bluuebunny/crossref_metadata_2025 huggingface dataset:
Splitting, Covariation, and Their Role in the Development of Exponential Functions
Authors: Confrey Jere , Smith Erick
Published: August 2020

Exponential and logarithmic functions are typically presented as formulas with which students learn to associate the rules for exponents/logarithms, a particular algebraic form, and routine algorithms. We present a theoretical argument for an approach to exponentials more closely related to students' constructions. This approach is based on a primitive multiplicative operation labeled “splitting” that is not repeated addition. Whereas educators traditionally rely on counting structures to build a number system, we suggest that students need the opportunity to build a number system from splitting structures and their geometric forms. We advocate a “covariation” approach to functions that supports a construction of the exponential function based on an isomorphism between splitting and counting structures.
Biographical Sketch: Henry Van Engen: An Analysis of Meaning in Arithmetic
Authors: Steffe Leslie P.
Published: August 2020

It is fitting to call this short feature on Henry Van Engen “An Analysis of Meaning in Arithmetic” because his famous articles in the Elementary School Journal (Van Engen, l949) go by that title. In Van Engen's operational school of meaning in arithmetic, the meaning of a symbol is an intention to act.
Toward a Working Model of Constructivist Teaching: A Reaction to Simon
Authors: Steffe Leslie P. , D'ambrosio Beatriz S.
Published: August 2020

We start this discussion of the article by Martin A. Simon with two major conjectures that we will try to substantiate. The first is that there is a kind of teaching that can legitimately be called “constructivist teaching.” The second is that Simon's model of teaching his prospective elementary school teachers, if modified, would fit our understanding of constructivist teaching, even though his basic premise is that constructivism does not tell us how to teach mathematics (Simon, 1995).
Biographical Sketch: Kenneth B. Henderson: The Development of Pedagogical Theory
Authors: Cooney Thomas J.
Published: August 2020

Kenneth Henderson was a scholar of the first magnitude who focued his attention on the development of theoretical construcrs related to the teaching of mathematics. His chapter, “Problem-Solving in Mathematics,” in the 21st yearbook of NCTM, written with Robert Pingry (Henderson & Pingry, 1953), was an early and significant contribution to the dialogue about problem solving in the classroom. His Chapter, “Research on Teaching Secondary School Mathematics,” published in the first Handbook of Research on Teaching (Henderson, 1963), further exemplifies the kind of scholarship that characterized his work.
Numeracy as Cultural Practice: An Examination of Numbers in Magazines for Children, Teenagers, and Adults
Authors: Joram Elana , Resnick Lauren B. , Gabriele Anthony J.
Published: August 2020

Many have argued for the importance of numeracy, yet little is known about the opportunities for numeracy presented to people in their daily lives. In this study, we analyzed and compared the characteristics of rational numbers in magazines written for children, teenagers, and adults. Our analysis indicates that difficult mathematical concepts that appear in the media, such as fractions, percents, and averages, are much more prevalent in adults' magazines than in those written for children and teenagers. Adults are often presented with rational numbers that are related to each other. Numbers in teenagers' texts do not appear to form a transition to those found in adults' texts, despite the fact that through formal schooling teenagers have covered all the mathematical concepts that are frequently found in adults' texts. Implications for preparing students for the numeracy demands of everyday life are discussed.
Brief Report: Solution Processes and Interpretations of Solutions in Solving a Division-with-Remainder Story Problem: Do Chinese and U. S. Students Have Similar Difficulties?
Authors: Cai Jinfa , Silver Edward A.
Published: August 2020

During the past several decades, there has been considerable attention to crossnational comparisons of mathematics achievement. A number of studies have examined the performance in various mathematical topic areas by students from different countries (e.g., Lapointe, Mead, & Askew, 1992; Robitaille & Garden, 1989; Stevenson et al., 1990; Stigler, Lee, & Stevenson, 1987). In general, when crossnational studies in mathematics have included samples of Chinese and U.S. students, the findings have been that Chinese students perform mathematical tasks at much higher levels of proficiency than U.S. students (e.g., Lapointe et al., 1992; Stevenson et al., 1990).
Review: The Researcher'S Human Face
Authors: Kelly Anthony E.
Published: August 2020

Research is a human task often masked by the research report. In Invesitigating Mathematics Teaching: A Constructivist Enquiry, Barbara Jaworski removes this mask as she presents a research biography describing a series of ethnographic studies in mathematics education. A research biography provides a detailed, retrospective account of the “struggle of research”: how the process unfolded; the author's growing awareness of limitations of theory, method, logic, or evidence; and the choices that were made along the way to address these problems. The express PUrPOSe of a research biography is to establish methodological rigor by exposing all stages of the research to critical review. It also invites us into the mind of the researcher.
Talking about Rates Conceptually, Part II: Mathematical Knowledge for Teaching
Authors: Thompson Alba G. , Thompson Patrick W.
Published: August 2020

