Journal for Research in Mathematics Education

Online ISSN: 0021-8251
Publications
Chapter
The constructivist teaching experiment is used in formulating explanations of children’s mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations of theoretical constructs—that represent our understanding of children’s mathematical realities. However, the models must be distinguished from what might go on in children’s heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children’s construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher’s understanding of the children. KeywordsConstructivist teaching experiment-Model building-Clinical interview-Teaching episode-Counting scheme-Teacher as researcher
 
Article
A total of 602 students (59.5% female) from academically selected schools in Germany were tested at three time points--end of Grade 7, end of Grade 10, and middle of Grade 12--in order to investigate the relationships between academic interest and achievement in mathematics. In addition, sex differences in achievement, interest, and course selection were analyzed. At the end of Grade 10, students opted for either a basic or an advanced mathematics course. Data analyses revealed sex differences in favor of boys in mathematics achievement, interest, and opting for an advanced mathematics course. Further analyses by means of structural equation modeling show that interest had no significant effect on learning from Grade 7 to Grade 10, but did affect course selection--that is, highly interested students were more likely to choose an advanced course. Furthermore, interest at the end of Grade 10 had a direct and an indirect effect (via course selection) on achievement in upper secondary school. In addition, results suggest that, at least from Grade 7 to Grade 10, achievement affected interest--that is, high achievers expressed more interest than low achievers. Findings underline the importance of interest for academic choices and for self-regulated learning. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with teachers in Grades 1-5. Participating teachers generated a wider variety of student strategies, including more strategies that reflected the use of relational thinking, than did nonparticipating teachers. Students in participating classes showed significantly better understanding of the equal sign and used significantly more strategies reflecting relational thinking during interviews than did students in classes of nonparticipating teachers. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to examine teachers' perceptions and children's application of the model method. The subjects were 14 primary teachers from 4 schools and 151 Primary 5 children. The model method affords higher ability children without access to letter-symbolic algebra a means to represent and solve algebraic word problems. Partly correct solutions suggest that representation is not an all-or-nothing process in which model drawing is either completely correct or completely incorrect. Instead, an incorrect solution could be the consequence of misrepresentation of a single piece of information. Our findings offer avenues of support in word problem solving to children of average ability. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
The purpose of this article is to sketch a hypothetical descriptive framework of statistical knowledge for teaching. Because statistics is a discipline in its own right rather than a branch of mathematics, the knowledge needed to teach statistics is likely to differ from the knowledge needed to teach mathematics. Doing statistics involves many primarily nonmathematical activities, such as building meaning for data by examining the context and choosing appropriate study designs to answer questions of interest. Although there are differences between mathematics and statistics, the two disciplines do share common ground in that statistics utilizes mathematics. This connection suggests that existing research on mathematical knowledge for teaching can help inform research on statistical knowledge for teaching. I propose the use of research from the Learning Mathematics for Teaching (LMT) project to help shape the discussion. I conclude by identifying areas of needed research and suggesting directions for teacher education efforts in statistics. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
In this rejoinder to Baroody and colleagues (2007) (see record 2007-03311-003), I point out that there are several areas of agreement between my position and that of Baroody et al.--most notably that both procedural knowledge and conceptual knowledge are of critical importance in students' learning of mathematics. However, there are issues on which Baroody et al. and I do not agree. In particular, I elaborate on the idea that procedures can be known deeply, flexibly, and with critical judgment--positive learning outcomes that are exclusively about students' knowledge of procedures and not necessarily a result of connections to conceptual knowledge. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
This editorial written by Steve Williams is about research articles in the JRME. The author comments that as difficult as it can be (and usually is), he nevertheless feels strongly that there is great value in shortening, tightening, and focusing a research report before submitting it. It benefits editors, reviewers, readers, and authors, and results in more timely and interesting publications. Moreover, the author wants to assure readers that he feels your pain. He is currently in the process of reducing an accepted manuscript by 1,000 words of his choice, at the suggestion of a kind editor who he realizes (despite his occasional grumbling) has the best interests of him, the manuscript, and its readers at heart. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
 
Article
There has been increased engagement in studying discourse in the field of mathematics education. But what exactly is a discourse, and how do researchers go about analyzing discourses? This study examines 108 articles from 6 international journals in mathematics education by asking questions such as these: In which traditions and in relation to which kinds of epistemological assumptions are the articles situated? How is the concept of discourse used and defined? How are mathematical aspects of the discourse accentuated? The results of this study show that a variety of conceptualizations are used for analyzing discourses but also that many articles would benefit from strengthening those conceptualizations by explicitly defining the concept of discourse, situating the article in relation to epistemological assumptions, and relating the work to other discourse studies in mathematics education. (Contains 14 footnotes and 7 tables.)
 
