bluuebunny/crossref_metadata_2025 huggingface dataset:
Editorial: Needed: An Agenda for Equity in Mathematics Education
Authors: Secada Walter G.
Published: August 2020

The mathematics education research community needs to help in setting an agenda for equity in mathematics education. We must become more conscious about whether our studies incorporate tacit ways of viewing the world that legitimate the social arrangements by which diverse student populations fail, or whether they provide us with the vision and understandings that go beyond merely labeling failure in different ways.
Qualitative and Numerical Reasoning about Fractions and Rates by Seventh- and Eighth-Grade Students
Authors: Heller Patricia M. , Post Thomas R. , Behr Merlyn , Lesh Richard
Published: August 2020

This study examined the relationship between junior high school students' directional reasoning about rates and numerical reasoning on proportion-related word problems. Also explored was the extent to which the ability to solve context-free fraction exercises is related to the ability to solve mathematically similar word problems. Four hundred twenty-one seventh-grade and 492 eighth-grade students were given a test consisting of eight directional and eight proportion-related word problems and a fraction test that included 11 items that precisely paralleled the mathematical structure of the word problems. The correlation between the directional and numerical scales was .38 for seventh grade and .45 for eighth grade. Regression analysis indicated that a high directional score is related to greater numerical success on proportion-related problems. The low correlations between the mathematically similar problems on the fraction and word-problem tests indicate that students are not capitalizing on the structural similarities inherent in the problems, even when the numerical quantities are identical.
The Estimation of Set Size by Potentially Gifted Kindergarten-Age Children
Authors: Baroody Arthur J. , Gatzke Mary R.
Published: August 2020

The study explored young children's ability to estimate set sizes and use reference points such as 10. Eighteen children in a preschool-kindergarten program for the potentially gifted were asked to estimate the number of dots in a set (range 3 to 35); to gauge whether a set of dots was larger or smaller than a stated reference point of 5, 10, or 20; and to judge where a set fit in regard to two given reference points. With sets of 8, most estimates were within 25% of the actual value; with sets of 15 or larger, most estimates were not. A majority of children were successful on the task with a single reference point; performance varied on the task with two reference points. An error analysis indicated that many children could accurately place sets somewhat smaller than a reference point but had difficulty placing sets somewhat larger than a reference point. These results suggest that the children had not constructed accurate perceptual anchors but appeared to have an overexaggerated mental image of 5, 10, and 20.
A 10-Year Longitudinal Follow-up of Participants in a Fast-Paced Mathematics Course
Authors: Swiatek Mary Ann , Benbow Camilla Persson
Published: August 2020

Students who participated in a fast-paced mathematics course for highly mathematically talented students were surveyed 10 years later, at approximately age 23. Areas considered were (a) undergraduate experience, (b) graduate experience, (c) attitudes toward mathematics and science, and (d) self-esteem. Participants attended more prestigious undergraduate colleges than did nonparticipants. Participants were more likely to attend graduate school than were nonparticipants; this finding stemmed from differences among females. Self-esteem ratings, although high for both groups, were found to be higher for students who qualified for the class but did not participate. Attitudes toward math and science were equivalent between the two groups. Overall, participation in the fast-paced mathematics classes of the Study of Mathematically Precocious Youth (SMPY) was associated with stronger undergraduate education for all students and with more advanced education among females. The fast-paced classes caused gifted students no harm.
Review: Beyond Computational Skill
Authors: Leder Gilah
Published: August 2020

Mathematics is widely regarded as a crucial component ofthe school curriculum. Indeed, for the greater part of their schooling, students spend at least some time each day doing mathematics. Although there is no consensus about the exact nature of the content to be studied, there is much overlap in the curriculum to be found in different countries. At the elementary level, arithmetic exercise or calculations abound. Other branches such as algebra. calculus, or geometry are typically not encountered till later years. Marked similarities in the way the curriculum is delivered in many classrooms have also been noted (see, e.g., Romberg & Carpenter, 1986).
Number Sentences: Linking Addition and Subtraction Word Problems and Symbols
Authors: Carey Deborah A.
Published: August 2020

