bluuebunny/crossref_metadata_2025 huggingface dataset:
Some Implications of the International Study of Achievement in Mathematics for Mathematics Educators
Authors: Kilpatrick Jeremy
Published: August 2020

The report of the International Study of Achievement in Mathematics has been around for almost four years. During that time it has been damned and defended, distorted and dismissed. Now that the rest of the parade has moved on—the lEA to other curriculum subjects, the book reviewers to other books, the editorial writers and their readers to other diversionsit seems appropriate for mathematics educators to take stock of the study and its findings. In this article, I discuss some of the implications of the study in the hope that the reader will be moved to look at the report more closely, with a greater appreciation of what it does and docs not say.
Testing Students' Ability to Do Geometric Proofs: A Comparison of Three Objective Item Types
Authors: Hanna Gerald S.
Published: August 2020

In final geometry examinations and in standardized achievement tests, scoring convenience often precludes the use of formal student-produced proofs. This study sought objective item types that would as nearly as possible measure the same abilities as those measured by student-produced formal proofs. A test of conventional proofs was used to compare the concurrent validity of 3 objective item types. Significant differences among the experimental item types were found.
Critiques Of Articles: Comments on Fletcher's Efficiency Ratio
Authors: Romberg Thomas A. , Montgomery Mary
Published: August 2020

In his paper, “An Efficiency Reanalysis of the Results,” Fletcher (1971) uses an efficiency ratio as the basis for reexamining data from the Inte rnational Study (Husén, 1967). The purpose of this paper is to cla rify Fletcher's definition of an efficiency ratio and to comment upon his use of the ratio in his reanalysis.
Editorial Comment
Authors: Johnson David C.
Published: August 2020

This issue of JRME includes a selected annotated bibliography of research reported in 1970. The listing has been an annual feature in the Arithmetic Teacher; however, the Mathematics Teacher and the Arithmetic Teacher panels have established new policies regarding the publishing of research articles and indicated that the annotated listing more appropri ately belongs in JRME. To make the listing more comprehensive—the material previously reported in the Arithmetic Teacher was usually restricted to research in elementary education, K–8—Suydam and Weaver have extended the bibliography to include research pertaining to secondary school mathematics. With this expansion the final listing should be of considerable value to mathematics education researchers. On behalf of the JRME subscribers and the Editorial Board a special thank you is extended to the authors for their efforts. Present plans for JRME include provisions for the bibliography to appear as an annual feature, published in each November issue.
Verbal Factors and Mathematics Learning: A Review of Research
Authors: Aiken Lewis R.
Published: August 2020

Recent investigations pertaining to the importance of verbal factors in the learning of mathematics are reviewed. The paper is divided into 3 sections: (a) the relationships of mathematical ability to reading ability and general intelligence, (b) reading instruction and mathematics learning, and (c) student and teacher verbalizations. The importance of general intelligence for mathematics achievement is recognized, but the evidence for a group factor of mathematical ability is not convincing. Training in careful, analytical reading appears to have a beneficial effect on achievement in mathematics, but more controlled experiments with larger samples need to be conducted. The influence of verbalizing awareness of mathematical generalizations and the effects of teacher–student verbal interactions in mathematics classroom settings are other promising directions for research. Finally, an appeal is made for long–term multivariate investigations rather than piecemeal, one–shot studies.
Brief Reports: Preservice Teachers' Misconceptions in Solving Verbal Problems in Multiplication and Division
Authors: Graeber Anna O. , Tirosh Dina , Glover Roseanne
Published: August 2020

In recent years much has been written about children's and adolescents' misconceptions concerning the operations needed to solve multiplication amd division word problems (e.g., Bell, 1982; Bell, Fischbein, & Greer, 1984; Bell, Swan, & Taylor, 1981; Brown, 1981; Yergnaud, 1983). Fischbein, Deri, Nello, and Marino (1985) argued that the arithmetical operations generally remain linked to implicit primitive behavioral models. They noted that the primitive models for multiplication and division may be viewed as the source of misconceptions such as “the divisor must be a whole number” or “multiplication always makes bigger and division always makes smaller.” The purpose of our study was to explore whether preservice elementary reachers have the same misconceptions. If preservice teachers hold these misconceptions, they are not likely to recognize the related errors students make. In fact. their instruction might inadvertently contribute to perpetulating the misconceptions.
The Development of Concepts and Strategies Used in Computational Estimation
Authors: Sowder Judith T. , Wheeler Margariete M.
Published: August 2020