Article
Overall scale scores from the National Assessment of Educational Progress (NAEP) indicate that there was only minimal improvement in the mathematics performance of high school students between 1978 and 2004. Using recently released data from the Long-Term Trend (LTT) NAEP, this study describes the content covered on the LTT NAEP and the performance of 17-year-old students on that content. In addition, it demonstrates that although overall gains in performance were small, there were areas within mathematics in which performance improved substantially and others in which students in 2004 did not do as well as their counterparts of the 1970s and 1980s. Specifically, performance on 3 items involving multiplication of whole numbers by fractions deteriorated but performance improved on most tasks involving percents and geometry. Performance was stable on most items assessing algebraic reasoning and logical reasoning and was stable or improved modestly on items assessing estimation, interpretation of tables and graphs, and understanding of integers. (Contains 3 figures, 8 tables, and 9 footnotes.)
 
Article
This bibliography provides information and brief annotations for 148 papers reporting research into the effectiveness of self-paced instruction in mathematics. The citations are organized into three major categories: research summaries (8 papers), studies comparing the effectiveness of self-paced programs with that of more traditional programs (101 papers), and studies designed to analyze or evaluate specific components of self-paced programs (39 papers). The papers annotated deal with mathematics instruction at all levels from the primary grades through college, and with a variety of cognitive and affective criteria for judging the effectiveness of instruction. Each annotation notes the grade levels at which the study was performed and summarizes the major findings. (SD)
 
Article
This listing of research related to mathematics education provides brief annotations for articles and dissertations (K-12) published in 1975. Studies on college-level and other post-secondary school levels are also listed, but without annotations. (SD)
 
Article
This is the seventh annual listing of research on mathematics education published in JRME . The research for K–12 is organized alphabetically by author(s) within three categories—research summaries, journal-published reports, and dissertation abstracts. The subset of Piagetian-oriented research is listed separately in each of the latter two divisions. The K–12 listing is followed by references to articles and dissertations at the college and other postsecondary levels. All entries are annotated this year.
 
Article
An annual annotated listing of research on mathematics education is presented. The research is organized alphabetically by author(s) within three categories (research summaries, journal-published reports, and dissertation abstracts). Grade and age levels are indicated for each reference. An index of general topics is appended to help readers locate studies of particular interest. Included in the listing are studies in which mathematics education was not the sole or primary focus of the research. While most of these peripheral studies are not annotated, those studies specific to mathematics are annotated, and most annotations indicate one principal finding of the study. (MK)
 
Article
This is the 12th annual listing of research on mathematics education. The research noted is alphabetically organized by author(s) within the following three categories: (1) research summaries; (2) journal-published reports; and (3) dissertation abstracts. Grade or age level is indicated for each reference. Included in the listing are studies in which mathematics education was not the sole or primary focus of research. While most of these peripheral studies are not annotated, those specific to mathematics are. Most annotations indicate one principle finding of the study. A list of the journals searched is provided, and the number of references from each cited journal is noted. An index of general topics is appended to help readers locate studies of particular interest. (MP)
 
Article
This is the 13th annual listing of research on mathematics education. Annotated references are organized alphabetically by author within three categories: (1) research summaries; (2) journal-published reports; and (3) dissertation abstracts. An index is also provided to help locate references to designated mathematical topics. Topic areas include: achievement; algebra; arithmetic operations; attitudes/anxiety; calculators and computers; cognitive style; diagnosis and remediation; ethnic and social variables; geometry and measurement; learning; learning disabilities; mathematics materials; number and numeration; organizing for instruction; problem-solving; sequencing; sex differences; and test analysis. Accompanying the author's name in this index is a grade-level designation. In addition, each annotation listed in the three major categories also includes a grade-level (or age-level) designation. Annotations generally indicate one principal finding of a study, although most have additional findings. Therefore, the original report should be checked for other results as well as for limitations affecting the validity of the findings. Several studies in which mathematics education was not the primary focus are also included. Such studies are usually not annotated. (JN)
 