Part I of this report (Thompson & Thompson, 1994) described the attempts of one teacher, Bill, to teach concepts of speed to one child, Ann. We analyzed sources of their eventual dysfunctional communication, concluding that Bill had encapsulated his deep understandings of rate and proportionality within his language for numbers and operations. Part II presents the session of the researcher, Pat, with Ann, analyzing his instructional actions in light of his agenda for having Ann build a scheme of operations by which to understand distance, time, and speed. We contrast Pat's agenda with Bill's, focusing on the mathematical knowledge that guided their instructional decisions and actions and briefly discuss implications for the preparation of teachers with regard to content knowledge.
Review: The Mathematics of Popular Consumption
Authors: Damarin Suzanne K.
Published: August 2020

Various forms of constructivism have been widely accepted among mathematics educators as describing not only the processes by which individual students learn mathematics, but also the development of mathematic itself as a body of knowledge. Constructivism has led to a revival of research in the philosophy of mathematics as a social-cultural-historical construction (Hersh, 1994) and to paradigm shifts in mathematics education research as we seek to understand the ways mathematics is learned and taught most effectively. Constructivism notwithstanding, however, accepted ideas of what is comprised in mathematics as a body of knowledge have not been significantly altered within the community of researchers and teachers who study. practice, and teach malhematics. the philosophy of mathematics, or mathematics education.
Posing Mathematical Problems: An Exploratory Study
Authors: Silver Edward A. , Mamona-Downs Joanna , Leung Shukkwan S. , Kenney Patricia Ann
Published: August 2020

In this study, 53 middle school teachers and 28 prospective secondary school teachers worked either individually or in pairs to pose mathematical problems associated with a reasonably complex task setting, before and during or after attempting to solve a problem within that task setting. Written responses were examined to determine the kinds of problems posed in this task setting, to make inferences about cognitive processes used to generate the problems, and to examine differences between problems posed prior to solving the problem and those posed during or after solving. Although some responses were ill-posed or poorly stated problems, subjects generated a large number of reasonable problems during both problem-posing phases, thereby suggesting that these teachers and prospective teachers had some personal capacity for mathematical problem posing. Subjects posed problems using both affirming and negating processes; that is, not only by generating goal statements while keeping problem constraints fixed but also by manipulating the task's implicit assumptions and initial conditions. A sizable portion of the posed problems were produced in clusters of related problems, thereby suggesting systematic problem generation. Subjects posed more problems before problem solving than during or after problem solving, and they tended to shift the focus of their posing between posing phases based at least in part on the intervening problem-solving experience. Moreover, the posed problems were not always ones that subjects could solve, nor were they always problems with “nice” mathematical solutions.
An Analysis of Arithmetic Problem Posing by Middle School Students
Authors: Silver Edward A. , Cai Jinfa
Published: August 2020

The mathematical problems generated by 509 middle school students, who were given a brief written “story-problem” description and asked to pose questions that could be answered using the information, were examined for solvability, linguistic and mathematical complexity, and relationships within the sets of posed problems. It was found that students generated a large number of solvable mathematical problems, many of which were syntactically and semantically complex, and that nearly half the students generated sets of related problems. Subjects also solved eight fairly complex problems, and the relationship between their problem-solving performance and their problem posing was examined to reveal that “good” problem solvers generated more mathematical problems and more complex problems than “poor” problem solvers did. The multiple-step data analysis scheme developed and used herein should be useful to teachers and other researchers interested in evaluating students' posing of arithmetic story problems.
President's Report: Building Bridges of Mathematical Understanding for All Children
Authors: Price Jack
Published: August 2020

The theme “Building Bridges of Understanding” is especially important to the mathematics education community. It proclaims the next steps that we all must take to make implementing the NCTM's Standards a long-lasting, successful effort to enhance mathematics education.
Editorial: January 1997
Authors: Sowder Judith T.
Published: August 2020

With this issue my name is listed as editor of this journal. Some have asked me what changes I plan to make. The journal has now passed the quarter-century mark, and so one change that seemed appropriate was the one evident to you before you opened the cover of this issue. There will be other changes before the end of my term. By that time, an electronic version of the journal should be a well-established option for our readers. And I intend for the time lag between acceptance of manuscripts and publication to be much shorter than it is now. Steps have already been taken to reduce the backlog of accepted manuscripts, as was noted in the November 1996 editorial.
Children's Conceptual Structures for Multidigit Numbers and Methods of Multidigit Addition and Subtraction
Authors: Fuson Karen C. , Wearne Diana , Hiebert James C. , Murray Hanlie G. , Human Pieter G. , Olivier Alwyn I. , Carpenter Thomas P. , Fennema Elizabeth
Published: August 2020

