning and Instruction: A SERP Research Agenda The ability to verbally count using number words. This ability is initially developed as a sequencing of words (one, two, three …) without an understanding of the specific meaning attached to the words. Quantity is still understood nonnumerically as more or less, big or small. The ability to count with one-to-one correspondence. When this ability develops, children are able to point at objects as they count, mapping the counting words onto the objects so that each is tagged once and only once. This ability is initially developed as a sensorimotor activity, with an understanding of quantity still absent. Children who can successfully count four objects and five objects cannot answer the question, “Which is more, four or five?” The ability to recognize quantity as set size. With development of this ability, children do understand that “three” refers to a set with three members. Initially this understanding is concrete, and children will often use their fingers as indicators of set membership. The ability to “mentally simulate” the sensorimotor counting. When this ability is in place, children can carry out counting tasks as though they were operating with a mental number line. They understand that movement from one set size to the next involves the addition or subtraction of one unit. While children with middle and higher socioeconomic status generally come to school with the central conceptual structure in place, many children from disadvantaged backgrounds do not. When first grade math instruction assumes that knowledge, these children are less likely to succeed. Sharon Griffin and Robbie Case designed a curriculum called Number Worlds that deliberately puts the central conceptual structure for whole number in place in kindergarten (Griffin and Case, 1997). Additional activities extend the knowledge base through second grade. Developed and tested with classroom teachers and children, the program consists primarily of 78 games that provide children with ample opportunity for hands-on, inquiry-based learning. Number is represented in a variety of forms—on dice, with chips, as spaces on a board, as written numerals. An important component of the program is the Number Knowledge Test, which allows teachers to quickly assess each individual student’s current level of understanding, and to choose individual or class activities that will solidify fragile knowledge and take students the next step. The Number Worlds program has been tested with disadvantaged populations in numerous controlled trials in both the United States and Canada with positive results. One longitudinal study charted the progress of three groups of children attending school in an urban community in Massachusetts for three years: from the beginning of kindergarten to the end of second grade. Children in both the Number Worlds treatment group and in the control group were from schools in low-income, high-risk communities where about 79 percent of children were eligible for free or reduced-price lunch. A third normative group was drawn from a magnet school in the urban center that had attracted a large number of majority students. The student body was predominantly middle income, with 37 percent eligible for free or reduced-price lunch.

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Learning and Instruction: A SERP Research Agenda As Figure 3.2 shows, the normative group began kindergarten with substantially higher scores on the Number Knowledge Test than children in the treatment and the control groups. The gap indicated a developmental lag that exceeded one year, and for many children in the treatment group was closer to two years. By the end of the kindergarten year, however, the Number Worlds children narrowed the gap with the normative group to a small fraction of its initial size. By the end of the second grade, the treatment children actually outperformed the magnet school group. In contrast, the initial gap between the control group children and the normative group did not narrow over time. The control group children did make steady progress over the three years; however, they were never able to catch up. FIGURE 3.2 Mean developmental scores on number knowledge test at four time periods. administrators need to know what type of implementation of a specific curriculum produces what results, compared with the alternatives before them. Yet to provide the information that is most useful to practice is a major undertaking. These questions are answerable, but research carefully designed to provide those answers will take a substantial investment.

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Learning and Instruction: A SERP Research Agenda FIGURE 3.3 Income-to-needs and child cognitive ability: Deep poverty and math ability (PIAT-Math), NLSY-CS data set. SOURCE: Brooks-Gunn et al. (1999). The significance of these findings is suggested by the data in Figure 3.3 that plots Peabody Individual Achievement Test (PIAT) math scores against poverty level. Clearly the correlation is powerful; the deeper the level of poverty, the poorer the math scores. Importantly, as students move through school, the gap becomes more pronounced. Children ages 9-10 showed even larger score disparities than those ages 7-8. NAEP data indicate that in 1999 black 4- and 8-year-olds ranked in the 15th and 14th percentiles in math, respectively (Thernstrom and Thernstrom, in press). If the Number Worlds program can put poor children on a path to success in math, the contribution would be substantial. Checkpoints: Assessment The curricula described above have embedded assessments that allow teachers to track student learning. A key feature of the Number Worlds curriculum is the Number Knowledge Test, which allows teachers to closely link instructional activities for children to the assessment results. How well other curricula link assessment and instruction is an issue worth investigation.