Article
Lists references of 16 research summaries, 152 articles, and 220 dissertations alphabetically by author within each category. Provides the grade or age level for each reference. A brief abstract is presented for most of the references. Journals searched and author index by content are included. (YP)
 
Teacher Reports on How Their Lessons Reflect Current Ideas About Teaching and Learning Mathematics: 1995 and 1999 % of teachers
Average percentage of making connections problem statements per lesson solved by using processes of each type: 1995 and 1999
Article
Debates about the future of school mathematics in the United States often center on whether standards-based instruction is improving or undermining students' achievement. Critical for making progress in these debates is information about the actual nature of classroom practice in U.S. classrooms. This article focuses on one key element of classroom practice--teaching--and presents the results of two studies of randomly selected, nationally representative U.S. eighth-grade mathematics lessons that were videotaped as part of the TIMSS 1995 and 1999 Video Studies. (Contains 4 figures, 6 tables and 11 footnotes.)
 
Article
California is one of 4 states that have accelerated addition and subtraction basic-facts memorization. This article reports on teacher practices, first-grade achievement of the standard, and a broader conception of basic-facts competence. Even among students from the highest performing schools, fewer than 11% made progress toward the memorization standard equivalent to their progress through the school year. Several negative correlations between instructional strategies and student retrieval suggest that teachers may benefit from professional development targeted at basic-facts teaching and learning. Textbook reliance was negatively correlated with basic-facts retrieval, suggesting that educators and policymakers may want to reexamine assumptions about the efficacy of traditional first-grade textbooks. This study's findings may prove useful to teachers, professional development trainers, and textbook publishers as they consider ways to improve basic-facts learning among early elementary children. (Contains 5 figures, 11 tables and 12 footnotes.)
 
Article
This article is about 8 African American middle school boys who have experienced success in mathematics. Working within a phenomenological methodological framework, the researcher investigated the limitations these students encounter and the compensating factors they experience. Critical race theory was the theoretical framework for this study; counter-storytelling was utilized to capture the boys' experiences, which is in stark contrast to the dominant literature concerning African American males and mathematics. Five themes emerged from the data: (a) early educational experiences, (b) recognition of abilities and how it was achieved, (c) support systems, (d) positive mathematical and academic identity, and (e) alternative identities.
 
Article
Each of 20 high school algebra teachers taught a lesson on direct variation to one first-year algebra class. The students (N=455) had not previously been taught this topic in class. Before the lessons were taught, each teacher was given a list of lesson objectives. Immediately after each lesson, a posttest that focused on the lesson objectives was administered. The teachers were not shown the posttest before they taught their lessons. Correlations were found between the mean posttest scores for the classes and several variables pertaining to teacher discourse.
 
Article
A substantial amount of research in mathematics education seeks to document disparities in achievement between middle-class White students and students who are Black, Latina/Latino, First Nations, English language learners, or working class. I outline the dangers in maintaining an achievement-gap focus. These dangers include offering little more than a static picture of inequities, supporting deficit thinking and negative narratives about students of color and working-class students, perpetuating the myth that the problem (and therefore solution) is a technical one, and promoting a narrow definition of learning and equity. (Contains 1 footnote.)
 
Article
Using data from the Longitudinal Study of American Youth (LSAY), we examined the extent to which students' mathematics coursework regulates (influences) the rate of growth in mathematics achievement during middle and high school. Graphical analysis showed that students who started middle school with higher achievement took individual mathematics courses earlier than those with lower achievement. Immediate improvement in mathematics achievement was observed right after taking particular mathematics courses (regular mathematics, prealgebra, algebra I, trigonometry, and calculus). Statistical analysis showed that all mathematics courses added significantly to growth in mathematics achievement, although this added growth varied significantly across students. Regular mathematics courses demonstrated the least regulating power, whereas advanced mathematics courses (trigonometry, precalculus, and calculus) demonstrated the greatest regulating power. Regular mathematics, prealgebra, algebra I, geometry, and trigonometry were important to growth in mathematics achievement even after adjusting for more advanced courses taken later in the sequence of students' mathematics coursework. (Contains 4 tables, 8 figures, and 7 footnotes.)
 
