bluuebunny/crossref_metadata_2025 huggingface dataset:
Brief Reports: Multisensory Information Matching Ability and Mathematics Learning
Authors: Sawada Daiyo
Published: August 2020

The multisensory approach to learning has found wide acceptance in elementary school mathematics based, in part, on the writings of Bruner, Piaget, and Dienes. Basic knowledge about what kind of sensory experiences to provide for which pupils in what sequence has been the focus of considerable inquiry. A study by Sawada and Jarman (1978) identified and supported the usefulness of a particular paradigm for assessing sensory integration ability within and across sense modalities. The study reported here is the second of a series of studies using this paradigm.
The Van Hiele Levels of Geometric Thought in Undergraduate Preservice Teachers
Authors: Mayberry Joanne
Published: August 2020

This study operationally defined and studied the five learning levels hypothesized by P. M. van Hiele and D. van Hiele-Geldof. Tasks were designed for the first four levels using seven common geometric concepts and revised after review by 11 mathematics educators. Audiotaped interviews were given to 19 preservice elementary teachers, 13 of whom had studied high school geometry. Guttman's scalogram analysis showed that the tasks representing the levels formed a hierarchy. A test of consensus revealed that students were on different levels for different concepts. Responses, in general, showed that the typical student in the study was not ready for a formal deductive geometry course.
The Constructivist Researcher as Teacher and Model Builder
Authors: Cobb Paul , Steffe Leslie P.
Published: August 2020

The constructivist teaching experiment is used in formulating explanations of children's mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations of theoretical constructs--that represent our understanding of children's mathematical realities. However, the models must be distinguished from what might go on in children's heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children's construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher's understanding of the children.
Conceptual Understanding in Solving Two-Step Word Problems with a Ratio
Authors: Quintero Ana H.
Published: August 2020

When solving multistep word problems, children may have difficulty in either understanding concepts and relationships or organizing a solution method. Both factors of problem difficulty were studied by means of interviews with 36 children (ages 9 to 14) who solved problems having the same structure but varying in context and in the numbers used. The child's representation of a problem was probed by tasks involving repetition of the problem statement and selection of drawings. The meaning of concepts and relationships was a major source of difficulty, and levels were identified in understanding the concept of ratio. The solution method was also a source of difficulty, but its nature was less clear. The results underline the importance of a careful analysis of problem types in studying sources of difficulty.
Brief Report: The Development of the Number-Word Sequence in the Counting of Three-Year-Olds
Authors: Baroody Arthur J. , Price Joyce
Published: August 2020

Even before formal schooling begins, there are significant developments in children's counting ability (e.g., Baroody & Ginsburg, 1982; Fuson & Hall, 1983; Gelman & Gallistel, 1978; Ginsburg, 1982; Schaeffer, Eggleston, & Scott, 1974). The child begins to learn the number-word sequence (rote counting) and then uses this sequence to count sets (object counting). Recently, some resea rchers (e.g., Briars & Siegler, 1981; Fuson & Hall, 1983; Wagner & Walters, 1982) have questioned Gelman and Gallistel's (1978) assertion that these two counting abilities are governed by (implicit) principles.
Mathematical Thinking: The Struggle for Meaning
Authors: Burton Leone
Published: August 2020

This paper argues that mathematical thinking is not thinking about the subject matter of mathematics but a style of thinking that is a function of panicular operations, processes, and dynamics recognizably mathematical. It further suggests that because mathematical thinking becomes confused with thinking about mathematics, there has been little success in separating process from content in the classroom presentation of the subject. A descriptive model of mathematical thinking is presented and then used to provide a practical response to the questions, Can mathematical thinking be taught? In what ways? The reacher is encouraged to recognize both what constitutes mathematical thinking, whether in the mathematics class or some other, and what conditions are necessary to foster it.
Editorial
Authors: Kilpatrick Jeremy
Published: August 2020

In the recent grants competitions conducted by the National Science Foundation and by the National Institute of Education, a substantial number of proposals were submitted by researchers in mathematics education. Among the peer reviewers, too, were such researchers. In each case when the awards were announced, however, few or no researchers in mathematics education were named among the recipients. Several conjectures might be advanced to account for this underrepresentation.
Review: What Does the Textbook Say?
Authors: Radatz Hendrik
Published: August 2020

