bluuebunny/crossref_metadata_2025 huggingface dataset:
Talking about Rates Conceptually, Part I: A Teacher's Struggle
Authors: Thompson Patrick W. , Thompson Alba G.
Published: August 2020

Classroom discourse and communication are issues central to current reform in mathematics education. For reform to happen in classrooms, teachers will have to teach from a conceptual curriculum. To do so they must be sensitive to children's thinking during instruction and shape their instructional actions accordingly—to ensure that children hear what they intend them to hear. This article examines how one teacher's way of knowing mathematics was reflected in the language he used in teaching concepts of rate to one student during a teaching experiment. The teacher's conceptualizations of rate, although strong and elaborate, were encapsulated in the language of numbers and operations, and this undermined his effort to help the student understand rates conceptually. We examine the teacher's language in light of his intentions and the student's interpretations of the tasks. We discuss implications for instruction that is based on a conceptual curriculum and is sensitive and responsive to children's thinking.
One Preservice Secondary Teacher's Understanding of Function: The Impact of a Course Integrating Mathematical Content and Pedagogy
Authors: Wilson Melvin R.
Published: August 2020

This study examines the evolving knowledge and beliefs of a preservice secondary mathematics teacher as she participated in a mathematics education course that emphasized mathematical and pedagogical connections and applications of the function concept. Her conceptions were revealed during a 10-week period through interviews, observations, and written work. The teacher's initial understanding of functions as computational activities (e.g., function machines, point plotting, vertical line test) was consistent with her larger view of mathematics as a collection of “concrete” procedures. Although her understanding of function grew substantially during the study, her anticipated approach to teaching, which was dominated by her narrow view of mathematics, was less significantly affected by course activities.
Research on Mathematics Education Reported in 1993
Authors: Suydam Marilyn N. , Brosnan Patricia A.
Published: August 2020

This 24th annual listing 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. An index is included at the end to help readers locate studies on certain broad topics.
Educational Development and Developmental Research in Mathematics Education
Authors: Gravemeijer Koeno
Published: August 2020

In light of anticipated changes in mathematics education, an alternative for the well-known “research-development-diffusion” model is presented. It is based on an integration of curriculum research and design embedded in “educational development.” In this context curriculum development is described as purposeful and sensible tinkering. It is argued that the theory production implied in this process may be exploited in “developmental research.” However, developmental research is not yet well established as a research discipline. The core of this article is an effort to explicate the characteristics of developmental research and to discuss its methodological aspects.
Research on Affect and Mathematics Learning in the JRME: 1970 to the Present
Authors: McLeod Douglas B.
Published: August 2020

Studies of affective issues have always been central to the goals of mathematics education and to the pages of the Journal for Research in Mathematics Educmion (JRME). Over the past 25 years, approximately 100 of the JRME's articles have dealt with some aspect of the affective domain: although it is not possible to review the contributions of each of those articles in this paper, we will try to convey some of the flavor of research on affective issues as it has developed over the life of the journal.
Children's Concepts of Average and Representativeness
Authors: Mokros Jan , Jo Russell Susan
Published: August 2020

Whenever the need arises to describe a set of data in a succinct way, the issue of mathematical representativeness arises. The goal of this research is to understand the characteristics of fourth through eighth graders' constructions of “average” as a representative number summarizing a data set. Twenty-one students were interviewed, using a series of open-ended problems that called on children to construct their own notion of representativeness. Five basic constructions of representativeness are identified and analyzed. These approaches illustrate the ways in which students are (or are not) developing useful, general definitions for the statistical concept of average.
Review: A New Era in Collegiate Mathematics Education Research: A Mathematician's View of RCME
Authors: Wheeler Robert F.
Published: August 2020

For the mathematician interested in issues of learning and teaching, these are, ostensibly, the best of times. The American Mathematical Society has joined the Mathematical Association of America in making choice time slots available at national and regional meetings for discussions of educational issues. These sessions have been well attended, not only by young Ph.D.'s, but also by senior mathematicians, full professors with the protection of tenure, who have the potential to substantially modify the way in which college mathematics teaching is done.
Review: A Critical Look At Practice In Mathematics And Mathematics Education
Authors: Williams Steven R.
Published: August 2020

Scholarly work in mathematics education has recently begun to look deeply at what mathematics is. This increased interest in what was once the purview of philosophers of mathematics grows from a recognition that both teaching and learning mathematics are intimately connected with doing mathematics.
Factors Associated with Types of Mathematics Anxiety in College Students
Authors: Bessant Kenneth C.
Published: August 2020