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Learning and Instruction: A SERP Research Agenda A separate issue is the assessment over time of the five strands that constitute mathematical proficiency. The past decade has seen the emergence of a spate of new tests and measures. No consensus has emerged, however, on critical measures. While there are some standard and widely used assessment tools to appraise young children’s emergent reading and language skills and competence, no such tools are used on any comparable basis in primary mathematics. This type of assessment is required to evaluate the effectiveness of a particular curriculum and to make comparisons across curricula. For the most part, we lack sophisticated methods for tracking student learning over time or for examining the contribution of any particular instructional interventions, whether large or small, on students’ learning. TEACHER KNOWLEDGE Little is known about what it takes for teachers to use particular instructional approaches effectively, a necessary element of taking any approach to scale. The challenges can be substantial. The curricula mentioned above introduce major changes in approach to teaching mathematics, and effective implementation will require that teachers change their view of mathematics teaching and learning dramatically. In Everyday Mathematics, for example, teachers are expected to introduce topics that will be revisited later in the curriculum. Complete mastery is not expected with the first introduction. This has created some confusion for teachers, who are often unclear about when mastery is sufficient to move on to the next topic (Fuson et al., 2000). All of the curricula encourage building on students’ own strategies for problem solving and supporting engagement through dialogue about the benefits of alternative strategies. The change required on the part of the teacher to relinquish control of the answer in favor of a dialogue among students has proven difficult when it has been studied (Palincsar et al., 1989). Adequate opportunities to learn and the ongoing supports for an entirely different approach to teaching will be critical to the effectiveness of efforts to scale up the implementation of the curricula. This is clearly an area in need of further study. One clue regarding teacher knowledge requirements can be found in research pursued for the most part separately from the work on student learning and the design of curriculum ap-

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Learning and Instruction: A SERP Research Agenda forms affords learners greater insight. Notions of what counts as concrete or abstract remain vaguely and variously defined. The Vehicle: Curriculum and Pedagogy New curricular materials introduce algebra in various segments of the K-12 mathematics sequence using a variety of conceptions of algebra. Some take an equation-solving orientation; others take a function approach; still others take a mathematical modeling approach. Yet we do not know how these various curricular approaches affect students’ understanding and continued use of algebra. The hypothesis that algebra instruction moves to abstraction without first connecting the abstractions with the concrete instances that justify them has led to the development of new curricula that emphasize richly contextualized problems. Some research evidence suggests that student engagement is higher and that students work meaningfully with important mathematical ideas, outperforming students whose curricular experiences do not include such rich investigative problems (e.g., Boaler, 1997; Brenner et. al., 1997; Nathan et. al., 2002). But some caution that such problems, if taken seriously, demand close attention to the contexts; whereupon students may become preoccupied with the contextual particularities in ways that distract from the mathematical ideas entailed (Lubinski et al., 1998). Consequently, they may not develop abstract knowledge central to mathematical proficiency. Some instructional approaches look for a middle ground in which algebra knowledge is contextualized, but the context is kept simple, and a single context is used extensively to help students see through to the underlying mathematical functions (Kalchman et al., 2001; Kalchman and Koedinger, forthcoming). Much remains to be investigated about how students develop the ability to work effectively with abstract ideas and notation, as well as about the relationships between abstraction and concrete experiences in learning. A focus on algebra would afford opportunities to probe how different instructional uses of technology interact with the development of symbol manipulation skills. With the increased availability of technology, what is meant by “symbolic fluency” raises new questions. What is the role that graphing calculators