Researchers from 4 projects with a problem-solving approach to teaching and learning multidigit number concepts and operations describe (a) a common framework of conceptual structures children construct for multidigit numbers and (b) categories of methods children devise for multidigit addition and subtraction. For each of the quantitative conceptual structures for 2-digit numbers, a somewhat different triad of relations is established between the number words, written 2-digit marks, and quantities. The conceptions are unitary, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate conceptions. Conceptual supports used within each of the 4 projects are described and linked to multidigit addition and subtraction methods used by project children. Typical errors that may arise with each method are identified. We identify as crucial across all projects sustained opportunities for children to (a) construct triad conceptual structures that relate ten-structured quantities to number words and written 2-digit numerals and (b) use these triads in solving multidigit addition and subtraction situations.
Reflective Discourse and Collective Reflection
Authors: Cobb Paul , Boufi Ada , McClain Kay , Whitenack Joy
Published: August 2020

The analysis in this paper focuses on the relationship between classroom discourse and mathematical development. We give particular attention to reflective discourse, in which mathematical activity is objectified and becomes an explicit topic of conversation. We differentiate between students' development of particular mathematical concepts and their development of a general orientation to mathematical activity. Specific issues addressed include both the teacher's role and the role of symbolization in supporting reflective shifts in the discourse. We subsequently contrast our analysis of reflective discourse with Vygotskian accounts of learning that also stress the importance of social interaction and semiotic mediation. We then relate the discussion to characterizations of classroom discourse derived from Lakatos' philosophical analysis.
Increased Knowledge in Geometry and Instructional Practice
Authors: Swafford Jane O. , Jones Graham A. , Thornton Carol A.
Published: August 2020

This study examined the effects on instruction of an intervention program designed to enhance teachers' knowledge of geometry and their knowledge of research on student cognition in geometry. Forty-nine middle-grade (4-8) teachers participated in a 4-week program consisting of a content course in geometry and a research seminar on van Hiele theory. The pretest and posttest results showed significant gains in content knowledge and in van Hiele level. The analysis of a lesson-plan task revealed a significant shift in goals and expectations to the next higher van Hiele level. Follow-up observations of 8 teachers found marked changes in what was taught, how it was taught, and the characteristics teachers displayed. Teachers attributed these changes to increased geometrical content knowledge and research-based knowledge of student cognition.
Editorial
Authors: Sowder Judith T.
Published: August 2020

Each year at the NCTM Annual Meeting, the JRME Editorial Panel holds a session for those interested in publishing in the JRME. Many prospective authors cannot attend, and so I decided to share the information from the Minneapolis session with JRME readers.
Teaching Realistic Mathematical Modeling in the Elementary School: A Teaching Experiment With Fifth Graders
Authors: Verschaffel Lieven , De Corte Erik
Published: August 2020

Recent research has convincingly documented elementary school children's tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10-and 11-year-old pupils—compared to the results in two control classes—provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.
Open and Closed Mathematics: Student Experiences and Understandings
Authors: Boaler Jo
Published: August 2020

This paper reports on 3-year case studies of 2 schools with alternative mathematical teaching approaches. One school used a traditional, textbook approach; the other used open-ended activities at all times. Using various forms of case study data, including observations, questionnaires, interviews, and quantitative assessments, I will show the ways in which the 2 approaches encouraged different forms of knowledge. Students who followed a traditional approach developed a procedural knowledge that was of limited use to them in unfamiliar situations. Students who learned mathematics in an open, project-based environment developed a conceptual understanding that provided them with advantages in a range of assessments and situations. The project students had been “apprenticed” into a system of thinking and using mathematics that helped them in both school and nonschool settings.
An Investigation of African American Students' Mathematical Problem Solving
Authors: Malloy Carol E. , Gail Jones M.
Published: August 2020

In this study we examined the problem-solving characteristics, strategy selection and use, and verification actions of 24 African American 8th-grade students. Students participated in individual, talk-aloud problem-solving sessions and were interviewed about their problem solutions and attitudes about learning mathematics. Students displayed approaches attributed to African American learners in the literature, regularly using holistic rather than analytic reasoning; their display of confidence and high self-esteem did not appear to be related to success. Students' problem-solving actions matched previously reported characteristics of good mathematical problem solvers: successful use of strategies, flexibility in approach, use of verification actions, and ability to deal with irrelevant detail. Success was highly correlated with strategy selection and use and moderately correlated with verification actions.
Kindergarten Students' Organization of Counting in Joint Counting Tasks and the Emergence of Cooperation
Authors: Wiegel Heide G.
Published: August 2020