Twenty-four first-grade children were asked to write number sentences and select appropriate alternative number sentences for addition and subtraction word problems. Responses were qualitatively different across five clusters of children. Clusters were characterized by the degree of flexibility in accepting alternative number sentences for word problems. Children in all clusters could write and select open number sentences, such as a+□ =b and □ −a=b, that matched the semantic structure of problems for word problems with small and large numbers. Only the more advanced children could identify standard number sentences, a+b=□ and a−b=□, as appropriate representations for all addition and subtraction word problems. Flexibility in selecting alternative number sentences was related to number size, suggesting that knowledge of number relationships plays a role in the development of a general understanding of part-whole relationships.
Secondary School Calculus: Preparation or Pitfall in the Study of College Calculus?
Authors: Ferrini-Mundy Joan , Gaudard Marie
Published: August 2020

This study investigated the effects of various levels of secondary school calculus experience on performance in first-year college calculus, with focus on student performance on conceptual and procedural exam items. Analysis of covariance, with mathematics SAT score as a covariate, was employed to explore differences among four groups of students. Students who had a year of secondary school calculus, advanced placement or otherwise, differed significantly in performance from students who had either no calculus or a brief introduction to calculus prior to college. A brief secondary school introduction to calculus, in comparison with no secondary school calculus, provided an initial advantage in the college course. This slight advantage reappeared on the final exam and on the procedural subscale of the final exam. Students who had studied a full year of secondary school calculus performed significantly better than other groups throughout the first-semester course. The advantage was revealed more strongly in procedural than in conceptual items. There were no significant differences among the four groups of students on outcome measures in the second-semester course. Students with more secondary school calculus background were more likely to continue into the second semester of college calculus.
Gender Differences in a Psychological Model of Mathematics Achievement
Authors: Ethington Corinna A.
Published: August 2020

This study examined the validity of the Eccles et al. (1983) theoretical model of achievement behaviors using data from the Second International Mathematics Study. With the achievement behavior defined as performance (mathematics achievement), the dominant effects of the value of mathematics and expectations for success that are hypothesized by the model were not apparent, and effects differed for males and females. There were more direct and indirect influences on achievement from the psychological constructs for females, suggesting that the explanation of mathematics achievement within the context of this model was more complicated for females than for males.
Applying Personal Construct Psychology to the Study of Teachers' Knowledge of Fractions
Authors: Lehrer Richard , Franke Megan Loef
Published: August 2020

Personal construct psychology provides a coherent theoretical and methodological framework for the examination of teachers' knowledge. We report case studies of two teachers who varied in their knowledge about fractions and mathematical pedagogy. We used personal construct psychology and the logic of fuzzy sets to elucidate the content and organization of the teachers' knowledge of fractions. The approach proved especially useful for describing conditional relationships among content, general pedagogical, and pedagogical content knowledge frames. We also explored associations between teachers' personal constructions and their classroom teaching. These associations suggested that personal construct psychology shows considerable promise as a way of addressing issues of teacher knowledge in the context of the classroom.
Experiments and Relational Studies in Problem Solving: A Meta-Analysis
Authors: Hembree Ray
Published: August 2020

Results in 487 reports were integrated by meta-analysis to study four regions of problem solving: characteristics of problem solvers, conditions for harder and easier problems, effects of different instructional methods on problem-solving performance, and effects of classroom-related conditions on problem-solving performance. Direct significant links were found between problem solving and various measures of basic performance, especially skills in basic mathematics. Weaker correlations appeared between problem solving and IQ measures. A format consisting of full problem statements supported by diagrams, figures, or sketches directly related to better performance; training for skill in such representations provided the largest performance improvement. During the early grades K–5, no method of problem-solving instruction emerged as superior. Heuristics in middle grades 6–8 seemed mildly better than other approaches and gained a distinctly superior status in high school. A positive impact on students' performance also resulted from teachers especially trained in heuristical methods.
Response to Critique of Meta-Analysis
Authors: Hembree Ray
Published: August 2020

Criticisms by Goldin (1992) of my report on problem solving (Hembree. 1992) relate to five areas structured as follows: (a) general skepticism regarding research integration by meta-analysis, (b) lack of a theoretical framework, (c) perceived violations of meta-analytic assumptions, (d) perceptions of questionable study conclusions, and (e) possible misuses of findings. This structure seems highly appropriate; I have retained it in my response. Following those specifics, I shall offer general views on the value of Goldin's critique.
Radical Constructivism and the Pragmatics of Instruction
Authors: Schoenfeld Alan H.
Published: August 2020

Radical Consfructivism in Mathematics Education contains a dozen attempts to deal with one of the most difficult philosophical and pedagogical issues of our time, the reconciliation of a radical constructivist perspective with the demands of individual and classroom instruction. To see why these problems are so difficult, we begin by identifying what distinguishes the radicals represented in this book from their mainstream counterparts.
Teacher Benefits from Using Impromptu Writing Prompts in Algebra Classes
Authors: Miller L. Diane
Published: August 2020