Article
Materials were written for an entire year's course in geometry in which transformations were used to develop the concepts of congruence, similarity, and symmetry, as well as being a vehicle for proof. This paper presents a study involving 413 students using these materials and 483 control students, comparing performance on standard geometry content and attitudes. Preliminary studies were done on perceptual, arithmetic, and algebraic skills. Student comprehension of transformation-related concepts, the problem of implementation, and attitudes of teachers were informally studied. While the data indicated that the experimental Ss could learn the material, few differences were detected with the measures employed--a significant difference ($p) in favor of the C group was found on the posttest of standard geometry content. Pre- (September) and posttest (June) student attitude data indicated a change in mean score, less positive, for both E and C groups ($p for the C group). Attitude differences between E and C were not significant. Informal feedback from a teacher questionnaire indicated a favorable reaction to the experimental materials.
 
Article
In this article we describe gender gaps in mathematics achievement and attitude as measured by the U.S. National Assessment of Educational Progress (NAEP) from 1990 to 2003. Analyzing relationships among achievement and mathematical content, student proficiency and percentile levels, race, and socioeconomic status (SES), we found that gender gaps favoring males (1) were generally small but had not diminished across reporting years, (2) were largest in the areas of measurement, number and operations (in Grades 8 and 12) and geometry (in Grade 12), (3) tended to be concentrated at the upper end of the score distributions, and (4) were most consistent for White, high-SES students and non-existent for Black students. In addition, we found that female students' attitudes and self-concepts related to mathematics continued to be more negative than those of male students. (Contains 6 footnotes, 3 figures, and 4 tables.)
 
Article
This retrospective study examined the impact of prior mathematics achievement on the relationship between high school mathematics curricula and student postsecondary mathematics performance. The sample (N = 4,144 from 266 high schools) was partitioned into 3 strata by ACT mathematics scores. Students completing 3 or more years of a commercially developed curriculum, the University of Chicago School Mathematics Project curriculum, or National Science Foundation-funded curriculum comprised the sample. Of interest were comparisons of the difficulty level and grade in their initial and subsequent college mathematics courses, and the number of mathematics courses completed over 8 semesters of college work. In general, high school curriculum was not differentially related to the pattern of mathematics grades that students earned over time or to the difficulty levels of the students' mathematics course-taking patterns. There also was no relationship between high school curricula and the number of college mathematics courses completed. (Contains 9 tables and 1 footnote.)
 
Article
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences between the work of these less successful students and the traditional problem-solving patterns of less successful students. These less successful students used the graphing software to obtain a broader view, to confirm conjectures, and to complete difficult operations. However, they delayed using symbolic formalism, and most of their solution attempts focused on numeric and graphic representations. Their process of reaching a solution was found to be relatively long, and the graphing software tool was often not used at all because it did not support symbolic formulation and manipulations. (Contains 12 figures and 1 table.)
 
Article
Currently, there are more theories of learning in use in mathematics education research than ever before (Lerman & Tsatsaroni, 2004). Although this is a positive sign for the field, it also has brought with it a set of challenges. In this article, I identify some of these challenges and consider how mathematics education researchers might think about, and work with, the multiple theories available. I present alternatives to views of the competition or supersession of theories and indicate the kinds of discussions that will support effective theory use in mathematics education research. I describe the potential for mathematics education researchers to make informed, justified choices of a theory or theories to address particular research agendas. (Contains 10 footnotes.)
 
Article
Although students of all levels of education face serious difficulties with proof, there is limited research knowledge about how instruction can help students overcome these difficulties. In this article, we discuss the theoretical foundation and implementation of an instructional sequence that aimed to help students begin to realize the limitations of empirical arguments as methods for validating mathematical generalizations and see an intellectual need to learn about secure methods for validation (i.e., proofs). The development of the instructional sequence was part of a 4-year design experiment that we conducted in an undergraduate mathematics course, prerequisite for admission to an elementary (Grades K-6) teaching certification program. We focus on the implementation of the instructional sequence in the last of 5 research cycles of our design experiment to exemplify our theoretical framework (in which cognitive conflict played a major role) and to discuss the promise of the sequence to support the intended learning goals. (Contains 5 tables, 4 figures, and 15 footnotes.)
 
Article
In this article, we use data from a large-scale Brazilian national assessment to discuss the relation between reform teaching and equity in mathematics education. We study the dimensionality of teaching style to better qualify what reform teaching means. We then use hierarchical linear models to explore whether reform teaching is associated with student achievement in mathematics and with student socioeconomic status (SES). Our results indicate that reform and traditional teaching are not opposite sides of a one-dimensional axis. They also emphasize both that reform teaching is related to higher school average achievement in mathematics and that the dissemination of reform teaching contributes to minimize the achievement gap between students who attend schools with low average SES and students who attend schools with high average SES. However, our results also show that reform is associated with an increase in within-school inequality in the social distribution of achievement.
 