Twelve students at each of Grades 3, 5, 7, and 9 were individually given tasks that presented problems with solutions from hypothetical students, accompanied by questions requiring students to contrast and compare the solutions. These tasks were followed by open-response estimation problems. The older children understood better than the younger children what was asked but were uncomfortable with estimation processes and outcomes. Acceptance of multiple estimates and rounding-then-computing rather than computing-then-rounding were both slow to develop. Recognition of the need to compensate for rounding errors increased with grade level. Schooling factors such as emphasis on unique answers and instruction on rounding and computional procedures seemed to influence results. Careful development of foundational concepts is recommended to prevent learning computational estimation as a set of algorithmic rules
Effects of Teaching Strategy, Relevant Knowledge, and Strategy Length on Learning a Contrived Mathematical Concept
Authors: Stiff Lee V.
Published: August 2020

The effects of teaching strategy, relevant knowledge, and strategy length on students' ability to learn a contrived mathematical concept were studied with 326 high school students randomly assigned to 1 of 18 treatment groups characterized by the type and length of teaching strategy and students' level of exposure to subordinate concepts and relationships. The teaching strategy and relevant knowledge effects on concept acquisition were significant; the strategy-length effect was not. Specifically, characterization strategies were more effective than exemplification strategies, and students' test performance increased as levels of relevant knowledge increased. Furthermore, exemplification moves were more effective for students with low relevant knowledge, and characterization moves were more effective for students with high relevant knowledge. The results supported John Kolb's model of concept-teaching strategies.
Allowing for Student Cognitive Levels in the Teaching of Fractions and Ratios
Authors: Harrison Bruce , Brindley Selwyn , Bye Marshall P.
Published: August 2020

In the context of fractions and ratios, assessments were made of cognitive ability level, achievement, facility with general mathematical strategies, computational ability, and attitude of 435 12-year-olds. Approximately half the students were taught fraction and ratio topics by means of a concrete, process-oriented method, whereas the other half were taught in the regular way. The concrete, process-oriented approach resulted in significantly improved achievement in, and attitude toward, fractions and ratios. Also, the development of general mathematical strategies was enhanced and computational facility maintained.
Explorations of Students' Mathematical Beliefs and Behavior
Authors: Schoenfeld Alan H.
Published: August 2020

A questionnaire with 70 closed and 11 open questions was administered to 230 mathematics students enrolled in Grades 10 through 12, the majority of whom were enrolled in the traditional year-long 10th-grade course in plane geometry. Sections of the questionnaire dealt with the students' attributions of success or failure; their comparative perceptions of mathematics, English, and social studies; their view of mathematics as a discipline; and their attitude toward mathematics. The data, which are closely tied to a series of classroom and protocol studies, suggest the resolution of contradictory patterns of data in other attitude surveys, where students simultaneously claim that “mathematics is mostly memorizing” but that mathematics is a creative and useful discipline in which they learn to think.
Developmental Differences in Children's Understanding of Number Word Conventions
Authors: Saxe Geoffrey B. , Becker Joseph , Sadeghpour Mojdeh , Sicilian Steven
Published: August 2020

The four studies reported here document developmental differences in children's understanding that symbols for number are arbitrary conventions. Children between 3 and 12 years were presented with tasks in which they had to make judgments about (a) the adequacy of puppets' counting activities when puppets used standard as opposed to nonstandard number words in counting and (b) the adequacy of puppets' token exchanges when the values of the same tokens varied across numerical systems. In both tasks, the findings revealed developmental shifts in children's ability to distance numerical meanings from conventional symbols. In the number words tasks, children increasingly appreciated that, provided the principle of one-to-one correspondence between symbol and counted item is maintained, any symbol list may be used for counting. In the token-exchange tasks, children increasingly appreciated that the same tokens could represent different values in the different systems.
Effects of Manipulative Materials in Mathematics Instruction
Authors: Sowell Evelyn J.
Published: August 2020

Using meta-analysis, the results of 60 studies were combined to determine the effectiveness of mathematics instruction with manipulative materials. Students ranged in age from kindergarteners to college-age adults and studied a variety of mathematics topics. Results showed that mathematics achievement is increased through the long-term use of concrete instructional materials and that students' attitudes toward mathematics are improved when they have instruction with concrete materials provided by teachers knowledgeable about their use. Instruction with pictures and diagrams did not appear to differ in effectiveness from instruction with symbols.
The Nature, Effects, and Relief of Mathematics Anxiety
Authors: Hembree Ray
Published: August 2020

Results of 151 studies were integrated by meta-analysis to scrutinize the construct mathematics anxiety. Mathematics anxiety is related to poor performance on mathematics achievement tests. It relates inversely to positive attitudes toward mathematics and is bound directly to avoidance of the subject. Variables that exhibit differential mathematics anxiety levels include ability, school grade level, and undergraduate fields of study, with preservice arithmetic teachers especially prone to mathematics anxiety. Females display higher levels than males. However, mathematics anxiety appears more strongly linked with poor performance and avoidance of mathematics in precollege males than females. A variety of treatments are effective in reducing mathematics anxiety. Improved mathematics performance consistently accompanies valid treatment.
Review: The Research Agenda Project as Prologue
Authors: Fisher Charles W.
Published: August 2020