The purpose of this study was to examine the benefits to teachers who used impromptu writing prompts in first- and second-year algebra classes. The study attempted to answer two questions: (a) What can teachers learn about their students' understanding of school mathematics from reading their responses to in-class, impromptu writing prompts; and, (b) Are instructional practices influenced as a result of reading students' responses to in-class, impromptu writing prompts? An interpretive research methodology was used to collect and analyze data. The researchers concluded that the teachers' assessment of students' understanding of school mathematics was enhanced by reading their students' responses to impromptu writing prompts, and that the teachers in the study perceived that their instructional practices were influenced as a result of reading their students' responses to impromptu writing prompts.
Research on Mathematics Education Reported in 1991
Authors: Suydam Marilyn N. , Brosnan Patricia A.
Published: August 2020

This 22nd annuallisting of mathematics education research to appear in the JRME contains research summaries, articles, and dissertations Listed alphabetically by author within each category. For each reference, the grade or age level is indicated (when the researcher has provided this information or it can be inferred). An index is included at the end to help readers locate studies on certain broad topics.
Do the Standards Go Far Enough? Power, Policy, and Practice in Mathematics Education
Authors: Apple Michael W.
Published: August 2020

Although NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and Professional Standards for Teaching Mathematics (1991) are generating considerable interest, there has been little discussion of their ideological and social grounding and effects. By placing the Standards within the growing conservative movement in education, this paper raises a number of crucial issues about the documents, including the depth of the financial crisis in education and its economic and ideological genesis and results; the nature of inequality in schools; the role of mathematical knowledge in our economy in maintaining these inequalities; the possibilities and limitations of a mathematics curriculum that is more grounded in students' experiences; and the complicated realities of teachers' lives. Without a deeper understanding of these issues, the Standards will be used in ways that largely lend support only to the conservative agenda for educational reform.
Simulating the Understanding of Arithmetic: A Response to Schoenfeld
Authors: Ohlsson Stellan
Published: August 2020

In our article “The cognitive complexity of doing and learning arithmetic,” Andreas Ernst, Ernest Rees, and I compare two computer models that learn arithmetic from one-on-one tutoring. One model simulates rote performance, whereas the other model simulates performance based on understanding of arithmetic concepts such as place value. Our results show that the conceptual model has to work more than the rote model to learn subtraction. Also, both models have to work more to learn the regrouping algorithm than the alternative augmenting (equal addition) algorithm.
Editorial
Authors: Lester Frank K.
Published: August 2020

In the final issue (November 1992) of his term as editor of the Journal for Research in Mathematics Education, Tom Carpenter noted that research in mathematics education has been undergoing a paradigm shift. Evidence of this shift is seen in the substantial increase in recent years in the number of research reports representing methodologies that have previously not been common in mathematics education. This increase is due in large part to Tom's receptiveness to publishing research based on alternative paradigms. Mo reover, as open as he was to increasing the scope of the journal, he insisted on doing so without lowering the standards of excellence that we have come to expect. Under Tom's capable guidance the journal has continued to grow in stature, and as a result, mathematics education research has continued to mature as a field of inquiry. As the new editor I will strive to uphold Tom's standards of editorial excellence while continuing to encourage a wide range of ideologies and methodologies.
Arithmetic from a Problem–Solving Perspective: An Urban Implementation
Authors: Villaseñor Albert , Kepner Henry S.
Published: August 2020

Twelve first-grade teachers participated in a staff development effort designed to provide them with opportunities to examine ways to teach the first-grade mathematics curriculum from a problem-solving perspective using a Cognitively Guided Instruction framework. Another group of 12 first-grade teachers and their classrooms comprised the comparison group. The percentage of minority students in these 24 urban classrooms ranged from 57% to 99%. Teachers in the experimental group taught arithmetic through the use of word problems, and their students spent considerably less time on skill worksheet drills. Instruction in experimental classes focused on the process that students used in their solutions rather than on the production of written answers to exercises. Students in experimental classes performed significantly better in solving word problems as well as completing number facts.
Brief Reports: An Examination of Relationships between the 1990 NAEP Mathematics Items for Grade 8 and Selected Themes from the NCTM Standards
Authors: Silver Edward A. , Kenney Patricia Ann
Published: August 2020

