bluuebunny/crossref_metadata_2025 huggingface dataset:
Invariance of Ratio: The Case of Children's Anticipatory Scheme for Constancy of Taste
Authors: Harel Guershon , Behr Merlyn , Lesh Richard , Post Thomas
Published: August 2020

In this article we study the concept of invariance of ratio through an investigation of children's understanding of constancy of taste—that is, the notion that random samples of a given mixture taste the same—using a device that does not resort to conventional symbolism. The paper begins with a definition of constancy of taste and other quantitative analogues. Then it presents a theoretical analysis of how constancy of taste may emerge from the child's additive world and grow into a conception where taste becomes an intensive quantity. The analysis suggests that one's conception of taste constancy is linked in a fundamental way to one's conception of invariance of ratio. Following this analysis, the paper reports a study that demonstrates the absence of taste constancy among sixth-grade children. More specifically, the study shows that sixth-grade children base their judgment of the relative strength of the taste of two samples from the same mixture on at least one of three (extraneous) variables: the relative volumes of the samples to be tasted, whether the mixture is thought of as consisting of a single ingredient or more than one ingredient, and the relative amount of the ingredients stated in the problem.
Academically Talented Students' Achievement in a Flexibly Paced Mathematics Program
Authors: Mills Carol J. , Ablard Karen E. , Gustin William C.
Published: August 2020

Third- through sixth-grade mathematically talented students (n = 306) enrolled in a flexibly paced mathematics course made achievement gains far beyond the normative gains expected over a one-year period. When compared to students several grade levels higher, these highly able students gained as much as 46 percentile points from pre- to posttesting. Above-grade-level testing revealed that the students possessed a wide range of mathematics knowledge prior to entering the course with some students scoring at exceptionally high levels. With an individualized learning pace, some students as young as fourth grade completed the arithmetic/prealgebra sequence in their first year and returned the second year to successfully complete the beginning algebra sequence. Restricting such students to a rigid instructional pace and a “grade-appropriate” curriculum may place them at risk for declines in motivation and achievement.
Biographical Sketch: William Brownell and the Search for Meaning
Authors: Noddings Nel
Published: August 2020

William Brownell had a distinguished career as a mathematics educator, arithmetic textbook writer, and educational researcher. He was an active member of NCTM and many other professional organizations and served for some years as Director of the National Society for the Study of Education and as president of AERA in 1938.
Musings about Mathematical Problem-Solving Research: 1970-1994
Authors: Lester Frank K.
Published: August 2020

On extended constructed-response tasks, which required students to solve problems requiring a greater depth of understanding and then explain, at some length, specific features of their solutions, the average percentage of students producing satisfactory or better responses was 16 percent at grade 4, 8 percent at grade 8, and 9 percent at grade 12. (Dossey, Mullis, & Jones, 1993, p. 2)
Radical Constructivism and Mathematics Education
Authors: Steffe Leslie P. , Kieren Thomas
Published: August 2020

Our intention in this article is to provide an interpretation of the influence of constructivist thought on mathematics educators starting around 1960 and proceeding on up to the present time. First, we indicate how the initial influence of constructivist thought stemmed mainly from Piaget's cognitive-development psychology rather than from his epistemology. In this, we point to what in retrospect appears to be inevitable distortions in the interpretations of Piaget 's psychology due primarily to its interpretation in the framework of Cartesian epistemology. Second, we identify a preconstructivist revolution in research in mathematics education beginning in 1970 and proceeding on up to 1980. There were two subperiods in this decade separated by Ernst von Glasersfeld's presentation of radical constructivism to the Jean Piaget Society in Philadelphia in 1975. Third, we mark the beginning of the constructivist revolution in mathematics education research by the publication of two important papers in the JRME (Richards & von Glasersfeld, 1980; von Glasersfeld, 1981). Fourth, we indicate how the constructivist revolution in mathematics education research served as a period of preparation for the reform movement that is currently underway in school mathematics.
Reconstructing Mathematics Pedagogy from a Constructivist Perspective
Authors: Simon Martin A.
Published: August 2020

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.
Editorial
Authors: Lambdin Diana V. , Kloosterman Peter
Published: August 2020

At NCTM annual meeting the JRME editorial staff always organizes a session at which potential author and reviewers can learn about the journal. News at this year's meeting included a change to use of the recently published fourth edition of the Publication Manual of the American Psychological Association as the standard for preparing JRME manuscripts (see inside from cover for information on how to obtain the PublicaTion Manual).
Cultural Tools and Mathematical Learning: A Case Study
Authors: Cobb Paul
Published: August 2020