Of all the aids to teaching and learning in the history of schooling, the textbook is the most venerable and the most disputed. In the school textbook, we meet fundamental problems of didactics. The textbook can aid the reacher in making decisions about instructional content and about pedagogical intention and methodology. The teacher may adopt ideas straight from the book or may modify, or possibly reject, them on the basis of his or her experience, knowledge, or personal concept of “good” mathematics instruction. In any case, the textbook presents essential guidel ines and has a considerable impact on the teacher's activities. At the same time, the textbook is expected to serve as a working manual for the pupil. It ought to motivate pupils, give them a chance to use and experiment with mathematics, and allow them to work out mathematical concepts or problems on their own. Most modern mathematics textbooks indeed claim to be pupils' books, but a closer view shows that they nre really addressed to the reacher. These and other problems concerning how to conceive of, and work with, school mathematics textbooks have seldom been discussed in teacher training programs.
Is It Farfetched That Some of Us Remember Our Arithmetic Facts?
Authors: Ashcraft Mark H.
Published: August 2020

Baroody (1985) suggests repeatedly that rules and procedures can provide the answers to the basic number combinations in simple arithmetic. My research with older children and adults indicates they do not, at least in the general case. After correcting Baroody's description of my network retrieval model, I indicate five weaknesses in his argument. The debate over fact retrieval versus rule- and procedure-generated answers is an empirical one. I invite Baroody to report data that buttress his position.
Computer-Video Instruction in Mathematics: Field Test of an Interactive Approach
Authors: Henderson Ronald W. , Landesman Edward M. , Kachuck Iris
Published: August 2020

Field tests were conducted to evaluate theory-based computer-video instructional modules designed to teach or review concepts and operations for factors, prime numbers, and fractions. Task analysis and demand specification procedures were used to sequence instructional objectives within and across topics. Cognitive social learning theory provided a framework for the incorporation of attentional, retentional, and motivational features into the video sequences. Field testing of the factors and prime numbers modules with ninth graders from five mathematics classes revealed statistically and practically significant effects on achievement and on the attribution of outcomes to effort. More limited testing of the fractions modules with a small group of high school students who had failed a basic skills competency test required for graduation also provided evidence supporting the efficacy of the approach.
A Beginning Teacher's View of Problem Solving
Authors: Cooney Thomas J.
Published: August 2020

This study focused on a mathematics teacher and examined his beliefs about problem solving while he was finishing his preservice training and during his first 3 months of teaching. His beliefs were revealed through interviews in which he responded to various types of open-ended situations, through his reaction to a report written about his beliefs, and through observations of his teaching and subsequent interviews. Analyses revealed conflicts between his idealism and the reality of classroom practice, as his students were not always receptive to his problem-solving teaching strategy.
Parent Attitudes and Student Career Interests in Junior High School
Authors: Pedersen Katherine , Elmore Patricia , Bleyer Dorothy
Published: August 2020

Data were collected on the mathematics achievement, spatial visualization ability, student attitudes, sex, parent attitudes, and student career interests of 1982 seventh and eighth graders from 13 midwestern junior high schools. Parent attitudes and student career interests were significant predictors of mathematics achievement over and above the other variables. Sex was the only variable that did not make a unique contribution to the variance. Three canonical variates were identified between student attitudes and parent attitudes; three between student attitudes and student career interests; and one between student career interests and parent attitudes.
Editorial
Authors: Kilpatrick Jeremy
Published: August 2020

In January at a conference in Berkeley, California, on shaping the research base for science and mathematics education, Lauren Resnick noted that cognitive psychology has begun to amass substantial evidence to support the proposition that humans are builders, not recorders, of knowledge. She used the term constructivism to characterize that thesis, acknowledging at the same time that cognitive psychology has few strong statements to make about how learning occurs. In her opinion, cognitive psychology offers not so much theories as metaphors.
A Critique of Teacher Effectiveness Research in Mathematics Education
Authors: Confrey Jere
Published: August 2020

A dialogue between Socrates and a researcher who claims to know what constitutes effective teaching is used to critique the implicit assumptions held by researchers on teaching, highlighting their differences with construcrivist researchers. Teacher effectiveness research is criticized for its empiricism; its disregard of the interaction between content and instruction; and its conceptions of mathematics, teaching, and learning. Potential bridges for communication are identified from the two communities' shared interest in how students perceive different examples and explanations and how they might be intrinsically motivated to learn mathematics. In Socratic style, the dialogue ends by emphasizing that the question of what constitutes effective teaching remains unresolved.
Children's Inferencing Behavior
Authors: Behr Merlyn J. , Khoury Helen Adi
Published: August 2020