There may be good news for precollege mathematics. For those who are alarmed by recent results of the National Assessment of Educational Progress, the widening gap between supply and demand for mathematically literate graduates (including new mathematics teachers), and discouraging comparisons in international studies, this possibility may come as a surprise. The good news comes in the form of the Research Agenda Project (RAP) for mathematics education that was conceived and implemented by NCTM's Research Advisory Committee.
Cognitive Aspects of Sharing
Authors: Davis Gary Ernest , Pitkethly Anne
Published: August 2020

To probe certain logical aspects of preschoolers' dealing and counting we showed edited highlights of three preschool children engaged in sharing activities to 17 second-grade children in an interview situation. In the videotaped episodes, the preschoolers routinely counted or checked the heights of the stacks when asked how they knew there was a fair share; the second graders, with one notable, and another possible, exception, expressed the view that counting after dealing was essential to tell if there was a fair share. One second grader gave us articulate explanations about the procedures used by the preschoolers, and another apparently changed her mind during the course of the interview about the need to count after dealing out.
Brief Reports: The Microcomputer as a Demonstration Tool for Instruction in Mathematics
Authors: Ganguli Aparna B.
Published: August 2020

A number of educators have indicated that for successful implementation of computer-assisted instruction (CAI), much more attention is needed in the area of teachers' use of this technology (Allinger, 1985; Lawton & Gerschner, 1982; Jay, 1981; Simmons, 1975). Although studies on CAI for large-group instruction demonstrate favorable results (Sherwood & Hasselbring, 1986; Berger, 1984; Phillips, Burkhardt, Coupland, & Fraser, 1984), relatively little research has been conducted on the use of CAI with teachers as the instructional leaders.
Review: Hot and Cold Responses to Good Mathematics
Authors: Kifer Edward
Published: August 2020

This is a tough book for me, both to read and to review. Part of its difficulty is format-22 different authors work with and write about specific aspects of a general topic; part is the wide variation in content, theory, and research methodology employed across the volume; and part is a demand placed on the reader or reviewer to be aware not only of the issues addressed in the past when looking at relationships between achievement and affect, broadly construed, but also to be aware of new perspectives and issues raised by cognitive scientists, again broadly defined.
Criterion-Referenced Reliability Indices Associated with the van Hiele Geometry Test
Authors: Crowley Mary L.
Published: August 2020

Recently the work of Pierre M. van Hiele and Dina van Hiele-Geldofhas gained prominence in the study of the teaching and learning of geometry. Their three-part model (a) describes five sequential and discrete levels learners pass through as geometric thought develops, (b) discusses the nature of insight into geometric concepts, and (c) presents a guide to the development of geometric lessons. A detailed description of the complete model can be found in Crowley (1987).
Towards a Problématique for Research on Mathematics Teaching
Authors: Balacheff Nicolas
Published: August 2020

This article presents the main features of the theoretical framework of French research known as recherches en didactique des mathématiques. The foundation of this approach consists mainly of the relationships between two hypotheses and two constraints, which are presented together with some specific key words. Outlines are given of Brousseau's théorie des situations didactiques (theory of didactical situations). An example is given that presents in some detail the rationale for the construction of a didactical situation and its analysis. This article ends with some questions addressed to research on mathematics teaching.
High School Mathematics Review Lessons: Expert-Novice Distinctions
Authors: Livingston Carol , Borko Hilda
Published: August 2020

This investigation contrasts two review lessons of two secondary mathematics student teachers with those of their high school cooperating teachers. The data included lesson descriptions constructed from field observations and audiotapes, pre- and post-lesson interviews, and planning materials. Differences between the review lessons were analyzed by interpreting teaching as a complex cognitive skill and as improvisational performance. Despite extensive preparation, the novices' review lessons were less comprehensive than those of the experts, and their explanations were less conceptual. The experts more skillfully improvised activities and explanations around student questions and comments. These differences are explained by the assumption that novices' cognitive schemata for content and pedagogy are less elaborated, interconnected, and accessible than those of the experts.
Critique: The Case for Instruction in the Use of General Problem-Solving Strategies in Mathematics Teaching: A Comment on Owen and Sweller
Authors: Lawson Michael J.
Published: August 2020