Article
This article reports on an investigation of how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in proving. Confirming what was hypothesized, participants found it remarkable that a teacher would allow a student to make an assumption while proving. But they perceived this episode in various ways, casting the episode as one of as many as 10 different stories. Those different castings of the episode make use of intellectual resources for professional practice that practitioners could use to negotiate the norms of a situation in which they had made a tactical but problematic move. This collection of stories attests to the effectiveness of the technique used for eliciting the rationality of mathematics teaching: By confronting practitioners with episodes of teaching in which some norms have been breached, one can learn about the rationality underlying the norms of customary teaching. ["Seeing a Colleague Encourage a Student to Make an Assumption while Proving: What Teachers Put in Play when Casting an Episode of Instruction" was written with Gloriana Gonzalez.] (Contains 14 footnotes and 3 figures.)
 
Article
The Mathematics Attitude Inventory, designed to measure the attitudes toward mathematics of secondary students, and its accompanying user's manual, are described. The six scales measure perception of mathematics teachers, value of mathematics, self-concept in mathematics, and anxiety toward, enjoyment of, and motivation in mathematics. (MK)
 
Article
Response to an earlier article in JRME in which the authors propose a constructivist alternative to the representational view of mind. Argues that the original article misinterprets the postepistemological perspective, confuses ontological and epistemological issues, and mistakes the pragmatic force of the constructivist argument. (45 references) (Author/MKR)
 
Article
In this experimental study, prospective elementary school teachers enrolled in a mathematics course were randomly assigned to (a) concurrently learn about children's mathematical thinking by watching children on video or working directly with children, (b) concurrently visit elementary school classrooms of conveniently located or specially selected teachers, or (c) a control group. Those who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not. (Contains 5 figures, 6 footnotes and 6 tables. Appended are: IMAP's Seven Beliefs Assessed; and Odds Ratios.)
 
Article
Remarks adapted from the Presidential Address at the 74th Annual Meeting of the National Council of Teachers of Mathematics in April 1996 charge educators with believing that every student can learn mathematics, every teacher must have adequate support and professional development opportunities, and every parent must have a vested interest in achieving higher standards for mathematics education. (AIM)
 
Article
Eight sixth-grade students received individualized instruction on the addition and subtraction of fractions in a one-to-one setting for 6 weeks. Instruction was specifically designed to build upon the student's prior knowledge of fractions. It was determined that all students possessed a rich store of prior knowledge about parts of wholes in real world situations based upon whole numbers. Students related fraction symbols and procedures to prior knowledge in ways that were meaningful to them; however, there was a danger of this prior knowledge interfering when it reflected algorithmic procedures rather than fraction ideas in real world situations. (Author/PK)
 
Article
This article examines a calculus graphing schema and the triad stages of schema development from Action-Process-Object-Schema (APOS) theory. Previously, the authors studied the underlying structures necessary for students to process concepts and enrich their knowledge, thus demonstrating various levels of understanding via the calculus graphing schema. This investigation built on this previous work by focusing on the thematization of the schema with the intent to expose those possible structures acquired at the most sophisticated stages of schema development. Results of the analyses showed that these successful calculus students, who appeared to be operating at varying stages of development of their calculus graphing schemata, illustrated the complexity involved in achieving thematization of the schema, thus demonstrating that thematization was possible.
 
Article
This article discusses the challenge of improving the interrelationships between research and practice in mathematics education, and it outlines actions being taking to respond to that challenge. The need for improvement is bidirectional. The practice of classroom mathematics teaching needs to be better informed by an understanding of the implications of existing bodies of research, and researchers need to learn more from the insights and knowledge of practitioners. Building on its series of initiatives designed to use research to guide mathematics teaching and learning, NCTM has made a new commitment to a flexible, nimble, and sustainable initiative that will strengthen the bidirectional link between research and practice. This initiative includes the development of Research Analyses, Briefs, and Clips (ABCs), research syntheses designed through collaboration of teacher leaders and researchers to inform instructional leaders and policymakers about research perspectives on critical issues of practice. (Contains 1 figure and 8 footnotes.)
 