This article investigates the interrelatedness of various types of mathematics anxiety with attitudes toward mathematics, learning preferences, study motives, and strategies. An 80-item version of the Mathematics Anxiety Rating Scale (MARS) was completed by 173 university students enrolled in one of three introductory statistics courses offered by the departments of mathematics, psychology, or sociology. Mathematics attitude scales were also administered in conjunction with the Study Process Questionnaire (SPQ), which depicts students' motivations to use study methods associated with contrasting learning approaches. Factor analysis of the MARS identified six factors labeled as General Evaluation Anxiety, Everyday Numerical Anxiety, Passive Observation Anxiety, Performance Anxiety, Mathematics Test Anxiety, and Problem-Solving Anxiety.
Learning to Solve Addition and Subtraction Word Problems through a Structure-plus-Writing Approach
Authors: Rudnitsky Al , Etheredge Susan , Freeman Sue J. M. , Gilbert Therese
Published: August 2020

The aim of this study was to design and field-test instruction intended to help students construct knowledge about addition and subtraction story problems and determine if this knowledge would transfer to actually solving problems. The study tests two related hypotheses: (a) structure-plus-writing instruction will result in improved word-problem solving, and (b) this improvement will be more enduring than that resulting from a more traditional heuristic and practice-based approach.
Identification of Multiplicative Thinking in Children in Grades 1–5
Authors: Clark Faye B. , Kamii Constance
Published: August 2020

Textbooks present multiplication as merely a faster way of doing repeated addition. However, research has shown that multiplication requires higher-order multiplicative thinking, which the child develops out of addition. Three hundred thirty-six children in grades 1–5 were interviewed individually using a Piagetian task to study their development from additive to multiplicative thinking. Multiplicative thinking was found to appear early (45% of second graders demonstrated some multiplicative thinking) and to develop slowly (only 48% of fifth graders demonstrated consistently solid multiplicative thinking). It was concluded that the introduction of multiplication in second grade is appropriate but that educators must not expect all children to use multiplication, even in fifth grade.
Review: One Very Complete View (Though Only One) of How Children Learn Mathematics
Authors: Davis Robert B.
Published: August 2020

Children's Mathematical Developmem is a monumental book and an excjting onean encyclopedia that will end one era and begin another. From now on, anyone concerned with understanding what is involved in thinking and learning in the area of mathematics will want to be familiar with the contents of Geary's awesome compendium. From now on there is a foundation, or at least part of one, on which all of us can build-a comprehensive summary of what is known about mathematical behavior as seen by an American psychologist in 1994.
The Development of Unitizing: Its Role in Children's Partitioning Strategies
Authors: Lamon Susan J.
Published: August 2020

The goal of this study was to gain a better understanding of the intuitive activity that precedes the construction of quantity, in particular, to make inferences concerning developmental trends and needs in children's unitizing processes. Using a cross-sectional design, the study analyzed the partitioning strategies of 346 children from grades four through eight in terms of a framework that translated economy in the number or size of pieces and the use of perceptual cues into sophistication in unitizing. At each grade level, a greater percentage of students used economical partitioning strategies than used less economical cut-and-distribute strategies. As grade level increased, the percentage of students using economical strategies increased, indicating a shift away from the distribution of singleton units toward the use of more composite units. Strategies were heavily influenced by social practice related to the commodity being shared and, to a lesser degree, by the numerical portion of the given extensive quantities.
The Development of the Idea of Mathematical Proof: A 5-Year Case Study
Authors: Maher Carolyn A. , Martino Amy M.
Published: August 2020

This longitudinal case study presents a sequence of episodes that document the mathematical thinking of one child, Stephanie, over a 5-year period. Her development of the idea of mathematical justification spans grades 1–5. Stephanie worked on several combinatorics tasks in small-group, whole-class, and individual interview settings. The documented events indicate Stephanie's progress in classifying, organizing, and reorganizing data. The study provides some significant insight into the process by which Stephanie learned to make proofs, within a setting that encouraged the development of her ideas.
Gender-Related Differences in Self-Referenced Cognitions in Relation to Mathematics
Authors: Seegers Gerard , Boekaerts Monique
Published: August 2020

