bluuebunny/crossref_metadata_2025 huggingface dataset:
Supporting Latino First Graders' Ten-Structured Thinking in Urban Classrooms
Authors: Fuson Karen C. , Smith Steven T. , Lo Cicero Ana Marie
Published: August 2020

Year-long classroom teaching experiments in two predominantly Latino low-socioeconomicstatus (SES) urban classrooms (one English-speaking and one Spanish-speaking) sought to support first graders' thinking of 2-digit quantities as tens and ones. A model of a developmental sequence of conceptual structures for 2-digit numbers (the UDSSI triad model) is presented to describe children's thinking. By the end of the year, most of the children could accurately add and subtract 2-digit numbers that require trading (regrouping) by using drawings or objects and gave answers by using tens and ones on various tasks. Their performance was substantially above that reported in other studies for U.S. first graders of higher SES and for older U.S. children. Their responses looked more like those of East Asian children than of U.S. children in other studies.
Computer-Intensive Algebra and Students' Conceptual Knowledge of Functions
Authors: O'Callaghan Brian R.
Published: August 2020

This article describes a research project that examined the effects of the Computer-Intensive Algebra (CIA) and traditional algebra curricula on students' understanding of the function concept. The foundation for the research is a proposed conceptual framework that describes function knowledge in terms of component competencies. The results indicated that the CIA students achieved a better overall understanding of functions and were better at the components of modeling, interpreting, and translating. No significant differences were found for reifying, which emerged as the most difficult component in the proposed function model. Further, the CIA students showed significant improvements in their attitudes toward mathematics, were less anxious about mathematics, and rated their class as more interesting. A higher percentage of students successfully completed the CIA course.
Editorial
Authors: Sowder Judith T.
Published: August 2020

Mathematics education research is no longer a young discipline. One way of measuring our age is in terms of years. Jeremy Kilpatrick (1992) traces our history back to the turn of the century, but he notes that a 1967 conference held at the University of Georgia seemed “for many participants … to mark the beginnings of true interdisciplinarity and community among researchers in mathematics education (p. 25).”
Preschoolers' Counting and Sharing
Authors: Pepper Kristine L. , Hunting Robert P.
Published: August 2020

Both counting and sharing require action on discrete elements, entailing the logic of one-to-one correspondence. How counting and sharing relate to one another was the focus of an experiment conducted to examine strategies preschool children used to subdivide items. We designed tasks in which applications of counting skills, of visual cues such as subitizing, and of informal measurement skills were made more difficult. Children exhibited alternative strategies, suggesting use of a recipient as a mental cycle marker and an adjacent recipient strategy, with pauses between allocations suggesting a re-presentation of lots corresponding to the number of recipients. Results supported the view that dealing competence does not relate directly to counting skill.
Conceptualizing Belief Structures of Preservice Secondary Mathematics Teachers
Authors: Cooney Thomas J. , Shealy Barry E. , Arvold Bridget
Published: August 2020

This is a study of the beliefs and belief structures of 4 preservice secondary mathematics teachers as they progressed through a 4-quarter sequence in mathematics education including student teaching. We considered the notions of centrally and peripherally held beliefs and whether beliefs were held dualistically or contextually. The various ways in which the teachers structured their beliefs helped account for the fact that some beliefs were permeable whereas others were not. The nature of the evidence supporting the teachers' beliefs was considered particularly as that evidence related to the voices of significant others or to what the individuals valued. A scheme for conceptualizing the professional development of preservice teachers is posited.
Use of Multiplicative Commutativity by School Children and Street Sellers
Authors: Schliemann Analúcia D. , Araujo Cláudia , Angela Cassundé Maria , Macedo Suzana , Nicéas Lenice
Published: August 2020

We analyzed use of the commutative property for solving multiplication problems by children who learn about multiplication in schools and by street vendors who solve multiplication problems through repeated addition. Subjects were Brazilian street vendors with irregular school attendance, who had received no, or very little, school instruction on multiplication, and 1st- to 3rd-grade Brazilian school children. Results from 2 studies show that use of commutativity to solve multiplication problems is closely related to use of multiplication. Street sellers who rely exclusively on repeated addition to solve such problems may, however, use multiplicative commutativity if it represents a clear reduction in the number of computational steps needed to reach a solution. But their justifications for its use are often based on knowledge about multiplication.
The Empty Number Line in Dutch Second Grades: Realistic Versus Gradual Program Design
Authors: Klein Anton S. , Beishuizen Meindert , Treffers Adri
Published: August 2020