The purpose of the study was to investigate and document possibilities for and manifestations of collaborative work with pairs of kindergarten students while they worked on tasks designed to promote early number development. Ten students, paired to be compatible with respect to their development of counting, were taught weekly for a period of 4 months. The students were addressed as pairs and provided with only one set of counting materials. The students generated 4 strategies to organize their counting: counting parts side by side, counting all at the same time, taking turns, and counting cooperatively. Cooperative counting was defined as a counting episode in which the counting acts of both students merged into a single activity with the partners working toward a common goal. Three themes emerged from the analysis of the cooperative solutions: (a) the relation between cooperation and the specific requirements of a counting task, (b) the relation between the ability to work cooperatively and the students' development of counting, and (c) the students' need to complement a cooperative venture with a solution of their own.
The Developmental Nature of Ability to Solve One-Step Word Problems
Authors: Christou Constantinos , Philippou George
Published: August 2020

Recently many researchers have focused on analyzing the structure of one-step word problems and the solution strategies employed by pupils. We have investigated the effect of mental schemes corresponding to additive and multiplicative situations in the process of interpreting and solving problems. The relative difficulties of problems classified according to their situations is considered through a written test administered to pupils in Grades 2, 3, and 4. The results seem to support the assumption that there is a developmental pattern in pupils' thinking, depending on the problem situation.
Students' Spatial Structuring of 2D Arrays of Squares
Authors: Battista Michael T. , Clements Douglas H. , Arnoff Judy , Battista Kathryn , Auken Borrow Caroline Van
Published: August 2020

We define spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. It is an essential mental process underlying students' quantitative dealings with spatial situations. In this article, we examine in detail students' structuring and enumeration of 2-dimensional (2D) rectangular arrays of squares. Our research indicates that many students do not “see”the row-by-column structure we assume in such arrays. We describe the various levels of sophistication in students' structuring of these arrays and elaborate the nature of the mental process of structuring.
Identification of Some Components of Computer Programming Aptitude
Authors: Alspaugh Carol Ann
Published: August 2020

This study was designed to determine what student characteristics seem to influence proficiency in symbolic and algebraic computer programming. 50 students in a beginning university computer programming course were administered the Thurstone Temperament Schedule, IBM Programmer Aptitude Test, and the Watson-Glaser Critical Thinking Appraisal. These test scores, along with SCAT scores and a measure of the students' mathematical background, were correlated with measures of student programming proficiency. Those students who possessed a mathematical background of approximately 2 college calculus classes, who scored low on “impulsiveness” and “sociability,” and who scored high on “reflectiveness,” appeared to have “aptitude” for computer programming.
Editorial Comment
Authors: Johnson David C.
Published: August 2020

The Editor has received correspondence in which recommendations were made for improving JRME—i.e., how the Journal might provide a greater service to the mathematics education community. Two of these letters, Bright and Romberg, each with an Editor's reply, appear in this issue. In each letter a rationale is presented for the inclusion of additional “types” of articles. Such letters are appreciated and readers are encouraged to communicate reactions and suggestions for improvement to the Editor and/or Editorial Board members.
The Consistency of Strategies in the Learning of Mathematical Structures
Authors: Branca Nicholas A. , Kilpatrick Jeremy
Published: August 2020

To validate and extend the results of previous research by Dienes and Jeeves, 100 adolescent girls were given 3 games to learn in individual interviews. 2 games had the structure of a Klein group; the third had a network structure. Subjects' retrospective evaluations of the group tasks conformed to the relative frequencies and hierarchical order found by Dienes and Jeeves, but evaluations were not related to strategy scores derived from the sequence of moves. Evaluations and performance showed some consistency across tasks, but strategies tended not to be consistent, probably owing to inadequacies in the strategy scoring system.
The Effects of Teaching Euclidean Geometry via Transformations on Student Achievement and Attitudes in Tenth-Grade Geometry
Authors: Usiskin Zalman P.
Published: August 2020

Materials were written for an entire year's course in geometry in which transformations were used to develop the concepts of congruence, similarity, and symmetry, as well as being a vehicle for proof. This paper presents a study involving 413 students using these materials and 483 control students, comparing performance on standard geometry content and attitudes. Preliminary studies were done on perceptual, arithmetic, and algebraic skills. Student comprehension of transformation-related concepts, the problem of implementation, and attitudes of teachers were informally studied. While the data indicated that the experimental Ss could learn the material, few differences were detected with the measures employed--a significant difference (p<.01) in favor of the C group was found on the posttest of standard geometry content. Pre-(September) and posttest (June) student attitude data indicated a change in mean score, less positive, for both E and C groups (p<.01 for the C group). Attitude differences between E and C were not significant. Informal feedback from a teacher questionnaire indicated a favorable reaction to the experimental materials.
The Effect of Organizers and Knowledge of Behavioral Objectives on Learning a Mathematical Concept
Authors: Peterson John C. , Thomas H. Laverne , Lovett C. J. , Bright George W.
Published: August 2020

