Showing papers in "Mathematical Thinking and Learning in 1999"
TL;DR: In this article, the role emergent models can play in the process of constituting formal mathematics is discussed, and the design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent model and what role they play in fostering the constitution of formal mathematics.
Abstract: This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initi...
402 citations
TL;DR: In this paper, the authors present episodes from a recently completed classroom teaching experiment that focused on statistics and develop the themes of change, diversity, and equity in the context of mathematics education.
Abstract: In the first part of this article, I clarify how we analyze students' mathematical reasoning as acts of participation in the mathematical practices established by the classroom community. In doing so, I present episodes from a recently completed classroom teaching experiment that focused on statistics. Against the background of this analysis, I then broaden my focus in the final part of the article by developing the themes of change, diversity, and equity.
305 citations
TL;DR: In this paper, a learning environment for teaching and learning how to model and solve mathematical application problems was developed and tested in 4 classes of 5th graders, where pupils were taught a series of heuristics embedded in an overall metacognitive strategy.
Abstract: Recent research has shown that many upper elementary school children do not master the skill of solving mathematical application problems. In this design experiment, a learning environment for teaching and learning how to model and solve mathematical application problems was developed and tested in 4 classes of 5th graders. Pupils were taught a series of heuristics embedded in an overall metacognitive strategy for solving mathematical application problems. Meanwhile, pupils of 7 control classes followed regular mathematics classes. The implementation and effectiveness of the experimental learning environment were tested in a study with a pretest-posttest-retention test design with an experimental and a control group. The results indicate that the intervention had a positive effect on different aspects of pupils' mathematical modeling and problem-solving abilities.
248 citations
TL;DR: In this article, a new concept of curriculum is proposed, organized in three strands: literacy, matheracy, and technoracy, which see education and scholarship as pursuing a major, comprehensive goal of building up a new civilization that rejects arrogance, inequity and bigotry.
Abstract: This article focuses on the relations between mathematics and mathematics education on the one hand and human behavior, societal models, and power on the other. Based on a critical analysis of school systems and of mathematical thinking, its history and its sociopolitical implications, anew concept of curriculum is suggested, organized in 3 strands: literacy, matheracy, and technoracy. This new concept sees education and scholarship as pursuing a major, comprehensive goal of building up a new civilization that rejects arrogance, inequity, and bigotry. Because the development of mathematics has been intertwined with all forms of human behavior in the history of human- kind, it is relevant to discuss mathematics and mathematics education with this major goal in mind.
133 citations
TL;DR: Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment as discussed by the authors.
Abstract: Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their who...
117 citations
TL;DR: In this paper, the authors analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them, and make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.
Abstract: In this article, we analyze the visual and symbolic strategies developed by students to express generalizations of number patterns and the connections they make between them. By analysis of a series of case studies, we compare the approaches adopted by students working through parallel task sequences, which integrate different computer tools in different ways. Finally, we make suggestions as to how students might be encouraged to exploit visual reasoning alongside the symbolic and draw out implications for curriculum design.
99 citations
TL;DR: This paper investigated the effects of higher-order and lower-order questions on student learning in 8th-grade mathematics classrooms in Germany, Japan, and the United States, and found that higher order questions did not improve student learning.
Abstract: Teachers in Germany, Japan, and the United States pose many questions to their students, and we assumed that the kinds of questions teachers asked influenced students' opportunities to think and communicate mathematically during lessons. However, previous research comparing the effects of higher and lower order questions on student learning have reported mixed results. Asking more higher order questions does not simply improve student learning. This article reports our attempt to go beyond simply counting the number of questions students are asked and to make sense of these inconsistent results. Two studies were conducted in which teachers' use of questions in 8th-grade mathematics classrooms in the 3 cultures was investigated. The 1st study, which employed primarily quantitative analyses, asked 2 questions: (a) How much do teachers and students talk during mathematics lessons in the 3 cultures, and (b) what kinds of things do teachers and students say? The 2nd study, which integrated qualitative descript...