In this study, a number of learner variables that are related to mathematics achievement in actual learning situations were examined. The dynamic model of the learning process as developed by Boekaerts was taken as a starting point. Both trait-like self-referenced cognitions (viz., academic self-concept of mathematics ability, goal orientation, and attribution style) and situation-specific variables were included. In a group of 8th graders (ages 11–12; N=186), marked differences between boys and girls on a mathematics test were found. These differences were parallelled by differences in both trait-like self-referenced cognitions and task-specific appraisals. It is concluded that boys experience learning situations where they are confronted with a mathematics test in a more positive way than girls do.
Young Children's Perceptions of Mathematics in Problem-Solving Environments
Authors: Franke Megan Loef , Carey Deborah A.
Published: August 2020

Thirty-six first graders from 2 different school systems participated in individual interviews to determine the children's stated perceptions regarding what it means to engage in mathematics and the rationale and conditions under which they held such perceptions. These children were in classrooms that reflected the spirit of the current reform movement in mathematics education. Generally, the children perceived of mathematics as a problem-solving endeavor in which many different strategies are considered viable and communicating mathematical thinking is an integral part of the task. The children recognized and accepted a variety of solution strategies, with many of the children valuing all solutions equally and assuming a shared responsibility with the teacher for their mathematics learning. Children had varying perceptions of what it meant to succeed in mathematics, but success was not determined by speed and accuracy. As we begin to understand children's perceptions of mathematics as they participate in reform-minded mathematics classrooms, we become aware of issues concerning the impact of the children's perceptions on both the development of their future perceptions and their mathematics learning.
Brief Report: Results of Third-Grade Students in a Reform Curriculum on the Illinois State Mathematics Test
Authors: Carroll William M.
Published: August 2020

Over the past decade, there has been a call for major reforms in mathematics education, from classrooms where students memorize facts and practice algorithms to classrooms in which reasoning and understanding are given more emphasis (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 1995). In response, a number of curricula have been developed that attempt to meet this vision. One reform curriculum in widespread usage is the University of Chicago School Mathematics Project's (UCSMP) elementary curriculum, Everyday Mathematics. In the UCSMP curriculum, students work in small groups exploring mathematics in real-life contexts, using calculators, manipulatives, and other mathematical tools from kindergarten onward. In contrast to traditional curricula, students are encouraged to use these tools or to “invent” their own computational algorithms to solve problems, and the sharing of their alternative solution methods is a regular part of class discussions. Additionally, problems are nearly always application-based and never presented as sets of symbolic problems. For example, in a second-grade UCSMP lesson, students are given a picture depicting various animals and their heights or lengths. During this activity, students work in small groups to construct number stories that compare the animal heights and then to find a solution method. In the follow-up discussion, students share their stories and solution procedures and offer alternative methods.
Listening for Differences: An Evolving Conception of Mathematics Teaching
Authors: Davis Brent
Published: August 2020

The question of how to teach mathematics has become increasingly problematic in recent years as critics from diverse perspectives have offered wide-ranging, and often seemingly incommensurate, challenges to conventional conceptions of the teacher's task. This article represents an effort to “bring into dialogue” some of the varied commentaries on mathematics teaching, using an enactivist framework to interpret and to propose an alternative way of framing mathematics teaching. In this report, the manner in which the teacher listens is offered as a metaphoric lens through which to reinterpret practice, as a practical basis for teaching action, and as a means of addressing some of the critics' concerns. The report is developed around an extended collaborative research project with a middle school mathematics teacher.
Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning
Authors: Henningsen Marjorie , Kay Stein Mary
Published: August 2020

In order to develop students' capacities to “do mathematics,” classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.
Race-Ethnicity, SES, Gender, and Lanuage Proficiency Trends in Mathematics Achievement: An Update
Authors: Tate William F.
Published: August 2020

The purpose of this article is to document changes in U.S. mathematics achievement by reviewing national trend studies, college admissions examinations, and Advanced Placement tests. This article examined this quantitative research literature to determine trends in mathematics achievement of various social groups defined along lines of race, class, gender, ethnicity, and language proficiency. The findings of this review indicate that over the last 15 years all demographic groups have improved in mathematics achievement—specifically, in basic skills. Moreover, the mathematics achievement gap is slowly closing between White students and students of color on assessments of basic skills. Males tended to outperform females on standardized measures; however, gender differences were small and generally not significant. Consistent with past reviews of mathematics achievement, course taking was a powerful variable, often resulting in similar achievement gains across diverse groups. This finding has serious implications for equity-related policy. The article concludes with two other recommendations: the need for fiscal and cultural policy to support standards-based reform.
Children's Problem Posing Within Formal and Informal Contexts
Authors: English Lyn D.
Published: August 2020

