Learning and Instruction A SERP Research Agenda (2004) / Chapter Skim
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3 Mathematics
3 Mathematics
Pages 66-101
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From page 66... ...
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.
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From page 67... ...
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; Proceduralfluency: 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.
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From page 68... ...
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~.
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From page 69... ...
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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From page 70... ...
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.
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From page 71... ...
, several full-scare 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~.
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From page 72... ...
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.
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From page 73... ...
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.
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From page 74... ...
2.0 ~ .0 FIGURE 3.2 Mean developmental o.o scores on number knowledge test at four time periods.
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From page 75... ...
· ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 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.
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From page 76... ...
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 ap76 LEARNING AND INSTRUCTION
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From page 77... ...
Following this work, some materials for use in teachers' professional instruction have been developed.3 Modules and other curriculum materials contain focused work aimed at helping teachers learn the sort of mathematical knowledge of whole 2Base ten blocks are a common material used to model place value concepts, and operations that rely centrally on place value. The materials consist of a unit cube, a ten-stick built of 10 cubes, a flat square built of 100 cubes or 10 ten-sticks, and a block composed of 1,000 cubes, or 10 flats, or 100 ten-sticks.
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From page 78... ...
Many teachers in both countries believed students needed a conceptual understanding, but within this group there were considerable differences. Some teachers wanted children to think through what they were doing, while others wanted them to understand core mathematical concepts.
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From page 79... ...
Rather, it suggests that the procedural knowledge and skills be organized around the core concepts. Ma describes the set of Chinese teachers who emphasize core concepts as seeing the knowledge in "packages" in which the concepts and skills are related.
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From page 80... ...
focus on the teacher knowledge requirements to comfortably and effectively use curricula that are built on research-based findings regarding student learning.
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From page 81... ...
Unlike many other areas of the curriculum, early mathematics has the theoretical and conceptual models, as well as supporting empirical data, on which to build quality assessments. Substantial work has already been done to specify critical concepts and skills in this domain, providing assessment developers with substantial resources on which to draw in drafting the elements of a measurement strategy.
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From page 82... ...
While there are several possible approaches to developing such a system of student assessments in early mathematics, one obvious place to begin is with a review of the assessment materials in existing widely used and exemplary curricular programs for formative and summative assessments, as well as state and national tests for policy making and accountability. These can be reviewed in light of cognitive theories of mathematical understanding, including empirical data regarding the validity of specific assessments.
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From page 83... ...
A strand of research focused on implementation issues should address the set of questions critical to successful use of quality assessments: · What teacher knowledge is necessary to support effective use of assessments in their instructional practice? These include teacher understanding of the assessments and their purpose, as well as practical considerations of the time to administer, score, and interpret results; What forms of technology support are needed to assist teachers in the administration, scoring, and interpretation of a range of standards-based and theory-based assessments; and How, and to what extent, does the process of implementing curriculum-based and standards-based assessments lead to changes in teachers' instructional practices, and how do these changes affect student learning outcomes?
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From page 84... ...
There are two approaches to this teacher learning that could strategically build on work that has already been done. The first emphasizes teachers' understanding of mathematical concepts and the connections among them, and the second focuses on the knowledge required for teachers to use promising curricula comfortably and effectively.
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From page 85... ...
The second exploration of teacher knowledge should build on research that suggests that professional development is more productive when it is tied to specific curricula or instructional programs that teachers will then incorporate into their practice (Cohen and Hill, 2000~. This research should begin with a clear articulation of the principles and assumptions about student learning that the curriculum incorporates, and comparing these to carefully solicited understandings of teachers.
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From page 86... ...
This initial core would be expanded over time to include other theoretically and practically important alternatives. Many of the evaluations of the curricula set out to answer the question "Does the curriculum improve student achievement?
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From page 87... ...
In still others, teachers might be given time and be provided incentives to spend time planning with the ample teacher guides. The work could be conducted in carefully controlled, Tongitudinal studies carried out in SERP field sites.
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From page 88... ...
Learning to make sense of and operate meaningfully and effectively with these tools is a central goal of instruction. This power involves both moving from contexts to abstract models and, conversely, interpreting abstract ideas skillfully in concrete situations.
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From page 89... ...
Trade-offs become necessary only when the limits on instructional time force them. Currently there is little understanding of the affordances of different instructional emphases.
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From page 90... ...
How do students develop the capacities to move from contexts to abstract models and, conversely, to interpreting abstract ideas skillfully in concrete situations? Beliefs abound about the directionality of these connections in learning: Some argue that all meaningful learning must move from the concrete to the abstract; others insist that the power of the generalized, abstract 90 LEARNING AND INSTRUCTION
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From page 91... ...
Consequently, they may not develop abstract knowledge central to mathematical proficiency. Some instructional approaches Took 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)
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From page 92... ...
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)
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From page 93... ...
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.
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From page 94... ...
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 held 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.
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From page 95... ...
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.
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From page 96... ...
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.
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From page 97... ...
Teacher knowledge; 3. Developing algebra assessments and instruments; 4.
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From page 98... ...
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.
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From page 99... ...
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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From page 100... ...
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.
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From page 101... ...
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?
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