In this study we compare 2 experimental programs for teaching mental addition and subtraction in the Dutch 2nd grade (N = 275). The goal of both programs is greater flexibility in mental arithmetic through use of the empty number line as a new mental model. The programs differ in instructional design to enable comparison of 2 contrasting instructional concepts. The Realistic Program Design (RPD) stimulates flexible use of solution procedures from the beginning by using realistic context problems. The Gradual Program Design (GPD) has as its purpose a gradual increase of knowledge through initial emphasis on procedural computation followed by flexible problem solving. We found that whereas RPD pupils showed a more varied use of solution procedures than the GPD pupils, this variation did not influence the procedural competence of the pupils. The empty number line appears to be a very powerful model for the learning of addition and subtraction up to 100.
The Value (and Convergence) of Practices Suggested by Motivation Research and Promoted by Mathematics Education Reformers
Authors: Stipek Deborah , Salmon Julie M. , Givvin Karen B. , Kazemi Elham , Saxe Geoffrey , MacGyvers Valanne L.
Published: August 2020

In this study we discuss convergence between instructional practices suggested by research on achievement motivation and practices promoted in the mathematics instruction reform literature, and we assess associations among instructional practices, motivation, and learning of fractions. Participants included 624 fourth- through sixth-grade students and their 24 teachers. Results indicated that the instructional practices suggested in literature in both research areas positively affected students' motivation (e.g., focus on learning and understanding; positive emotions, such as pride in accomplishments; enjoyment) and conceptual learning related to fractions. Positive student motivation was associated with increased skills related to fractions
Editorial
Authors: Sowder Judith T.
Published: August 2020

Many JRME readers are aware of NCTM's Standards 2000 initiative, but I suspect that some are not. This work will have important ramifications for future schooling in the United States and will probably have some influence in other countries as well. By the time this editorial is in print, a draft of the Principles and Standards for School Mathematics will be available for reading and comments. This seems to be a good time to provide some background on this initiative and to urge you to read the draft document and provide comments to NCTM's Commission on the Future of the Standards
The Portrayal of Parents in the School Mathematics Reform Literature: Locating the Context for Parental Involvement
Authors: Peressini Dominic D.
Published: August 2020

In this article, using reform recommendations that call for parental involvement as a springboard, I provide an analysis of the positioning of parents in the school mathematics reform literature. Employing Foucault's (1980) conception of “regimes of truth,” I demonstrate how the literature has created the accepted discourse for mathematics education reform. I then argue that the professionalization of teachers has distanced parents from schools and led to conflict between parents and mathematics educators and that to reconcile this conflict, ways in which parents can be included in mathematics education must be considered. It is essential first, however, to understand issues central to involving parents in mathematics education. A research agenda for parental involvement in mathematics education is presented.
Review: Complexity and Insight
Authors: Kieran Carolyn
Published: August 2020

Members of the English-speaking mathematics education research community have long been awaiting the day when they could become better acquainted with Guy Brousseau's Theory of Didactical Situations. Brousseau, who is a professor of didactique des mathématiques at the Institut de Formation des Maîtres d'Aquitaine of the University of Bordeaux in France, is one of mathematics education research's pioneers. From the early 1960s (when he gave birth to an ambitious project aimed at scientifically determining the best way of teaching mathematics to elementary school children) to the present (when he continues to work with pupils and teachers from the preschool to university levels and with research collaborators both at home and abroad), his impact has been considerable. Even though Brousseau began developing key concepts of his Theory as long as 30 years ago, direct access to his writings and those of his colleagues and students has been possible only for those who could read the French language. For several decades, non–francophone researchers have been wanting to get to the original sources that introduced concepts such as didactical contract, adidactical situation, institutionalization, and so on–concepts that have been referred to and interpreted in countless international research texts. In response to this need, the editors of this volume have selected from among Brousseau's papers those written between 1970 and 1990 that best exemplify the components of the Theory. But they have not only translated these papers, they have also edited, annotated, and organized them so as to provide a coherent and comprehensive presentation of the Theory of Didactical Situations.
Ways Mathematics Teachers Help Students Organize Knowledge
Authors: Cooney Thomas J. , Henderson Kenneth B.
Published: August 2020

To find out how mathematics teachers help students organize their knowledge, audiotapes were made of 44 instances of classroom teaching of mathematics, ranging from the seventh to the twelfth grade. The classes were taught by 10 teachers. The transcripts made from these audiotapes were analyzed and 9 relations employed by the teachers in helping students organize their knowledge were identified. Each of these relations is explicated.
The Influence of Teacher Behavior and Dyad Compatibility on Clinical Gains in Arithmetic Tutoring
Authors: Schultz Edward W.
Published: August 2020