This study investigates the role that four second graders' use of the hundreds board played in supporting their conceptual development over a 10-week period. Particular attention is given to the transition from counting by ones to counting by tens and ones. The analysis indicates that the children's use of the hundreds board did not support the construction of increasingly sophisticated concepts of ten. However, children's use of the hundreds board did appear to support their ability to reflect on their mathematical activity once they had made this conceptual advance. The constructivist perspective exemplified in the analysis is contrasted with a sociocultural perspective on mathematical development. The differing roles attributed to cultural tools are clarified, and potentially complementary aspects of the two perspectives are discussed.
Editorial
Authors: Lester Frank K. , Galindo Enrique
Published: August 2020

Like any organism that is alive and well, the JRME has been growing and changing throughout its 25 years. A recent editorial (May 1995) discussed changes in the number, type, and length of research reports that have been submitted in recent year. In this editorial we want to share ideas about some potentially even more radical changes in the journal—related to recent unprecedented growth of the world-wide “information superhighway.”
The Tensions and Contradictions of the School Mathematics Tradition
Authors: Gregg Jeff
Published: August 2020

The ubiquitousness and robustness of traditional practices in the teaching of school mathematics raise two fundamental and interrelated questions: Will the fate of the current reform effort in mathematics education be any different from that of previous reform efforts? Why have these traditional practices been so constant and durable? Certainly these are questions that cannot be answered in a single study. However, in an effort to develop a research basis that could aid in thinking about these questions, I conducted an ethnographic case study of a beginning high school mathematics teacher's acculturation into the school mathematics tradition (i.e., the beliefs and practices that characterize the traditional approach to school mathematics).
Intention and Significance in the Teaching and Learning of Mathematics
Authors: Brown Tony
Published: August 2020

This article discusses the role of language in mathematical understanding and how the boundaries assigned to language determine its status. It focuses on a classic debate between two leading writers in hermeneutics, Gadamer and Habermas, and it shows how this debate can guide us in drawing the boundaries of mathematical learning. In particular, the article suggests that personal learning of mathematics is inseparable from the social practices within which learning takes place. Nevertheless, the teacher's task can be seen in terms of enabling the student to move between stressing the personal and stressing the social aspects of his or her mathematical learning. In this way the initiation of the student into the socially conventional ways of doing mathematics can be more effectively grounded in his or her personal experience.
Brief Report: Results of Research Studies on Causal Predominance between Achievement and Attitude in Junior High School Mathematics of Japan
Authors: Minato Saburo , Kamada Tsugio
Published: August 2020

Data from the Second International Mathematics Study indicated that the very high mathematics achievement of Japanese students was accompanied by a very low attitude toward mathematics in the distribution of the scores of the educational systems (Travers & Westbury, 1989). This perplexing result is worrisome for mathematics teachers in Japan, because both high achievement in mathematics and the development of positive attitudes toward mathematics are important goals of teaching mathematics. Thus, it is urgent to make teachers aware of the relationship between achievement and attitude.
Intersubjectivity in Mathematics Learning: A Challenge to the Radical Constructivist Paradigm?
Authors: Lerman Stephen
Published: August 2020

Radical constructivism is currently a major, if not the dominant, theoretical orientation in the mathematics education community, in relation to children's learning. There are, however, aspects of children's learning that are challenges to this perspective, and what appears to be “at least temporary states of intersubjectivity” (Cobb, Wood, & Yackel, 1991, p. 162) in the classroom is one such challenge. In this paper I discuss intersubjectivity and through it offer an examination of the limitations of the radical constructivist perspective. I suggest that the extension of radical constructivism toward a social constructivism, in an attempt to incorporate intersubjectivity, leads to an incoherent theory of learning. A comparison of Piaget's positioning of the individual in relation to social life with that of Vygotsky and his followers is offered, in support of the claim that radical constructivism does not offer enough as an explanation of children's learning of mathematics.
Letter to the Editor
Authors: Liedtke Werner
Published: August 2020

I wish to offer one practitioner's reaction to some of the content of the very informative July 1995 issue of JRME.
Students' Understanding of Three-Dimensional Rectangular Arrays of Cubes
Authors: Battista Michael T. , Clements Douglas H.
Published: August 2020