This paper reports the findings of three independent replications of Peterson's (1971) study on the effect of organizers and the knowledge of a behavioral objective on the learning and retention of a mathematical concept. The purpose of these studies was to test the hypothesis that the learning and retention of a mathematical concept can be facilitated by providing the learner with an advance organizer (AO), a post organizer (PO), or knowledge of the behavioral objective (KBO).
Research on Mathematics Education (K—12) Reported in 1972
Authors: Suydam Marilyn N. , Weaver J. F.
Published: August 2020

An annual annotated listing of research pertaining to elementary and secondary school mathematics.
Editorial: Manuscript Review as Scholarly Work
Authors: Heid M. Kathleen , Zbiek Rose Mary
Published: August 2020

The tempting answer at first glance is to make a decision about whether the piece should be published. That answer is too simple. Peer review of scholarly papers serves a much more important purpose. Reviews are a way for the field to monitor its progress and refine its products, and strong reviews are themselves scholarly work. In this editorial, we reflect on the effects the hundreds of reviews we have received over the past 2 years have had both on decisions made and on the journeys of good manuscripts to published articles.
Brief Report: Teaching Practices Associated With the Authoritative Proof Scheme
Authors: Harel Guershon , Rabin Jeffrey M.
Published: August 2020

This Brief Report summarizes case studies of the classroom teaching practices of 2 algebra teachers. The data consist of videotaped classroom observations during 1 academic year. The study identifies and characterizes specific teaching practices that establish the norm that the teacher is the sole arbiter of mathematical correctness in the classroom. The authors suggest that these practices are likely to promote the development of the authoritative proof scheme in students. The results can provide a basis for future research investigating the prevalence of these teaching practices and their impact on student learning, and they can be used as parameters to investigate teacher change.
Mathematics Skills of 17-Year-Olds in the United States: 1978 to 2004
Authors: Kloosterman Peter
Published: August 2020

Overall scale scores from the National Assessment of Educational Progress (NAEP) indicate that there was only minimal improvement in the mathematics performance of high school students between 1978 and 2004. Using recently released data from the Long-Term Trend (LTT) NAEP, this study describes the content covered on the LTT NAEP and the performance of 17-year-old students on that content. In addition, it demonstrates that although overall gains in performance were small, there were areas within mathematics in which performance improved substantially and others in which students in 2004 did not do as well as their counterparts of the 1970s and 1980s. Specifically, performance on 3 items involving multiplication of whole numbers by fractions deteriorated but performance improved on most tasks involving percents and geometry. Performance was stable on most items assessing algebraic reasoning and logical reasoning and was stable or improved modestly on items assessing estimation, interpretation of tables and graphs, and understanding of integers.
Problems Dealing With Unknown Quantities and Two Different Levels of Representing Unknowns
Authors: Filloy Eugenio , Rojano Teresa , Solares Armando
Published: August 2020

This article centers on studying the progress of algebraic syntax once students have surmounted initial obstacles found in the transition toward symbolic algebra. It specifically analyzes students' progress concerning the operation on the unknown, when the latter is represented by an expression that involves a 2nd unknown. In curricular terms, students face such a situation when they learn the classic methods to solve systems involving 2 linear equations with 2 unknowns. Results from this empirical study show that extending previous algebraic syntax to the 2nd tier of representation of the unknown requires fundamental reconceptualizations such as that of algebraic equalization and that of mathematical unknown.
Nonstandard Student Conceptions About Infinitesimals
Authors: Ely Robert
Published: August 2020

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.
The Impact of Prior Math Achievement on the Relationship Between HS Math Curricula and Postsecondary Math Performance, Course-Taking, and Persistence
Authors: Post Thomas R. , Medhanie Amanuel , Harwell Michael , Norman Ke Wu , Dupuis Danielle N. , Muchlinski Thomas , Andersen Edwin , Monson Debra
Published: August 2020