This study investigated the problem-posing abilities of third-grade children who displayed different profiles of achievement in number sense and novel problem solving. The study addressed (a) whether children recognize formal symbolism as representing a range of problem situations, (b) whether children generate a broader range of problem types for informal number situations, (c) how children from different achievement profiles respond to problem-posing activities in formal and informal contexts, and (d) whether children's participation in a problem-posing program leads to greater diversity in problems posed. Among the findings were children's difficulties in posing a range of problems in formal contexts, in contrast to informal contexts. Children from different achievement profiles displayed different response patterns, reflected in the balance of structural and operational complexity shown in their problems.
Supporting Innovation: The Impact of a Teacher's Conceptions of Functions on His Implementation of a Reform Curriculum
Authors: Lloyd Gwendolyn M. , Wilson Melvin
Published: August 2020

In this study we investigate the content conceptions of an experienced high school mathematics teacher and link those conceptions to their role in the teacher's first implementation of reform-oriented curricular materials during a 6-week unit on functions. The teacher communicated deep and integrated conceptions of functions, dominated by graphical representations and covariation notions. These themes played crucial roles in the teacher's practice when he emphasized the use of multiple representations to understand dependence patterns in data. The teacher's well-articulated ideas about features of a variety of relationships in different representations supported meaningful discussions with students during the implementation of an unfamiliar classroom approach to functions.
Using Transactional Reading Strategies to Support Sense-Making and Discussion in Mathematics Classrooms: An Exploratory Study
Authors: Borasi Raffaella , Siegal Marjorie , Fonzi Judith , Smith Constance F.
Published: August 2020

In this study we explore the potential for mathematics instruction of four reading strategies grounded in transactional reading theory. On the basis of the descriptive study of 18 instructional episodes developed in 4 secondary mathematics classes as a result of collaborative action research, we show that encouraging mathematics students to talk, write, draw, and enact texts can provide them with concrete ways to construct and negotiate interpretations of what they read. In addition to helping students better understand the text read, acting on and acting out a text allow students to use that text as a springboard for sense-making and discussion of important mathematical ideas and issues about the nature of mathematics, especially when these reading experiences are supported by compatible classroom norms and values.
Learning to Estimate in the Mathematics Classroom: A Conversation-Analytic Approach
Authors: Forrester Michael A. , Pike Christopher D.
Published: August 2020

In contrast to contemporary estimation researchers who have focused primarily on children's computational estimation abilities, we examined the ideas surrounding the teaching and learning of measurement estimation in the classroom. Employing ethnomethodologically informed conversation analysis, we focused on 2 teachers' instructions during estimation lessons and on pupils' (aged 9-11 years) talk during small-group follow-up activities. The results indicated that estimation is understood as discursively interdependent with measurement and is associated both with teacher-formulated accountability and with vagueness, ambiguity, and guessing. Furthermore, the meaning of what it is to estimate is embedded in practical action. In concluding comments we consider the advantages of using conversational analysis as a method for highlighting the relationships between language and mathematics in the classroom.
Supporting Students' Mathematical Inquiries Through Reading
Authors: Siegal Marjorie , Borasi Raffaella , Fonzi Judith
Published: August 2020

The purpose of this article is to identify specific functions that reading, in combination with writing and talking, can serve in mathematical inquiries and thus to contribute to a better understanding of how inquiry experiences can be planned and supported in mathematics classrooms. This purpose is achieved through an analysis of 3 classroom experiences in which secondary mathematics students engaged in “inquiry cycles” on quite different topics. These instructional experiences were developed by a collaborative team of mathematics teachers, mathematics education researchers, and a reading researcher in the context of action research and teacher research. Analysis of the data led to the identification of 30 functions of reading that are specific to distinct elements of an inquiry cycle. On the basis of these findings we suggest that reading can serve multiple roles in inquiry-based mathematics classes and, in doing so, can afford students unique opportunities for learning mathematics
Longitudinal Evaluation of Mathematical Computational Abilities of New Hampshire's Eighth and Tenth Graders, 1963-1967
Authors: Austin Gilbert R. , Prevost Fernand
Published: August 2020

New Hampshire Grade Eight Statewide Testing Program results for 1964 and 1965 showed a noticeable drop in Arithmetic Computation scores. Studies in 1965 and 1967 showed 3 groups, designated modern, transitional, and traditional, by type of text used, significantly different in intellectual ability. In 1965, the modern group scored highest on a general ability measure, but lowest on Arithmetic Computation. In 1967 this was no longer true. In 1967, all scores were lower than the 1965 or 1963 scores. The study further compared achievement among the 3 groups at tenth grade on the Stanford High School Mathematics Test, Part A. The modern and transitional groups' scores suggest performance (transfer) in algebra and geometry is relatively greater for those students who have studied modern mathematics in earlier grades.
Intellectual Growth and Understanding Mathematics
Authors: Lovell Kenneth R.
Published: August 2020