Owen and Sweller (1989) question the wisdom of recent moves to allocate time in mathematics teaching to instruction in the use of general problem-solving strategies because they doubt that such instruction will help overcome problems in the transfer of learning. According to Owen and Sweller transfer failure is more likely to be the result of a lack of appropriate schema or insufficient automation of rules. They imply that attention allocated to general problem-solving strategies would be more appropriately diverted to instruction concerned with domain-specific knowledge and practice with worked examples and goal-modified problems. Because curricula in several countries are in the process of being modified to incorporate explicit consideration of the nature of general problem-solving strategies, Owen and Sweller's view that the evidence on the efficacy of such instruction is “very sparse” deserves examination.
President's Report: Maintaining Momentum as we Reach New Heights
Authors: Frye Shirley M.
Published: August 2020

Recently I went back into my files to reminisce over the exciting and challenging years that have passed so quickly since Iris Carl, the chair of the Election Committee, invited me to accept the honor of a nomination for president-elect. There I found and read the position statement that I had written as a candidate in 1986 and decided that it was a most appropriate opening for these remarks.
Assessment of a Problem-Centered Second-Grade Mathematics Project
Authors: Cobb Paul , Wood Terry , Yackel Erna , Nicholls John , Wheatley Grayson , Trigatti Beatriz , Perlwitz Marcella
Published: August 2020

Ten second-grade classes participated in a year-long project in which instruction was generally compatible with a socioconstructivist theory of knowledge and recent recommendations of the National Council of Teachers of Mathematics. At the end of the school year, the 10 project classes were compared with 8 nonproject classes on a standardized achievement test and on instruments designed to assess students' computational proficiency and conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computational performance were comparable, but there were qualitative differences in arithmetical algorithms used by students in the two groups. Project students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success. Responses to a questionnaire on pedagogical beliefs indicated that the project teachers' beliefs were more compatible with a socioconstructivist perspective than were those of their nonproject colleagues.
Effects of Computer Algebra Systems on Concept and Skill Acquisition in Calculus
Authors: Palmiter Jeanette R.
Published: August 2020

This study involving 78 subjects compared the performance of university students taught calculus using a computer algebra system to the performance of students using paper-and-pencil computations. Students who were taught calculus using a computer algebra system had higher scores on a test of conceptual knowledge of calculus than the students taught by traditional methods. Students in the computer class also had higher scores on a calculus computational exam using the computer algebra system than students in the traditional class using paper and pencil.
Models of Limit Held by College Calculus Students
Authors: Williams Steven R.
Published: August 2020

This study documents 10 college students' understanding of the limit concept and the factors affecting changes in that understanding. Common informal models of limit were identified among the 10 students, who were then presented with alternative models of limit and with anomalous limit problems. The problems were designed to encourage students to make changes in their own models to reflect a more formal conception. Individual models of limit varied widely even among students who initially described limits in similar ways. The dynamic aspect of these models was extremely resistant to change. This resistance was influenced by students' belief in the a priori existence of graphs, their prior experiences with graphs of simple functions, the value they put on conceptually simple and practically useful models, and their tendency to view anomalous problems as minor exceptions to rules. These factors combined to inhibit students' motivation to adopt a formal view of limit.
Task-Related Verbal Interaction and Mathematics Learning in Small Groups
Authors: Webb Noreen M.
Published: August 2020

The past fifteen years have shown a resurgence of interest in small-group, peer-directed learning in the classroom. This article reviews and analyzes the research linking task-related verbal interaction to learning in small groups in mathematics classrooms, as well as factors that have been shown to predict peer interaction in mathematics groups, and discusses strategies for shaping group interaction. Critical features of group interaction include the level of elaboration of help given and received and the responsiveness of help to the needs of students. Important predictors of group interaction included student ability, gender and personality, and group composition on ability and gender. Possible strategies for promoting effective small-group interaction include using certain group compositions, altering the reward structure, providing training in desirable verbal behavior, and structuring the group activity to require students to give explanations to each other.
Small-Group Interactions as a Source of Learning Opportunities in Second-Grade Mathematics
Authors: Yackel Erna , Cobb Paul , Wood Terry
Published: August 2020

Small-group problem solving was used as a primary instructional strategy for all aspects of second-grade mathematics, including computation, for the entire school year. This gave rise to learning opportunities that do not typically occur in traditional classrooms, including those that arise from collaborative dialogue as well as from the resolution of conflicting points of view. The nature of these learning opportunities is elaborated and illustrated. The manner in which the teacher used paradigm cases as she initiated and guided discussion of obligations and expectations to make possible the mutual construction of classroom norms for cooperative learning is also illustrated. This and the use of cognitively based activities designed to be problematic for children at a variety of conceptual levels are the crucial features of a cooperative learning environment in the absence of extrinsic rewards.
The Role of Cooperative Learning in Increasing Problem-Solving Ability in a College Remedial Course
Authors: Dees Roberta L.
Published: August 2020