This retrospective study examined the impact of prior mathematics achievement on the relationship between high school mathematics curricula and student postsecondary mathematics performance. The sample (N = 4,144 from 266 high schools) was partitioned into 3 strata by ACT mathematics scores. Students completing 3 or more years of a commercially developed curriculum, the University of Chicago School Mathematics Project curriculum, or National Science Foundation-funded curriculum comprised the sample. Of interest were comparisons of the difficulty level and grade in their initial and subsequent college mathematics courses, and the number of mathematics courses completed over 8 semesters of college work. In general, high school curriculum was not differentially related to the pattern of mathematics grades that students earned over time or to the difficulty levels of the students' mathematics course-taking patterns. There also was no relationship between high school curricula and the number of college mathematics courses completed.
Are There Any Places That Students Use Their Heads? Principles of High-Quality Japanese Mathematics Instruction
Authors: Corey Douglas L. , Peterson Blake E. , Lewis Benjamin Merrill , Bukarau Jared
Published: August 2020

Previous research gives evidence that Japanese mathematics teachers “may have a more detailed and widely shared theory about how to teach effectively” when compared to their U.S. counterparts (Jacobs & Morita, 2002). This study explores the conceptions and cultural scripts of a group of Japanese mathematics teachers by analyzing the conversations between cooperating teachers and student teachers. It describes 6 principles of high-quality instruction that arose in at least half the conversations we analyzed. Each of these principles is examined in detail. Finally, some advantages of having a strong, shared conception of high-quality instruction and focusing on widely applicable instructional principles are presented.
On Mathematicians' Proof Skimming: A Reply to Inglis and Alcock
Authors: Inglis Matthew , Alcock Lara
Published: August 2020

We recently reported a study in which undergraduate students and research mathematicians were asked to read and validate purported proofs (Inglis & Alcock, 2012). In our eye-movement data, we found no evidence of the initial skimming strategy hypothesized by Weber (2008). Weber and Mejía-Ramos (2013) argued that this was due to a flawed analysis of eye-movement data and that a more fine-grained analysis led to the opposite conclusion. Here we demonstrate that this is not the case, and show that their analysis is based on an invalid assumption.
High Versus Low Geometry Achievers in the NLSMA Y-Population
Authors: Sowder Larry
Published: August 2020

Discriminant analyses with several NLSMA measures were run on 2 groups of geometry students selected from the NLSMA Y-population, one designated as high geometry achievers (n = 310), the other as low geometry achievers (n = 310). The analyses correctly classified from 75 to 95% of the groups, depending on the measures chosen. Best discriminators, in the sense of entering the analyses early, were 4 achievement measures, 3 cognitive measures, and a debilitating-anxiety scale.
The Correlation of Selected Nonmathematical Measures with Mathematics Achievement
Authors: Cathcart W. George
Published: August 2020

Most studies that have investigated achievement in mathematics have found that it is correlated with numerous nonmathematical variables. This study examined the relative contribution of intelligence, conservation, socioeconomic status, age, listening ability, vocabulary level, and sex as correlates of mathematics achievement with second- and third-grade pupils. Listening ability and vocabulary levels were significant variables. Intelligence was a significant variable for Grade 3 but not for Grade 2 pupils. Sex and the ability to conserve were not significant at either grade level.
An Analysis Of Content And Task Dimensions Of Mathematics Items Designed To Measure Level Of Concept Attainment
Authors: Harris Margaret L. , Romberg Thomas A.
Published: August 2020

Content and task dimensions of mathematics items, developed to measure concept attainment using a crossed design involving 30 concepts and 12 tasks, were studied using factor-analytic techniques. Conventional factor analyses were performed separately for boys (N = 195) and girls (N = 196) for concept scores and for task scores. Also, 3-mode factor analyses were performed.

The main conclusions drawn from the results of the conventional factor analyses are that all 30 of the concepts are measures of a single functional relationship existing among the concepts, and that all 12 tasks are measures of a single underlying ability or latent trait. The 3-mode results indicate that there are no important concept-task interactions for the idealized persons, thus it is reasonable to regard the concepts and the tasks as being 2 independent modes.
Negative Instances in Mathematical Concept Acquisition: Transfer Effects between the Concepts of Commutativity and Associativity
Authors: Shumway Richard J.
Published: August 2020

The effects of negative instances on the acquisition of the mathematical concepts of commutativity and associativity were examined. 2 treatment levels for commutativity (positive instances only, or positive and negative instances) and the same treatment levels for associativity were crossed to form a 2 × 2 factorial design with 16 ninth-grade subjects per cell. Criterion variables were numbers of correct responses, stimulus intervals, and post-feedback intervals. The results favored the treatments containing both positive and negative instances and supported the hypothesis that the effects for negative instances transferred from one concept to another. Near-significant time differences weaken effects observed for negative instances.
Logical Reasoning Ability and Teacher Verbal Behavior within the Mathematics Classroom
Authors: Gregory John W. , Osborne Alan R.
Published: August 2020