Third and fifth graders were given a sequence of number pairs and asked ro discover the function rule relating them, test the hypothesis, and generalize it to other instances. The fifth graders were better than the third graders on all performance criteria. Task difficulty depended on the funcuon. Some functions were easier when presented in a graphic mode, others in a symbolic mode. Whether inferences made use of abscissa-ordinate pairs, in contrast to a sequence of ordinate values, appeared to depend on both function and mode of presentation. Unlike adults, the children seemed ready to relinquish a hypothesis in the face of disconfirming evidence.
The Influence on Mathematics Test Scores, by Ethnicity and Sex, of Prior Achievement and High School Mathematics Courses
Authors: Jones Lyle V.
Published: August 2020

Data from a national sample of high school sophomores in 1980 from the High School and Beyond project show that senior-year mathematics test scores are highly dependent on the number of courses taken in mathematics, Algebra I or above. Within each level of course taking, senior test scores are predicted reasonably well from student socioeconomic status, sophomore-year verbal test scores, and sophomore-year mathematics test scores. The results support the theses that (a) whether black or white, female or male, sophomore students with similar levels of mathematics achievement may be expected to experience similar levels of improvement by taking additional mathematics courses and (b) the expected improvement is elevated for students with four or more credits of advanced mathematics or with three credits that include calculus.
Calculators and Instruction in Problem Solving in Grade 7
Authors: Szetela Walter , Super Doug
Published: August 2020

For an entire school year 24 seventh-grade classes were taught problem-solving strategies by teachers with special training. In 14 of the classes the instruction was supplemented by calculators. Each problem-solving group performed significantly (p<.05) better than a control group of 18 classes on two of five problem-solving tests. The calculator group scored significantly higher than the control group on attitude toward problem solving and as well as the other two groups on paper-and-pencil computation. Responses to a teacher questionnaire indicated a high degree of satisfaction with the program.
Editorial
Authors: Kilpatrick Jeremy
Published: August 2020

Since the founding of the JRME, the editors have, from time to time, sent the reviewers of a manuscript a copy of the editor's letter to the author concerning whether the manuscript was accepted, together with a copy of the comments that each reviewer provided for the author. This practice lets reviewers see how their judgments and comments compare with those of others. It serves both to educate and to calibrate. The novice reviewer lea rns something of the care and thoroughness needed to produce comments that are helpful. The maverick reviewer gets a better sense of the criteria that other reviewers are using in making their judgments.
Race, Sex, Socioeconomic Status, and Mathematics
Authors: Reyes Laurie Hart , Stanic George M. A.
Published: August 2020

Many black students, female students, and students of low socioeconomic status do not achieve up to their potential in mathematics. A model to explain group differences in performance is presented. Relationships are described among several groups of variables, including societal influences, school mathematics curricula, teacher attitudes, student attitudes and achievement-related behavior, classroom processes, and student achievement. Relevant research results are included, along with suggestions for future research to test the model.
Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research
Authors: Garfield Joan , Ahlgren Andrew
Published: August 2020

There is a growing movement to introduce elements of statistics and probability into the secondary and even the elementary school curriculum, as part of basic literacy in mathematics. Although many articles in the education literature recommend how to teach statistics better, there is little published research on how students actually learn statistics concepts. The experience of psychologists, educators, and statisticians alike is that a large proportion of students, even in college, do not understand many of the basic statistical concepts they have studied.
The Development of Children's Concepts of the Arithmetic Average
Authors: Strauss Sidney , Bichler Efraim
Published: August 2020

The study was conducted to (a) determine the development of children's understanding of seven properties of the arithmetic mean and (b) assess the effects of the material used in the testing (continuous, discontinuous) and the medium of presentation (story, concrete, and numerical). Twenty children were selected at each of the ages 8, 10, 12, and 14 years. Different developmental courses of the children's reasoning were found on some tasks measuring the properties of the average. No significant effects were found for the materials used or the medium of presentation. The findings are discussed in terms of their importance for developmental psychology and educational practice.
Brief Report: Sex Differences on New York State Regents Examinations: Support for the Differential Course-Taking Hypothesis
Authors: Smith Stuart E. , Walker William J.
Published: August 2020

In a comprehensive review of sex differences, Maccoby and Jacklin (1974) concluded that male students consistently outperformed female students on mathematics tests. Hyde (1981), using results from the pertinent studies reported by Maccoby and Jacklin, reported that although the differences consistently favored male students, the sizes of the differences were generally small.
The Ethnographic Research Tradition and Mathematics Education Research
Authors: Eisenhart Margaret A.
Published: August 2020