For about 20 years, the National Assessment of Educational Progress (NAEP) has reported on the status and progress of U.S. educational achievement in a variety of subject areas, including mathematics (Mullis, 1990). The 1990 NAEP mathematics assessment, which was the fifth in this subject area, was different from the previous four assessments in some important ways. For example, the 1990 NAEP assessment was the first NAEP for which it was possible to report state-level results for those states willing to participate. In fact. the 1990 NAEP consisted of two tests: one given to a national sample at grades 4, 8, and 12 as in prior assessments, and the other given only at grade 8 to a different sample drawn specifically for the stateby-state reporting of results.
The Relationship between the Order-Irrelevance Principle and Counting Skill
Authors: Baroody Arthur J.
Published: August 2020

This study was undertaken to reevaluate whether preschoolers understand that counting in different orders yields the same number before they learn tag-reassigning skills.
President's Report: Tides of Change—Teachers at the Helm
Authors: Lindquist Mary M.
Published: August 2020

Ten years ago. critics of our education system produced A Nation at Risk. Many charged that too little was being done to educate our youth. Standards were too low, and students were not prepared ro embrace the challenges and opportunities of the world arou nd us. As our nation rallied on the shoreline anticipatjng the tides of change. one message rang out loud and clear: “We must change the education we provide our students today if we want to e nsure their success in the future.”
Guest Editorial
Authors: Carpenter Thomas P.
Published: August 2020

It might seem a little strange for the editors of a journal whose mission is to publish the highest quality research in the field to be debating proactive and reactive stances of the journal. All the articles that appear in the journal are submitted for publication by the authors and are peer revjewed, and each of the editors has gone to great lengths to ensure that all submissions have received fair reviews. If the editor or the editorial panel were to take a strongly proactive stance in deciding which research is of most value, there is a danger that the openness and fairness of the review process would be compromised. to everyone's loss.
Guest Editorial
Authors: Kilpatrick Jeremy
Published: August 2020

The proactive versus the reactive role of the JRME in mathematics education—what might that mean? Everyone knows that journals are reactive. They serve as the collective memories of a field. reacting to what is being done and attempting to preserve the best of it for today's subscribers and tomorrow's searchers. The JRMM is a repository of much of the seminal work in mathematics education over the past quarter century, and one can scarcely imagine what the field would look like had the journal not been around to react to what was happening.
Building and Understanding Multiplicative Relationships: A Study of Prospective Elementary Teachers
Authors: Simon Martin A. , Blume Glendon W.
Published: August 2020

This study focused on understanding the evaluation of the area of a rectangular region as a multiplicative relationship between the lengths of the sides. The analysis of data from a whole-class teaching experiment involving prospective elementary teachers resulted in the development of a description of the quantitative reasoning involved. Important aspects of this reasoning include the anticipation of a rectangular array of units as the structure of the area quantity and the subsequent constituting of the units based on the linear measures. The analysis of the prospective teachers' understandings proved useful in generating additional problems. The teachers' engagement with these problems resulted in their developing more complete understandings of the constitution of area units.
President's Report: Linking Yesterday to Tomorrow
Authors: Lindquist Mary M.
Published: August 2020

“Linkages,” our theme for this year's annual meeting, represents more than lin king yeste rday to tomorrow. It is about linking today. We need to link in many ways—with members, with each of the Affiliated Groups, with our committees and task forces, with the Headquarters staff, and with other professional groups. Most important, we need to link with our students, who are facing a much different world from the one that many of us experienced as students. We need to help them link ideas within mathematics and between mathematical topics and link mathematics to real-world problems. We need to strengthen many of our links, forge new ones, and sever some links to the past.
Letter to the Editor
Authors: Liedtke Werner , Gray Eddie , Tall David
Published: August 2020

I enjoyed reading the article about the proceptual view of arithmetic by Gray and Tall (March 1994). The vivid “snapshot” presented by the authors gives the reader a glance at a few of the arithmetical abilities of young students in the UK. The article included a few interesting comments about how the difference between students classified as “less able” might be explained. Some of the data collected about the Jailer group led the authors to make the intriguing suggestion that “slow learner” may be a misnomer. since students who belong to this category do not learn techniques more s lowly—they develop different techniques. This suggestion can be of value to those of us who keep s truggling against the notion that any kind of a “deficit” seems to automatically imply that there exists a need for direct (procedurally oriented) instruction and practice. as opposed to instruction that is conceptually focused. I think the suggestion can also provide valuable food for thought to those who use the hypothetical construct “learning rate” to draw conclusions about, and design special programs for, groups of students.
Doing and Seeing Things Differently: A 25-Year Retrospective of Mathematics Education Research on Learning
Authors: Kieran Carolyn
Published: August 2020

