bluuebunny/crossref_metadata_2025 huggingface dataset:
Computational Estimation Performance and Strategies used by Fifth- and Eighth-Grade Japanese Students
Authors: Reys Robert E. , Reys Barbara J. , Nohda Nobuhiko , Ishida Junichi , Yoshikawa Shigeo , Shimizu Katsuhiko
Published: August 2020

Four hundred and sixty-six fifth- and eighth-grade Japanese students were administered a computational estimation test. The fifth-grade mean was 7.39 and the eighth-grade mean was 11.15 on the 39-item open-ended test. Interviews with 21 students who had scored in the top 5% revealed that the Japanese students employed the three general cognitive processes outlined in a theoretical model based on interviews with United States students: reformulation, translation, and compensation. They also used many of the same strategies (front-end, compatible numbers, flexible rounding) utilized by American students. Few of the Japanese students could recall being taught to estimate in school. Japanese students demonstrated a greater degree of mental computation ability than American students, less frequently made order-of-magnitude errors, and were more reluctant to accept error. Japanese students tended to apply algorithmic computational procedures. Their tendency to use paper-and-pencil procedures mentally often interfered with the estimation process.
Review: Toward a Cultural Psychology of Mathematics
Authors: Bishop Alan J.
Published: August 2020

We are all like the candy sellers—says Saxe. We all interact in socially defined situations that involve more, or fewer, mathematical ideas, some of which we might have been taught in school. (What's your personal method for calculating the amount of the tip to be added to the restaurant bill?) The cliche to help us think about this kind of situation used to be “First learn the technique in school, then apply it to the real world.” That maxim has been under siege for some time now, particularly from researchers with anthropological leanings.
Research on Mathematics Education Reported in 1990
Authors: Suydam Marilyn N. , Crocker Deborah A.
Published: August 2020

As in previous years, this 21st 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 (when the researcher has provided this information or it can be inferred). An index is included at the end to help readers locate studies on certain broad topics.
An Overview of Research on Cooperative Learning Related to Mathematics
Authors: Davidson Neil , Kroll Diana Lambdin
Published: August 2020

Increased use of cooperative learning methods is one of the most visible changes in (mathematics) education in the last decade. To illustrate, the National Counci l of Teachers of Mathematics has promoted cooperative work in its two recent Standards documents (National Council of Teachers of Mathematics [NCTM], 1989, 1991), and the entire December 1989 issue of Educational Leadership (Brandt, 1989) was devoted to cooperative learning. Given all this attention, it seems reasonable to ask what research evidence there is concerning the various types of cooperative learning, their effect on achievement. and their effect on other aspects of student learning.
Instruction Supporting Children's Counting on for Addition and Counting up for Subtraction
Authors: Fuson Karen C. , Fuson Adrienne M.
Published: August 2020

Children in the United States ordinarily invent a series of increasingly abbreviated and abstract strategies to solve addition and subtraction problems during their first 4 years in school (Carpenter & Moser, 1984; Fuson, 1988, in press–a, in press–b; Steffe & Cobb, 1988). Several studies have shown that instruction can help children learn specific strategies in this developmental sequence. Fuson (1986), Fuson and Secada (1986), and Fuson and Willis (1988) demonstrated that by the end of first grade children of all achievement levels could add and subtract single-digit sums and differences (sums to 18) by sequence counting on and sequence counting up. Sequence counting on and counting up are abbreviated counting strategies in which the number words present the addends and the sum. In both strategies the counting begins by saying the number word of the first addend. For example, to count on to add 8 + 6, a child would say, “8 (pause), 9, 10, 11, 12, 13, 14.” The same sequence of number words is used to find 14–8 by counting up, but the answer is the number of words said after the first addend word rather than the last word in the sequence. When the second addend is larger than 2 or 3, some method of keeping track of the words said for the second addend is required. In the studies above this method was one-handed finger panems that showed quantities l through 9 (the thumb is 5) so that children could hold their pencil in their writing hand all of the time. The counting-on and counting-up instruction related the counting words to objects showing the addends and the sum, thus focusing on conceptual prerequisites for these abbreviated counting procedures and enabling children to relate counting and cardinal meanings of number words (Secada, Fuson, & Hall, 1983). The countingup instruction provided interpretations of subtraction and the“–” symbol as adding on, as well as the usual take-away interpretation that leads children to count down for subtraction.
Two Generations of Feminist Thinking
Authors: Hart Laurie E.
Published: August 2020