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Learning and Instruction: A SERP Research Agenda and computational algebraic systems might play? What is the role of paper and pencil computation in developing understanding as well as skill? These are questions that appear at every level of school mathematics. Checkpoints: Assessment Algebra represents a major challenge for many students. If more students are to succeed in meeting that challenge, it will be important to identify the points of difficulty for individual students and provide effective instructional responses before they are lost. The difficulty factors assessments of algebra reading (Koedinger and Nathan, In Press) and algebra writing (Heffernan and Koedinger, 1997, 1998) are examples of efforts to provide assessment tools for this purpose. Two features of the subject make assessing individual progress very important. Algebra requires facility with much of the mathematics that has come before. If the mathematical foundation is weak in any of its components, algebra mastery will be undermined. Determining where students need to shore up the preparatory mathematics, as well as opportunities for doing so, are critical to success. Second, algebra instruction moves toward high-level abstraction. The readiness of individual students to move from one level of difficulty to the next will differ. If the movement comes before a bridge is effectively built to a student’s prior knowledge or before new knowledge is consolidated, the student will be lost. If movement toward greater difficulty does not come soon enough, a student will make less progress in higher level algebra than is possible. Indeed, precisely this is at the heart of opposing views of algebra pedagogy. If formative assessment were sophisticated enough to determine individual student readiness to move on, then trade-offs between attending to the needs and preparedness of different students would not be necessary. A research and development effort at Carnegie Melon University that generated the Algebra Cognitive Tutor has focused very productively on the second element of this problem (see Box 3.5). It began as a project to see whether a computational theory of thought, called ACT (Anderson, 1983), could be used as a basis for delivering computer-based instruction. The cognitive theory applies to problem solving more broadly. For pur-

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Learning and Instruction: A SERP Research Agenda poses of algebra teaching, it was the foundation for modeling the variety of different approaches—both correct and incorrect—that students typically take to solving algebra problems. A number of different approaches can lead to a correct solution, and the program does not favor one over another. However, some approaches lead the student astray. If the student is working effectively on a problem, there is no computer feedback. But when a student begins down an unproductive or erroneous path, the computer program recognizes this by a process called model tracing and provides hints and instruction to guide the student’s thinking. The Algebra Cognitive Tutor also assesses mastery of elements of the curriculum by a process called knowledge tracing. When a student’s problem-solving efforts suggest that the relevant knowledge or skill is not yet consolidated, the computer selects instruction and problems appropriate to where that student is in the learning trajectory. In studies of cognitive tutors more generally, it was found that under controlled conditions, students could complete the curriculum in about a third of the time generally required to master the same material, with about a standard deviation (approximately a letter grade) improvement in achievement (Anderson et al., 1995). In real classrooms, the impact has generally not been as large. A third-party evaluation of the tutors suggested that the scaffolding of learning that allowed students to experience success with challenging problems produced large motivational gains (Schofield et al., 1990). TEACHER KNOWLEDGE In the past, only teachers of high school students were thought to need knowledge of algebra. Although their preparation to teach does not include study of the objects and processes of high school algebra, little attention has been paid to whether or not secondary school teachers do in fact have adequate algebraic knowledge for teaching (Ferrini-Mundy and Burrill, 2002). Students preparing to be secondary school teachers typically take courses in abstract algebra and analysis, under the assumption that such mathematical background will serve them well as secondary school teachers. Yet the actual knowledge developed in such courses and its application by teachers in classrooms has not been thoroughly studied. Some research suggests that the