When we look back over the mathematics education research of the last 25 years as reflected in the pages of this journal and compare the per pectives of the past with those of the present, it is tempting to project onto the activities and intentions of the earlier researchers all of the notions that one finds in today's work. In fact. some current researchers seem quite prepared to argue that certain area of attention in mathematical learning. such as problem solving, constructivism, and individual differences, to name just a few, were with us even then and that nothing has changed very much. One could also say that we have always had teachers. studenrs, and mathematical content as well; but there is ample evidence to suggest that there have been changes over time, changes in the ways that we have researched not only these three general didactical components and the interactions among them but also particular areas such a those others referred to above. Some of these innovations have been quite subtle, others more pronounced. In the main, researchers today are looking at aspects of mathematical learning in ways that were, if not unthought of, at least not common 25 years ago.
Teachers' Interpretations of “CGI” after Four Years: Meanings and Practices
Authors: Knapp Nancy F. , Peterson Penelope L.
Published: August 2020

Twenty primary teachers were interviewed who, three or four years earlier, had participated in in-service workshops on Cognitively Guided Instruction (CGI). Three patterns of CGI use seemed related to the meanings teachers constructed for CGI itself. Teachers who reported developing their use of CGI until it formed the mainstay of their mathematics teaching saw CGI conceptually. They also reported learning mainly through their interactions with students and other teachers and developing beliefs about the conceptual nature of mathematics, the constructivist nature of learning, and the students' central role in that learning. Teachers who reported never having used CGI more than supplementally saw CGI as a group of procedures and espoused more traditional beliefs in these areas. Teachers who reported using CGI more at first, but less currently, showed a marked incongruity between their espoused beliefs and reported practices. The authors ask whether additional reseacher support, collegial interaction, or perhaps prescriptiveness in the intervention might have helped teachers in this third group enact their conceptually based beliefs.
Review: A New Era in Collegiate Mathematics Education Research: Good Learning of College Mathematics: Can We Recognize, Understand, Foster, and Document It?
Authors: Flores Alfinio
Published: August 2020

Research in college mathematics education is neither new (Suydam, 1975) nor unusual (Becker & Pence, 1994). However, the literature of this field is still very scattered, with many of the articles published in journals that mathematicians do not usually read. Although in recent years a few books have appeared that deal with college mathematics, books that collect papers devoted to research in the field are certainly needed.
Motivation and Ability as Factors in Mathematics Experience and Achievement
Authors: Schiefele Ulrich , Csikszentmihalyi Mihaly
Published: August 2020

This study examined relationships among interest, achievement motivation, mathematical ability, the quality of experience when doing mathematics, and mathematics achievement. One hundred eight freshmen and sophomores (41 males, 67 females) completed interest ratings, an achievement motivation questionnaire, and the Preliminary Scholastic Aptitude Test. These assessments were followed by 1 week of experience sampling. Mathematics grades were available from the year before the study started, from the same year, and from the following 3 years. In addition, a measure of the students' course level in mathematics was included. The results showed that quality of experience when doing mathematics was mainly related to interest. Grades and course level were most strongly predicted by level of ability. Interest was found to contribute significantly to the prediction of grades for the second year and to the prediction of course level. Quality of experience was significantly correlated with grades but not course level.
A Study of Intrinsic Motivation in the Mathematics Classroom: A Personal Constructs Approach
Authors: Middleton James A.
Published: August 2020

This article examines the relationship between teachers' and students' personal constructs with regard to intrinsic motivation in the mathematics classroom. The research focused on (a) the ways in which teachers attempted to build student motivation into their lessons and (b) the belief systems of teachers as compared to those of their students. In a repertory grid task, students and teachers were asked to distinguish what they believed makes mathematics motivating.
The Frequency of Arithmetic Facts in Elementary Texts: Addition and Multiplication in Grades 1–6
Authors: Ashcraft Mark H. , Christy Kelly S.
Published: August 2020

We tabulated the frequency with which simple addition and multiplication facts occur in elementary school arithmetic texts for grades 1–6. The results indicated a strong “small-fact bias” in both addition and multiplication. “Large” facts, with operands larger than 5, occurred up to half as frequently as those with operands in the 2–5 range. As was also found in an earlier tabulation for grades K–3, facts with operands of 0 and 1 occurred relatively infrequently; the ostensible exceptions to this pattern, high frequencies for combinations like 1+2 and 1×3, were caused by the small-fact bias in multicolumn problems.
Children's Understanding of the Principles of Covariation and Compensation in Part-Whole Relationships
Authors: Irwin Kathryn C.
Published: August 2020