This study was conducted in a college remedial mathematics course to determine whether a minimal level of teacher intervention could induce cooperative learning on the part of students and whether cooperative learning would help students increase their problem-solving skills in mathematics. The intact, four-credit semester course included high school algebra and geometry. Practical methods that induced cooperation, with only three direct teacher interventions during the semester, are described. Instruments for evaluation of achievement included multiple-choice skills tests and open-ended teacher-made tests. Students using cooperative learning performed as well as or better than the control on every measure. Outcome variables that showed significant differences in favor of cooperative learning were solving word problems in algebra and proof-writing in geometry. It was concluded that the resulting increase in students' problem-solving abilities justified use of the method.
Bilingualism and Mathematics Learning: Another Perspective
Authors: Clarkson Philip C. , Galbraith Peter
Published: August 2020

There has been continuing discussion in the literature as to the effects of bilingualism on cognitive tasks. Much of the research has been carried out with immigrant students in developed countries. This study is set in a developing country and was based on the theoretical work of Cummins. Grade 6 students from five urban Community Schools in Papua New Guinea completed several mathematics and language tests, a survey form eliciting information on home background, and a test of cognitive development. The data supported the thesis that students' level of competence in their original tongue and in English, the language of their regular schooling, were significant influences on mathematical performance. In particular these results supported Cummins's threshold hypothesis. They also support the notion that bilingualism should not be taken as a unidimensional entity, and that allowance should be made for its multidimensional nature when carrying out research and when making educational decisions.
Computational Estimation Strategies of Professional Mathematicians
Authors: Dowker Ann
Published: August 2020

Forty-four academic pure mathematicians were given Levine's (1982) computational estimation task, which involved estimating the answers to 20 multiplication and division problems and describing the strategies used. The mathematicians were accurate estimators, and they used a great variety of strategies, as many as 23 for a single problem. All but 2 of the mathematicians used some strategy that was not used by the other 43 for at least one problem. When 18 mathematicians were retested after an interval of several months, they used different strategies on the second test for 9 to 17 of the 20 problems. Some theoretical implications of these findings are discussed.
Links between Teaching and Learning Place Value with Understanding in First Grade
Authors: Hiebert James , Wearne Diana
Published: August 2020

Conceptually based instruction on place value and two-digit addition and subtraction without regrouping was provided in four first-grade classrooms, and more conventional textbook-based instruction was provided in two first-grade classrooms. An observer compiled extensive notes of 20 lessons in each kind of classroom. Students who received conceptually based instruction performed significantly better on items measuring understanding of place value and two-digit addition and subtraction with regrouping and used strategies more often that exploited the tens and ones structure of the number system. Content and pedagogical differences between the instruction lessons are linked to the learning differences and are used to explain between-group differences in levels of performance and understanding. Observations are offered on the complex interactions between instruction, understanding, and performance.
Korean Children's Single-Digit Addition and Subtraction: Numbers Structured by Ten
Authors: Fuson Karen C. , Kwon Youngshim
Published: August 2020

Korean children's ability to solve addition problems with sums of 10, single-digit addition problems with sums between 10 and 18, and single-digit subtraction problems with minuends between 10 and 18 was assessed in interviews given at the end of the first semester of first grade, before children had studied problems involving numbers larger than 10 in school. These children showed considerable competence with all three kinds of problems, solving correctly 95%, 85%, and 75% of these problems, respectively. Almost two-thirds of the solutions of the problems above 10 were addition or subtraction recomposition methods structured around ten or known facts. Korean children demonstrated two finger methods that allowed fingers to be reused to show sums over 10. These methods and the regular named-ten Korean number words for numbers between 10 and 18 (“ten, ten one, ten two, ten three,…, ten eight”) support Korean children's learning of three efficient recomposition methods structured around ten.
Editorial
Authors: Carpenter Thomas P.
Published: August 2020

Research in mathematics education and related fields is in the midst of a major paradigm shift. This has lead to a great deal of discussion in the research community about standards of evidence. Whatever the methodology, questions of standards of evidence ultimately boil down to whether the conclu ions drawn on the basis of a given study are warranted or whether there are plausible alternative hypotheses that are not consistent with the conclusions. In this case plausibility is evaluated not in terms of whether the conclusions themselves are plausible but whether a compelling case has been built that the evidence leads to the conclusions. In other words, we should not evaluate research in tenns of whether it tells a story that we resonate with, but whether the evidence presented leads to the conclusions drawn. On several occasions I have encouraged researchers to submit articles to the joumal based on discussions of findings that l found particularly compelling. Only when it was possible to examine a complete report of the studies did it become apparent that there were such serious flaws in methodology that the validity of the conclusions was called into question. The authors had an interesting story to tell. but they did not have evidence to support it that was convincing to the reviewers.
Further Thoughts on the Standards: A Reaction to Apple
Authors: Romberg Thomas A.
Published: August 2020