The purpose of this study was to investigate the relationship between the frequency of seventh-grade mathematics teachers' use of the language of conditional logic and their students' growth in conditional reasoning ability. Children of twenty classrooms were tested in September and again at the end of January using the Cornell Conditional Reasoning Test-Form X. Frequency of teacher use of conditional logic language was determined by analyzing the transcriptions of five 42-minute lessons for each teacher. Adjusted posttest performance on the conditional reasoning test was correlated significantly with these frequencies exhibited by the teachers.
Criteria for Assessing The Effectiveness of Teachers of Secondary School Mathematics
Authors: Robitaille David F.
Published: August 2020

The 167 teachers of secondary mathematics in an urban school system were rated by each of 4 judges. Those scoring in the upper and lower 20% formed the sample for the study. Discriminant Analysis was used to compare aspects of the teachers' backgrounds and preparation. Results showed that more than 95% of the teachers were correctly classified as effective or ineffective on the basis of these measures. Classroom observations showed significant differences in the teaching behavior of the 2 groups as indicated by the extent of student participation.
The Effect of Mathematics Instruction Using Manipulative Models on Attitude and Achievement of Prospective Teachers
Authors: Warkentin Gary
Published: August 2020

Students in several sections of a theory of arithmetic course at Arizona State University were given instruction with a laboratory approach based on structural manipulative models. Their attitudes and achievement were compared with those of students receiving instruction with lectures and a textbook. The students who used the manipulative models had more favorable attitudes toward mathematics than the other group. However, they were not able to cover as much content and did not have as much measured achievement.
A Comparison of Achievement and Attitudes of Students Using Conventional or Calculator-Based Algorithms for Operations on Positive Rational Numbers in Ninth-Grade General Mathematics
Authors: Gaslin William L.
Published: August 2020

This study investigated 3 strategies for performing operations on rational numbers in ninth-grade general mathematics. Performed at 2 Minneapolis area schools using samples of 48 and 53 students, operations on rational numbers were taught by (a) the conventional algorithmic approach, (b) the conventional algorithmic approach with calculators, and (c) an alternative algorithm approach through which students changed each fractional operand to a decimal on the calculator. The major variable examined was achievement (transfer of skill to seminovel situations and retention), and another variable considered was attitude. The results showed that the third treatment group performed significantly better (p <.01p<.01) than the other 2 treatment groups and retained that skill to a significantly greater extent (p<.01p<.01) after 2 weeks studying neutral material. The third treatment group also performed better on ordering rational numbers, estimating fractional values, and working with rational numbers involving more than 1 operand or more than 2 operations than the other 2 groups. No differences in attitude were found.
Comparison of Content Structure and Cognitive Structure in High School Students' Learning of Probability
Authors: Geeslin William E. , Shavelson Richard J.
Published: August 2020

The learning of the structure of elementary probability was investigated. High school students (N = 34) were assigned randomly to read either probability (experimental group) or negative number bases (control group). The structure of the probability material was represented by a directed graph that produced an interpretable map of the content structure. Cognitive structures were examined using a word association technique. The experimental students learned and retained a significant portion of the probability structure but the control students did not. A comparison of word association, achievement, and attitude data indicated that the learning of structure developed somewhat independently from learning to solve problems.
Interaction of Spatial Visualization and General Reasoning Abilities with Instructional Treatment in Quadratic Inequalities: A Follow-up Study
Authors: Webb Leland F. , Carry L. Ray
Published: August 2020

This study was the second 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. The purpose of the study was to replicate Carry's (1968) study and to extend the theoretical basis of the problem by analyzing the instructional treatments and criterion variables using a multiprocess theoretical model due to Melton (1967). 249 Ss were randomly assigned to one of two treatment groups: Group A, a verbal-symbolic numeric treatment; and Group G, a verbal-spatial-numeric treatment. The hypotheses were that spatial visualization would predict transfer from the analytic group and that general reasoning would predict transfer from the graphical group. When the results were analyzed using multiple linear regression, no interactions were found between any aptitude variable and the treatments
Systematic Errors in the Four Vertical Algorithms in Normal and Handicapped Populations
Authors: Cox L. S.
Published: August 2020

In a two-year study, frequencies and descriptions of systematic errors in four algorithms in arithmetic were studied in upper-middle income regular and special education classrooms involving 744 children. Children were screened for adequate knowledge of basic facts and for receiving prior instruction on the computational processes. Systematic errors contained a recurring incorrect computational process and were differentiated from careless errors and random errors. Errors were studied within levels of computational skill for each algorithm. Results showed that 5-6% of the children made systematic errors in the addition, multiplication, and division algorithms. The figure was 13% for the subtraction algorithm. One year later 23% of the children were making either the identical systematic error or another systematic error.
Further Work on Verbalization of Discoveries
Authors: Sowder Larry
Published: August 2020