The present study extends previous research in this area by providing a more elaborate and theoretical description of students' solution strategies and errors in dealing with 3-D cube arrays. It describes several cognitive constructions and operations that seem to be required for students to conceptualize and enumerate the cubes in such arrays, exploring in depth general cognitive operations such as coordination, integration, and “structuring” as they are manifested in a spatial context. It describes how, in dealing with 3-D rectangular arrays, students' spatial thinking is related to their enumeration strategies. The findings suggest that students' initial conception of a 3-D rectangular array of cubes is as an uncoordinated set of faces. Eventually, as students become capable of coordinating views, they see the array as space filling and strive to restructure it as such. Those who complete a global restructuring of the array use layering strategies. Those in transition use strategies that indicate that their restructuring is local rather than global. Finally, the data suggest that many students are unable to enumerate the cubes in a 3-D array because they cannot coordinate the separate views of the array and integrate them to construct one coherent mental model of the array.
Engaging Whole-Number Knowledge for Rational-Number Learning Using a Computer-Based Tool
Authors: Hunting Robert P. , Davis Gary , Pearn Catherine A.
Published: August 2020

We present data from a two-year teaching experiment involving 8- and 9-year-old children. The purpose of the teaching experiment was to investigate children's fraction learning and the role their whole-number knowledge might play in it. A major source for the children's experiences and experiments was an operator-like computer program called Copycat. In particular, we focus on the accomplishments of Elliot and Shannon as they solved fraction comparison problems. We identify and discuss three cognitive schemes that Elliot and Shannon used. These were an Equal Outputs scheme, limited in effectiveness to unit fraction comparisons; an Equal Inputs scheme that activated strategies for determining a common multiple; and a Scaling scheme in which fractions were scaled up or down using ratios. The major focus of the discussion is the role whole-number knowledge played in the application of these schemes. This study demonstrates an interdependence between development of rational-number knowledge and whole-number knowledge. Rational-number tasks in operator settings can help stimulate and extend children's whole-number knowledge. Facility with whole-number relationships enables students to solve fraction comparison problems.
Elementary Students' Construction and Coordination of Units in an Area Setting
Authors: Reynolds Anne , Wheatley Grayson H.
Published: August 2020

Much of students' numeric mathematical activity can be interpreted as the construction and coordination of units. This article reports evidence of construction and coordination of units in a measurement setting, using data gathered from 4 students during individual problem-solving sessions near the end of their fourth-grade year and augmented through knowledge gained with these students in the previous 2 years. Three students each developed different ways of solving the measurement task that reflected the construction and coordination of abstract units, whereas a fourth student did not. On the basis of this evidence we propose that the task of measuring 2-dimensional regions becomes possible when a student is able to construct iterable 2-dimensional units and coordinate those units.
Review: Equity and Mathematics: Not a Simple Issue
Authors: Noddings Nel
Published: August 2020

All four of the books reviewed here are deeply concemed with issues of equiry in mathematics education. I'll say a bit about each book in order to orient readers, and then I'll organize my remarks around the themes that arise again and again: the nature of mathematics. mathematics curriculum and pedagogy, and the philosophical and cultural factors inside and outside classroom that affect our educational efforts.
Brief Report: The Evolution With Age of Probabilistic, Intuitively Based Misconceptions
Authors: Fischbein Efraim , Schnarch Ditza
Published: August 2020

The purpose of this research was to investigate the evolution, with age, of probabilistic, intuitively based misconceptions. We hypothesized, on the basis of previous research with infinity concepts, that these misconceptions would stabilize during the emergence of the formal operation period. The responses to probability problems of students in Grades 5, 7, 9, and 11 and of prospective teachers indicated, contrary to our hypothesis, that some misconceptions grew stronger with age, whereas others grew weaker. Only one misconception investigated was stable across ages. An attempt was made to find a theoretical explanation for this rather strange and complex situation.
Developing Ratio and Proportion Schemes: A Story of a Fifth Grader
Authors: Lo Jane Jane , Watanabe Tad
Published: August 2020

There is a growing theoretical consensus that the concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this developmental process. One fifth-grade student, Bruce, was encouraged to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Bruce made during the teaching experiment and analyzes the challenges Bruce faced in attempting to schematize his method. Finally, the mathematical knowledge Bruce needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.
The Changing Role of the Mathematics Teacher
Authors: Clarke Doug M.
Published: August 2020