This paper, which reviews 25 years of Piagetian research in intellectual growth as it pertains to the learning and understanding of mathematics, was commissioned by the ERIC Information Analysis Center for Science and Mathematics Education. The paper was presented by Professor Lovell at a meeting of the Special Interest Group for Research in Mathematics Education at the annual convention of the American Educational Research Association on 5 February 1971.
The Relative Effectiveness of a Symbolic and a Concrete Model in Learning a Selected Mathematical Principle
Authors: Fennema Elizabeth H.
Published: August 2020

Mathematical ideas taught in the elementary school can be represented by at least two types of models: concrete (manipulative or en active) and symbolic. Although there has been much theoretical discussion concerning the value of concrete models in enhancing learning (Kieren, 1969), empirical data are just beginning to accumulate that give some information concerning the efficacy of concrete models (Kieren, 1971), and such data are far from conclusive.
The Effectiveness of Discovery and Expository Methods in the Teaching of Fourth-Grade Mathematics
Authors: Olander Herbert T. , Robertson Howard C.
Published: August 2020

The experiment involved fourth-grade pupils studying mathematics with emphasis upon principles and relationships. 1 group of 190 pupils was taught for 31 weeks through a Discovery Approach while another 184 pupils used an Expository Approach.

The 13 teachers were rated on a scale measuring the degree to which each succeeded in using the method to which she had been assigned. Pupils were tested on intelligence, attitudes toward mathematics, and performance in mathematics involving computation, concepts, applications, and principles and relationships.

Analysis of the data through multiple analysis of covariance showed Expository pupils were significantly better in computation, and Discovery pupils were significantly better in retention of ability to apply mathematics (p<.05p<.05).

When interaction effects were studied through the use of regression slopes, pupils scoring lower on the pretest on computation and applications improved more under the Expository Approach; those scoring lower on concepts gained more under the Discovery Approach. On principles and relationships, the Discovery pupils gained at a greater ratio throughout the study.
Study 3-2: An Exploration of Transitivities Formulated from a Set of Piagetian-Derived Operations and Their Implications in Traversing Learning Hierarchies
Authors: Hopkins Layne V.
Published: August 2020

In the Paradigms studies by Klein (1970) and Farris (1970), selected relationships among the three modes of representation (enactive, iconic, and symbolic) were carefully examined. These researchers conceptualized instructional objectives as ordered pairs, where the first element represented the mode of representation of the given component of an objective and the second element represented the mode of representation of the required performance of the objective. Further, they used an ordered-pair matrix that consisted of nine cells so that a systematic procedure could be established for investigating selected relationships among the cells of the matrix. This study, in conjunction with the one conducted by Sawada (1971), investigated selected relationships within cells of this modes-of-representation matrix.
A Comparison of Initially Teaching Division Employing the Distributive and Greenwood Algorithms with the Aid of a Manipulative Material
Authors: Kratzer Richard O. , Willoughby Stephen S.
Published: August 2020

Division of whole numbers is traditionally accepted as a part of the elementary school mathematics program and is considered to be the most difficult of the four basic operations with whole numbers. The Greenwood algorithm is employed in most contemporary elementary tex tbooks to initially develop the recording process. The common usage of the Greenwood algorithm for division of whole numbers today is largely due to the study by Van Engen and Gibb, General Mental Functions Associated with Division (1956). Later, in almost all textbook series, the standard algorithm is used for the final recording process. Van Engen and Gibb, however, did not recommend converting to the standard algorithm after learning the Greenwood algorithm. This trans ition stage is the source of substantial difficulty for many pupils. lf the standard algorithm is to be the terminal algorithm, then perhaps an approach that initially employs the standard algorithm should be used.
Editorial: Reviews That Advance the Community
Authors: Heid M. Kathleen
Published: August 2020