Article
The recent releases of two major international comparative studies that addressed mathematics--the Trends in International Mathematics and Science Study 2003 (TIMSS 2003) and the Program for International Student Assessment 2003 (PISA 2003)--provide opportunities and challenges for mathematics education researchers interested in using the findings, instruments, and conceptual and theoretical perspectives of these studies as catalysts for secondary analysis and additional research. In particular, a number of important questions in mathematics education in the United States can be pursued, using various resources from these studies as a base, by mathematics education researchers and mathematicians collaborating with statisticians and assessment experts. We highlight some of the main findings of TIMSS 2003 and PISA 2003, and suggest how some of the instruments that were used in these studies, as well as the conceptual frameworks and priorities that guided them, might be beneficial in pursuing pressing questions in such areas as the role of curriculum in mathematics performance, the ways in which social contextual variables interact with mathematics learning, and the challenges in measuring the ability to use mathematics in real-world situations.
 
Article
Presents remarks from the Presidential Address at the 75th Annual Meeting of the National Council of Teachers of Mathematics held in Minneapolis in April, 1997. Two major challenges addressed were the need to change classroom practice and the need to rethink the definition of "basic". (AIM)
 
Article
This case-study research investigated changing teacher roles associated with two teachers' use of innovative mathematics materials at Grade 6. Using daily participant observation and regular interviews with the teachers and the project staff member responsible for providing in-school support, a picture emerged of changing teacher roles and of those factors influencing the process of change. One teacher demonstrated little change in either espoused beliefs or observed practice over the course of the study. The second teacher demonstrated increasing comfort with posing nonroutine problems to students and allowing them to struggle together toward a solution, without suggesting procedures by which the problems could be solved. He also increasingly provided structured opportunities for students' reflection on activities and learning. Major influences on this teacher's professional growth appeared to be the provision of the innovative materials and the daily opportunity to reflect on classroom events in conversations and interviews with the researcher.
 
Article
The major objective of this study was to investigate differential performances among 5-year-old children when using transitivity of matching relations. Instruments were constructed to measure the subjects' (1) knowledge of matching relations, (2) ability to conserve the relations, and (3) proficiency in making inferences using the transitive property of the relations. Three specially designed tests, administered individually to 21 boys and 21 girls, assessed knowledge of relational terminology, level of conservation and performance on manipulative tasks (matching, comparing, judging quantitative relations among objects). A surprising result of this investigation was that subjects in the high category of conservation did not perform significantly better on transitivity than subjects in the low category of conservation for each relation. In general, differential performance on transitivity between equivalence and order relational groups occurred only within the high conservation level. An extensive comparison is made between present results and Smedslund's (1963a) data for conservation and transitivity of discontinuous quantity. (WY)
 
Number of Symmetry Groups Created by Each Student on the Posttest
Article
Twelve middle school students working in pairs used a computer microworld to explore an introductory curriculum in transformation geometry. The microworld linked a symbolic representation (a set of simple Logo commands) with a visual display that showed the effects of each transformation. Worksheets were designed with the objective of encouraging the students to find and express mathematical patterns in the domain. The students were successful in constructing an accurate working understanding of the transformations. There was a tendency for symbolic overgeneralization in some activities, but the students were able to use visual feedback from the microworld and discussions with their partners to correct their own errors.
 
Tree diagrams for the five 2-dimensional strategies.
Tree diagrams for the five three-dimensional strategies 
Article
The study investigated the strategies that 7- to 12-year-old children spontaneously apply to the solution of novel combinatorial problems. The children were individually administered a set of six problems involving the dressing of toy bears in all possible combinations of tops and pants (twodimensional) or tops, pants, and tennis rackets (three-dimensional). Two sets of solution procedures were identified, each comprising a series of five increasingly complex strategies ranging from trial-and-error approaches to sophisticated odometer procedures. Results suggested that experience with the two-dimensional problems enabled children to adopt and subsequently transform their efficient 2-D odometer strategy (where one item is held constant) into the most sophisticated 3-D odometer strategy, which involved working simultaneously with two constant items. The study highlights the importance of discrete mathematics as a source of problem-solving activities in which children are motivated to create, modify, and extend their own theories.
 