This study focused on the independent and combined influence of teacher facilitative behavior (accurate empathy, interpersonal respect) and interpersonal compatibility between teacher and student on selected educational outcomes in a clinical tutoring program for students diagnosed as having learning difficulties in arithmetic concepts and principles. The Ss were 20 teachers and 40 students drawn from a larger pool of teachers and students. In general, the results of ANOVA procedures failed to support several relationships between teacher facilitative behavior and interpersonal compatibility and many of the dependent dimensions of interest. The discussion section offers alternate explanations and suggestions for subsequent research.
Definitions and Examples as Feedback in a CAI Stimulus-Centered Mathematics Program
Authors: Keats John B. , Hansen Duncan N.
Published: August 2020

The relative effect of verbal definitions and numerical examples as corrective feedback in a computer-assisted learning task was investigated for 45 ninth-grade students. The task involved the application of rules to justify steps in mathematical proofs. Error analysis during learning indicated that when the learner was in an erroneous state, providing correctional messages in the form of verbal definitions increased his probability of being removed from that state more than the use of numerical examples. Furthermore, verbal definitional feedback enhanced the probability of success on later trials involving reapplication of the same rule or application of a similar rule. No conclusions could be drawn about the superiority of either feedback mode when acquisition/retention was used as a criterion.
Performance of Kindergarten Children on Transitivity of Three Matching Relations
Authors: Owens Douglas T. , Steffe Leslie P.
Published: August 2020

A sample of 42 kindergarten children was partitioned into 27 conservers and 15 nonconservers of matching relations “as many as,” “more than,” and “fewer than,” based on an 18-item conservation of matching relations test (CMRT), 6 items for each relation. An 18-item transitivity of matching relations test (TMRT) was then administered to each subject under each of 3 stimulus conditions; neutral stimulus, conflictive stimulus, and screened stimulus. Subjects were randomly assigned to relations. A 2× 3× 3 design was used to detect differences among means or interactions of the 2 conservation levels, the 3 relations, and the 3 stimulus conditions, where Stimulus Condition was a repeated factor. Stimulus Condition was significant (p<.05) with a significant interaction occurring (p<.05) between Conservation and Matching Relations. The only relation on which the high conserver performed significantly better on transitivity (p<.05) than the low conserver was the equivalence relation “as many as,” where the means were 77% and 40%, respectively.
Critiques of Articles: Comments on the Effect of Activity-Oriented Instruction
Authors: Holz Alan W.
Published: August 2020

Moody, Abell, and Bausell (1971) suggest that the results of their study contradict outcomes predicted by activity-oriented theories of instruction. This critique of their paper raises questions about the validity of that study for testing any established theory or activity-oriented instruction. Both the theoretical justification of the activity-oriented treatment used in the study and the validity of the tests for measuring outcomes predicted by activity-oriented theories of instruction are questioned.
Research on Mathematics Education (K–12) Reported in 1971
Authors: Suydam Marilyn N. , Weaver J. Fred
Published: August 2020

An annual annotated listing of research pertaining to elementary and secondary school mathematics.
Inductive and Deductive Learning Styles in Junior High School Mathematics: An Exploratory Study
Authors: Gawronski Jane Donnelly
Published: August 2020

Programmed materials were developed to investigate if 406 eighth grade subjects exhibited inductive and deductive learning styles in mathematics. Ss above the median on the posttest following a concept taught inductively and below the median on the posttest following a concept taught deductively were classified as Inductive Learners. Ss above the median on the posttest following a concept taught deductively and below the median on a posttest following a concept taught inductively were classified as Deductive Learners. 62 Ss scored above the median on both and 77 Ss scored below on both. 105 Ss scored at the median on 1 or both. The testing scheme resulted in the classification of 32 Deductive and 22 Inductive Learners. 2 additional programs were administered to these Ss. 1 utilized an inductive strategy and the other a deductive strategy. Posttests of achievement were administered for the additional programs. No significant difference between groups was found.
A Study of Transformational Geometry Instruction in the Primary Grades
Authors: Williford Harold J.
Published: August 2020

The purpose of this study was to assess the performance of Ss who were given instruction in transformational geometry as compared with Ss who received minimal introductory instruction on transformational geometry terminology and concepts. The criterion measures were geometry achievement and spatial ability. 63 second- and third-grade Ss were randomly assigned to experimental and control groups. The experimental Ss received a 12 lesson treatment and the control Ss a 1 lesson treatment concerning rigid motion and congruence concepts. Results indicated that after instruction the experimental Ss performed significantly better than the control Ss on the achievement test, but not on the space test. The results imply that children can learn transformational geometry skills, but the evidence also suggests they have difficulty in applying these skills to a more general task.
Individualized Instruction in Problem Solving in Elementary School Mathematics
Authors: Jerman Max
Published: August 2020

Some researchers have argued for the existence of general problem solving skills which when mastered are amenable to transfer to specific disciplines.
The Symmetric Property of the Equality Relation and Young Children's Ability to Solve Open Addition and Subtraction Sentences
Authors: Weaver J. Fred
Published: August 2020

