Showing papers in "Journal for Research in Mathematics Education in 2002"
TL;DR: This paper developed a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events and found that students appeared to have difficulty forming images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations.
Abstract: The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of high-performing 2nd-semester calculus students’ ability to reason about covarying quantities in dynamic situations. The study revealed that these students were able to construct images of a function’s dependent variable changing in tandem with the imagined change of the independent variable, and in some situations, were able to construct images of rate of change for contiguous intervals of a function’s domain. However, students appeared to have difficulty forming images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations. These findings suggest that curriculum and instruction should place increased emphasis on moving students from a coordinated image of two variables changing in tandem to a coordinated image of the instantaneous rate of change with continuous changes in the independent variable for dynamic function situations.
493 citations
TL;DR: This paper found that using reform-oriented mathematics curricula achieved a reduction in linguistic, ethnic, and class inequalities in their schools, suggesting that relational analyses of equity go beyond the curriculum to include the teacher and their teaching.
Abstract: Some researchers have expressed doubts about the potential of reform-oriented curricula to promote equity. This article considers this important issue and argues that investigations into equitable teaching must pay attention to the particular practices of teaching and learning that are enacted in classrooms. Data are presented from two studies in which middle school and high school teachers using reform-oriented mathematics curricula achieved a reduction in linguistic, ethnic, and class inequalities in their schools. The teaching and learning practices that these teachers employed were central to the attainment of equality, suggesting that it is critical that relational analyses of equity go beyond the curriculum to include the teacher and their teaching.
306 citations
TL;DR: This article examined 16 in-service secondary school mathematics teachers' conceptions of proof and found that teachers recognize the variety of roles that proof plays in mathematics; noticeably absent, however, was a view of proof as a tool for learning mathematics.
Abstract: Recent reform efforts call on secondary school mathematics teachers to provide all students with rich opportunities and experiences with proof throughout the secondary school mathematics curriculum--opportunities and experiences that reflect the nature and role of proof in the discipline of mathematics. Teachers' success in responding to this call, however, depends largely on their own conceptions of proof. This study examined 16 in-service secondary school mathematics teachers' conceptions of proof. Data were gathered from a series of interviews and teachers' written responses to researcher-designed tasks focusing on proof. The results of this study suggest that teachers recognize the variety of roles that proof plays in mathematics; noticeably absent, however, was a view of proof as a tool for learning mathematics. The results also suggest that many of the teachers hold limited views of the nature of proof in mathematics and demonstrated inadequate understandings of what constitutes proof.
272 citations
TL;DR: The authors compared the achievement of students using either commercial curricula (CC) for initial fraction learning with students using the Rational Number Project (RNP) fraction curriculum, which placed particular emphasis on the use of multiple physical models and translations within and between modes of representation-pictorial, manipulative, verbal, real-world, and symbolic.
Abstract: This study contrasted the achievement of students using either commercial curricula (CC) for initial fraction learning with the achievement of students using the Rational Number Project (RNP) fraction curriculum. The RNP curriculum placed particular emphasis on the use of multiple physical models and translations within and between modes of representation-pictorial, manipulative, verbal, real-world, and symbolic. The instructional program lasted 28-30 days and involved over 1600 fourth and fifth graders in 66 classrooms that were randomly assigned to treatment groups. Students using RNP project materials had statistically higher mean scores on the posttest and retention test and on four (of six) subscales: concepts, order, transfer, and estimation. Interview data showed differences in the quality of students' thinking as they solved order and estimation tasks involving fractions. RNP students approached such tasks conceptually by building on their constructed mental images of fractions, whereas CC students relied more often on standard, often rote, procedures when solving identical fraction tasks. These results are consistent with earlier RNP work with smaller numbers of students in several teaching experiment settings.
252 citations
TL;DR: This paper explored the challenges of learning to think about data as signal and noise in the context of three different statistical processes: repeated measures, measuring individuals, and dichotomous events, and made several recommendations about research and instruction.