How can we expand the perspectives used to examine issues of gender and mathematics? Writings from a feminist viewpoint (e.g., Belenky, Clinchy, Goldberger, & Tarule, 1986; Noddings, 1 990) may be helpful in fostering this expansion. Belenky et al. studied adult women to understand “women's ways of knowing”; on the basis of their data they provide a provocative analysis of teaching and learning for women. They found that “connected teaching” (instruction that uses the “midwife” model rather than the more typical “banking” model [Freire, 1971]) facilitated the learning of women. Noddings has also critiqued research and practice in education from a feminist perspective. She discussed three generations of feminism.
Notations, Conventions, and Constraints: Contributions to Effective Uses of Concrete Materials in Elementary Mathematics
Authors: Thompson Patrick W.
Published: August 2020

Twenty fourth-grade children were matched according to performance on a whole-number calculation and concepts pretest and assigned at random to one of two groups: wooden base-ten blocks and computerized microworld. Instruction in each group was designed to orient students toward relationships between notation and meaning. Instruction given the two groups was based upon a single script that extended whole number numeration to decimal numeration, and emphasized solving problems in concrete settings while inventing notational schemes to represent steps in solutions. Neither group changed in regard to whole-number notational methods. Blocks children understood decimal numerals as if they were funny whole numbers; Microworld children attempted to give meaning to decimal notational methods, but were largely in a state of disequilibrium at the end of the study.
Thinking More Politically about the Challenges before Us: A Response to Romberg
Authors: Apple Michael W.
Published: August 2020

I have a good deal of respect for Thomas Romberg. Over the past three decades, he has had a significant impact on how we think about educational research, policy, and practice. This is true not only in mathematics education but in the larger research community as well. His response to my analysis rightly reminds us of many of the educational and social goals and beliefs that stood behind the history of the Standards documents. These are valuable reminders, because all too often we forget the historical genesis of important arguments. We pull them out of the issues that generated them. thereby losing some of the power and insights they may possess. Yet this very sense of history points to a number of pressing concerns. I shall raise them in the context of what my friends in logic call the genetic fallacy.
What's in a Model? Issues in the Use of Simulation Models to Analyze Student Understanding: A Reaction to Ohlsson, Ernst, and Rees
Authors: Schoenfeld Alan H.
Published: August 2020

Ohlsson. Ernst, and Rees (this issue) have produced a wonderfully lucid description of their paradigmatic approach to issues of cognition and instruction. They illustrate their approach by presenting the details of a well worked out computational model. Then, on the basis of simulation runs on the model, they derive some implications for prac tice. The authors have also laid down some rather stringent constraints for commentary. Do not critique our paradigms, they say, unless you can offer a replacement that does better. Do not critique the choice of knowledge representation (production systems) or modeling assumptions (e.g., limitations on working memory) unless you have compelling data to offer in service of your argument and in contradiction of our assumptions. Argument about details is useful, they say, but that won't change the conclusions we draw. So what's a reviewer to do?
A Professional Self-Portrait
Authors: Bauersfeld Heinrich
Published: August 2020

The famous philosopher Karl Jaspers's book Von der Wahrheit [About truth] caused G. Benn to write: “Five hundred pages—the truth can't be that long!”
A Longitudinal Study of Students Completing Four Years of UCSMP Mathematics
Authors: Hirschhorn Daniel B.
Published: August 2020

This article reports on the achievement and attitude aspects of a study comparing students who had the first 4 years of the University of Chicago School Mathematics Project (UCSMP) secondary curriculum to two distinct groups of comparable students, an age cohort and a mathematics course level cohort. A case study design with schools in three different sites (one urban, two suburban) was used. Three instruments were given: a Mathematics Level I Achievement Test from the Educational Testing Service, a 30-item Applications Test, and a 25-item Student Opinion Survey. At two of the sites, the UCSMP students outperformed both the age and course level cohorts by substantial amounts on both the achievement and application tests. At the third site, both comparison cohorts outperformed the UCSMP students on the achievement test, but results on the application test were mixed. At all three sites, there was little difference in the attitude items except on items concerning calculator use.
Mathematics Performance Assessment: Technical Quality and Diverse Student Impact
Authors: Baxter Gail P. , Shavelson Richard J. , Herman Sally J. , Brown Katharine A. , Valadez James R.
Published: August 2020