Interviews were held with 107 children, aged 4 through 7, to ascertain their understanding of compensation and covariance in part-whole relationships. They were questioned about uncounted quantities, counted quantities, and numerical equations. The results showed that 4-year-old children could usually predict the effect of changes to one or more parts of an uncounted whole, but were less competent in predicting changes to counted quantities. The ability to predict changes to counted quantities and give reasons for the change increased with age. Some 6-year-old children were able to give reasons for these relationships that went beyond direct observation. Only 7-year-old children were able to use covariance and compensation in a purely numerical context, that of derived equations. These results have implications for teaching young children.
Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operation
Authors: Zaslavsky Orit , Peled Irit
Published: August 2020

The main objectives of this study were to identify difficulties encountered by mathematics teachers and student teachers associated with the concept of binary operation regarding the associative and commutative properties and to reveal possible sources for them. Thirty-six in-service mathematics teachers and 67 preservice mathematics teachers participated in the study. All participants were presented with a task calling for the generation of a counterexample, namely, a binary operation that is commutative but not associative. Responses to the task were analyzed according to four categories: correctness, productiveness, mathematical content, and underlying difficulties. The findings point to similarities and differences between the two groups. Both groups exhibited a weak concept by failing to produce a correct example and by using a limited content search-space. These findings suggest two main inhibiting factors: one related to the overgeneralization of the properties of basic binary operations and the other related to pseudo-similarities attributed to these properties, which seem to be created by the recurring theme of order. Teachers were superior to student teachers on the categories of correctness and productiveness.
Conceptual Knowledge in Introductory Calculus
Authors: White Paul , Mitchelmore Michael
Published: August 2020

Responses to word problems involving rates of change were collected on four occasions during and after 24 hours of concept-based calculus instruction given to a group of first-year university students, all of whom had studied calculus in secondary school. The number of students who could symbolize rates of change in noncomplex situations increased dramatically. However, there was almost no increase in the number who could symbolize rates of change in complex items or in items that required modeling a situation using algebraic variables. Detailed analysis revealed three main categories of error, in all of which variables are treated as symbols to be manipulated rather than as quantities to be related. We surmise that the students had developed an “abstract-apart” concept of a variable rather than the “abstract-general” concept that is needed for the successful study of calculus.
Intuitive Strategies and Preconceptions about Association in Contingency Tables
Authors: Batanero Carmen , Estepa Antonio , Godino Juan D. , Green David R.
Published: August 2020

The aim of this research was to identify students' preconceptions concerning statistical association in contingency tables. An experimental study was carried out with 213 preuniversity students, and it was based on students' responses to a written questionnaire including 2 × 2, 2 × 3, and 3 × 3 contingency tables. In this article, the students' judgments of association and solution strategies are compared with the findings of previous psychological research on 2 × 2 contingency tables. We also present an original classification of students' strategies, from a mathematical point of view. Correspondence analysis is used to show the effect of item task variables on students' strategies. Finally, we include a qualitative analysis of the strategies of 51 students, which has served to characterize three misconceptions concerning statistical association.
Assessment of a Problem-Centered Mathematics Program: Third Grade
Authors: Wood Terry , Sellers Patricia
Published: August 2020

Six classes received problem-centered mathematics instruction for 2 years in second and third grade. This instruction was generally reflective of a socioconstructivist theory of knowing and compatible with recommendations for reform in mathematics education. A class-by-instruction factorial design was used to compare students in problem-centered classes for 2 years with students in problem-centered classrooms for 1 year, and with students in textbook classes for 2 years on a standardized achievement test. In addition, classes using problem-centered instruction for 2 years were compared with students in problem-centered classes for 1 year on an instrument designed to assess students' conceptual development in arithmetic and an instrument developed to examine personal goals and beliefs about reasons for success in mathematics. The results of the analyses indicate that significant differences exist in arithmetic learning for students in problem-centered classes for 2 years on the standardized achievement test and the arithmetic test. The results indicate that after 2 years of instruction in reform-based classes, students score significantly higher on standardized measures of computational proficiency as well as conceptual understanding. Additionally, these students hold stronger beliefs about the importance of finding their own or different ways to solve problems.
Editorial
Authors: Lester Frank K. , Sowder Judith T.
Published: August 2020