As always, I enjoy and learn from the thoughtful analyses and well-argued comments about educational practices by my colleague Michael Apple. In reading his often flattering, yet critical, analysis of NCTM's Curriculum and Evaluation Standards/or School Mathematics (1989) and Professional Standards for Teaching Mathematics (1991), I must admit he is right on several counts. The lens through which he views the world is not one familiar to most mathematics educators. Relating any message about schooling to the patterns of differential economic, political, and cultural power is not common in our field. Furthermore. the five issues he raises—the financial crisis in education, the nature of inequality in schools, the role of mathematical knowledge as a filter that maintains those inequalities. the possibilities and limitations of a curriculum grounded in problem situations, and the complex realities of teachers' lives—are critical issues facing all educators interested in changing the current system. Clearly. the Standards documents by themselves are not well situated with respect to these broader ideological and political issues. In fact, none of the five issues, although considered, were the driving forces behind the NCTM's reform documents.
The Cognitive Complexity of Learning and Doing Arithmetic
Authors: Ohlsson Stellan , Ernst Andreas M. , Rees Ernest
Published: August 2020

A theory of the cognitive processes involved in doing and learning place value arithmetic is proposed. The theory is embodied in a computer model that simulates the learning of multicolumn subtraction under one-on-one tutoring. The model is used to measure the relative difficulty of two different methods for subtraction, with either a conceptual or a procedural representation. The model predicts that regrouping is more difficult to learn than an alternative method, particularly in a conceptual representation, a result that contradicts current practice in U. S. schools.
Conceptual Knowledge Falls through the Cracks: Complexities of Learning to Teach Mathematics for Understanding
Authors: Eisenhart Margaret , Borko Hilda , Underhill Robert , Brown Catherine , Jones Doug , Agard Patricia
Published: August 2020

In this article we focus on two interrelated aspects of the process of learning to teach mathematics for understanding: (a) ideas and practices for teaching procedural knowledge and (b) ideas and practices for teaching conceptual knowledge. We explore one student teacher's ideas and practices, together with the messages about teaching for procedural and conceptual knowledge that were presented by the teacher education program in which the student teacher was enrolled and the placement schools in which she student taught. We reveal a pattern in which the student teacher, her mathematics methods course instructor, her cooperating teachers, and the administrators of her placement schools expressed a variety of strong commitments to teaching for both procedural and conceptual knowledge; but with these commitments, the student teacher taught, learned to teach, and had opportunities to learn to teach for procedural knowledge more often and more consistently than she did for conceptual knowledge. We find that the actual teaching pattern (what was done) was the product of unresolved tensions within the student teacher, the other key actors in her environment, and the learning-to-teach environment itself. We hypothesize that situational supports constructed to emphasize more consistently teaching for conceptual knowledge might help resolve at least some of the tensions, and we suggest that such supports should be developed if the national goal to increase the teaching of mathematics for understanding is to be achieved.
Ratio and Proportion: Connecting Content and Children's Thinking
Authors: Lamon Susan J.
Published: August 2020

Twenty-four sixth-grade children participated in clinical interviews on ratio and proportion before they had received any instruction in the domain. A framework involving problems of four semantic types was used to develop the interview questions, and student thinking was analyzed within the semantic types in terms of mathematical components critical to proportional reasoning. Two components, relative thinking and unitizing, were consistently related to higher levels of sophistication in a student's overall problem-solving ability within a semantic type. Part-part-whole problems failed to elicit any proportional reasoning because they could be solved using less sophisticated methods. Stretcher/shrinker problems were the most difficult because students failed to recognize the multiplicative nature of the problem situations. Student thinking was most sophisticated in the case of associated sets when problems were presented in a concrete pictorial mode.
Review: Cognitive Snapshots of the Stochastic River
Authors: Shaughnessy J. Michael
Published: August 2020

A tour through seven different perspectives on the learning of probability offered by the chapter authors of Chance Encounters is simultaneously enlightening and frustrating. It is enlightening because each perspective provides some conceptual morsels for researchers in probability to assimilate or to reject. It is frustrating because no synthesis of the perspectives, no careful distillation of the major ideas in the book, was attempted either by the editors or by any chapter author. The book can be thought of as a series of cognitive photographs from a river tour of stochastic thinking. Like real photos, some of the pictures we receive are in crystalline focus, whereas others are either underdeveloped or overdeveloped. Each reader will have to construct an overall perspective for himself or herself in order to bring the main ideas into focus.
Brief Reports: Some Factors That Impede or Enhance Performance in Mathematical Problem Solving
Authors: Hart Lynn C.
Published: August 2020