Reform in mathematics education calls for a closer tie between instruction and assessment. This study developed performance assessments from hands-on instructional activities and examined their reliability and validity for obtaining individual achievement data in large-scale surveys. The major source of unreliability was the tasks, not the raters. Many tasks are needed to get a dependable measure of a student's mathematics achievement. With respect to validity, results suggested that the performance assessments measure different aspects of mathematics achievement than do traditional multiple-choice tests. Moreover, the performance assessments, but not the multiple-choice test, distinguished the performance of students in hands-on and traditional curricula with the former scoring higher, on average, than the latter. Ethnic group comparisons indicated that Anglos scored higher, on average, than Latinos on all achievement measures. The magnitude of the difference varied by the curricular experience of the student. For students in traditional curricula, qualitative analysis indicates that Anglo and Latino students approached the problems similarly, made the same types of errors, and employed the same strategies in solving the mathematics problems.
Children's Strategies for Solving Two– and Three–Dimensional Combinatorial Problems
Authors: English Lyn D.
Published: August 2020

The study investigated the strategies that 7-to 12-year-old children spontaneously apply to the solution of novel combinatorial problems. The children were individually administered a set of six problems involving the dressing of toy bears in all possible combinations of tops and pants (two-dimensional) or tops, pants, and tennis rackets (three-dimensional). Two sets of solution procedures were identified, each comprising a series of five increasingly complex strategies ranging from trial-and-error approaches to sophisticated odometer procedures. Results suggested that experience with the two-dimensional problems enabled children to adopt and subsequently transform their efficient 2-D odometer strategy (where one item is held constant) into the most sophisticated 3-D odometer strategy, which involved working simultaneously with two constant items. The study highlights the importance of discrete mathematics as a source of problem-solving activities in which children are motivated to create, modify, and extend their own theories.
A Forum for Researchers: What Is Research in Mathematics Education, and What Are Its Results?
Authors: Sierpinska Anna , Kilpatrick Jeremy , Balacheff Nicolas , Howson A. Geoffrey , Sfard Anna , Steinbring Heinz
Published: August 2020

As mathematics education has become better established as a domain of scienti fic research (if not as a scientific discipline), exactly what this research is and what its results are have become less clear. The hi story of the past three International Congresses on Mathematical Education demonstrates the need for greater clarity. At the Budapest congress in 1988, in particular, there was a general feeling that mathematics educators from different parts of the world. countries, or even areas of the same country often talk past one another. There seems to be a lack of consensus on what it means to be a mathematics educator. Standards of scientific quality and the criteria for accepting a paper vary considerably among the more than 250 journals on mathematics education published throughout the world.
Review: Yes, Virginia, Some Children Do Grow up to Be Mathematicians
Authors: Thompson Patrick W.
Published: August 2020

If in the distant future an archaeologist were to build an image of mathematics ed ucation based on artifacts from the educational research community, she might conclude that, as late as 1990, mathematics had not progressed past proportional reasoning.
Research on Mathematics Education Reported in 1992
Authors: Suydam Marilyn N. , Brosnan Patricia A.
Published: August 2020

This 23rd ann ual listing of mathematics education research to appear in the JRME contains research summaries, articles, and di ssertations 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.
Contextual Factors Affecting Errors in Proportional Reasoning
Authors: Lawton Carol A.
Published: August 2020

Although instruction in proportions generally begins in the middle school years. proportional reasoning remains problematic for many college students (Adi & Pulos, 1980; Niaz, 1989; Renner & Paske, 1977; Thornton & Fuller, 1981). A review of the proportional reasoning literature by Tourniaire and Pulos (1985) suggests that a number of factors relating to the context of the problem are responsible for variability in performance. Such fac tors include the type of problem (e.g., rate, mixture, or physical problems), presence of discrete versus continuous quantities, and familiarity of problem content. It is apparent that most students' understanding of proportion concepts is relatively fragile and easily influenced by structural variations in the problem. Recent research in mathematics education has emphasized the importance of the interaction between intuitive knowledge (i.e., knowledge about a subject matter gai ned through natural expe rience) and the acq uisition of formal mathematics concepts (see Behr, Hare!, Post, & Lesh, 1992). It would be useful to determine exactly which factors in proportion problems elicit an intuitive understanding of the underlying concept; instruction in proportional reasoning could then use that knowledge as the basis for developing a more generalizable system of symbols and algorithms.
Guest Editorial
Authors: Wilson James W.
Published: August 2020

Editor's Note, Interest in a mathematics education journal devoted exclusively to research began to develop at least as long ago as the mid-1950s, but it wasn't until the publication of the NCTM monograph. Research in Mathematics Education, in 1967 that sufficient impetus existed to establish one. The first issue of the Journal for Research in Mathematics Education appeared in January 1970 with David C. Johnson as its editor.
Michael's Fraction Schemes
Authors: Saenz-Ludlow Adalira
Published: August 2020