Although in theory ethnography has been put forward as a powerful naturalistic methodology, in practice it has rarely been used by educational researchers because of differences in assumptions, goals, and primary research questions. From my perspective as an educational anthropologist, I describe the research tradition of ethnography—its underlying assumptions, its heritage in holistic cultural anthropology, its goals and research questions, and the organization of its research methods. Throughout, I compare elements of this ethnographic tradition with more common educational research practices. In the final section, I discuss the advantages of improved communication for future research in both mathematics education and educational anthropology.
Affective Issues in Mathematical Problem Solving: Some Theoretical Considerations
Authors: McLeod Douglas B.
Published: August 2020

Mathematics students often report feelings of frustration or satisfaction when they work on nonroutine problems. These affective responses are an important factor in problem solving and deserve increased attention in research. Mandler's theory of emotion is suggested as a framework for investigating affective issues in problem solving. Several dimensions of the emotional states of problem solvers are specified, including the magnitude and direction of the emotions, their duration, and the students' level of awareness and level of control of the emotions. The implications of this framework for research on affective issues in problem solving are also discussed.
Brief Reports: Effective College Instruction in Basic Mathematics
Authors: Friedman Mordechai , Stomper Connie
Published: August 2020

One of the designs that researchers use to investigate effective teaching is the field-based process-product paradigm. Investigators who have conducted process-product studies at the precollege level have been successful in identifying behaviors of mathematics teachers that are associated with student achievement (Evertson, Emmer, & Brophy, 1980; Fey, 1970; Good & Grouws, 1977; McDonald & Elias, 1976). Good and Grouws's study of fourth-grade mathematics classrooms and Evertson, Emmer, and Brophy's study of secondary mathematics classrooms inspired us to extend the paradigm to the college classroom and to investigate the behavior of mathematics professors as they taught a basic course.
Review: Down with the Barricades
Authors: Lester Frank K.
Published: August 2020

Within the past 25 years or so a new discipline has evolved from the confluence of research in such disparate disciplines as computer science, linguistics, neuroscience, philosophy, and psychology.
Review: Goals, Goals, and More Goals
Authors: Lappan Glenda
Published: August 2020

In 1968 in the Netherlands, the Commission for Modernization of rhe Mathematics Curriculum started a new project—Wiskobas—whose goal was the improvement of mathematics education for ages 6–12 through instruction at reacher training colleges. Three Dimensions reviews the first 10 years of the Wiskobas effort. In particular, it focuses on an elaboration of the interplay between curriculum development at Wiskobas, mathematics education issues in Europe and the United States, and the setting of goals that guide and explain the view of mathematics teaching and learning that evolved at Wiskobas between 1968 and 1978.
Editorial
Authors: Stanic George M. A.
Published: August 2020

“Smart women like sports.” That, at least, is the opinion of J. E. Vader, a writer for Sports lllustrated. “ I'm convinced,” says Vader, “sports help make girls smart.” She is quick to add that “because virtually every American boy is programmed to play and watch sports, it's nonsense to state ‘smart men like sports.’ If you doubt this, simply attend any game in Yankee Stadium.” Vader even posits a causal connection between familiarity with sports and skill in mathematics.
Computer Experiences in Learning Composition of Functions
Authors: Ayers Thomas , Davis George , Dubinsky Ed , Lewin Philip
Published: August 2020

Computer instantiations of abstract concepts could assist mathematics education by providing concrete experiences and by inducing the mental process of construction that Piaget called reflective abstraction. Two classes of college students (n = 13) given 6 weeks of computer experiences to help induce reflective abstraction scored higher on a test of their understanding of functions and compositions than one class of students (n = 17) who were taught according to traditional methods. The comparison was based on questions intended to indicate whether reflective abstraction had taken place.
Review: Teaching Adapted to Thinking
Authors: Fuson Karen C.
Published: August 2020

Children's Mathematical Thinking is an excellent book for preservice or in-service preschool, primary, and special education teachers. Much recent research on children's mathematics learning is presented in an interesting and assimilable manner. Two features used throughout the book contribute to its effectiveness with its intended audience: Short case studies are described, and two general theories of learning, absorption theory and cognitive theory, are contrasted.
Brief Report: Hints: An Aid to Diagnosis in Mathematical Problem Solving
Authors: Trismen Donald A.
Published: August 2020

The diagnosis of student difficulties in mathematical problem solving requires more than a knowledge of the kinds of problems that are correctly and incorrectly solved. Meaningful diagnosis requires insight into the faulty cognitive processes that lead to incorrect results.
A Cognitive Approach to Meaningful Mathematics Instruction: Testing a Local Theory Using Decimal Numbers
Authors: Wearne Diana , Hiebert James
Published: August 2020