Two studies that sought further information on the question of whether verbalizing a discovered generalization influences one's subsequent ability to use the generalization are reported. Verbalizing methods considered were writing, reading, and no-verbalizing. A total of 140 college subjects were given 3 shortcuts to discover. Proportions of correct responses on immediate and 1-week retention tests made up of items amenable to the shortcuts were compared. Only isolated significant differences appeared. The bulk of the research indicates that how or whether one verbalizes a discovery does not affect subsequent ability to use the discovery.
The Development of the Concept of a Standard Unit of Measure in Young Children
Authors: Carpenter Thomas P. , Lewis Ruth
Published: August 2020

The purpose of this study was to find the degree to which young children recognize the importance of maintaining a standard unit of measure in a measurement operation and how they acquire the knowledge that the number of units measured is inversely related to the size of the unit.
The Effect of Instructional Gaming on Absenteeism: The First Step
Authors: Allen Layman E. , Main Dana B.
Published: August 2020

For Bertrand Russell, the essential features of the good life are affective and cognitive: The good life is one inspired by love and guided by knowledge. We submit that these dimensions are also at the core of good learning, which is a central part of human life. This study focuses on the affective dimension as it is influenced by a learning environment organized a round instructional gaming.
Retroactive Interference of Similar Methods to Teach Translation of Base Systems in Mathematics
Authors: Barszcz Edward L. , Gentile J. Ronald
Published: August 2020

It is commonly assumed that a person who can solve a mathematical problem via alternative methods understands the tested concept better than someone who can solve the problem in only one way. There are good psychological reasons for making this assumption, not the least of which are data from conservation studies stimulated by Piaget's developmental theory (Flavell, 1963; Piaget, 1970, pp. 703-732). This line of evidence shows, in general, that a child becomes increasingly capable of dealing with more than one dimension (e.g., height and width) or logical argument (e.g., identity, reversibility, and compensation of relations) at the same time in conservation problems. Furthermore, ability to conserve using these cogni tive functions is assumed to be necessary for the kind of reasoning required in mathematical exercises. Thus it seems entirely reasonable for a teacher to be pleased with the progress of a student who can invent new solutions to mathematical problems, while at the same time being concerned about a student who can use only one method.
Verbal-Nonverbal Conservation and Primary Mathematics
Authors: Riggs F. T. , Nelson L. D.
Published: August 2020

The purposes of the present study were (a) to measure length conservation by using a verbal and nonverbal test of conservation, and (b) to compare the verbal and nonverbal results as predictors of mathematics achievement. The sample consisted of 100 children ranging in ages from 74 to 94 months. One half of the sample was given a nonverbal conservation test. The other half of the sample was given a verbal conservation test. The total sample was given a mathematics achievement test and an intelligence test. Significant correlations were found between mathematics achievement and verbal conservation scores, nonverbal conservation scores, and intelligence scores. By a partial correlation procedure it was found that verbal conservation scores are superior to nonverbal and intelligence scores as predictors of mathematics achievement. The results indicated that a nonverbal length conservation test enabled more first-grade children to express their knowledge of conservation than did the verbal length conservation test.
A Forum for Researchers
Authors: Kenneth J Travers
Published: August 2020

In 1964, the International Association for the Evaluation of Educational Achievement (IEA) surveyed mathematics achievement in the schools of 12 countries, including the United States (Husén, 1967). Subsequently, surveys were conducted in six other school subjects involving some 23 nations(e.g., Comber & Keeves, 1973; Walker, 1976). In the United States, a special issue of J RM £ was devoted to the IEA mathematics survey (Volume 2, November 1971), and articles on the survey were written for NCTM publications (Willoughby, 1968) as well as for the press. Some countries, Australia for example, prepared national reports on the findings of the survey (Keeves & Radford, 1969). Recently, critiques of the IEA studies (Coleman, 1975) and (Freudenthal, 1975) have generated considerable discussion in some quarters.
The Effects of Instruction in Sentential Logic on Selected Abilities of Second- and Third-Grade Children
Authors: McGinty Robert L.
Published: August 2020

The main purpose of the study was to determine the effect of different types of instruction on the logical abilities of children. Sixteen classes, eight Grade 2 and eight Grade 3, were randomly assigned to four treatments (three experimental and one control)--two classes per treatment at each grade level. Class size was 25. The instruction in logic was provided once a week, approximately 50 minutes, for 13 weeks. The analysis of variance from three different posttests indicated that instruction has positive effects on certain logical abilities (p = .05). It would appear that children could profit by such instruction as early as the second grade.