The field of mathematics education bas witnessed dramatic changes in its nature and complexion during recent years and there is every indication that the pace of change is likely to accelerate in the future. Naturally, the JRME has been changing as well, most notably with respect to the types of articles being published and the move toward an electronic version of the journal. Collectively, most evidence indicates that the journal is thriving, that it has been adapting well to changes in the field. But, unfortunately, not all of the changes have been for the good! rn particular, one negative change has been an increase in the size of the backlog of accepted manuscripts.
Divisibility and Multiplicative Structure of Natural Numbers: Preservice Teachers' Understanding
Authors: Zazkis Rina , Campbell Stephen
Published: August 2020

This study contributes to a growing body of research on teachers' content knowledge in mathematics. The domain under investigation was elementary number theory. Our main focus concerned the concept of divisibility and its relation to division, multiplication, prime and composite numbers, factorization, divisibility rules, and prime decomposition. We used a constructivist-oriented theoretical framework for analyzing and interpreting data acquired in clinical interviews with preservice teachers. Participants' responses to questions and tasks indicated pervasive dispositions toward procedural attachments, even when some degree of conceptual understanding was evident. The results of this study provide a preliminary overview of cognitive structures in elementary number theory.
The Implementation of a “Function” Approach to Introductory Algebra: A Case Study of Teacher Cognitions, Teacher Actions, and the Intended Curriculum
Authors: Haimes David H.
Published: August 2020

This article reports a case study set in the context of a “function” curriculum approach to introductory algebra and involving a teacher and her ninth-grade class in a high school in Western Australia. The expectation that the teacher's actions would reflect the teaching philosophy implied in the curriculum was not supported by the study's findings. Instead, the teacher gave priority to curriculum content coverage, emphasized methods and procedures, and adopted teacher-focused pedagogical practices.
Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task
Authors: Behr Merlyn J. , Khoury Helen A. , Harel Guershon , Post Thomas , Lesh Richard
Published: August 2020

This study explores preservice teachers' understanding of the operator construct of rational number. Three related problems, given in 1-on-1 clinical interviews, consisted of finding 3/4 of a pile of 8 bundles of 4 counting sticks. Problem conditions were suggestive of showing 3/4 of the number of bundles (duplicator/partition-reducer [DPR] subconstruct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] subconstruct). This study provides confirming instances that students use these 2 rational number operator subconstructs. The SS strategies are identified when the rational number, as an operator, is distributed over a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less often than the DPR strategies. Detailed cognitive models of these strategies in terms of the underlying conceptual units, their structures, and their modifications, were produced, and a “mathematics of quantity” notational system was used as an analytical tool to describe and model the embedded abstractions.
Students' Development of Length Concepts in a Logo-Based Unit on Geometric Paths
Authors: Clements Douglas H. , Battista Michael T. , Sarama Julie , Swaminathan Sudha , McMillen Sue
Published: August 2020

We investigated the development of linear measure concepts within an instructional unit on paths and lengths of paths, part of a large-scale curriculum development project funded by the National Science Foundation (NSF). We also studied the role of noncomputer and computer interactions in that development. Data from paper-and-pencil assessments, interviews, and case studies were collected within the context of a pilot test of this unit with 4 third graders and field tests with 2 thirdgrade classrooms. Three levels of strategies for solving length problems were observed: (a) apply general strategies such as visual guessing of measures and naive guessing of numbers or arithmetic operations; (b) draw hatch marks, dots, or line segments to partition lengths to serve as perceptible units to quantify the length; (c) no physical partitioning—use an abstract unit of length, a “conceptual ruler,” to project onto unsegmented objects. Those students who had connected numeric and spatial representations evinced different and more powerful problem-solving strategies in geometric situations than those who had forged fewer such connections.
Deepening the Analysis: Longitudinal Assessment of a Problem-Centered Mathematics Program
Authors: Wood Terry , Sellers Patricia
Published: August 2020

Longitudinal analyses of the mathematical achievement and beliefs of 3 groups of elementary pupils are presented. The groups consist of those students who had received 2 years of problem-centered mathematics instruction, those who had received 1 year, and those who had received textbook instruction. Comparisons are made for the groups using a standardized norm-referenced achievement test from first through fourth grade. Next, student comparisons are made using instruments developed to measure conceptual understanding of arithmetic and beliefs and motivation for learning mathematics. The results of the analyses indicate that after 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than do those in textbook instruction. In addition, these differences remain after problem-centered students return to classes using textbook instruction. Comparisons of pupils in problem-centered classes for only 1 year reveal that after returning to textbook instruction, these students' mathematical achievement and beliefs are more similar to the textbook group. Also included are exploratory analyses of the pedagogical beliefs held by teachers before and after teaching in problem-centered classes, and those held by teachers in textbook classes. The results of the student and teacher analyses are interpreted in light of research on children's construction of nonstandard algorithms and the nature of classroom environments.
Assessment and Grading in High School Mathematics Classrooms
Authors: Senk Sharon L. , Beckmann Charlene E. , Thompson Denisse R.
Published: August 2020