A theory is proposed for how students develop competence with the written symbols of decimal fractions. The theory views competence as the cumulative and sequential mastery of four separate cognitive processes in working with written symbols. The first two processes develop for the user the semantics of the symbol system, and the second two elaborate its syntax. Initial predictions from the theory were tested by instructing small groups of students for about 2 weeks. As predicted, (a) most students were able to acquire the first two processes, (b) these processes generated correct performance and transferred to novel tasks, and (c) it was more difficult for students to acquire the semantic-based processes once they had routinized syntactic processes. The theory and data are interpreted in the context of some persistent issues in mathematics learning.
Teachers' Pedagogical Content Knowledge of Students' Problem Solving in Elementary Arithmetic
Authors: Carpenter Thomas P. , Fennema Elizabeth , Peterson Penelope L. , Carey Deborah A.
Published: August 2020

This study investigated 40 first-grade teachers' pedagogical content knowledge of children's solutions of addition and subtraction word problems. Most teachers could identify many of the critical distinctions between problems and the primary strategies that children used to solve different kinds of problems. But this knowledge generally was not organized into a coherent network that related distinctions between problems, children's solutions, and problem difficulty. The teachers' knowledge of whether their own students could solve different problems was significantly correlated with student achievement.
Subtracting by Counting up: More Evidence
Authors: Fuson Karen C. , Willis Gordon B.
Published: August 2020

To find any subtraction facts that children did not know (e.g., 14–8), 10 teachers successfully taught first- and second-grade children of all ability levels to count up from the smaller to the larger number while keeping track with one-handed finger patterns. Second graders at and above grade level and first graders above grade level in mathematics counted up with finger patterns to solve large multidigit subtraction problems that required regrouping. Learning to subtract by counting up greatly improved children's performance on take-away, compare, and equalize word problems, did not interfere with their understanding of take-away problems, and enabled them to accelerate by as much as 3 years their learning of subtraction topics. Other evidence indicated that subtracting by counting up is natural for children if they learn that one meaning of subtraction (and of the – symbol) is a compare/equalize meaning.
A Technique for Studying Concept Formation in Mathematics
Authors: Collis K. F.
Published: August 2020

A card sorting task was designed to enable a teacher to gain an insight into an individual child's structuring of the mathematical principles involved in an experimental course in mathematics. A pilot experiment in which the task was performed on 6 occasions over a period of 7 months by 20 first year secondary school girls suggests that the card sorting task offers a useful tool to mathematics teachers for use either in the normal classroom evaluation program or in research projects involving experimental courses.
Further Interpret Allons of Achievement and Attitude Results
Authors: Coxford Arthur F.
Published: August 2020

The release of the International Study of Achievement in Mathematics evoked strong criticism in leading American news media of the mathematical education provided United States' students (see The New York Times, March 12, 1967; Time, March 17, 1967; Education U.S.A., March 13 and 20, 1967).
An Efficiency Reanalysis of the Results
Authors: Fletcher Harold J.
Published: August 2020

Easily one of the most ambitious educational projects ever undertaken, this TEA Study (Husén, 1967) not only assessed mathematical achievement internationnlly, but, more importantly, attempted to ascertain those variables related to that mathematical achievement. For their pioneering effort and generation of invaluable data, the committee that supervised this study must be gratefully commended. This paper will argue, however, that the published data analyses are almost totally inappropriate and, therefore, new analyses must be made if the results are to be interpreted in any meaningful way.
The Effect of Activity-Oriented Instruction upon Original Learning, Transfer, and Retention
Authors: Moody William B. , Abell Roberta , Bausell R. Barker
Published: August 2020

Activity-oriented, rote, and rote-with-word problems instruction was compared on degree of original learning, transfer, and retention elicited with a sample of 90 third-grade subjects with no previous experience in multiplication. There was no difference in original learning between instructional methods, although all Ss displayed original learning when contrasted to a control treatment. All treatments failed to produce transfer of original learning to word problems dealing with multiplication, and there was no differential retention.
Brief Reports: Cognitive Restructuring Ability, Teacher Guidance, and Perceptual Distracter Tasks: An Aptitude-Treatment Interaction Study
Authors: Cramer Kathleen A. , Post Thomas R. , Behr Merlyn J.
Published: August 2020

The aptitude-treatment interaction (ATI) study reported here explored the relationship between cognitive restructuring ability, as measured by the Group Embedded Figures Test (GEFT), and treatments varying in amounts of teacher guidance. It specifically investigated how these two variables affected performance on rational number tasks involving perceptual distracters.
Classroom Processes, Sex of Student, and Confidence in Learning Mathematics
Authors: Hart Laurie E.
Published: August 2020