Based upon the equation form a□ b=c, coupled with addition and subtraction facts having sums between 10 and 18, an inventory consisting of 4 8-item tests was developed to investigate 1st-, 2d-, and 3d-grade pupils' ability to solve open sentences which differed with respect to several mathematical factors, only 1 of which is the focus of this informal report. Data from 135 classes in 23 schools were included in the analysis. Differences were observed between performance levels on open addition and subtraction sentences generated from the equation form a□ b=c and performance levels on similar sentences generated from the equivalent symmetric form c=a□ b. It is suggested that school mathematics programs should give more explicit attention to variations in equation or open-sentence form associated with the symmetric property of the equality relation.
Editorial Comment
Authors: Johnson David C.
Published: August 2020

The January 1971 JRME editorial included a discussion of the problem of incorporating or pulling together the large number of seemingly unrelated studies reported in the Journal. The suggestion was made for “Coordinated efforts in designing large ‘blocks’ of complementary studies in an area, under some common direction [p. 3].” An invitation was extended to researchers to submit papers that represent a coordinated research effort. The invitation was accepted by a group of researchers at the Pennsylvania State University, and this issue of JRME includes a collection of their papers, which were generated by just such a research program, entitled Toward a Theory of Sequencing: An Integrated Program of Research.
The Effect of an Advance Organizer, Cognitive Set, and Post Organizer on the Learning and Retention of Written Materials
Authors: Romberg Thomas A. , Wilson James W.
Published: August 2020

This study investigated the effect of an advance organizer (AO), cognitive set (CS), and post organizer (PO) in learning and retention of written mathematics materials. Self-instructional booklets were designed so that the pages containing the AO, CS, and PO varied according to experimental treatment in a 2 × 2 × 2 factorial design. Booklets were randomly assigned within 9 Algebra 2 classes.

A significant (p<.05p<.05) interaction between the AO and the PO and a significant main effect (p<.05p<.05) for CS was found on the retention test. Examination of the cell means indicated (a) a facilitating effect for CS, and (b) either an AO or a PO was facilitating, but used together they were not facilitating.
Study 1-3: An In-Depth Exploration of the Role of Iconic Representations in the Study of Congruence of Triangles
Authors: Paquette Gerard A.
Published: August 2020

The studies conducted by Klein (1970) and Farris (1970) opened, in a somewhat indirect way, a multitude of questions about how to more adequately conceptualize the various modes of representation. The descriptions of the modes, as stated by Bruner (1964), are clearly much too loose for conducting instructional research of the type envisioned by the Paradigms Project. What appears to be needed are definitions that at least come close to permitting consistency of classification. In this connection, it was thought that more information about the individual modes—if even in their more pure forms—was a first order of business. In any event, it was this sort of thinking that gave rise to this investigation as well as to the companion study by Bowers (1971).
Study 1-4: The Effect of Triangle-Pair Configuration Variation on Achievement of Selected Classes of Instructional Objectives in Plane Geometry
Authors: Bowers Robert Garth
Published: August 2020

This investigation was designed in conjunction with the study conducted by Paquette (1971) to test the effects of varying referent triangle-pair configurations (where the instruction was mediated by an IBM 1500 Instructional System) on the achievement of selected instructional objectives related to the study of congruent triangles. The configuration classification (CC) matrix, as described in Figure 1 of the article by Heimer and Lottes, was used to generate and classify all congruent triangle-pair referents employed in this study. The primary purpose of the study was concerned with selected effects of multiconfiguration instructional treatments, as defined below, on the achievement of instructional objectives over collections of configuration classes for which instruction was not provided.
Educational Research in Mathematics at the University of Wisconsin Research and Development Center for Cognitive Learning
Authors: Harvey John G. , Romberg Thomas A.
Published: August 2020

This overview will relate the three papers that follow to the past, the current, and the projected research related to the learning and teaching of mathematics and conducted under the auspices of the University of Wisconsin Research and Development Center for Cognitive Learning. Regular readers of JRME are already aware that several Center studies have been reported here. (Fennema, 1972; King, 1973; Romberg & Gilbert, 1972; Romberg & Shepler, 1973; Shepler, 1970; Sowder, 1971; and Steffe, 1970). No doubt in the future other Center studies will be reported in JRME, since it is a natural dissemination point for communicating selected results to the mathematics education research community.
The Interaction of Three Levels of Aptitude Determined by a Teach-Test Procedure with Two Treatments Related to Area
Authors: Montgomery Mary E.
Published: August 2020