A longitudinal teaching experiment was conducted with six third graders to analyze the itinerary of their ways of operating while solving fraction tasks. These children's quantitative reasoning with fractions was based on their quantitative reasoning with natural numbers. They solved fraction tasks in similar ways, though with different degrees of sophistication, depending on their sophistication in working with natural numbers. Their consistent method of operating indicated that they constructed fraction schemes that had the characteristic of being developmental, in the sense that prior schemes were the cornerstones of new ones. This article presents the constructive itinerary of one of the most advanced children in the group, Michael.
Guest Editorial
Authors: Johnson David C.
Published: August 2020

In his guest editorial in the January issue, James Wilson (JRME Editor, 1976–1982) looked at how JRME has assumed, in various ways, both proactive and reactive stances over the years. But is the issue really one of the journal being reactive or proactive? Before addressing the question, one might ask, “Exactly who or what is supposed to be proactive?” Is it the NCTM Research Advisory Committee, JRME's editors, its Editorial Board, its reviewers or authors, or some subset, or even all of these? Furthermore, how does one establish what we are to be proactive about or for?
Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic
Authors: Gray Eddie M. , Tall David O.
Published: August 2020

In this paper we consider the duality between process and concept in mathematics, in particular, using the same symbolism to represent both a process (such as the addition of two numbers 3 + 2) and the product of that process (the sum 3 + 2). The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema. Symbolism that inherently represents the amalgam of process/concept ambiguity we call a “procept.” We hypothesize that the successful mathematical thinker uses a mental structure that is manifest in the ability to think proceptually. We give empirical evidence from simple arithmetic to support the hypothesis that there is a qualitatively different kind of mathematical thought displayed by the more able thinker compared to that of the less able one. The less able are doing a more difficult form of mathematics, which eventually causes a divergence in performance between them and their more successful peers.
Assessment: What's New. And Not so New. A Review of Six Recent Books
Authors: Marshall Sandra P. , Thompsonc Alba G.
Published: August 2020

The field of educational assessment is rapidly changing. Calls for reform have been heard for some time, and they are now being heeded in unprecedented ways. One outcome is that we are suddenly undergoing a deluge of books and articles about assessment in mathematic education, ranging in topics from broad impassioned tatements about the need for change, to specific hypotheses about how to change and what the anticipated outcomes might be, to actual implementations representing specific efforts to change, to sharp criticisms about some of the unexpected results of change. The current situation leaves us, as reviewers, in something of a quandary. On the one hand, there is certainly a need to keep abreast of important contribution to the literature, and we want to review a many of the new books as is feasible. On the other hand, it seems somewhat unfair to review books that attempt to deal with ongoing and urgent changes in educational assessment. becaue they are all too often outdated by the time they appear. All researchers are aware of the lag between the time a manuscript is written and the time it reaches its audience. Sometimes the lag is great, and sometime it is s mall. For rapidly changing fields—like educational assessment—a large lag is especially problematic because what seems new and revolutionary at the time of writing may no longer appear so new by the time a reader encounters it. Such is the case with some of the books we review here.
The Development of, and Interaction between, Intuitive and School-Taught Ideas about Percent
Authors: Lembke Linda O. , Reys Barbara J.
Published: August 2020

The objective of this study was to determine how students' strategies for solving percent problems change over grades 5, 7, 9, and 11. The questions addressed included the following: What intuitive knowledge do students bring to the study of percent? Do students use this intuitive knowledge in solving percent problems? What processes do students use to solve percent problems? Does choice of solution method differ after formal instruction on percent? The researcher employed a cross-sectional research design whereby 31 students representing two ability levels (middle and high) from grades 5, 7, 9, and 11 were interviewed. The responses were compared and contrasted by ability and grade level. The 5th and 7th graders, who had little or no formal instruction on percent, used a variety of strategies including benchmark, ratios, and fractions, to solve percent problems. The 9th graders made extensive use of the school-taught equation strategy. The 11th graders, who had been away from formal instruction on percent for at least a year, also used the equation strategy extensively, but also employed a variety of other strategies and were more reflective in their choice of strategy.
Textbooks, Teachers, and the SIMS Test
Authors: Flanders James R.
Published: August 2020

This article is a report of a study of relationships between intended, implemented, and tested curricula of 84 classes of United States eighth-grade mathematics students in 1981–82. It is based on data from the Second International Mathematics Study (SIMS, Population A) and the SIMS test: 180 multiple-choice questions covering topics in arithmetic, algebra, geometry, measurement, and statistics. The study focused on three areas: (a) coverage of the SIMS test items in six nonalgebra textbooks, (b) teacher reports of whether students had the opportunity to learn the SIMS items, and (c) teacher expectations for student success on the SIMS items. Of special concern was the challenge presented to students as measured by textbook and teacher coverage of new, versus reviewed, mathematics topics.