The purpose of this study was to compare the classroom processes of girls and boys who differed in confidence in their ability to learn mathematics. The students were observed daily during their seventh-grade mathematics classes for 3 to 4 weeks. All 93 students observed (i.e., the target students) scored at or above the mean on a standardized test of mathematics achievement. Trained observers recorded characteristics of each verbal interaction between a target student and the teacher and the amount of time each target student spent on task in mathematics. Boys were involved in more public interactions with their teacher than girls were. High- and low-confidence students differed very little in their interactions with the teacher. Differences between boys and girls in their interactions with the teacher varied from classroom to classroom.
Images and Definitions for the Concept of Function
Authors: Vinner Shlomo , Dreyfus Tommy
Published: August 2020

Images held by 271 college students and 36 junior high school teachers for the concept of a mathematical function were compared to the definitions they gave for the concept. A questionaire was designed to exhibit the cognitive schemes for the function concept that become active in identification and construction problems and to make possible the comparison of these schemes with the definition. Many of the definitions and even more of the images were primitive among all but the mathematics majors and the teachers. Discrepancies between image and definition were frequent for all subjects who gave the Dirichlet-Bourbaki definition.
Learning of Geometric Concepts in a Logo Environment
Authors: Clements Douglas H. , Battista Michael T.
Published: August 2020

To investigate the effects of computer programming in Logo on specific geometric conceptualizations of primary grade children, 48 third graders were randomly assigned to either a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo environment. The children were then interviewed to ascertain their conceptualizations of angles, shapes, and motions. In both groups children's notions of angle and angle measure were multifaceted and included a number of misconceptions, although performance was uniformly higher for the Logo group. The Logo children were more aware than the control children of the components of geometric shapes and were more likely to conceptualize geometric objects in terms of the actions or procedures used to construct them.
Learning Fractions with Understanding: Building on Informal Knowledge
Authors: Mack Nancy K.
Published: August 2020

Eight sixth-grade students received individualized instruction on addition and subtraction of fractions in a one-to-one setting for six weeks. Instruction was designed to build on the student's informal knowledge of fractions. All students possessed a rich store of informal knowledge of fractions that was based on partitioning units and treating the parts as whole numbers. Students' informal knowledge was initially disconnected from their knowledge of fraction symbols and procedures. Students related fraction symbols and procedures to their informal knowledge in ways that were meaningful to them; however, knowledge of rote procedures frequently interfered with students' attempts to build on their informal knowledge.
Assessing Students' Theories of Success in Mathematics: Individual and Classroom Differences
Authors: Nicholls John G. , Cobb Paul , Wood Terry , Yackel Erna , Patashnick Michael
Published: August 2020

Scales assessing Task Orientation (where the goal is to gain understanding) and Ego Orientation (where the goal is to be superior to others) in mathematics were administered to six second-grade classes with a total of 102 students. Task Orientation was associated with the beliefs that success in mathematics is fostered by effort, attempts to make sense of things, and cooperation with one's peers. Ego Orientation was associated with the beliefs that success depends on superior ability and attempts to beat others. Perceived ability was not appreciably associated with these motivational orientations. One second-grade class that had experienced, for one year, mathematics instruction consistent with a constructivist view rated higher than traditional classes on Task Orientation, beliefs that success depends on attempts to understand, and cooperation with peers. This class was lower on the desire for superiority over one's friends (an aspect of Ego Orientation), desire to avoid work, and beliefs that success depends on superior ability. It did not differ from others on perceived ability. The results suggest the validity of the present scales and the value of employing a wider than usual range of indices when assessing the outcomes of mathematics teaching practices.
Prospective Elementary and Secondary Teachers' Understanding of Division
Authors: Ball Deborah Loewenberg
Published: August 2020

This article reports an analysis of 19 prospective elementary and secondary teachers' understanding of division. Interview questions probed the prospective teachers' understanding of division in three contexts. Although many of the teacher candidates could produce correct answers, several could not, and few were able to give mathematical explanations for the underlying principles and meanings. The prospective teachers' knowledge was generally fragmented, and each case of division was held as a separate bit of knowledge.
High School Calculus: Comparing the Content of Assignments and Tests
Authors: Nagy Philip , Traub Ross E. , MacRury Kathryn , Klaiman Roslyn
Published: August 2020