This study examined the interaction of children's ability to learn unit-of-length concepts with 2 treatments based on area and unit-of-area concepts. 59 second and third grade Ss were randomly assigned to the 2 treatments: 1 treatment emphasized the unit of area whereas the other did not. No significant interactions were found between aptitude and treatment on measures of achievement, retention, and transfer. Significant main effects due to aptitude and treatment on the achievement and retention measures were found favoring the higher aptitude level and the treatment emphasizing the unit. The Ss needed exposure to unit concepts before mastering the associated behaviors but did learn the area concepts presented.
Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts
Authors: Oehrtman Michael
Published: August 2020

This study investigated introductory calculus students' spontaneous reasoning about limit concepts guided by an interactionist theory of metaphorical reasoning developed by Max Black. In this perspective, strong metaphors are ontologically creative by virtue of their emphasis (commitment by the producer) and resonance (support for high degrees of elaborative implication). Analysis of 120 students' written and verbal descriptions of their thinking about challenging limit concepts resulted in the characterization of 5 clusters of strong metaphors. These clusters were based on the objects, relationships, and logic related to intuitions about (a) a collapse in dimension, (b) approximation and error analyses, (c) proximity in a space of point-locations, (d) a small physical scale beyond which nothing exists, and (e) the treatment of infinity as a number. Students' reasoning with these metaphors had significant implications for the images they formed and the claims and justifications they provided about multiple limit concepts.
Research Commentary: Amidst Multiple Theories of Learning in Mathematics Education
Authors: Simon Martin A.
Published: August 2020

Currently, there are more theories of learning in use in mathematics education research than ever before (Lerman & Tsatsaroni, 2004). Although this is a positive sign for the field, it also has brought with it a set of challenges. In this article, I identify some of these challenges and consider how mathematics education researchers might think about, and work with, the multiple theories available. I present alternatives to views of the competition or supersession of theories and indicate the kinds of discussions that will support effective theory use in mathematics education research. I describe the potential for mathematics education researchers to make informed, justified choices of a theory or theories to address particular research agendas.
Editorial: Research Articles Can Be Helpful to Practitioners
Authors: Blume Glendon W.
Published: August 2020

Because JRME is a research journal, its value to those who conduct research in mathematics education is obvious. What may not be as obvious, however, is that JRME articles also have the potential to benefit another audience, namely, mathematics education practitioners. Research articles in JRME (and those in other mathematics education research journals, as well) can offer to practitioners helpful information and a variety of tools that have the potential to be useful in their work. The variety of “practitioners” who can benefit from research articles in JRME includes those who teach mathematics at the prekindergarten through collegiate levels, teacher educators who work with prospective mathematics teachers at any of those levels, mathematics coaches or supervisors who serve as school- or district-based leaders for groups of mathematics teachers, teacher educators who engage in-service mathematics teachers in professional development, and even researchers who teach others about mathematics education research.
Book Review: … And All the Children Are Above Average: A Review of The End of Ignorance: Multiplying Our Human Potential
Authors: Kirshner David
Published: August 2020

The End of Ignorance (TEOI), laced with anecdote, is published for a lay audience, by a nonacademic press, with no real foundation in learning theory. Yet every mathematics education researcher—perhaps every educational researcher—needs to have John Mighton's book clearly within his or her sights. Mighton's curriculum approach, JUMP Math, threatens to leapfrog over the head of the mathematics education establishment, delivering mathematical learning to every child, from the profoundly learning disabled on up, at up to three or four grade levels beyond their current grade placement.
A Letter from the NCTM President: Linking Research and Practice: The Research Agenda Project
Authors: Shaughnessy J. Michael
Published: August 2020

About the same time that this issue of JRME arrives at your doorstep or in your computer, NCTM will release the report Linking Research and Practice: The NCTM Research Agenda Report. This publication summarizes the work and recommendations from a conference of practitioners and researchers that was held in summer 2008. The conference was charged with identifying and prioritizing a set of guiding research questions that could help to focus researchers' attention on critical problems of practice. With this letter I have an opportunity to share information and recommendations from the Research Agenda Conference Report, and point to some next steps for NCTM to implement the vision shared in the Report.
Research Committee: Strengthening Research by Designing for Coherence and Connections to Practice NCTM Research Committee
Authors: Boerst Timothy , Confrey Jere , Heck Daniel , Knuth Eric , Lambdin Diana V. , White Dorothy , Baltzley Patricia C. , Quander Judith Reed
Published: August 2020