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Learning and Instruction: A SERP Research Agenda BOX 3.5 The Algebra Cognitive Tutor The Algebra Cognitive Tutor is one of a set of cognitive tutors developed at Carnegie Mellon. Of great relevance to the SERP vision, the tutors are a good illustration of how to make the transition from the laboratory to the classroom. The work at Carnegie Mellon began as a project to see whether a computational theory of thought, called ACT (Anderson, 1983), could be used as a basis for delivering computer-based instruction in algebra. The ACT theory of problem-solving cognition is the basis for modeling students’ algebra knowledge. These models can be captured in a computer program that can generate or identify a range of characteristic approaches to solving an algebra problem. These cognitive models enable two sorts of instructional responses that are individualized to students: By a process called model tracing, the program will infer how a student is going about problem solving and generate help and instruction appropriate to where that student is in the problem. By a process called knowledge tracing, the program will infer where a student falls in the learning trajectory and select instruction and problems appropriately. Developing cognitive models that accurately reflect competence and developing appropriate instructional responses is an iterative process. The success of the tutors depends on a design-test-redesign effort in which models are assessed for how well they capture competence and instructional responses are assessed for how effective they are. In studies of cognitive tutors more generally, it was found that in controlled laboratory condition students using a cognitive tutor could go through a curriculum in a third of the time, and in carefully managed classrooms students would show about a standard deviation (approximately a letter grade) improvement in achievement compared with students receiving standard instruction (Anderson et al., 1995). In real classroom situations, the impact of the tutors tends not to be as large, varying from 0 to 1 standard deviation across more than 13 evaluations. Another third-party evaluation, focusing on the social consequences of the tutors, documented large motivational gains resulting from the active engagement of students and the successful experiences on challenging problems (Schofield et al., 1990). However, unlike many such small-scale success stories in cognitive science, this project was able to grow to the point at which the cognitive tutors now are used in over 1,200 schools, 46 of 100 largest school districts, and interacting with about 170,000 students yearly. A number of features were critical to making this successful transition:

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Learning and Instruction: A SERP Research Agenda While the ACT theory provided the technology, there was a concerted effort to identify a curriculum that educators wanted to be taught in the classroom. In particular, the project recognized that it was a priority for the schools to teach a curriculum that was in compliance with the NCTM standards (National Council of Teachers of Mathematics, 2000) and designed a curriculum around this. A curriculum was designed that teachers would accept and could implement. A full-year curriculum was developed rather than an enrichment program to be inserted into an existing curriculum. The curriculum was designed with the critical help of teachers with experience in urban classrooms. The computer tutors were used as a support rather than a replacement for the teachers. In this curriculum students spend 40 percent of their time with the computer tutors and 60 percent of their time with other activities. These classroom activities help them transition to their lessons with the tutor and transition those lessons to mathematics that they will have to do without the tutor on paper and in the real world. A structure was set up for supporting the use of the curriculum and tutors. Before introducing the tutors into a classroom, it has been important to provide professional development time to enable teachers to prepare for the change they are about to experience. A center at Carnegie Mellon was set up for responding to teacher and school problems. As the adoptions grew, a separate company, Carnegie Learning, was created to perform this function and maintain and adapt the materials. Ultimately, such a curriculum must be financially self-sustaining and it was developed from the beginning with a plausible financial model in mind. In particular, by offering a full grade 9-11 curriculum, it was possible to earn in sales the kind of income that is necessary to sustain this activity. While the cognitive tutor enterprise illustrates what needs to be done to transition research ideas into the American classroom, it does not represent a complete solution to even high school algebra. Early in the development of the Algebra Cognitive Tutor, a decision was made to place a heavy emphasis on contextualizing algebra to help students make the transition to the formalism. The course has been demonstrated to raise student achievement in urban schools and to reduce the number of students dropping out. However, high-achieving students may not be fully achieving the desired fluency in symbol manipulation and abstract analysis. There is no reason why the cognitive tutors could not more fully address these topics and, in fact, many tutor units do, particularly in the algebra 2 course. However, a more accelerated course may yield better results for high-achieving students.