The analysis indicated that the SIMS test for eighth graders was not representative of the curriculum defined by students' texts. The books were heavily biased toward a review of arithmetic and away from algebra or geometry items. Items new to texts were among the least covered. Teacher expectations for student success mirrored the textbook coverage of SIMS items and indicated a high degree of reliance on textbooks for topic sequencing. Teachers using nonbasal texts had higher expectations than teachers of basal texts. Teachers reported that they taught more old items than new items and more arithmetic and algebra items than geometry, measurement, or statistics items.
Exploring the Nature of Street Mathematics
Authors: Millroy Wendy L.
Published: August 2020

For many years, mathematics education researchers have been fascinated by the mathematics that is generated and used outside of institutions of learning. This is the mathematics that allows unschooled and sometimes illiterate people to practice crafts and trades, conduct business transactions, and make their livings in a variety of ways that depend on knowledge of what we call mathematics.
Guest Editorial
Authors: Weaver J. Fred
Published: August 2020

In this editorial I continue the discussion—begun in the previous two issues of JRME by James Wilson and David Johnson—of the proactive and reactive positions taken by the journal over the years.
Report of the Executive Director
Authors: Gates James D.
Published: August 2020

During the five years since the release of the Curriculum and Evaluation Standards for School Mathematics (1989), NCTM membership has increased from 77 000 (54 000 individuals) to 106 000 (79 000 individuals). These totals include both individual and institutional members. The most recent issues of the Arithmetic Teacher, Mathematics Teacher, and Journal for Research in Mathematics Education had circulations of 67 000, 61 000, and 12 000, respectively.
Gender and the JRME
Authors: Fennema Elizabeth , Hart Laurie E.
Published: August 2020

Researchers must assume the responsibility for the impact that their work will have on society as a whole, and work in the sensitive area of sex-related differences does have an impact. Any author who writes or any journal editor who makes publication decisions must consider the social implications of all publications. (Fennema, 1981, p. 384)
An International Perspective on Research through the JRME
Authors: D'ambrosio Ubiratan , D'ambrosio Beatriz
Published: August 2020

As the JRME reaches its 25th year as the research journal of the American community of mathematics educators, clearly its reach has become much broader and extends into the international realm of mathematics education. As the most prestigious journal in the field, the JRME undoubtedly influences research in mathematics education all over the world. It sets styles, priorities, and, above all, standards. In all three of these areas there are resentments, reactions, and dissent. We hope in this paper to discuss these and related issues from an international perspective.
A Discourse on Methods
Authors: Schoenfeld Alan H.
Published: August 2020

It is widely accepted that cultures are reflected by, and are reflections of, their technologies, and also that changes in technology may both reflect and catalyze significant changes in the societies that employ them. Perhaps the generic case in point is agriculture, where advances in crop cultivation led to fundamental changes in the organization of society. Crop cultivation “encouraged settlement of stable farm communities, some of which grew to be towns and city states. … Early agricultural implements—the digging stick, the scythe, the hoe, and the plow—developed slowly over the centuries, each innovation … causing profound changes in human life.” (Bridgwater & Kurtz, 1963, p. 29.) Similar effects can be traced to the current era: The existence of techniques for mass agricultural production is a necessary condition for the existence of a highly urbanized society, as well as a contributor to the decrease in the percentage of the farming population, and the kind of life associated with it.
E Pluribus Unum: Challenges of Diversity in the Future of Mathematics Education Research
Authors: Silver Edward A. , Kilpatrick Jeremy
Published: August 2020

Our task in preparing this article on the occasion of the 25th anniversary of the Journal for Research in Marhematics Education (JRME) was to look ahead to the future of research in the field and to identify and discuss issues that might be important for the next decade or two in the life of the journal. Rather than merely offering our own opinions and speculations. we decided to interview a number of other researchers, some from the United States and some from other countries, to sample their views regarding the current state of research in mathematics education, the issues that may affect the future of the field, and the role of the JRME in the current and future scene. In particular, we asked these researchers to identify examples of work they considered significant and to comment on its imponant characteristics. We probed their definitions of the field by asking them to identify the types of work (e.g., empirical studies, historical or theoretical analyses) they judged could legitimately be called research in mathematics education. We explored their visions of the future of research over the next few decades. And we questioned them about the role and place of the JRME in the research community and about its impact on the field. In addition, we participated in a conference on Research in Mathematics Education and Its Results in May 1994 that was part of a study conducted by the International Commission on Mathematical Instruction (ICMI). The discussion document framing the study (Sierpinska et al., 1993) and many of the presentations and conversations at the ICMI Research Conference should be acknowledged as sources of ideas for the article. Naturally, we have included our own opinions, analyses, and perspectives.
Elaborating Models of Mathematics Teaching: A Response to Steffe and D'Ambrosio
Authors: Simon Martin A.
Published: August 2020