The purpose of this study was to compare the content of what is taught, as inferred from the content of seatwork and homework questions assigned, with what is tested in a high school calculus course. Data were obtained from a sample of 17 teachers, who provided lists of the homework and seatwork assigned each day and the criteria used to arrive at student grades for the course. Results showed differences in content coverage across teachers and differences in the overlap between content taught and content tested. The discussion focuses on methodological issues in the investigation of teacher grading practices, on the importance of such investigation, and on steps that might be taken to reduce discrepancies.
An Alternative Paradigm to Evaluate the Acquisition of the van Hiele Levels
Authors: Gutiérrez Angel , Jaime Adela , Fortuny José M.
Published: August 2020

This article presents an alternative way of analyzing the van Hiele level of students' geometrical reasoning. We evaluate the students' answers, taking into account the van Hiele level they reflect and their mathematical accuracy. This gives us a description of how accomplished the students are in applying the procedures associated with each of the van Hiele levels and allows us to determine the students' degree of acquisition of the van Hiele levels. In this way we obtain a clearer picture of the students' geometrical reasoning than with the traditional assignment of one van Hiele level to the learners. An example of the application of this method is provided: We describe a test that evaluated students' ability to reason in three-dimensional geometry, some responses of students (9 eighth-grade pupils and 41 future primary school teachers), and the classification of their responses using our method.
Factors and Influences on High School Students' Career Choices
Authors: Dick Thomas P. , Rallis Sharon F.
Published: August 2020

Women continue to be disproportionately underrepresented in science and engineering fields. A model for career choice is proposed that includes both the direct and indirect effects that socializers can play in determining career choices. A sample of 2213 high school seniors from nine schools in Rhode Island were surveyed about their academic and career choices and the perceived influences on those choices. Parents and teachers were perceived to be influences on career choice more often for students (both men and women) choosing careers in engineering and science than for those not choosing such careers. Pay was a more important factor in career choice for men in general, and genuine interest was a more important factor for women not choosing careers in engineering or science. However, these gender differences do not appear among students with extremely strong mathematics and science coursework backgrounds, even though there remains a marked disparity in the proportion of men to women planning careers in engineering or science. Teachers may play a particularly important role in influencing the career choice of some of these women. Equity of access and encouragement in mathematics and science is certainly a necessary, but insufficient, condition for improving the representation of women in science and engineering.
A Constructivist Alternative to the Representational View of Mind in Mathematics Education
Authors: Cobb Paul , Yackel Erna , Wood Terry
Published: August 2020

The representational view of mind in mathematics education is evidenced by theories that characterize learning as a process in which students modify their internal mental representations to construct mathematical relationships or structures that mirror those embodied in external instructional representations. It is argued that, psychologically, this view falls prey to the learning paradox, that, anthropologically, it fails to consider the social and cultural nature of mathematical activity and that, pedagogically, it leads to recommendations that are at odds with the espoused goal of encouraging learning with understanding. These difficulties are seen to arise from the dualism created between mathematics in students' heads and mathematics in their environment. An alternative view is then outlined and illustrated that attempts to transcend this dualism by treating mathematics as both an individual, constructive activity and as a communal, social practice. It is suggested that such an approach might make it possible to explain how students construct mathematical meanings and practices that, historically, took several thousand years to evolve without attributing to students the ability to peek around their internal representations and glimpse a mathematically prestructured environment. In addition, it is argued that this approach might offer a way to go beyond the traditional tripartite scheme of the teacher, the student, and mathematics that has traditionally guided reform efforts in mathematics education.
Building a Foundation
Authors: Steffe Leslie P.
Published: August 2020

At a time when “social constructivism,” “radical constructivism,” and “social constructionism” are as commonplace as “mathematics education,” a carefully articulated treatise like The Philosophy of Mathematics Education is timely if not overdue. I recommend it highly for anyone who may value the contributions of philosophers in mathematics education as well as for seasoned critics of constructivism. Paul Ernest offers a remarkably clear critique of the major schools of the philosophy of mathematics in his development of social constructivism. His philosophical position emerges from these critiques as well as from radical constructivism.
Learning to Teach Hard Mathematics: Do Novice Teachers and Their Instructors Give up Too Easily?
Authors: Borko Hilda , Eisenhart Margaret , Brown Catherine A. , Underhill Robert G. , Jones Doug , Agard Patricia C.
Published: August 2020