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Learning and Instruction: A SERP Research Agenda assumption that secondary school teachers have strong and flexible knowledge of algebra is unfounded (Ball, 1988). In fact, evidence suggests that secondary school teachers may often have rule-bound knowledge of procedures but lack conceptual, connected understanding of the domain. That some signals existed that high school teachers’ knowledge may not be as robust as had been suspected is not so surprising, since they are educated in the very mathematics classrooms that many seek to improve. Still, this result signals a more significant problem. First, the changes in and expansion of what is meant by algebra mean that secondary school teachers are increasingly being called on to teach algebraic ideas and connections that they have not themselves studied or have not studied in such ways. Analyses of what the new curricula demand of teachers could make visible the mathematical demands of those curricula and permit investigation of teachers’ current knowledge to teach those materials. Second, the movement of algebraic ideas into the middle school and especially the elementary school curriculum means that teachers who have not in the past taught algebra are now being called on to teach ideas and processes for which they have not in the past been responsible. Prospective elementary school teachers’ knowledge of algebra may be based largely on their own experiences as high school students. The new requirements of elementary school teaching raise important and pressing needs for research on teacher knowledge, teaching, and teacher learning specifically in algebra. Not only do teachers need knowledge of the mathematical content, however. Equally important (and related) is knowledge of how students think about and develop algebraic ideas and processes. What ideas or procedures are particularly difficult, both in reading and writing mathematical relationships, for many students? As algebra shifts to being a K-12 subject, rather than a pair of high school courses, new questions emerge that warrant investigation: If students learn about variables and equations sooner and engage earlier in algebraic reasoning (Carpenter and Franke, 2001), how will these earlier experiences shape the development of students’ algebraic proficiency over time? How do students of different ages manage and use symbolic notation, both in reading and writing mathematical relationships? What supports the development of meaningful and skilled fluency with mathematical symbols and syntax? In fac-

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Learning and Instruction: A SERP Research Agenda ing diverse classes of students, teachers also need to understand better the mathematical resources and difficulties that their students bring from their own environments, as well as how to make productive use of and mediate those (see Moses and Cobb, 2001, for a robust example of designing strong connections between the domain of algebra and students’ out-of-school activities and knowledge use). RESEARCH AGENDA There is little agreement at this point on what algebra should be taught or how it should be taught. As in other areas of the curriculum, the questions are in part a matter of valued outcomes for algebra instruction and the instructional time allocation across algebra and other subjects. But a study of the outcomes of different instructional choices can make the decisions far more rational than they can be in the absence of high-quality data. We propose research and development on four major initiatives in this area: Alternatives in the teaching and learning of algebra; Teacher knowledge; Developing algebra assessments and instruments; Students’ development over time and the effects of different curricular choices. Initiative 1: Research and Development on Alternatives in the Teaching and Learning of Algebra Work supported by the National Science Foundation as well as by private foundations has generated a variety of curriculum materials for schools that constitute different perspectives on algebra, different ideas about what is important for students to learn, and different ideas about how students can most effectively be taught that can be contrasted with the best traditional approaches to teaching algebra. Since these curricula are already developed and in use, they provide an opportunity for understanding the consequences of the choices made. For example, in some materials a functions approach to algebra is central, while in others, algebra is treated more as generalized arithmetic, and the solving of equations is more

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Learning and Instruction: A SERP Research Agenda prominent. In some approaches, students are engaged in using the tools of algebra to model situations and problems, while, in others, algebra as an abstract language is stressed. While much controversy surrounds the worth and merit of these different perspectives on the subject, additional debates center on the contribution of calculators and other technology, the structure of lessons, and the role of the teacher. Because curricula have already been developed that represent these different perspectives on the subject and on how it might best be taught, one important initiative of SERP might be to design comparative studies of how these curricula are taught in classrooms and what and how diverse students learn algebra over time. In this initiative, cohorts of students could be followed longitudinally. Studies could gather information about the instruction they receive, exposure to curriculum, information on the teachers, and their use of the curriculum and other tools. This initiative will depend on the development of effective assessments (see Initiative 3). As with elementary mathematics, however, knowing why particular curricular interventions produce particular outcomes will require companion controlled experiments at the level of particular program features to test for causality. This kind of research is necessary not only to advance scientific understanding, but also because it provides critical knowledge for teachers who adapt curricula and allows developers to improve curricula or design alternatives that are responsive to research findings. Simultaneous with this effort, SERP can support curriculum development that extends existing curricula in promising directions. The Algebra Cognitive Tutor, for example, emphasizes highly contextualized problem solving. While many fewer students drop out and students master the material covered more quickly and effectively, the curriculum may not achieve the fluency in symbol manipulation and abstract analysis expected for high-achieving students. The developers suggest that the curriculum could quite easily be strengthened in this respect, and a separate accelerated algebra course is likely to yield even better results for high-achieving students. In studying the set of curricula as they are being implemented, SERP as a third-party entity would be well positioned to identify and support promising areas like this for further development.

