An investigation of teachers’ beliefs of students’ algebra development. Arithmetic Teacher, 34(8), 18–25. Hatano, G. (1988, Fall). New York: Simon & Schuster. Knowledge of students and how they learn mathematics includes general knowledge of how various mathematical ideas develop in children over time as well as specific knowledge of how to determine where in a developmental trajectory a child might be. Novices would see these problems as unrelated; experts would see both as involving five choices between two things: red and green, or with and without.27. The sentences could also provide opportunities for discussions about how to resolve disagreement and develop a mathematical argument. Particular emphasis is on high-speed 2D/3D optical sensing, embedded & reconfigurable processing architectures, single photon avalanche devices (SPAD) and design optimization techniques. (2000). In the last two sections, we discuss four programs for developing proficient teaching and then consider how teachers might develop communities of practice. Just as students who have acquired procedural fluency can perform calculations with numbers efficiently, accurately, and flexibly with minimal effort, teachers who have acquired a repertoire of instructional routines can readily draw upon them as they interact with students in teaching mathematics. To represent a problem accurately, students must first understand the situation, including its key features. Lund, Sweden: Lund University Press. As a result of their work in this program, the teachers became more likely to bring out students’ reasoning in discussions and to invite both public and private reflection on the students’ ideas. The currency of value in the job market today is more than computational competence. Problem-solving transfer. Box 10–3 Investigating Mathematical Tasks Using Cases from Real Practice. In those programs, teachers engage in analyses in which they are asked to provide evidence to justify claims and assertions. Over time, people tend to forget the reasons a procedure works or what is entailed in understanding or justifying a particular algorithm. The instructor has challenged them to consider why they are getting what seems to be an answer that is larger than either of the numbers in the original problem ( and ). Problem solving in context(s). Reasoning about operations: Early algebraic thinking in grades K-6. Second, college mathematics courses do not provide students with opportunities to learn either multiple representations of mathematical ideas or the ways in which different representations relate to one another. [July 10, 2001]. NAEP 1996 mathematics report card for the nation and the states. Washington, DC: National Academy Press. The data do not indicate, however, whether the students thought they could make sense out of the mathematics themselves or depended on others for explanations. Washington, DC: Author. Not only do they develop more elaborated conceptions of how students’ mathematical thinking develops by listening to their students, but they also learn mathematical concepts and strategies from their interactions with students. (1998). Review of Educational Research, 69(3), 287–314. Leinhardt, G., & Smith, D.A. Mathematics for all? Students develop procedural fluency as they use their strategic competence to choose among effective procedures. The difficulty of integrating such courses is compounded when they are located in different administrative units. Nunes, T. (1992a). As a result of that education, teachers may know the facts and procedures that they teach but often have a relatively weak understanding of the conceptual basis for that knowledge. (2000). However, students, especially those in the fourth and eighth grades, had difficulty with more complex problem-solving situations. Available: http://books.nap.edu/catalog/9832.html. Lewis and Tsuchida, 1998; Shimahara, 1998; Stigler and Hiebert, 1999. They also learn that solving challenging mathematics problems depends on the ability to carry out procedures readily and, conversely, that problem-solving experience helps them acquire new concepts and skills. She points out that it is not obvious what the value of 2.50 means in the algebraic expression of the function. Brian claims that from 1980 to 1990 the populations of the two towns grew by the same amount. Journal of Educational Psychology, 87, 18–32. Becoming strategically competent involves an avoidance of “number grabbing” methods (in which the student selects numbers and prepares to perform arithmetic operations on them)23 in favor of methods that generate problem models (in which the student constructs a mental model of the variables and relations described in the problem). At first, several. Stereotype threat and the intellectual test performance of African-Americans. Fuson, K.C., & Burghardt, B.H. Several are overheard to be discussing features of this problem that seem particularly fruitful and that have them thinking about how they frame problems for their students. Silver, E.A., & Stein, M. (1996). But perhaps surprisingly, it is students who have historically been less successful in school who have the most potential to benefit from instruction designed to achieve proficiency.70 All will benefit from a program in which mathematical proficiency is the goal. What do new views of knowledge and thinking have to say about research on teacher learning? In R.J.Sternberg & T.Ben-Zee (Eds. Instructional innovation: Reconsidering the story. Capturing teachers’ generative change: A follow up study of teachers’ professional development in mathematics. (1999). The prospective teachers set to work, some in pairs, some alone. ), Assessment in transition: Monitoring the nation’s educational progress (Background Studies, pp. Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. They then need to generate a mathematical representation of the problem that captures the core mathematical elements and ignores the irrelevant features. Concepts associated with the equality symbol. (1996). ), Conceptual and procedural knowledge: The case of mathematics (pp. With examples and illustrations, the book presents a portrait of mathematics learning: The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics. Understanding the mathematics of the domain being studied is a prerequisite to understanding children’s thinking in that domain. Novice problem solvers are inclined to notice similarities in surface features of problems, such as the characters or scenarios described in the problem. Some groups divide their work into three major phases, each taking about one third of the school year. Cognition and Instruction, 14, 251–283. In D.Grouws (Ed. New York: Free Press. Journal for Research in Mathematics Education, 16, 337–355. (1992). Journal of Experimental Psychology: General, 124, 83–97. [July 10, 2001]. This package includes MyLab Math. ematics and children’s thinking are set in a context that relates to their practice. They monitor what they remember and try to figure out whether it makes sense. We consider below examples of four such program types that represent an array of alternative approaches to developing integrated proficiency in teaching mathematics.39, Some teacher preparation and professional development programs attempt to enhance prospective and practicing teachers’ knowledge of mathematics by having them probe more deeply fundamental ideas from elementary school. In the course of their work as teachers, they must understand mathematics in ways that allow them to explain and unpack ideas in ways not needed in ordinary adult life. In turn, as a procedure becomes more automatic, the child is enabled to think about other aspects of a problem and to tackle new kinds of problems, which leads to new understanding. Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. (1991). 73– 106). Young children using counters solve the first problem by putting 24 counters in piles of 6 counters each. In Asian countries, perhaps because of cultural traditions encouraging humility or because of the challenging curriculum they face, eighth graders tend to perceive themselves as not very good at mathematics. Sociology of Education, 70, 256–284. (1981). The incidence of satisfactory responses was greater than 25% on only two tasks.61. After the students observe that the successful problems— involving the sesame crackers and the liters of water—are measurement problems, she asks them to try to develop a problem situation for that represents a sharing division. See English, 1997b, for an extended discussion of these ideas. Using the lesson as the unit of analysis and improvement, the teachers are encouraged to improve their knowledge of all aspects of teaching within the context of their own classrooms—knowledge of mathematics, of students’ thinking, of pedagogy, of curriculum, and of assessment. Dweck, C. (1986). Becoming mathematically proficient is necessary and appropriate for all students. By renaming the fractions so that they have the same denominator, the students might arrive at a common measure for the fractions, determine the sum, and see its magnitude on the number line. In W.D.Reeve (Ed. The central notion that strands of competence must be interwoven to be useful reflects the finding that having a deep understanding requires that learners connect pieces of knowledge, and that connection in turn is a key factor in whether they can use what they know productively in solving problems. A second kind of item that measures adaptive reasoning is one that asks students to justify and explain their solutions. (1993). They have read a case of a teacher teaching a lesson on functions. 17–24). this error.15 Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. They begin to describe and name the different moves she makes. But they also have to know how to use both kinds of knowledge effectively in the context of their work if they are to help their students develop mathematical proficiency. As one researcher put it, “The human ability to find analogical correspondences is a powerful reasoning mechanism.”30 Analogical reasoning, metaphors, and mental and physical representations are “tools to think with,” often serving as sources of hypotheses, sources of problem-solving operations and techniques, and aids to learning and transfer.31, Some researchers have concluded that children’s reasoning ability is quite limited until they are about 12 years old.32 Yet when asked to talk about how they arrived at their solutions to problems, children as young as 4 and 5 display evidence of encoding and inference and are resistant to counter suggestion.33 With the help of representation-building experiences, children can demonstrate sophisticated reasoning abilities. For teachers who have already achieved some mathematical proficiency, separate courses or professional development programs that focus exclusively on mathematics, on the psychology of learning, or on methods of teaching provide limited opportunities to make these connections. These findings, which surprised most teachers, have led them to begin to listen to their students, and a number of teachers have engaged their students in a discussion of the reasons for their responses. ties that students have with certain mathematical concepts and procedures, and it encompasses knowledge about learning and about the sorts of experiences, designs, and approaches that influence students’ thinking and learning. Not surprisingly, these teachers gave the students little assistance in developing an understanding of what they were doing.20 When the teachers did try to provide a clear explanation and justification, they were not able to do so.21 In some cases, their inadequate conceptual knowledge resulted in their presenting incorrect procedures.22, Some of the same studies contrasted the teaching practices of teachers with low levels of mathematical knowledge with the teaching practices of teachers who had a better command of mathematics. The power of computational algorithms is that they allow learners to calculate without having to think deeply about the steps in the calculation or why the calculations work. For example, Inhelder and Piaget, 1958; Sternberg and Rifkin, 1979. How people learn: Brain, mind, experience, and school. [July 10, 2001]. For work in psychology, see Baddeley, 1976; Bruner, 1960, pp. The contents of a given cluster may be summarized by a short sentence or phrase like “properties of multiplication,” which is sufficient for use in many situations. Behr, M., Erlwanger, S., & Nichols, E. (1976). Research with older students and adults suggests that a phenomenon termed stereotype threat might account for much of the observed differences in mathematics performance between ethnic groups and between male and female students.49 In this phenomenon, good students who care about their performance in mathematics and who belong to groups stereotyped as being poor at mathematics perform poorly on difficult mathematics problems under conditions in which they feel pressure to conform to the stereotype.