The assessment and grading practices in 19 mathematics classes in 5 high schools in 3 states were studied. In each class the most frequently used assessment tools were tests and quizzes, with these determining about 77% of students' grades. In 12 classes other forms of assessment, such as written projects or interviews with students, were also used, with performance on such instruments counting for about 7% of students' grades averaged across all 19 classes. Test items generally were low level, were stated without reference to a realistic context, involved very little reasoning, and were almost never open-ended. Most test items were either neutral or inactive with respect to technology. Written projects usually involved more complex analyses or applications than tests did. The teachers' knowledge and beliefs, as well as the content and textbook of the course, influenced the characteristics of test items and other assessment instruments. Only in geometry classes did standardized tests appear to influence assessment.
Young Children's Intuitive Models of Multiplication and Division
Authors: Mulligan Joanne T. , Mitchelmore Michael C.
Published: August 2020

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children's intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it.
Guest Editorial
Authors: Lindquist Mary , Ferrini-Mundy Joan , Kilpatrick Jeremy
Published: August 2020

In two spirited, well-attended sessions at the research presession of the annual NCTM meeting in Minneapolis in April, colleagues shared with each other and us their thoughts about perspectives from theory, research, and practice that ought to serve as the basis for revising the NCTM Standards documents. As the December 1996 NCTM News Bulletin detailed, the Commission on the Future of the Standards is overseeing a recasting of the current documents into a single entity for the twenty-first century that will strengthen relationships among the curriculum, teaching, and assessment standards and that will address new challenges in these areas. Writing teams have been assembled and are meeting this summer. A draft document is scheduled to be released in the fall of 1998, with the final document (possibly available in an electronic format) scheduled for release in 2000.
Designing Representations: Reasoning About Functions of Two Variables
Authors: Yerushalmy Michal
Published: August 2020

One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematics. The ability to generalize, especially when the generalization requires a major breakthrough in habits of mind, is one indication of algebraic reasoning. In this article, I describe generalization activity as an opportunity to learn about seventh graders' understanding of functions. A group of students who had studied functions modeled a multivariable situation. Through individual and group work, they designed, described, and discussed various representations for functions of 2 variables. Their modeling efforts allowed them to analyze their understanding of representations of quantities, relationships among quantities, and relationships among the representations of quantities in both single- and multivariable functions.
Mathematics, Multicultural Education, and Professional Development
Authors: Sleeter Christine E.
Published: August 2020

Multicultural education grew out of the civil rights movement and, as such, is grounded in a vision of democracy, social justice, pluralism, and equality—ideals that have yet to be realized in U.S. society and its schools. For the past 25 years, multicultural education has served as a mobilizing focus for struggles to articulate visions of schooling that are consistent with the ideals of the U.S. and for the development of theory and research that offer a countervision to the way that schooling is usually conducted, particularly for children from historically marginalized groups. As this body of theory and research has grown so also have the implications for restructuring various dimensions of the education enterprise. Mathematics is one such dimension and is the focus of this article. First, however, I contextualize the discussion that follows within a vision of what multicultural education means.
It Doesn't Add Up: African American Students' Mathematics Achievement
Authors: Ladson Billings Gloria
Published: August 2020

Mathematics education has been heralded for its leadership role in the U.S. school reform effort (Stein, Grover,&Henningsen, 1996; Grant, Peterson, & Shojgreen- Downer, 1996). Prominent in the reform of mathematics education is the call for students not merely to memorize formulas and rules and apply procedures but rather to engage in the processes of mathematical thinking, that is, to do what mathematicians and other professional users of mathematics do. The revamped mathematics education program is based on engaging students in problem posing and problem solving rather than on expecting rote memorization and convergent thinking. These changes in mathematics education suggest that mathematics teaching must build on students' learning and on their ability to pose and solve problems previously considered too difficult for their age-grade levels (Carpenter&Fennema, 1988; Fennema, Franke, Carpenter,&Carey, 1993).