The National Council of Teachers of Mathematics (NCTM) is committed to strengthening relations between research and practice and to the development of a coherent knowledge base that is usable in practice. The fifth of NCTM's strategic priorities states, “Bring existing research into the classroom, and identify and encourage research that addresses the needs of classroom practice” (NCTM, 2008). The need to work toward connection and coherence is not unique to the field of mathematics education. Fields such as medicine (e.g., Clancy, 2007), software engineering (e.g., Gorschek, Garre, Larsson, & Wohlin, 2006), and social work (e.g., Hess & Mullen, 1995) routinely attend to these issues. Researchers in many fields strive to find new ways or to engage more effectively through existing means to enhance coherence and connection. In a sense, this is not a goal that can be achieved definitively, but one that requires persistent engagement. In education, the constant flux of variables in the system, such as curriculum, goals for student learning, and school contexts, requires that new connections between research and practice be investigated and that old connections be reexamined. Changes in educational contexts open new territory in need of study and also challenge the coherence of explanations grounded in previous research. In this way, attention of the field to connection and coherence is neither unique to mathematics education nor an effort due solely to inadequacies of research efforts in the past.
The Relative Effectiveness of Four Strategies for Teaching Disjunctive Concepts in Mathematics
Authors: Dossey John A. , Henderson Kenneth B.
Published: August 2020

Several studies have been made of the relative effectiveness of different strategies for teaching concepts. Those studies in mathematics have provided a little information, but the majority of the findings have indicated no significant differences exist between the different strategies studied (Henderson & Rollins, 1967; Rector & Henderson, 1970). One possible reason for this is that these studies have not gone beyond the point of studying strategies for teaching concepts, as compared to strategies for teaching different types of concepts. That is to say, they have neglected to take into account the different logical forms of concepts.
A Forum for Researchers: Is the (Psychological) World Flat?
Authors: Columbus Christopher
Published: August 2020

The purpose of this note is to question the almost exclusive use of linear models in mathematics education research and to urge thoughtful exploration of alternate models when a linear model fits the data of a study poorly. (Here “linear model” refers to a model linear in one or more continuous independent variables, not to a model linear in the parameters to be estimated.)
A Reply to General Questions about Jerman's Study
Authors: Jerman Max E.
Published: August 2020

It is true that a writer must compromise something when a 91-page report is condensed to some 13 pages, which still exceeds the editorial policy of a journal. In these cases, the writer must assume that an interested reader will reference the original documents for details not included in a shortened version.
The Influence Of Professional Reference Groups On Decisions Of Preservice Secondary School Mathematics Teachers
Authors: Bright George W. , Ray Carry L.
Published: August 2020

Preservice secondary school mathematics teachers were studied relative to 2 hypotheses: (a) Mathematicians and educators influence decisions in projected classroom situations; (b) Mathematicians exert more influence than educators. Ss were presented with projected classroom situations, each accompanied by 3 plausible resolutions. For each situation, labels of “mathematicians” and “educators” were attached to different resolutions to represent the alleged consensus of the respective professional group. The sample was split randomly into control (N = 33) and experimental (N = 28) groups. Randomization was used in producing the test instrument to control content effects. Kolmogorov-Smirnov statistics indicated significant influence of the label “mathematicians” (p < .03) and the label “educators” (p < .01) on the resolution choices of experimental Ss. A Wilcoxon matched-pairs, signed-rank statistic indicated no significant differential influence between the professional groups.
Initial Attitude Differences among Successful, Procrastinating, and “Withdrawn-from-Course” Students in a Personalized System of Statistics Instruction
Authors: Newman Frederick L. , Ball Stanley E. , Young Dennis L. , Smith Clarence C. , Purtle Ronald B.
Published: August 2020

A Personalized System of Instruction (PSI) with a mastery learning component was used to teach introductory statistics to nonmathematics majors at two universities. At the beginning of the semester 250 students were given a mathematics attitude questionnaire (standardized on 938 students) to determine if certain attitudes were characteristic of the students who procrastinate (exhibiting spurts of effort immediately prior to a deadline) or those who withdrew their course registration. The attitude-factor scores of these two groups were contrasted with a success group. Initial feelings of anonymity in traditionally taught mathematics courses and a diminutive interest in the acts of doing mathematics problems consistently characterized the procrastinating and the “withdrawn-from-course” groups. Initially perceived difficulty or usefulness of the mathematical subject matter did not differentiate the groups.
The Effects of Instruction on Length Relations on the Classification, Seriation, and Transitivity Performances of First- and Second-Grade Children
Authors: Johnson Martin L.
Published: August 2020

A sample of 81 first- and second-grade children from 2 Atlanta elementary schools was randomly partitioned into experimental and control groups. The experimental children were given experiences in sorting and seriating linear objects. The main effects of treatment and grade were significant (p < .01) on a seriation test. Grade and school effects were significant (p < .05) on a conservation test, with only school effect significant (p < .05) on a transitivity test. No significant relationship could be detected between classification and transitive ability using “same length as”; or between seriation and transitive ability using “shorter than” and “longer than.”
Mathematics Learning and the Sexes: A Review
Authors: Fennema Elizabeth
Published: August 2020

It has long been accepted as true that boys learn mathematics better than girls do. To determine the validity of this belief, 36 studies concerned basically or tangentially with sex differences in mathematics achievement were reviewed and two others were analyzed in depth. The data from one study (Parsley, et al., 1964), which often has been quoted as supportive of boys' mathematics superiority, was reevaluated with the conclusion that the data from this study do not support the idea that boys are superior to girls in mathematics achievement. Data concerned with sex differences in achievement from the National Longitudinal Study of Mathematics Achievement were also presented.