I recently attended a meeting of research journal editors at which we shared some of our practices and procedures. The editors represented a range of educational fields. Impressed by the nature and number of reviews that the JRME editorial office requests and receives, one of the editors of a non-mathematics-education journal offered this explanation: “That's because the mathematics education group is really a community.” Thinking about how good it made me feel to be a member of such a well-regarded community, I reflected on the remarkable work I had seen over the past year and a half from members of our community. During that time, we received 1687 reviews (give or take a few). I continue to be impressed by the careful and insightful reviews that constantly populate the JRME mailbox.
Brief Report: Measuring Curriculum Implementation
Authors: Huntley Mary Ann
Published: August 2020

Using curriculum-specific tools for measuring fidelity of implementation is an essential yet often overlooked aspect of examining relationships among textbooks, teaching, and student learning. This “Brief Report” describes the variety of ways that curriculum implementation is measured and argues that there is an urgent need to develop curriculum-sensitive tools for analyzing classroom practice. The report outlines the use of the Concerns-Based Adoption Model (CBAM) theory to develop analytical tools for measuring implementation of two middle-grades reform mathematics curricula: Connected Mathematics and MathThematics. The report also presents next steps in this program of research.
Reconceiving Fidelity of Implementation: An Investigation of Elementary
Authors: Brown Stacy A. , Pitvorec Kathleen , Ditto Catherine , Kelso Catherine Randall
Published: August 2020

Recent research on mathematics reforms in the United States indicates that the reforms are not yet widely implemented. Generally, this claim results from looking at the extent to which teachers use curricular materials or engage in particular classroom practices. This article moves beyond disparate questions of use and practice to examine interactions between teachers and curricula as evidenced by their enactments of whole-number lessons from a Standards-based curriculum. Specifically, we analyze videorecorded 1st- and 2nd-grade classroom lessons in terms of students' opportunities to reason and communicate about mathematics. This analysis indicates that the level of fidelity to the written curriculum differs from the level of fidelity to the authors' intended curriculum during lesson enactments. Drawing on this analysis, this article explores how curricula support and hinder teachers as they engage students in opportunities to learn mathematics and how teachers' instructional moves and choices impact the enactment of curricula.
Book Review: All You Need Is Math: A Review of What's Math Got to Do With It?: Helping Children Learn to Love Their Least Favorite Subject-and Why It's Important for America
Authors: Secada Walter G.
Published: August 2020

In What's Math Got to Do With It? Jo Boaler joins a heady list of public intellectuals such as Jo Sanders (1986) and Sheila Tobias (1994) who seek to engage public policy by explaining extant research to “just plain folk” (Lave, 1988) and to empower their audiences to “do something about it.” Boaler's audience is parents; the “it” is the dreary mathematics teaching that is provided to their children. In fact, one girl told Boaler: “In math, you have to remember; in other subjects, you can think about it” (p. 40). For parents and their children, Boaler presents alternatives, such as Emily Moskam's classroom at Greendale High School, in which mathematics is taught using complex problems that engage students and that result in authentic mathematical activity.
Book Review: Exploring Sfard's Commognitive Framework: A Review of Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing
Authors: Felton Mathew D. , Nathan Mitchell
Published: August 2020

Discourse as a medium of learning and instruction has gained tremendous ground among educational researchers and cognitive scientists. Yet earlier notions of cognition as computation have not been reconciled with views of knowing and learning as socially mediated processes. In Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing, Anna Sfard asks us to re-imagine thinking as communication, with the hopes that this will resolve many of the current dilemmas facing research on thinking in general, and in mathematics education in particular. In doing so, she argues that we should move beyond the metaphor of learning as acquiring knowledge—for example, treating knowledge of something like counting as an object that is “held” by the mind and applied when needed—to conceptualizing learning as participating in discourse—for instance, participating in a discourse that engages in counting when asked “which box has more?” Sfard begins with five quandaries facing paradigms that treat learning metaphorically as acquisition—such as, if a child “possesses” counting, then why would he or she not count when asked “which box has more?” Sfard then provides a thorough yet accessible review of previous learning paradigms, and lands finally on a redefinition of thinking as internalized communication. To break out of older modes of talking about thinking, she coins the term commognition as a mix of cognition and communication. By the end of the book, the new perspective of commognition is offered as a way to avoid the quandaries facing paradigms that treat learning as acquisition.
Editorial: Where's the Math (in Mathematics Education Research)?
Authors: Heid M. Kathleen
Published: August 2020

Twenty-five years ago, a fast-food TV ad initiated a catchphrase, “Where's the beef?” The phrase, originally intended to point out the small amount of beef in fastfood hamburgers, evolved into a way to question the amount or substance of an idea or product. Adopted in popular culture, the phrase made its way into the 1984 vicepresidential debate, and reappeared in popular TV shows. It is now time for the phrase to make its way into discussions about mathematics education research. While reading all the manuscripts submitted to JRME over the past 2 years, a paraphrase of that statement frequently passed through my mind: “Where's the math?”
Research Commentary: Reflections of an Accidental Theorist
Authors: Shoenfeld Alan H.
Published: August 2020