82 citations
TL;DR: In this paper, a case study was conducted with precalculus students enrolled in a moderate-sized private research university, where they were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation.
Abstract: in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to mak...
53 citations
TL;DR: The human mind inevitably comprehends the world in mathematical terms (among others) and teachers should attempt to uncover the mathematical ideas contained in their students' thinking because much, but not all, of the mathematics curriculum is immanent in children's informal and invented knowledge.
Abstract: The human mind inevitably comprehends the world in mathematical terms (among others). Children's informal and invented mathematics contains on an implicit level many of the mathematical ideas that teachers want to promote on a formal and explicit level. These ideas may be innate, constructed for the purpose of adaptation, or picked up from an environment that is rich in mathematical structure, regardless of culture. Teachers should attempt to uncover the mathematical ideas contained in their students' thinking because much, but not all, of the mathematics curriculum is immanent in children's informal and invented knowledge. This mathematical perspective requires a focus not only on the child's constructive process but also on the mathematical content underlying the child's thinking. Teachers then can use these crude ideas as a foundation on which to construct a significant portion of classroom pedagogy. In doing this, teachers should recognize that children's invented strategies are not an end in themselv...
40 citations
TL;DR: This paper analyzed video-taped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms in terms of a recurrent pattern in public discourse among a teacher and students.
Abstract: Videotaped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms were analyzed in terms of a recurrent pattern in public discourse among a teacher and students. This pattern—called inquiry, response, feedback—occurs when a teacher initiates discourse (mostly with an inquiry), a student or students respond (often with an answer to the teacher inquiry), and the teacher provides feedback to the student's response. We found2 approaches to the teaching-learning of the criteria for evaluating mathematical arguments. In the Japanese classroom, students were encouraged to offer their own argument to the whole class and evaluate arguments proposed by other students. They seldom were given direct evaluation by their teacher. In contrast, American teachers often gave individual elaboration as well as direct evaluation to the student's responses, and some of the teachers offered their own opinions about mathematics, about valid ways of argumentation, or about both. The Japanese appr...
23 citations
TL;DR: This paper observed that students react similarly to a wide variety of conceptually unrelated situations and that many responses that literature describes as alternative conceptions are interpreted as if they evolved from a small number of intuitive rules.
Abstract: Our work in science and mathematics education has led us to observe that students react similarly to a wide variety of conceptually unrelated situations. Our work suggests that many responses that literature describes as alternative conceptions are interpreted as if they evolved from a small number of intuitive rules. Two such rules are manifested in comparison tasks. The first, more A-more B, is reflected in students' responses to tasks in which 2 objects that differ in a certain, salient quantity A are described (A1 > A2). Students are then asked to compare the 2 objects with respect to another quantity B (B1 = B2 or B1 < B2). In these cases, a substantial number of students responded inadequately according to the more A (the salient quantity)-more B (the quantity in question) rule. The second, same amount of A-same amount of B, is activated in situations in which A1 = A2, but B1 ≠ B2. In such situations, students often incorrectly claim that B1 = B2 because A1 = A2. In this article, we demonstrate the ...
TL;DR: In this article, the relation between cognitive psychology and mathematics education is analyzed, and it is argued that mathematics education should not simply "borrow" from cognitive psychology; rather, our discipline should provide its own psychological research problems, its adapted investigation strategies, and even, in certain circumstances, its adequate original concepts.
Abstract: This article analyzes the relation between cognitive psychology, as a broad theoretical framework, and the psychology of mathematics education. It is argued that mathematics education should not simply "borrow" from cognitive psychology; rather, our discipline should provide its own psychological research problems, its adapted investigation strategies, and even, in certain circumstances, its adequate original concepts. It is argued that the didactical orientation of its research endeavors highlights new, original theoretical and applicative perspectives, perspectives that cognitive psychology cannot provide by itself. Some examples are described that emphasize the difference between the broad cognitive approach and that of the psychology of mathematics education.