This case-study research investigated changing teacher roles associated with two teachers' use of innovative mathematics materials at Grade 6. Using daily participant observation and regular interviews with the teachers and the project staff member responsible for providing in-school support, a picture emerged of changing teacher roles and of those factors influencing the process of change. One teacher demonstrated little change in either espoused beliefs or observed practice over the course of the study. The second teacher demonstrated increasing comfort with posing nonroutine problems to students and allowing them to struggle together toward a solution, without suggesting procedures by which the problems could be solved. He also increasingly provided structured opportunities for students' reflection on activities and learning. Major influences on this teacher's professional growth appeared to be the provision of the innovative materials and the daily opportunity to reflect on classroom events in conversations and interviews with the researcher.
Facilitating Student Interactions in Mathematics in a Cooperative Learning Setting
Authors: Leikin Roza , Zaslavsky Orit
Published: August 2020

This article discusses effects on different types of students' interactions while learning mathematics in a particular cooperative small-group setting. Three low-level ninth-grade classes engaged in learning mathematics in an experimental cooperative method. Data were collected through classroom observations, students' written self-reports, and an attitude questionnaire. A group of four students from one of the classes was observed more closely. Analysis of classroom observations and students' self-reports focused on students' activeness, interactions, and attitudes toward the experimental method. Findings for the cooperative small-group setting indicated (a) an increase in students' activeness, (b) a shift toward students' on-task verbal interactions, (c) various opportunities for students to receive help, and (d) positive attitudes toward the cooperative experimental method. A fourth class served as a control group to ascertain whether student achievement increased or decreased as a result of the experimental method.
Teachers' Evolving Conceptions of One-to-One Tutoring: A Three-Tiered Teaching Experiment
Authors: Lesh Richard , Kelly Anthony E.
Published: August 2020

This article describes a three-tiered teaching experiment in which teachers were studied over a protracted period of time as they attempted to understand and improve their approaches to one-to-one tutoring. In a three-tiered teaching experiment model, emphasis is placed on (a) establishing a collaborative relationship between the research staff and the teachers, (b) careful choice of tasks for teachers and their students, (c) the development of “learning environments” for the study, and (d) the use of continuous and diverse dependent measures. The study documented the initial and revised strategies of teachers as they tutored children over a period of 10 weeks. Various indices of teacher change are reported, and some of the strengths and limitations of the methodology are discussed.
Families and Mathematics: A Study of Parent-Child Interactions
Authors: Anderson Ann
Published: August 2020

A group of 21 middle-class parents and their 4-year-old children participated in this study of mathematics and parent-child interactions. Each family chose 4 separate 15-minute periods over 2 days to share multilink blocks, a child's book, blank paper, and preschool worksheets with their child at home. Each 15-minute session was audiotaped, all artifacts generated during the sessions were kept, and each parent was interviewed. The results of the study indicate that a wide range of mathematics was evident, with counting being the most prevalent activity. All parents succeeded in injecting some mathematics in most sessions; some did so by explicitly setting mathematics as a goal, whereas others injected the mathematics as a means of carrying out, or as an aside to, the play activity they coconstructed. Questioning children's knowledge was the main way parents elicited mathematics, although some requested explanations, or clarifications, and some mediated patterns, relationships, and strategies.
Inconsistency Between a Beginning Elementary School Teacher's Mathematics Beliefs and Teaching Practice
Authors: Raymond Anne M.
Published: August 2020

This study investigates relationships between a beginning elementary school teacher's beliefs and mathematics teaching practices. A proposed model of relationships between beliefs and practice provided a conceptual framework for the examination of factors that influence beliefs, practice, and the level of inconsistency between them. Data were gathered over 10 months through audiotaped interviews, observations, document analysis, and a beliefs survey. Analyses included the categorization and comparison of beliefs and practice and the development of a revised model of relationships between beliefs and practice. Findings indicate that this teacher's beliefs and practice were not wholly consistent. Rather, her practice was more closely related to her beliefs about mathematics content than to her beliefs about mathematics pedagogy. Her beliefs about mathematics content were highly influenced by her own experiences as a student and her beliefs about mathematics pedagogy were primarily influenced by her own teaching practice. However, the extent to which her teacher preparation program influenced either her beliefs or practice was limited.
President's Report: Choices and Challenges
Authors: Burrill Gail
Published: August 2020

Life has changed enormously since I grew up in Fisher, Minnesota. Over the years, we faced many challenges and made choices that helped us meet those challenges as we kept up with changes in agriculture, manufacturing, transportation, entertainment, and education.
Culturally Relevant Mathematics Teaching in a Mexican American Context
Authors: Gustein Eric , Lipman Pauline , Hernandez Patricia , los Reyes Rebeca de
Published: August 2020