In my article, I framed a developing model of mathematics teaching. As such, the model presented was neither complete nor elaborated in detail The article was designed to generate discussion that can contribute to the further development of this and other models. Steffe and D'Ambrosio's response contributes in important ways to this discussion. They indicate an acceptance of many of the components of the model and elaborate several of them.
Biographical Sketch: Glenadine Gibb: A Researcher for All Teachers
Authors: Montgomery Lindquist Mary
Published: August 2020

lf there were an award for researchers whose efforts have provided extraordinary benefit to mathematics teachers, there is no doubt that Glenadine Gibb would have received it at some point during her career. An active member of NCTM who served as president of the Council from 1974 to 1976. Gibb was a visionary—ahead of her times in many of her research and service endeavors.
Ockham's Razor And Plato's Beard: (Or, “The possible relevance of the philosophy of mathematics, and the problem of universals in particular, to the philosophy of mathematics education, and the problem of constructivism in particular”)
Authors: Of Ockham William , Orton Robert E.
Published: August 2020

William of Ockham responds from the dead to an article appearing in the January 1992 issue of the Journal for Research in Mathematics Education, in which Paul Cobb, Erna Yackel, and Terry Wood propose a “constructivist alternative to the representational view of mind.” Ockham, now a convert to Platonism, argues three points.
The Relevance of Practice: A Response to Orton
Authors: Cobb Paul
Published: August 2020

In this response to Robert Orton, I address each of the major points he raises and attempt to clarify the discrepancies in our positions. I give particular attention to his approach of reframing issues in terms of the categories of traditional academic philosophy. In adopting this stance, Orton (a) presents a highly idiosyncratic interpretation of the Cartesian dualism, (b) creates a gulf between theory and practice, and (c) implies that the social and cultural aspects of mathematical activity can be dismissed. I discuss each of these points and further develop my position by outlining Putnam's (1987) pragmatic realism, clarifying why Rorty's (1979) work might be of interest to mathematics educators, and revisiting Bereiter's (1985) learning paradox. I then conclude by exploring the relationship between the resulting nondualist approach and John Dewey's philosophy and pedagogy.
Review: Investigating the Development of Multiplicative Reasoning
Authors: Simon Martin A.
Published: August 2020

If the current mathematics education reform is to have significant impact on the mathematical development of students, the mathematical understandings encompassed by the multiplicative conceptual field are paramount. These mathematical understandings range from initial conceptions of multiplication and division through concepts of ratio and proportion. fractions, and linear and nonlinear functions. Not only is an understanding of multiplicative relationships important in the modeling of real world situations, but such relationships form the basis of the mathematics of probability, of similarity of geometric figures, and of rate of change.
Editorial
Authors: Lester Frank K.
Published: August 2020

In the November 1971 issue of this journal, then-editor David Johnson drew the readers' attention to a new feature of the journal, “a selected annotated bibliography of research” prepared by Marilyn Suydam and Fred Weaver. Johnson went on to state that “present plans for JRME include provisions for the bibliography to appear as an annual fearure, published in each November issue.” The bibliography: which focused on research pertaining to precollege mathematics education, appeared annually in the November issue until 1976, when the addition of a July issue devoted to the bibliography expanded the journal from four to five issues per volume to accommodate the growing demand for more space to publish research reports: In that same issue the scope of the listing was extended to include postsecondary-level research.
Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8
Authors: Reys Robert E. , Reys Barbara J. , Nohda Nobuhiko , Emori Hideyo
Published: August 2020

This study assessed attitude, computational preferences, and mental computational performance of 176, 187, 186, and 206 Japanese students in grades 2, 4, 6, and 8, respectively. A sample of students in grades 4 and 8 scoring in the upper and middle quintiles on the mental computation test was interviewed to identify strategies used to mentally compute. All data were collected during the last month of the school year. A wide range of performance on mental computation was found with respect to all types of numbers (whole numbers, decimals, and fractions) and operations at every grade level; the mode of presentation (visual or oral) significantly affected performance levels, with visual items generally producing higher performance; and the range of strategies (initial and alternative) used to do mental computation was narrow, with the most popular approach reflecting a mental version of a learned “paper/pencil” algorithm.
Confounding Whole-Number and Fraction Concepts When Building on Informal Knowledge
Authors: Mack Nancy K.
Published: August 2020