Abstract: The idea of data as a mixture of signal and noise is perhaps the most fundamental concept in statistics. Research suggests, however, that current instruction is not helping students to develop this idea, and that though many students know, for example, how to compute means or medians, they do not know how to apply or interpret them. Part of the problem may be that the interpretations we often use to introduce data summaries, including viewing averages as typical scores or fair shares, provide a poor conceptual basis for using them to represent the entire group for purposes such as comparing one group to another. To explore the challenges of learning to think about data as signal and noise, we examine the "signal/noise" metaphor in the context of three different statistical processes: repeated measures, measuring individuals, and dichotomous events. On the basis of this analysis, we make several recommendations about research and instruction.
241 citations
TL;DR: The authors made a distinction between mathematical reasoning and scientific reasoning in mathematics, on the basis of the criteria used to accept or reject reasoning in each domain, and made a contribution toward clarifying what mathematical reasoning is and what it looks like in school contexts.
Abstract: This article makes a contribution toward clarifying what mathematical reasoning is and what it looks like in school contexts. It describes one pattern of reasoning observed in the mathematical activity of students in a Grade 5 class and discusses ways in which this pattern is or is not mathematical in order to clarify the features of a pattern of reasoning that are important for making such a judgment. The pattern involves conjecturing a general rule, testing that rule, and then either using it for further exploration, rejecting it, or modifying it. Each element of reasoning in this pattern is described in terms of the ways of reasoning used and the degree of formulation of the reasoning. A distinction is made between mathematical reasoning and scientific reasoning in mathematics, on the basis of the criteria used to accept or reject reasoning in each domain.
135 citations
TL;DR: In this paper, the authors describe a novel assessment method used to examine Japanese and American teachers' ideas about what constitutes effective mathematics pedagogy, and find that U.S. teachers have more culturally sanctioned options for teaching mathematics; however, Japanese teachers may have a more detailed and widely shared theory about how to teach effectively.
Abstract: This article describes a novel assessment method used to examine Japanese and American teachers’ ideas about what constitutes effective mathematics pedagogy. Forty American and 40 Japanese teachers independently evaluated either an American or Japanese mathematics lesson captured on videotape. Their comments were classified into over 1600 idea units, which were then sorted into a hierarchy of categories derived from the data. Next, the authors hypothesized underlying ideal instructional scripts that could explain the patterns of responses. Whereas the U.S. teachers were supportive of both traditional and nontraditional elementary school mathematics instruction and had different scripts for the two lessons, the Japanese teachers had only one ideal lesson script that was closely tied to typical Japanese mathematics instruction. The findings suggest that U.S. teachers may have more culturally sanctioned options for teaching mathematics; however, Japanese teachers may have a more detailed and widely shared theory about how to teach effectively.
123 citations
TL;DR: This article investigated the arithmetical and algebraic problem-solving strategies and skills of preservice primary school and secondary school teachers in Flanders, Belgium, both at the beginning and at the end of their teacher training.
Abstract: The study reported here investigated the arithmetical and algebraic problem-solving strategies and skills of preservice primary school and secondary school teachers in Flanders, Belgium, both at the beginning and at the end of their teacher training. The study then compared these aspects of the preservice teachers' own problem-solving behavior with the way in which they evaluated students' arithmetical and algebraic solutions to problems. Future secondary school teachers clearly preferred the use of algebra, both in their own solutions and in their evaluations of students' work, even when an arithmetical solution seemed more evident. Some future primary school teachers tended to apply exclusively arithmetical methods, leading to numerous failures on difficult word problems, whereas others were quite adaptive in their strategy choices. Taken as a whole, the evaluations of the preservice primary school teachers were more closely adapted to the nature of the task than those of their secondary school counterparts.
112 citations
TL;DR: This article used a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof.
Abstract: This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contract and double bind to uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching-what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition-are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.
110 citations
TL;DR: This article found that broadly similar work could be interpreted differently by different teachers and that the formation of teachers' views of students and evaluation of their mathematical attainments appeared to be influenced by surface features of students' work and behavior and by individual teachers' prior expectations.