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Learning and Instruction: A SERP Research Agenda Initiative 2: Research and Development on Teacher Knowledge Important questions remain unanswered about the knowledge of mathematics needed to teach algebra effectively. As with elementary mathematics, the existence of different curricular approaches and efforts to study them, as outlined above in Initiative 1, provide the opportunity to investigate the demands for teachers in teaching different curricular approaches to algebra. For example, specifically what mathematical demands arise for teachers in teaching approaches to algebra that emphasize symbolic fluency compared with approaches that emphasize modeling and connections to situations? What sort of representational and notational fluency do teachers need? How do teachers need to understand the connections between algebra and other domains of mathematics, and what is demanded of teachers with respect to mathematical reasoning under different approaches to algebra? The movement of algebra into the elementary school curriculum, as recommended both by the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000) and Adding It Up (National Research Council, 2001a), creates the opportunity to examine what elementary teachers need to know with respect to algebra. Typically regarded as a secondary school subject, algebra has not played a central role in the preparation of elementary school teachers. Studies of teachers engaged with the new curricula that include elementary school skills and ideas of algebra could provide insight into the kinds of algebra knowledge useful to the teaching of young children. Where and how do ideas and skills of algebra surface in younger children’s learning, and what sorts of knowledge would help teachers address and develop those? As in other areas of the curriculum, it will be particularly important to identify the issues with which teachers struggle most, the conceptions that make effective teaching more difficult. As in Initiative 1, the study of teacher knowledge requirements would provide the basis for research and development on effective teacher education interventions. The development efforts would be expected to target a variety of teacher learning opportunities, including pre-service education in teaching mathematics, teacher support materials, and in-service education associated with the use of particular curricula.

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Learning and Instruction: A SERP Research Agenda Initiative 3: Developing Algebra Assessments and Instruments Efforts to improve algebra instruction, as well as to evaluate the effectiveness of alternative approaches to the teaching of algebra, will depend on the development of new assessments of students’ and teachers’ learning. The development of formative assessments for instructional purposes will need to test hypotheses about what is difficult for students to learn, as well as hypotheses about the kinds of scaffolds that provide support for learning when students are struggling. For classroom effectiveness, these assessments must be closely tied to instructional materials. An investment in the development of algebra assessments that capture all aspects of algebra proficiency, including the robustness and flexibility of conceptual and procedural knowledge and the ability to transfer learning to novel problems, will need to be developed if outcomes of alternative approaches to instruction are to be meaningfully compared. Assessments will also be needed that can discriminate different kinds and levels of knowledge for the teaching of algebra. These should include both the knowledge of subject matter and pedagogical content knowledge. Moreover, in order to compare differences in students’ opportunities to learn in circumstances in which the teacher or the curriculum changes, instruments to gather information about instruction itself will be important. For example, the representations and tools that are used and the type, frequency, and duration of their use needs to be captured. Measures of fidelity of implementation and of teacher support will be required as well. These investments in instrumentation and assessment tools at the start will allow for subsequent work to be far more powerful for guiding instructional practice. Initiative 4: Students’ Development Over Time and the Effect of Different Curricular Choices Because algebra is increasingly seen as a K-12 strand of a mathematics curriculum, not merely as a high school course or pair of courses, the timing is right to design studies that track students across their school careers, investigating the develop-

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Learning and Instruction: A SERP Research Agenda ment of proficiency in algebra. Such longitudinal studies of algebra learning could be designed to examine how particular configurations of curricular and pedagogical choices affect what students learn. For example, do students whose experiences with number and operations are designed to develop deep conceptual understanding and procedural fluency fare differently in algebra than those whose opportunities to learn emphasize applications and modeling? How do differences in the development of arithmetic fluency affect the development of students’ algebraic proficiency? Initially, the work involved will be to design careful procedures for longitudinal data collection. Doing so will hinge on Initiative 3, in which input and outcomes measures are tested and developed. While the fruits of this research would not be expected in the early years of the program, designing the data collection effort early and carefully will be critical to high-quality analysis further down the road.