This study examined the development of students' understanding of fractions during instruction with respect to the ways students' prior knowledge of whole numbers influenced the meanings and representations students constructed for fractions as they built on their informal knowledge of fractions. Four third-grade and three fourth-grade students received individualized instruction on addition and subtraction of fractions in a one-to-one setting for 3 weeks. As students attempted to construct meaning for symbolic representations of fractions, they overgeneralized the meanings of symbolic representations for whole numbers to fractions, and they overgeneralized the meanings of symbolic representations for fractions to whole numbers.
President's Report: Selling and Buying Reform: If We Build It, Will They Come?
Authors: Price Jack
Published: August 2020

People who effect change do persevere. Now is not the time to be faint of heart. We have not reached all the stakeholders. The Standards present a guide, not a blueprint.
Editorial
Authors: Lester Frank K.
Published: August 2020

In a Forum for Researchers article in the May 1993 issue of this journal, a committee of the International Commission on Mathematical Instruction noted that “we in mathematics education have not been very reflective about the growing body of research we have been producing” (Sierpinska et al., 1993, p. 278). The committee insisted that, in order for the field of mathematics education to become more coherent. it will be necessary for the communjty to give much more attention to discussing the criteria that should be used to evaluate the results of our research efforts.
Multidigit Number Sense: A Framework for Instruction and Assessment
Authors: Jones Graham A. , Thornton Carol A. , Putt Ian J. , Hill Kevin M. , Mogill A. Timothy , Rich Beverly S. , Zoest Laura R. Van
Published: August 2020

This research extended the validation of a framework for assessing and describing children's thinking in multidigit number situations and used this framework to generate and evaluate different versions of an instructional program. The key constructs of the framework—counting, partitioning, grouping, and number relationships—appeared to be highly stable within each of the five levels and across the full range of thinking exhibited by 12 case studies. Results suggest that the levels may generate a hierarchy of thinking. Teachers were effective in implementing two versions of the framework-driven instructional program. Any differences could be attributed to the quality of problem-solving experiences, the level of student interactions, and, in essence, to differences in teachers' familiarity with the program.
Review: Constructing the Individual Knower-A Review of Radical Constructivism
Authors: Smith Erick
Published: August 2020

For over 20 years, Ernst von Glasersfeld has eloquently and consistently described a theoretical model of the individual knower. In doing so, he has become one of the best-known contemporary educational theorists, particularly among the mathematics education community.1 In addition, as he states in the preface to his newest book, the name he gave to his approach, “radical constructivism,” has become a catch word among educators. Today it seems everyone is a constructivist and many attach the descriptor radical to their orientation. However, as with any popular movement, there are many interpretations, many critiques, and much passion associated with radical constructivism. Glasersfeld2 sees this volume as an opportunity to tell the whole story, or as he says, “to lay out the main constructivist ideas as I see them” (p. xiii).
A Convergence of Transformative Multicultural and Mathematics Instruction? Dilemmas of Group Deliberations for Curriculum Change
Authors: Murtadha-Watts Khaula , D'Ambrosio Beatriz S.
Published: August 2020

In the work described here we report on our collaborative efforts to produce a multicultural mathematics curriculum for Grades K—6 that was socially transformative. We drew insights and direction from the following discourses: multicultural education, socially proactive mathematics, and preservice and in-service teacher deliberations. Through group deliberations, the analysis of existing mathematics and multicultural curricula, and the sharing of personal histories, we planned our teacher-research activities. Our goal was to define mathematics as a tool for social analysis. In this article we describe the perplexity of issues related to the definition of multicultural curricula and mathematics curricula for social transformation, the complexities of the group deliberative process, and the demands involved in the teacher-research process.
A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction
Authors: Carpenter Thomas P. , Franke Megan L. , Jacobs Victoria R. , Fennema Elizabeth , Empson Susan B.
Published: August 2020

This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1—3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.
The Construction of the Social Context of Mathematics Classrooms: A Sociolinguistic Analysis
Authors: Atweh Bill , Bleicher Robert E. , Cooper Tom J.
Published: August 2020

This study employed sociolinguistic perspectives developed by Michael Halliday to investigate the social context of 2 mathematics classrooms that differed in the socioeconomic backgrounds and genders of their students. Our analysis focused on the effects of these 2 factors on teacher perceptions of student needs and abilities and on how these perceptions shaped the discourse in the classroom. We argue that as mathematical knowledge is being constructed in the classroom, the student participants are being constructed by the teachers according to their ability in and need for different dialects of mathematics that have different values in our Western culture.
Making Sense of Instructional Devices: The Emergence of Transparency in Mathematical Activity
Authors: Meira Luciano
Published: August 2020