Considerable emphasis in problem-solving research has been placed on studying the processes used by expert problem solvers or above-average students, often with the goal of identifying processes that might be taught to average-and below-average students (Dalton, 1974; Knippenberg, 1978; Simon, 1986). But, as Lesh (1981) points out, “the qualitatively different systems of thought used by gifted problem solvers may be … inaccessible to … average-ability children” (p. 239). Lesh's comments indicate that it may be important to identify factors that enhance or impede the problemsolving progress of average-ability students, rather than focusing attention on experts. Lester (1987) also calls for research in th is area. Identifying some of these factors was the goal of this study (a more complete report of which can be found in Hart, 1985).
Review: Arithmetic as Mathematics
Authors: Kouba Vicky L.
Published: August 2020

Too often in the past “arithmetic” has been viewed as simple computing or simple acquisition of procedures. In contrast, the chapters in Analysis of Arithmetic for Mathematics Teaching portray arithmetic as a thoughtful building of a flexible knowledge of our complex number system, rich in concepts, connections, and patterns that are crucial for understanding most of the rest of mathematics. This book provides research foundations and conceptual analyses of the standard topics of arithmetic: understanding of and operations on whole numbers, fractions, and decimals. The authors provide updates on the results, theories, and thoughts about children's learning of arithmetic concepts and processes, a field that has coalesced over the past twenty years.
Diagnosing Knowledge States in Algebra Using the Rule-Space Model
Authors: Birenbaum Menucha , Kelly Anthony E. , Tatsuoka Kikumi K.
Published: August 2020

This paper illustrates the use of rule space as a tool to support cognitive analyses of students' mathematical behavior. The rule-space approach is explained and is then used to classify students into one of two methods for solving linear algebraic equations in one unknown and to diagnose their knowledge states in this topic. A 32-item test with open-ended questions was administered to 231 eighth and ninth graders. Two outcomes of the rule-space model are presented: (a) a classification of examinees into knowledge states resulting from the two solution approaches at the group level along with individual examples and (b) tree diagrams of the transitional relationships among the states for each strategy. Implications for using the feedback provided by the rule-space model in the context of instruction and assessment are discussed.
Developing Number Sense: An Intervention Study in Grade 7
Authors: Markovits Zvia , Sowder Judith
Published: August 2020

Few students exhibit number sense when solving arithmetic problems in school. This study examined the effects of an intervention in the instruction of seventh-grade students for the purpose of developing number sense. Students were taught by the classroom teacher from experimental units on number magnitude, mental computation, and computational estimation. Instruction was designed to provide rich opportunities for exploring numbers, number relationships, and number operations and to discover rules and invent algorithms. Written measures and interviews before instruction, immediately after instruction, and several months later revealed that after instruction students were more likely to elect to use strategies that reflected number sense and that this was a long-term change. It appeared to the investigators that the students reorganized and used existing knowledge rather than acquiring new knowledge structures.
A Forum for Researchers: Some Further Observations of the Environment/Model Metaphor
Authors: Greeno James G.
Published: August 2020

I am grateful to Michael Battista (see the preceding article, pp. 86–94) for the complime nt of his interest in ideas that I began to develop (Greeno, 1991a) and for his advancing those ideas in his very interesting and useful discusion. For a metaphor of the kind I proposed to be scientifically productive. it has to be developed in further depth and detail and be applied in additional domains. and Battista's discussion does this with the metaphor of a conceptual domain as a spatial environment. I am also grateful to Frank Lester for the invitation to comment on Battista's discussion and to add orne remarks that reflect further research, discussions, and thinking of mine subsequent to the paper that Battista discussed so generously.
Using Retelling Data to Study Elementary School Children's Representations and Solutions of Compare Problems
Authors: Verschaffel Lieven
Published: August 2020

In an attempt to explain subjects' difficulties with particular types of compare problems, Lewis and Mayer (1987) have presented a model that simulates the comprehension processes when solving these problems. The crucial component of their model is the “consistency hypothesis,” according to which students are more likely to experience representational difficulties—and consequently to make errors—when the presentation of the terms in the relational statement of the compare problem is not consistent with their preferred format. To test this model, an experiment was carried out in which 40 fifth graders were asked to solve and retell a set of one-step compare problems. The accuracy data, solution times, and, especially, the retelling data yielded strong evidence in favor of the consistency hypothesis.
Capitalizing on Errors as “Springboards for Inquiry”: A Teaching Experiment
Authors: Borasi Raffaella
Published: August 2020