This “Research Commentary” explores the roles of theoretical and empirical work in the development of productive lines of research. It elaborates on 3 main points. First, the dialectic between theory and empirical practice is highly productive for both: Theory is enriched by close attention to data, and one's understanding of empirical issues is deepened when one attends closely to theory. Second, an effective mechanism for making systematic progress is to focus on cases at the edge of one's (and the field's) collective theoretical understanding. Third, close observations of empirical phenomena often raise issues that have theoretical import and that give rise to new lines of work. These 3 points, and implications, are illustrated with examples from my research.
Editorial: What Is the Purpose of Publishing Papers in a Mathematics Education Research Journal?
Authors: Blume Glendon W. , Heid M. Kathleen , Zbiek Rose Mary
Published: August 2020

The Journal for Research in Mathematics Education has the reputation of being one of the top journals worldwide that publishes mathematics education research. The task of analyzing the viability of a manuscript as a potential JRME article naturally leads to the question: What is the purpose of publishing papers in a mathematics education research journal?
Mathematical Caring Relations in Action
Authors: Hackenberg Amy J.
Published: August 2020

In a small-scale, 8-month teaching experiment, the author aimed to establish and maintain mathematical caring relations (MCRs) (Hackenberg, 2005c) with 4 6th-grade students. From a teacher's perspective, establishing MCRs involves holding the work of orchestrating mathematical learning for students together with an orientation to monitor and respond to energetic fluctuations that may accompany student–teacher interactions. From a student's perspective, participating in an MCR involves some openness to the teacher's interventions in the student's mathematical activity and some willingness to pursue questions of interest. In this article, the author elucidates the nature of establishing MCRs with 2 of the 4 students in the study and examines what is mathematical about these caring relations. Analysis revealed that student–teacher interaction can be viewed as a linked chain of perturbations; in student–teacher interaction aimed toward the establishment of MCRs, the linked chain tends toward perturbations that are bearable (Tzur, 1995) for both students and teachers.
The Nature of Arguments Provided by College Geometry Students With Access to Technology While Solving Problems
Authors: Hollebrands Karen F. , Conner AnnaMarie , Smith Ryan C.
Published: August 2020

Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.
The Nature and Predictors of Elementary Teachers' Mathematical Knowledge for Teaching
Authors: Hill Heather C.
Published: August 2020

This article explores elementary school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. The Learning Mathematics for Teaching project administered a multiple-choice assessment covering topics in number and operation to a nationally representative sample of teachers (n = 625) and at the same time collected information on teacher and student characteristics. Performance did not vary according to mathematical topic (e.g., whole numbers or rational numbers), and items categorized as requiring specialized knowledge of mathematics proved more difficult for this sample of teachers. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.
The Teaching of Heuristic Problem-Solving Strategies in Elementary Calculus
Authors: Lucas John F.
Published: August 2020

Research in the area of computer simulation of human thought processes indicates that the solution of every problem involves the element of search in a space of many alternatives (Newell, Shaw, & Simon, 1958). For human problem solving, the space may conceivably include infinitely many alternatives; however, humans appear to exhibit more efficient performance than exhaustive scanning or purely random trial and error. Human problem solving is characterized by heuristic strategies. These higher-order processes (heuristics) guide the search, enabling the problem solver to select from a reduced set of alternatives and to order his solution process in a sequence of steps; they are tentative rules of thumb that are based on experience or plausible assumptions and that apply generally to problems (Pylyshyn, 1963).
The Relationship of Area Conservation to Area Measurement as Affected by Sequence of Presentation of Piagetian Area Tasks to Boys and Girls in Grades One through Three
Authors: Taloumis Thalia
Published: August 2020

Three area conservation tasks and two area measurement tasks, all representative models of area tasks reported by Piaget, were administered in one sitting to each of 168 middle-class children, ages six years five months to nine years four months. Statistical procedures included an analysis of variance, a Pearson product-moment correlation, and multiple-regression correlation for subgroups. The findings indicated (1) that whichever group of area tasks (area conservation tasks or area measurement tasks) was presented second, that group of tasks resulted in higher scores; (2) that there was a significant correlation between area conservation scores and area measurement scores; (3) that boys and girls did not score significantly differently on area conservation tasks and area measurement tasks; and (4) that scores in area conservation tasks and area measurement tasks showed no significant difference in amount of increase from grade to grade.
Learning Environments, Academic Self-Concepts, and Achievement in Mathematics
Authors: House Peggy A.
Published: August 2020