This article examines the mathematical sense-making of children as they use physical devices to learn about linear functions. The study consisted of videotaped problem-solving sessions in which pairs of 8th graders worked on linear function tasks using a winch apparatus, a device with springs, and a computerized input-output machine. The following questions are addressed: How do children make sense of physical devices designed by experts to foster mathematical learning? How does the use of such devices enable learners to access selected aspects of a mathematical domain? The concept of transparency is suggested as an index of access to knowledge and activities rather than as an inherent feature of objects. The analysis shows that transparency is a process mediated by unfolding activities and users' participation in ongoing sociocultural practices.
Prospective Teachers' Use of Computing Tools to Develop and Validate Functions as Mathematical Models
Authors: Mary Zbiek Rose
Published: August 2020

This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated.
Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan
Authors: Reys Robert E. , Yang Der-Ching
Published: August 2020

This research provides information on the number sense of Taiwanese students in Grades 6 and 8. Data were collected with separate tests on written computation and number sense. Seventeen students were interviewed to learn more about their knowledge of number sense. Taiwanese students' overall performance on number sense was lower than their performance on written computation. Student performance on questions requiring written computation was significantly better than on similar questions relying on number sense. There was little evidence that identifiable components of number sense, such as use of benchmarks, were naturally used by Taiwanese students in their decision making. This research supports the need to look beyond correct answers when computational test results are reported.
Editorial
Authors: Sowder Judith T.
Published: August 2020

As a mathematics education researcher, my biggest surprise this past year has been to learn that behaviorism continues to be alive and well and on the way to having profound effects on mathematics education in this country in the coming years. Once again, California is leading the way. The California State Board of Education is taking seriously the law (California Education Code 60200c-3) demanding that standards and frameworks be research based. In order to make certain that this law is carried out, the Board last summer funded a project to identify research findings in mathematics education relevant to the development of mathematics standards and a mathematics framework. One cannot argue with this decision. It is the details of the decision that concern me.
Editorial Comment
Authors: Johnson David C.
Published: August 2020

In Volume XIX of the SMSG monographs Studies in Mathematics (1969), Fey reviews the research in the area of classroom teaching of mathematics. A shortened version of this paper was also published in the October 1969 Review of Educational Research (RER).
Diagnosing Selected Behavior Characteristics of Teachers of Secondary School Mathematics
Authors: Flora Ben V.
Published: August 2020

This paper reports on the development and initial exploratory research with an instrument designed for diagnostic use with preservice teachers of secondary school mathematics. The structure of the “Teaching Situation Reaction Test for Teachers of Secondary School Mathematics” (TSRT-TSSM), the criteria purported to be measured by the TSRT-TSSM, and the nature of the samples involved in the exploratory studies involving the TSRT-TSSM are presented. The exploratory studies indicated that total test scores for teachers predicted to be highly effective were significantly higher than those for teachers predicted to minimally effective and for a number of the teacher behavior characteristics being measured the profiles of the 2 groups were significantly different. Use of the instrument in a diagnostic setting suggests potential for communication of useful information to a preservice teacher of his behavior characteristics. Further revision of the instrument is suggested.
Representational Modes as They Interact with Cognitive Development and Mathematical Concept Acquisition of the Retarded to Promote New Mathematical Learning
Authors: Armstrong Jenny R.
Published: August 2020

A critical goal of mathematics instruction is to gradually enable pupils to transverse from the enactive to the symbolic mode of internal representation of mathematical ideas. Nevertheless, during certain periods of the pupil's mental and cognitive development, the use of selected representational modes in the instructional presentation of ideas would seem to be better than others in promoting mathematical learning. In 2 related studies the mathematical learning of Trainable Mentally Retarded (TMR) and Educable Mentally Retarded (EMR) was examined under 2 modes of representational presentation: manipulative (enactive) and nonmanipulative (pictorial-symbolic). The TMR sample did significantly better under the manipulative representational mode. The EMR sample also learned better under the manipulative mode, but only on certain types of mathematical ideas.
Editorial Comment
Authors: Sension Donald B.
Published: August 2020

Large-scale educational assessment programs are the vogue in education today. The National Assessment of Educational Progress represents a new, creative approach to the longitudinal assessment problem. In addition, approximately three-fourths of the fifty state departments of education maintain some sort of statewide testing or assessment program. Many of the statewide programs, unlike the National Assessment Project, have long traditional histories. The New Hampshire statewide testing program (Austin and Provost, JRME, J anuary 1972) is representative of these statewide programs.