This article examines mathematics instruction and its intersection with culturally relevant teaching in an elementary/middle school in a Mexican American community. The findings are based on a collaborative-research and school-change project involving university researchers, teachers, and the school's principal. On the basis of ethnographic data and an interdisciplinary theoretical framework, we propose a three-part model of culturally relevant mathematics instruction. The 3 components are (a) building on students' informal mathematical knowledge and building on students' cultural and experiential knowledge, (b) developing tools of critical mathematical thinking and critical thinking about knowledge in general, and (c) orientations to students' culture and experience.
Editorial: January 1998
Authors: Sowder Judith T.
Published: August 2020

With this issue of the Journal for Research in Mathematics Education we are pleased to announce that we are launching our electronic version of the journal. In addition to this issue, the March and May issues of the JRME will be accessible on the NCTM Web site, www.nctm.org, at no charge. Beginning with the July 1998 issue, the electronic version will become a fee-based offering with its own URL. A fee structure for subscribing to one or both forms of the journal is still being developed. Watch the Web site for information on subscribing to the electronic version of the journal. Each subscriber to the electronic version will receive a log-in password to access the journal. A password will not be necessary, however, for features of the journal now on-line, including tables of content, article abstracts, the list of accepted manuscripts, and the telegraphic reviews.
Review: Mathematics, Culture, and Power
Authors: Kitchen Richard S. , Rossi Becker Joanne
Published: August 2020

Arthur B. Powell and Marilyn Frankenstein's new book, Ethnomathematics: Challenging Eurocentrism in Mathematics Education, illuminates for our consideration a body of very practical mathematical knowledge largely discounted in the traditional mathematical community when compared with the abstract, theoretical mathematical knowledge typically valued highly by mathematicians. Ethnomathematics has caused us to call into question which mathematical knowledge really counts and thus has come to signify more than just “the study of mathematical ideas of nonliterate peoples” (a definition first offered by Marcia and Robert Ascher in the early 1980s in their paper, “Ethnomathematics,” reprinted as chapter 2 of this volume, p. 26). Editors Powell and Frankenstein use, instead, the broader definition of ethnomathematics provided in the book's opening chapter, “Ethnomathematics and Its Place in the History and Pedagogy of Mathematics,” by Ubiratan D'Ambrosio, a Brazilian mathematics educator whom many consider the intellectual progenitor of ethnomathematics. D'Ambrosio defines ethnomathematics as the mathematics that all cultural groups engage in, including “national tribal societies, labor groups, children of a certain age bracket, professional classes, and so on” (p. 16). Each group, including mathematicians, has its own mathematics. From D'Ambrosio's perspective, ethnomathematics exists at the confluence of the history of mathematics and cultural anthropology, overcoming the Egyptian/Greek differentiation between practical and academic mathematics.
Using Concept Maps to Assess Conceptual Knowledge of Function
Authors: Williams Carol G.
Published: August 2020

In this study I examine the value of concept maps as instruments for assessment of conceptual understanding, using the maps to compare the knowledge of function that students enrolled in university calculus classes hold. Twenty-eight students, half from nontraditional sections and half from traditional sections, participated in the study. Eight professors with PhDs in mathematics also completed concept maps. These expert maps are compared with the student maps. Qualitative analysis of the maps reveals differences between the student and expert groups as well as between the 2 student groups. Concept maps proved to be a useful device for assessing conceptual understanding.
Attitudes of Elementary School Pupils and Their Parents toward Mathematics and Other Subjects of Instruction
Authors: Levine George
Published: August 2020

A forced-choice questionnaire was administered to a sample of 144 third-, fourth-, and sixth-grade pupils, and a similar questionnaire was completed by the parents of these pupils, to investigate the attitudes of pupils and their parents toward mathematics, English, science, and social studies. Analysis of the data was made on the basis of the grade level and sex of the children. Among pupils, mathematics was ranked higher than the other 3 areas with respect to importance, enjoyment, success, and additional criteria related to pupils' conception of parental attitudes. Parents recalled enjoying and doing best in English, but they ranked mathematics and English equally high in importance, and as the subjects in which they hoped their children would do best.
Prospective Elementary Teachers' Intensity and Ambivalence of Beliefs about Mathematics and Mathematics Instruction
Authors: Collier C. Patrick
Published: August 2020

A formal and an informal dimension of belief about mathematics and mathematics instruction were defined. Scales designed to measure these beliefs were administered to 264 prospective elementary teachers. Ambivalence was defined and ambivalence quotients reported. Students in later stages of their preparation had more informal views of mathematics and mathematics instruction, and their beliefs were less ambivalent than students beginning their programs. High achievers viewed mathematics as being less formal than did low achievers. Low achievers had a more ambivalent view of mathematics instruction than did high achievers. The results suggest that it is important to consider both intensity and ambivalence in investigating attitudes.
Educational Theory Testing: A Reply to Holz
Authors: Moody William B. , Bausell R. Barker
Published: August 2020