Previous research studies have established that students' academic achievement can be predicted from their self-concepts of academic ability and from selected characteristics of the learning environment. The present study provides evidence for the relationships between learning environments and academic self-concepts, and it has demonstrated that, for ninth-grade mathematics students, both general and mathematical self-concepts of academic ability can be predicted from combinations of classroom environment characteristics. This in turn provides a rationale for additional research to determine if it is possible to enhance self-concepts through providing suitable learning environments and thereby to increase academic achievement.
A Test with Selected Topological Properties of Piaget's Hypothesis concerning the Spatial Representation of the Young Child
Authors: Martin J. Larry
Published: August 2020

One task of mathematics educators is to introduce various mathematical concepts into a comprehensive mathematics program designed for students in kindergarten through grade 12. The sequencing of these concepts is sometimes determined by the mathematics involved, sometimes by the experiences deemed appropriate for a student's needs in his particular physical surroundings, sometimes by the cognitive readiness of the student, sometimes by tradition, and sometimes by mere chance. In the case of geometry, the sequence has traditionally followed the order of the historical development of geometry, that is, from Euclidean geometry to the more general geometries.
Models and Applications as Advanced Organizers
Authors: Lesh Richard A. , Johnson Howard
Published: August 2020

This study involved 240 fourth graders and 240 seventh graders from four North Chicago schools. It was a replication and extension of an unpublished doctoral dissertation that was conducted by Johnson at Northwestern University in 1973. The objective of the study was to investigate the relative effectiveness of four different types of advanced organizers when they were used to introduce a 30-minute self-instructional unit about elementary concepts in motion geometry. One type of organizer used several simple applications to introduce intuitively the ideas that were in the unit; another type of organizer used only a single application: another used several concrete models; and the fourth used a single concrete model. In addition to comparing the four basic types of organizers, experimental treatments compared children who received an organizer before reading the basic unit (i.e., an advanced organizer) with children who received an organizer after they had completed the basic unit (i.e., a post organizer): and it compared children who worked on the organizer in small groups with children who worked individually.
Brief Reports: Fennema-Sherman Mathematics Attitudes Scales: Instruments Designed to Measure Attitudes Toward the Learning of Mathematics by Females and Males
Authors: Fennema Elizabeth , Sherman Julia A.
Published: August 2020

The importance of affective factors in partially explaining individual differences in the learning of mathematics is well recognized.
The Difference in Level of Anxiety in Undergraduate Mathematics and Nonmathematics Majors
Authors: Ohlson E. Lamonte , Mein Lillian
Published: August 2020

Anxiety levels of mathematics and nonmathematics majors, as measured by the State-Trait Anxiety Inventory (STAI), were investigated using a multiple linear regression model. The results confirm that mathematics majors are no more or no less anxious than nonmathematics majors, and that anxiety levels of mathematics majors did not increase from freshman through senior year. Finally, the research indicates that being in a mathematics classroom created no more anxiety than being in a nonmathematics classroom.
Number Patterns: Discovery versus Reception Learning
Authors: O'Brien Thomas C. , Shapiro Bernard J.
Published: August 2020

Subjects were randomly assigned to one of three treatment groups that differed in terms of the manner in which initial learning of number patterns took place. In one group, the presentation was expository. In the other two, discovery learning--either in groups or individually--was the mode. Results were assessed in terms of immediate recall, longer-term retention, and transfer. Data analyses indicated that the expository presentation was the most effective for immediate recall. In terms of retention and transfer, however, the group discovery approach was clearly superior to both other treatments.
Critique: Teaching Problem-Solving Heuristics: A Response
Authors: Brennan Michael L. , Post Thomas R.
Published: August 2020

The improvement of problem-solving ability continues to be an elusive, yet important, goal of mathematics education. A variety of individual attempts have been made to improve student performance in this area. The Journal for Research in Mathematics Education (JRME) has performed a valuable service to the profession by facilitating interaction between individuals who have made or are contemplating such attempts. Only through such professional interaction will old ideas be reshaped and new ones generated. In this manner perhaps we shall someday provide definitive answers to questions we know to be of mutual concern. The following comments constitute our response to Kulm's critique of our previous JRME article (Post % Brennan, 1976).