Although teachers and researchers have long recognized the value of analyzing student errors for diagnosis and remediation, students have not been encouraged to take advantage of errors as learning opportunities in mathematics instruction. The study reported here was designed to explore how secondary school students could be enabled to capitalize on the potential of errors to stimulate and support mathematical inquiry. The article provides a case study of the proposed strategy of “using errors as springboards for inquiry” in action, identifies some important variations within the strategy, and discusses its potential contributions to mathematics instruction.
Functional Thinking with Graphs
Authors: Biehler Rolf
Published: August 2020

The representation of functions by curves and areas, however, has opened a new world of notions and taught one of the most fruitful methods by which the human mind has enhanced its own powers, … Between two arbitrary quantities, one of which may be conceived of as being dependent on the other, no relationship, not even the most complicated one, is conceivable which would not be representable by a curve, a fact instructively documented by Quetelelt's representation of the penchant towards crime, literary talent, etc., by curves as a function of the individual's age. This way of imagin ing the connection between things thus is just as useful for the administrator, the national economist, the physicist and the meteorologist. (Emil du Bois-Reymond, my translation. R. B.)
Introduction
Authors: Wilson James W. , Brown Catherine A. , Kieran Carolyn , Lester Frank K.
Published: August 2020

This special issue of the Journal for Research in Mathematics Education was prepared to help celebrate the 25th anniversary year of the journal. President Mary Lindquist appointed an ad hoc task force to develop activities to mark this 25th year. Input was solicited from former editorial board members and editors and from others throughout mathematics education. We came to a recognition that doing something to reflect on the journal's journey over the past 25 years, while underscoring the scholarship that guides our work, would be a vehicle to help look ahead to the next 25 years.
The Origins of the JRME: A Retrospective Account
Authors: Johnson David C. , Romberg Thomas A. , Scandura Joseph M.
Published: August 2020

This article presents a hitory of the origins of the Journal for Research in Marhematics Education (JRME) published by the National Council of Teachers of Mathematics (NCTM). Written by three of the many persons responsible for initiating the journal, this account is based on minutes of meetings, notes from files, and personal recollections of the events and persons involved in the deliberations over a quarter of a century ago. David C. Johnson, then at the University of Minnesota, was the first editor of the JRME (Volumes 1 to 4). Thomas A. Romberg was chair of NCTM's Research Advisory Commillee, which wrote the proposal arguing-that a research journal was both needed and an appropriate publication for NCTM to undertake. Joseph M. Scandura was the editor of Research in Mathematics Education, a special NCTM publication designed to determine the feasibility of publishing a research journal. We are confident that this retrospective account in general captures the issues and events leading to the decisions to publish the JRME.
Research and Teacher Education: In Search of Common Ground
Authors: Cooney Thomas J.
Published: August 2020

Most mathematics educators are involved in the practice of teacher education at some level. Indeed, the field of mathematics education is predicated on the assumption that someone has to be educated to teach mathematics in our schools. This raises the question of what it means to be educated in o rder to become a teacher of mathematics. What kinds of knowledge do teachers need to become effective teachers of mathematics? What sorts of experiences are needed for teachers to acquire this knowledge? A fundamental question for mathematics teacher educators is how the field of teacher education can be conceptualized so that programs and activities can be created to assist in the acquisition of this knowledge. Given the high visibility of standards developed by the National Council of Teachers of Mathematics (NCTM) (NCTM, 1988, 1991, in press), a question of interest to many is, What does it take to develop teachers who can move the field toward realizing these standards?
Technology in Mathematics Education Research: The First 25 Years in the JRME
Authors: Kaput James J. , Thompson Patrick W.
Published: August 2020

The first 25 years of the JRME overlap with the first decades of electronic technologies in education. Hence the growth of technology and research in mathematics education have tended to occur in parallel. But the interactions among mathematics education research, developments in technology. and the evolving nature of school mathematics and learning are complex. To some extent, the technology was superimposed on both school practice and research in mathematics education. On the other hand, it has become increasingly evident that the technology altered the nature of the activity using it.
Students' Use of Part-Whole and Direct Comparison Strategies for Comparing Partitioned Rectangles
Authors: Armstrong Barbara E. , Novillis Larson Carol
Published: August 2020

Thirty-six students, twelve each at the fourth-, sixth-, and eighth-grade levels, were asked to solve 21 comparison-of-area tasks during a clinical interview. The areas were partitioned and shaded so that the tasks could be solved by using rational number knowledge. The types of strategies students used to compare the areas were identified and classified into categories. Initially, fractional terms and symbols were not introduced into the tasks in order to compare the types of strategies students used without and with symbols introduced. Most students tended to ignore the part-whole relationships inherent in the tasks and used a Direct Comparison strategy when they compared the areas. The use of a Part-Whole strategy increased with the introduction of fractional terms and symbols, especially at the eighth-grade level.