Holz appears to center his critique of “The Effect of Activity-oriented Instruction upon Original Learning, Transfer, and Retention” along two major dimensions. One dimension questions the validity of the measures employed to test the three hypotheses; the other dimension questions the fit between the instructional procedures and the theoretical justification for selecting the hypotheses in the first place. In general, the two chief methodological criticisms dealing with the mode of analysis and the degree of original learning are justified. Even though examination of Table 1 (p. 2 L 0) of the manuscript indicates that a modest increase in power and stability could in no way have changed the study's conclusions, analysis of covariance should have been employed instead of difference scores. The degree of original learning was, in the present authors' opinion, discussed in sufficient detail in the original manuscript. It is interesting that the original JRME reviewers were not unaware of these criticisms, but they accepted the study chiefly on heuristic grounds. They were probably not aware, as the authors were not aware, that such absolute dictums existed as: “such low reliabilities (.50) would insure the invalidity of the retention and transfer tests,” or that “since there was little original learning for any group in the study, no significant differences in retention or transfer could exist regardless of the measures employed [italics added].”
Letter to the Editor
Authors: Steffe Leslie P.
Published: August 2020

Effective working relations between university researchers and public school personnel are crucial if university-based research is to have an impact on problems encountered by public school people. For this reason, l was pleased to note that Donald B. Sension has been appointed Associate Editor of JRME. Various aspects of Dr. Sension's Editorial Comment in the March 1972 issue of JRME (vol. 3, no. 2), however, do need a reply.
Editorial Comment
Authors: Johnson David C.
Published: August 2020

There is an important question that needs regular consideration with any publication—Is the publication serving the needs and interests of the readers and the host organization? The charge to JRME from the NCTM Board of Directors is.
Study 2-1: An Exploration of the Effect of Selected Sequence Variables on Student Choice in the Use of Algorithms
Authors: Hostetler Robert P.
Published: August 2020

The general aim of this investigation was to initiate a search for sequence variables that influence the choice-behavior of a learner when he is placed in a problem-solving situation.
Study 3-1: An Assessment of a Selected Set of Piagetian-Derived Operators for the Generation of Effective Learning Hierarchies
Authors: Sawada Daiyo
Published: August 2020

When Klein's study (reported in this journal) is viewed so that the forest can be seen as well as the trees, one realizes that the problems of sequencing can be analyzed as two types: (a) problems of transfer within a cluster (intracluster transfer) and (b) problems of transfer between clusters (intercluster transfer). Klein limited his study to an examination of intracluster transfer by focusing on the ramifications of the MR matrix as applied to a given objective, such application giving rise to a cluster of objectives. The problem of intercluster transfer (going from one cluster to another) was not attended to by Klein.
Critiques of Articles: Some Remarks on Levine's Study of Attitudes
Authors: Bright George W.
Published: August 2020

Levine (1972) reported a “study to investigate and further explore the relationship between the attitudes of elementary school pupils and their parents toward mathematics and other areas of instruction [p. 51].” Yet he later admitted that “this exploratory study did not delve into parental or student attitudes toward mathematics; the primary emphasis was the attitudes of parents and pupils toward mathematics relative to three other curriculum areas [p. 58].” The confusion about the objective of the study was accompanied by confusion in other aspects of the work. The following comments are intended to point out some of these weaknesses.
Editorial Comment
Authors: Weaver J. F.
Published: August 2020

In addition to a paper by Richard O. Kratzer and Stephen S. Willoughby, this issue of the JRME features a set of related papers that pertain to a research program relevant to mathematics education that is associated with the Wisconsin Research and Development Center for Cognitive Learning; the annual annotated research listing for 1972 (principally the work of Marilyn N. Suydam); and the biennial Index (compiled by Jon L. Higgins and F. Joe Crosswhite).
The Relative Effectiveness of Two Different Instructional Sequences Designed to Teach the Addition and Subtraction Algorithms
Authors: Wiles Clyde A. , Romberg Thomas A. , Moser James M.
Published: August 2020