Article
In this study, a researcher-teacher examined seventh-graders' experiences with a problem-centered curriculum and pedagogy. Analyses of interviews, surveys, teaching journal entries, and audio recordings focused on differences between lower- and higher-socioeconomic status (SES) students' reactions to learning mathematics through problem solving. While higher-SES students displayed confidence and solved problems with an eye toward the intended mathematical ideas, the lower-SES students preferred more external direction and sometimes approached problems in a way that allowed them to miss their intended mathematical point. The lower-SES students more often became "stuck" in the open and contextualized nature of the problems. An examination of sociological literature revealed ways in which these patterns in the data could be related to more than individual differences in temperament or achievement between the children. This study suggests that class cultural differences could relate to students' approaches to learning mathematics through solving open, contextualized problems. This paper discusses equity-related challenges to be faced as well as interventions that could be helpful in our attempts to center mathematics instruction around problem solving. Contains 32 references. (Author)
 
Article
The purpose of this study was to compare the classroom processes of girls and boys who differed in confidence in their ability to learn mathematics. The students were observed daily during their seventh-grade mathematics classes for 3 to 4 weeks. All 93 students observed (i.e., the target students) scored at or above the mean on a standardized test of mathematics achievement. Trained observers recorded characteristics of each verbal interaction between a target student and the teacher and the amount of time each target student spent on task in mathematics. Boys were involved in more public interactions with their teacher than girls were. High- and low-confidence students differed very little in their interactions with the teacher. Differences between boys and girls in their interactions with the teacher varied from classroom to classroom.
 
Article
This article reports on models of teaching that developed as outgrowths of a study of middle-grades mathematics classes. Grounded theory methodology and sociolinguistic tools were used to move from classroom observations and interviews to line-by-line coding of classroom discourse, to mapping the flow of talk and verbal assessment moves, to a multilevel analysis of the relationships of forms of talk and verbal assessment, and, ultimately, to models of teaching that promote discourse on a continuum from univocal (conveying meaning) to dialogic (constructing meaning through dialogue). Three specific cases are highlighted that represent deductive (associated with univocal), inductive (associated with dialogic), and mixed (a hybrid of deductive and inductive) models of teaching. Teaching practices associated with each model are illustrated and discussed.
 
Article
Examines the changes that have been evolving in teaching and learning practices in mathematics classrooms by reflecting on issues in the past and present. Discusses what should be done in order to prepare students for the future. Contains 16 references. (ASK)
 
Article
This article considers the question of what specific actions a teacher might take to create a culture of inquiry in a secondary school mathematics classroom. Sociocultural theories of learning provide the framework for examining teaching and learning practices in a single classroom over a two-year period. The notion of the zone of proximal development (ZPD) is invoked as a fundamental framework for explaining learning as increasing participation in a community of practice characterized by mathematical inquiry. The analysis draws on classroom observation and interviews with students and the teacher to show how the teacher established norms and practices that emphasized mathematical sense-making and justification of ideas and arguments and to illustrate the learning practices that students developed in response to these expectations.
 
Article
Mathematics teachers' selection and implementation of instructional tasks were analyzed before, during, and after their participation in a professional development initiative that focused on selecting and enacting cognitively challenging mathematical tasks. Data collected from 18 secondary mathematics teacher participants included tasks and student work from teachers' classrooms, lesson observations, and interviews. Ten secondary mathematics teachers who did not participate in the professional development initiative served as the contrast group and participated in 1 lesson observation each. Analysis of the data indicated that, following their participation in the professional development initiative, project teachers more frequently selected high-level tasks as the main instructional tasks in their classrooms and had improved the maintenance of high-level cognitive demands. Significant differences existed between project teachers and the contrast group in task selection and implementation. These differences were not influenced by the use of Standards-based or conventional curricula in project teachers' classrooms. (Contains 1 footnote, 8 tables, and 3 figures.)
 
Article
This study investigated introductory calculus students' spontaneous reasoning about limit concepts guided by an interactionist theory of metaphorical reasoning developed by Max Black. In this perspective, strong metaphors are ontologically creative by virtue of their emphasis (commitment by the producer) and resonance (support for high degrees of elaborative implication). Analysis of 120 students' written and verbal descriptions of their thinking about challenging limit concepts resulted in the characterization of 5 clusters of strong metaphors. These clusters were based on the objects, relationships, and logic related to intuitions about (a) a collapse in dimension, (b) approximation and error analyses, (c) proximity in a space of point-locations, (d) a small physical scale beyond which nothing exists, and (e) the treatment of infinity as a number. Students' reasoning with these metaphors had significant implications for the images they formed and the claims and justifications they provided about multiple limit concepts. (Contains 4 tables and 1 figure.)
 
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