No significant differences between boys' and girls' mathematics achievement were found before boys and girls entered elementary school or during early elementary years. In upper elementary and early high school years significant differences were not always apparent. However, when significant differences did appear they were more apt to be in the boys' favor when higher-level cognitive tasks were being measured and in the girls' favor when lower-level cognitive tasks were being measured. No conclusion can be reached concerning high school learners.
Five Forms of Written Feedback to Homework in a Mathematics Course for Elementary Teachers
Authors: Schoen Harold L. , Kreye Bettibel C.
Published: August 2020

The form of teacher feedback to students' written work has been an independent variable in previous research. The value of written feedback to student errors on test papers and homework has been established in several studies (Hedl, 1969; Karabinus, 1966; Page, 1958). In one investigation, written feedback specific to student errors on test papers was found to be superior to general feedback as measured by a subsequent achievement test (Page, 1958). On the contrary, in a study by one of the investigators, achievement scores favored general feedback over specific feedback in a CAI unit (Schoen, 1972). Attitude toward the CAI unit in the same study strongly favored groups whose feedback contained first names over the absence of the name.
An Experimental Study of the Effects of Three Instructional Methods in Basic Probability and Statistics
Authors: Dan Austin Joe
Published: August 2020

The effects of 3 instructional methods for teaching probability and statistics were investigated with university-level Ss. The instructional units were based on Bruner's model of cognitive development. They were called MP, manipulative-pictorial (enactive); P, pictorial (iconic); and S, symbolic. 71 students were assigned to the 3 treatments, and Ss studied a unit for 1 month. Examination items were stratified into 4 cognitive levels: comprehension, computation, application, and analysis. Data were analyzed with ANOVA and significant differences (p ≤ .05) were found between treatment groups for selected subtests and generally favored MP and P over S. Differences between MP and P were not significant on any variable.
The Influence of Verbalization of Discovered Numerical- or Sorting-Task Generalizations on Short-Term Retention in Connection with the Hendrix Hypothesis
Authors: Sowder Larry
Published: August 2020

Experimentation dealing with the influence of verbalizing a discovered generalization on retention of the ability to use the generalization has not given consistent results. The reported study sought information on whether the effects of 5 verbalizing methods (subject speaks, reads, writes, listens, or does not verbalize) were different on short-term retention of the ability to use discovered generalizations. During individual interviews, 50 college subjects gave evidence of having formed generalizations on 5 numerical or sorting tasks by giving successive correct responses, and then verbalized their discoveries according to one of the verbalizing methods. A retention test consisting of items amenable to the generalizations was administered 10 minutes later. Analyses of retention-test scores indicated no significant differences among the effects of the verbalizing methods, between effects of modes of verbalizing (oral, written), or between effects of sources of verbalizing (subject, external source).
Editorial Comment
Authors: Weaver J. F.
Published: August 2020

It is good to have a positive reaction to recent research in mathematics education come from a person of Professor Wittrock's professional stature. His view of learning as a generative process (set forth in his article in this issue) should be very thought-provoking in its implications for potentially fruitful lines of future investigation pertaining to mathematical learning.
Determining the Best Predictors of Success and of Time of Completion or Dropout in an Individualized Course in Arithmetic at a Community College
Authors: Dahlke Richard M.
Published: August 2020

In 1969-70, the author conducted an exploratory study of an individualized, self-paced, mastery-oriented course in arithmetic (Math 031) at Washtenaw Community College (WCC), Ann Arbor, Michigan (Dahlke, 1971). A portion of the study, which is the substance of this report, was devoted to determining the best predictors of student success and of time of completion or dropout.
The Cloze Procedure as a Measure of Mathematical English
Authors: Hater Mary A. , Kane Robert B.
Published: August 2020

A method of adapting cloze tests for use with mathematical passages is discussed. The validity of the cloze procedure as a measure of comprehensibility and difficulty of mathematical selections is tested. For 5 passages, 25 cloze tests and 5 comprehension tests were administered to subjects, grades 7 through 10. Results supported the hypotheses that adapted cloze tests are reliable and valid predictors of comprehensibility and that rankings of cloze tests are reliable. Evidence supports the hypothesis that cloze tests are valid measures of difficulty, but this conclusion needs further verification due to the number of passages used in the study.
Interaction of Spatial Visualization and General Reasoning Abilities with Instructional Treatment in Quadratic Inequalities: A Further Investigation
Authors: Eastman Phillip M. , Carry L. Ray
Published: August 2020