The purpose of this study was to compare the relative effectiveness of two instructional sequences designed to teach the addition and subtraction algorithms for two-digit whole numbers. One of these sequences is the traditional sequence of addition followed by subtraction. The other sequence is an integrated p resentation of the two tasks. Each sequence was embodied in a set of instructional activities that were used with a randomly selected group of second-grade children.
Mathematical Knowledge for Teaching in Planning and Evaluating Instruction: What Can Preservice Teachers Learn?
Authors: Morris Anne K. , Hiebert James , Spitzer Sandy M.
Published: August 2020

The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an essential starting point for studying and improving one's teaching. Thirty K–8 preservice teachers completed 4 written tasks. Each task specified a learning goal and then asked the preservice teachers to complete a teaching activity with this goal in mind. For example, preservice teachers were asked to evaluate whether a student's responses to a series of mathematics problems showed understanding of decimal number addition. The results indicate that preservice teachers can identify mathematical subconcepts of learning goals in supportive contexts but do not spontaneously apply a strategy of unpacking learning goals to plan for, or evaluate, teaching and learning. Implications for preservice education are discussed.
Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions
Authors: Speer Natasha M. , Wagner Joseph F.
Published: August 2020

Using case study analysis and a cognitive theoretical orientation, we examine elements of knowledge for teaching needed by a mathematician to orchestrate whole-class discussions in an undergraduate mathematics classroom. The instructor, an experienced teacher and mathematics researcher, used an inquiry-oriented curriculum to teach a differential equations course for the first time after teaching it with traditional lecture methods for many years. Examples of classroom teaching and interview data demonstrate that, despite having extensive teaching experience and possessing strong content knowledge, some instructors may still face challenges when trying to provide analytic scaffolding to move whole-class discussions toward a lesson's mathematical goals. We also hypothesize several component practices necessary for the successful use of analytic scaffolding. Our analysis focuses on the relationship between the instructor's pedagogical content knowledge and specialized content knowledge and his capacity to enact these component practices during whole-class discussions.
Brief Report: The Older of Two Trees: Young Children's Development of Operational Time
Authors: Kamii Constance , Russell Kelly A.
Published: August 2020

Piaget (1971) made a distinction between intuitive (preoperational) time and operational (logico-mathematical) time. According to Piaget, operational time develops around 7–8 years of age and is characterized by children's ability to deduce, for example, that if A was born before B, A will always be older than B. When time is still intuitive, children base their judgments of age on what is observable (e.g., people's height). With the aid of 11 pictures of an apple tree and a pear tree taken on 6 consecutive birthdays, 184 children in grades K–5 were individually asked, at a specific time, if two trees were the same age or if one was older than the other. Operational time was demonstrated by 79% of these children by grade 3.
Book Review: Mathematical Modeling on the Catwalk: A Review of Modelling and Applications in Mathematics Education: The 14th ICMI Study
Authors: Dossey John
Published: August 2020

A frequently heard query concerns why students receive relatively high grades in their mathematical school work and yet fail to be able to apply their mathematical knowledge and skills in contextual settings that they encounter in their other studies and their daily lives, not to mention meeting the mathematical challenges presented in subsequent mathematical courses. Many reform documents call for students to have opportunities to apply their mathematical knowledge in modeling settings and to reason and make sense of quantitative, spatial, or data-intensive situations (NCTM, 2000, 2009). The acquisition of such modeling expertise was a central focus of the 2004 ICMI conference at the University of Dortmund and this ICMI volume that documents the plenary presentations and issue papers related to themes discussed at or emerging from that meeting. What are the barriers that stand in the way of developing the adaptive expertise called for by modeling challenges?
Professional Noticing of Children's Mathematical Thinking
Authors: Jacobs Victoria R. , Lamb Lisa L. C. , Philipp Randolph A.
Published: August 2020

The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.
Research Commentary: Toward Clarifying the Meanings of Gender in Mathematics Education Research
Authors: Damarin Suzanne , Erchick Diana B.
Published: August 2020

Attention to gender in mathematics education research can be characterized by a lack of clarity (cf. Glasser & Smith, 2008). The importance of clarity in definitions of gender is discussed, and several conceptual models of gender are presented. Four of these models begin with biological sex differences but draw attention to other aspects of gender. Four models set biology aside and are based on social and cultural theories. Some of the advantages of the latter for mathematics education researchers are identified.
Enacting Proof-Related Tasks in Middle School Mathematics: Challenges and Opportunities
Authors: Bieda Kristen N.
Published: August 2020