Abstract: This paper discusses fairness and equity in assessment of mathematics. The increased importance of teachers’ interpretative judgments of students’ performance in highstakes assessments and in the classroom has prompted this exploration. Following a substantial theoretical overview of the field, the issues are illustrated by two studies that took place in the context of a reformed mathematics curriculum in England. One study is of teachers’ informal classroom assessment practices; the other is of their interpretation and evaluation of students’ formal written mathematical texts (i.e., responses to mathematics problems). Results from both studies found that broadly similar work could be interpreted differently by different teachers. The formation of teachers’ views of students and evaluation of their mathematical attainments appeared to be influenced by surface features of students’ work and behavior and by individual teachers’ prior expectations. We discuss and critique some approaches to improving the quality and equity of teachers’ assessment.
80 citations
TL;DR: Noss, Noss, and Pozzi as mentioned in this paper explored nurses' conceptions of concentration in a more detailed way than is possible during in situ observations and examined the degree of situatedness of nurses' knowledge and reasoning.
Abstract: ion in Expertise: A Study of Nurses' Conceptions of Concentration Richard Noss, Celia Hoyles, and Stefano Pozzi University of London, United Kingdom Building on an ethnographic study of nurses' working practices on the ward (Hoyles, Noss, & Pozzi, 2001), we elaborate the notion of situated abstraction as an analytical tool for understanding nurses' conceptions of the intensive quantity of drug concentration. Data were gathered through interviews based on simulations of tasks observed to be problematic in the earlier study. The methodology was designed to explore nurses' conceptions in a more detailed way than is possible during in situ observations and to undertake a pointed examination of the degree of situatedness of nurses' knowledge and reasoning. Analysis of the data demonstrated that nurses' conceptions were abstracted within their practice when they were able to coordinate their mathematical knowledge with their professional expertise, yet their conceptions were situated, as evidenced by their difficulties in realizing this coordination in tasks too distant from their practice.
TL;DR: The Learning Principle propounded in Principles and Standards for School Mathematics (NCTM, 2000) rehearses the familiar distinction between facts/procedures and understanding as a central guiding principle of teaching reform.
Abstract: The Learning Principle propounded in Principles and Standards for School Mathematics (NCTM, 2000) rehearses the familiar distinction between facts/procedures and understanding as a central guiding principle of teaching reform. This rhetorical stance has polarized mathematics educators in the "math wars," (Becker & Jacob, 1998), while creating the discursive space for mathematics teaching reform to be reified into a unitary "reform vision" (Lindquist, Ferrini-Mundy, & Kilpatrick, 1997)-a vision that teachers can all too easily come to see themselves as implementing rather than authoring. Crossdisciplinarity is a strategy for highlighting the discrete notions of learning that psychology thus far has succeeded in coherently articulating. This strategy positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities, at the same time marshaling accessible, theory-based guidance toward realization of its diverse possibilities.
TL;DR: Sixty Japanese children between the ages of 3 years 4 months and 7 years 5 months were individually interviewed to investigate the relationship between their levels of abstraction and their level of representation.
Abstract: Sixty Japanese children between the ages of 3 years 4 months and 7 years 5 months were individually interviewed to investigate the relationship between their levels of abstraction (as assessed by a task involving conservation of number) and their levels of representation (as assessed by a task asking for their graphic representation of small groups of objects). The investigation concluded that abstraction and representation are closely related and that children can represent at or below their level of abstrac- tion but not above this level. The educational implication is that educators need to focus more on the mental relationships children make (i.e., their abstraction) because the meaning children can give to conventional symbols depends on their level of abstraction.
TL;DR: This article examined children's understanding of the mathematical significance of unit size through problems that involved making judgments about the number of larger objects versus the smaller objects that would fit in a container or bounded space.
Abstract: Preschool children's understanding of the mathematical significance of unit size was examined through problems that involved making judgments about the number of larger objects versus the number of smaller objects that would fit in a container or bounded space. Children's judgments about effects of object size were elicited both before and after the presentation of a series of demonstration trials. The accuracy of children's judgments about the effects of object size improved significantly from pretest to posttest, and the degree of improvement was similar for 3-year-olds versus 4-year-olds and for Head Start children versus children who attended private preschools. The Head Start children, however, were much less likely than their peers attending private preschools to be able to articulate relevant quantitative features when asked to explain the outcomes that they observed during the training trials.