This study was the third in a series of investigations dealing with aptitude × treatment interactions--using aptitude measures of spatial visualization and general reasoning and a learning task of quadratic inequalities. 80 Ss were randomly assigned to one of two treatment groups: Group A, a deductively structured verbal-symbolic-numeric treatment; and Group B, an inductively structured verbal-spatial-numeric treatment. The hypotheses were that spatial visualization would predict success on the graphical treatment, and that general reasoning would predict success on the analytic treatment. Results were analyzed using multiple linear regression. A significant disordinal interaction was statistically confirmed using a homogeneity-of-regression test, and the hypotheses were confirmed.
Problems in Empirical Research on Individualized Mathematics Programs
Authors: Golladay Mary A. , Devault M. Vere , Fox G. Thomas , Skuldt Karen
Published: August 2020

Suggests a conceptual framework having implications for future research on individualized mathematics instruction.
An Instructional Strategy Taxonomy: Theory and Applications
Authors: Merrill M. David , Wood Norman D.
Published: August 2020

The authors have proposed that instruction consists of four relatively independent facets: learner aptitudes, content structure, delivery systems, and instructional strategies. Instructional strategies consist of sequences of informational displays. Eight variables were proposed that characterize a given informational display as follows: content type, content mode, content representation, mathemagenic prompting, response conditions, response mode, response representation, and mathemagenic feedback. For each variable several principal parameter values were suggested. The instructional strategy system described and its accompanying flow chart conventions should have value in the construction of instructional theory, the review of instructional research, the analysis of existing instructional strategies, and the design of instructional materials and systems.
Transfer Effects as a Function of Variety of Method and Ability Level in Elementary Mathematics
Authors: Burton John K. , Lemke Elmer A. , Williams Jeral R.
Published: August 2020

A 2 × 3 design was used to examine the relationship between fifth-grade mathematics abilities (high and low) and variety of methods (one, two, and four) for teaching fractions. On a transfer task, the interaction of abilities with variety of method was significant. The high-ability students' performances decreased with increases in variety of method. The low-ability students' performances increased with increases in variety of methods. The results were interpreted as evidence in support of Ausubel's cognitive theory.
An Experimental Study of the Effectiveness of a Formal versus an Informal Presentation of a General Heuristic Process on Problem Solving in Tenth-Grade Geometry
Authors: Post Thomas R. , Brennan Michael L.
Published: August 2020

It was the purpose of this study to investigate the effects of various modes of presentation of general heuristic processes on students' problem-solving ability. It was felt that if students are to become effective problem solvers they must be formally instructed in the established techniques of, and approaches to, problem solving. It was hypothesized that these techniques would both faci litate students' efficiency in solving problems and improve their ability to solve various kinds of problems. This study, then, was an attempt to formally instruct students in a particular heuristic problemsolving process so that awareness and ability could be developed, and subsequently to evaluate the effects of this instruction on their ability to solve problems.
An Interpretation of Advanced Organizers
Authors: Lesh, Richard A.
Published: August 2020

The term advanced organizer refers to a type of instructional material that has been hypothesized to be eiTective in introducing meaningful topics. Meaningful topics are those in which the new material that is to be learned is related in a nonarbitrary fashion to ideas that have already been mastered by the learner. Using this detlnition, most mathematics instruction has to do with potentially meaningful learning materials, since most mathematical concepts occu r as part of a system of ideas and since it is typical that such concepts are inherently related to other ideas in the system that are already understood by the learner.
An Analysis of Children's Written Solutions to Word Problems
Authors: Knifong J. Dan , Holtan Boyd
Published: August 2020

The relative influence of different factors in children's word problem difficulties has not been established. Nevertheless, it is persistently assumed that poor reading contributes substantially to these difficulties. The written solutions of 35 sixth-graders to the word problem portion of the Metropolitan Achievement Test were assigned to nine disjoint groups. Poor reading could not have been a factor in the 52% of the problems that were incorrectly solved solely because of computational and clerical mistakes. Reading may or may not have been a factor in the remaining 48% of the incorrectly solved problems, but computational difficulties were clearly the major deterrent to success.
An Analysis of the Fragion Concept into a Hierarchy of Seleged Subconcepts and the Testing of the Hierarchical Dependencies
Authors: Novillis Carol F.
Published: August 2020

The fraction concept was analyzed into a set of 16 selected subconcepts, which were arranged into a hypothesized hierarchy. A Fraction Concept Test of 68 items, constructed to measure students' attainment of the 16 selected subconcepts was administered to 279 students in grades 4, 5, and 6. The data were analyzed by using two ratios to determine sequencing validity. 18 of the 23 dependencies in the hypothesized hierarchy were supported. On the basis of these results, it was concluded that certain stated subconcepts were prerequisite to others.