This article analyzes from several vantage points a classroom lesson in which a student teacher was unsuccessful in providing a conceptually based justification for the standard division-of-fractions algorithm. We attempt to understand why the lesson failed, what it reveals about learning to teach, and what the implications are for mathematics teacher education. We focus on (a) the student teacher's beliefs about good mathematics teaching, her knowledge related to division of fractions, and her beliefs about learning to teach; and (b) the treatment of division of fractions in the mathematics methods course she took. The student teacher's conception of good mathematics teaching included components compatible with current views of effective mathematics teaching. However, these beliefs are difficult to achieve without a stronger conceptual knowledge base and a greater commitment to use available resources and to engage in hard thinking than she possessed. Further, the mathematics methods course did not require the student teacher to reconsider her knowledge base, to confront the contradictions between her knowledge base and at least some of her beliefs, or to reassess her beliefs about how she would learn to teach. These findings suggest that mathematics teacher education programs should reconsider how they provide subject matter knowledge and opportunities to teach it, and whether and how they challenge student teachers' existing beliefs.
Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept
Authors: Even Ruhama
Published: August 2020

This article investigates teachers' subject-matter knowledge and its interrelations with pedagogical content knowledge in the context of teaching the concept of function. During the first phase of data collection, 152 prospective secondary teachers completed and open-ended questionnaire concerning their knowledge about function. In the second phase, an additional 10 prospective teachers were interviewed after responding to the questionnaire. The analysis shows that many of the subjects did not have a modern conception of function. Appreciation of the arbitrary nature of functions was missing, and very few could explain the importance and origin of the univalence requirement. This limited conception of function influenced the subjects' pedagogical thinking. Therefore, when describing functions for students, many used their limited concept image and tended not to employ modern terms. In addition, many chose to provide students with a rule to be followed without concern for understanding.
Sense Making and the Solution of Division Problems Involving Remainders: An Examination of Middle School Students' Solution Processes and Their Interpretations of Solutions
Authors: Silver Edward A. , Shapiro Lora J. , Deutsch Adam
Published: August 2020

In this study, about 200 middle school students solved an augmented-quotient division-with-remainders problem, and their solution processes and interpretations were examined. Based on earlier research, semantic-processing models were proposed to explain students' success or failure in solving division-with-remainder story problems on the basis of the presence or absence of an adequate interpretation provided by the solver after obtaining a numerical solution. In this study, students' solutions and their attempts and failures to “make sense” of their answers were analyzed for evidence that supported or refuted the hypothesized semantic-processing models. The results confirmed that the models provide a solid explanation of students' failure to solve division-with-remainder problems in school settings. More generally, the results indicated that student performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their mathematical thinking and reasoning.
Cognitive Models Underlying Students' Formulation of Simple Linear Equations
Authors: MacGregor Mollie , Stacey Kaye
Published: August 2020

Data are presented to show that errors in formulating algebraic equations are not primarily due to syntactic translation, as has been assumed in the literature. Furthermore, it is shown that the reversal error is common even when none of the previously published causes of the error is applicable. A new explanation is required and is proposed in this paper. An examination of students' errors leads us to suggest that students generally construct from the natural language statement a cognitive model of compared unequal quantities. They formulate equations by trying to represent the model directly or by drawing information from it. This hypothesis is supported by research on the comprehension of relationships by linguists, pyscholinguists and psychologists. Data were collected from 281 students in grade 9 in free response format and from 1048 students in grades 8, 9, and 10 who completed a multiple-choice item.
Prospective Elementary Teachers' Knowledge of Division
Authors: Simon Martin A.
Published: August 2020

Prospective teachers' knowledge of division was investigated through an open-response written instrument and through individual interviews. Problems were designed to focus on two aspects of understanding division: connectedness within and between procedural and conceptual knowledge and knowledge of units. Results indicated that the prospective teachers' conceptual knowledge was weak in a number of areas including the conceptual underpinnings of familiar algorithms, the relationship between partitive and quotitive division, the relationship between symbolic division and real-world problems, and identification of the units of quantities encountered in division computations. The research also characterized aspects of individual conceptual differences. The research results suggest conceptual areas of emphasis for the mathematical preparation of elementary teachers.
A Forum for Researchers: On Greeno's Environmental/Model View of Conceptual Domains: A Spatial/Geometric Perspective
Authors: Battista Michael T.
Published: August 2020

Greeno (1991) has recently proposed an alternative to the information processing view of conceptual understanding and reasoning in mathematics. In his “environmental/model” metaphor, a conceptual domain is thought of as an “environment with spatial properties” in which reasoning is accomplished by interacting with mental models (p. 211). Because spatial-like ideas play a prominent role in both the environmental metaphor and mental models, I will highlight the spatial aspects of Greeno's theory. I will then argue that relating the theory to research on spatial thinking (a) suggests some possible avenues for elaboration of the theory, (b) allows us tQ apply the theory to geometry learning in a way that provides a different perspective on some important problems in that field, and (c) suggests an explanation for the often-pondered connection between spatial and mathematical thinking.