Discussions about school mathematics often address the importance of reasoning and proving for building students' understanding of mathematics. However, there is little research examining how teachers enact tasks designed to engage students in justifying and proving in the classroom. This article presents results of a study investigating the processes and outcomes of implementing proof-related tasks in the classroom. Data collection consisted of observations of 7 middle school classrooms during implementation of proof-related tasks—tasks providing opportunities for students to produce generalizations, conjectures, or proofs—in the Connected Mathematics Project (CMP) curriculum by teachers experienced in using the materials. The findings suggest that students' experiences with such tasks are insufficient for developing an understanding of what constitutes valid mathematical justification.
One Hundred Years of Elementary School Mathematics in the United States: A Content Analysis and Cognitive Assessment of Textbooks From 1900 to 2000
Authors: Baker David , Knipe Hilary , Collins John , Leon Juan , Cummings Eric , Blair Clancy , Gamson David
Published: August 2020

A content analysis of over 28,000 pages from 141 elementary school mathematics textbooks published between 1900 and 2000 shows that widely used mathematics textbooks have changed substantially. Textbooks from the early part of the century were typically narrow in content but presented substantial amounts of advanced arithmetic and also asked students simultaneously to engage with material in effortful and conceptual ways. A period of change marked the middle of the century, when less advanced topics were presented and problem-solving tasks were simplified. From the mid-1960s onward, however, the trend reversed, and 3 major changes occurred in primary school mathematics curricula over the next 4 decades: (a) expansion of topics and the number of pages devoted to each topic; (b) a shift of traditionally more advanced topics from higher to lower grades; and, (c) within arithmetic, an increase in the number, abstraction, and cognitive demand of problem-solving strategies. Implications of these findings are discussed in terms of the historical study of mathematics and curriculum in U.S. schools.
Measuring the Effects of Professional Development on Teacher Knowledge: The Case of Developing Mathematical Ideas
Authors: Bell Courtney A. , Wilson Suzanne , Higgins Traci , McCoach D. Betsy
Published: August 2020

This study examines the impact of a nationally disseminated professional development program, Developing Mathematical Ideas (DMI), on teachers' specialized knowledge for teaching mathematics and illustrates how such research could be conducted. Participants completing 2 DMI modules were compared with similar colleagues who had not taken DMI. Teacher knowledge was measured with multiple-choice items developed by the Learning Mathematics for Teaching project and open-ended items based on problems initially developed by DMI experts. After controlling for pretest scores, a hierarchical linear model identified statistically significant differences: The DMI group outperformed the comparison group on both assessments. Gains in teachers' scores on the more closely aligned measure were related to the degree of facilitator experience with DMI. This study adds to our understanding of the ways in which professional development program features, facilitators, and issues of scale interact in the development of teachers' mathematical knowledge for teaching. Study limitations and challenges are discussed.
Structural Models for Predicting the Difficulty of Multiplication Problems
Authors: Eugene Cromer Fred
Published: August 2020

The purpose of this investigation was to explore the relationship between problem characteristics and problem difficulty in simple, 2-factor multiplication problems and to determine whether an objective, parametric measure of problem difficulty could be found.

The difficulty levels of 168 randomly generated multiplication problems were obtained by testing 238 fifth-grade children. Variables concerning the number of operations, digits carried, and magnitude of the digits were defined, and a least-squares procedure was used to construct several structural models. The predictive power was quite good, with these models accounting for 60-78% of the observed variance of problem difficulty.

Possibilities for extension of this technique to other types of problems were also discussed.
A Generative Model of Mathematics Learning
Authors: Wiitrock M. C.
Published: August 2020

The learning of mathematics is presented as a cognitive process, rather than as a behavioristic one. A generative model of mathematics learning is described. Empirical research relevant to the model is discussed. Implications for teaching mathematics as a generative process are summarized.
Research on Mathematics Education (K–12) Reported in 1973
Authors: Suydam Marilyn N. , Weaver J. F.
Published: August 2020

Research reported during the 1973 calendar year is listed, accompanied by brief annotations of findings.
Developmental Aspects of Children's Ability to Understand Mathematical Proof
Authors: Lester Frank K.
Published: August 2020

Simple elements of mathematical proof were analyzed by considering the making of a proof as a complex problem-solving task. Quantitative measures of performance on certain problem-solving tasks were obtained in order to determine the relation between problem-solving ability and chronological age (6-18). A consideration of all variables suggested that, even though problem-solving ability does increase with age, certain aspects of problem solving that approximate mathematical proof can be dealt with successfully in the upper elementary grades. An analysis of problem-solving strategies is recommended as a further step in the investigation of problem solving. IQ, reading ability, and mathematical aptitude are suggested as possible covariates or blocking variables in future research.