bluuebunny/crossref_metadata_2025 huggingface dataset:
A Screening Procedure to Identify Children Having Difficulties in Arithmetic
Authors: Siders James A. , Siders Jane Z. , Wilson Rebecca M.
Published: August 2020

The study was designed to examine the potential of using repeated rate-of-response measures to identify students exhibiting difficulties in arithmetic. Thirty subjects were followed over 2 years. Pearson product-moment correlations were computed to determine the ability of rate-of-response measures to predict achievement in arithmetic. A follow-up evaluation indicated that the predictive accuracy of the rate of response was 97%.
Effects of Hand-Held Calculators in Precollege Mathematics Education: A Meta-Analysis
Authors: Hembree Ray , Dessart Donald J.
Published: August 2020

The findings of 79 research reports were integrated by meta-analysis to assess the effects of calculators on student achievement and attitude. Effect sizes were derived by the method invented by Glass and tested for consistency and significance with inferential statistics provided by Hedges. At all grades but Grade 4, a use of calculators in concert with traditional mathematics instruction apparently improves the average student's basic skills with paper and pencil, both in working exercises and in problem solving. Sustained calculator use in Grade 4 appears to hinder the development of basic skills in average students. Across all grade and ability levels, students using calculators possess a better attitude toward mathematics and an especially better self-concept in mathematics than students not using calculators.
Brief Reports: Abstraction Anxiety: A Factor of Mathematics Anxiety
Authors: Ferguson Ronald D.
Published: August 2020

The Mathematics Anxiety Rating Scale (MARS), designed by Richardson and Suinn (Suinn, Edie, Nicoleti, & Spenelli, 1972), has become a popular instrument in the investigation of mathematics anxiety. Several studies have contributed evidence supporting the validity and reliability of MARS (Brush, 1978; Morris, Kellaway, & Smith, 1978; Suinn et al., 1972).
Third Graders' Ability to Associate Foldout Shapes With Polyhedra
Authors: Bourgeois Roger D.
Published: August 2020

Thirty-four Grade 3 pupils were asked ro match 14 foldour shapes (nets) with 7 polyhedra. The pupils had more success associating nets containing triangular regions than nets composed entirely of rectangular regions. It was easier for the pupils to identify the nets of pyramids and of a triangular prism when the lateral faces were distributed around the base. After working with solids, the pupils demonstrated a marked improvement in their ability to recognize nets.
Research on Mathematics Education Reported in 1985
Authors: Suydam Marilyn N.
Published: August 2020

Every year since 1971, this introduction has said basically the same thing. This year will not be any different. Much as I have struggled to find something different to say, the fact remains that it is only an introduction: The references that follow are the reason for these pages.
Brief Report: Nonconservation of Multiplication and Division Involving Decimals
Authors: Greer Brian
Published: August 2020

It has been firmly established that a pupil's choice of operation for the solution of a single-operation word problem is often affected by the types of numbers in the problem (Bell, Fischbein, & Greer, 1984; Bell, Swan, & Taylor, 1981; Fischbein, Deri, Nello, & Marino, 1985; Hart, 1981). A more specific observation is that a choice of different operations occurs even when two problems differing only in the numbers involved are juxtaposed and attention is drawn to the similarity between them (Ekenstam ' Greger, 1983).
Sequencing Examples and Nonexamples to Facilitate Concept Attainment
Authors: Petty Osmond S. , Jansson Lars C.
Published: August 2020

The relative effects of instruction using two strategies for sequencing examples and nonexamples of the concept parallelogram were compared for sixth-grade students' performance at the classificatory and formal levels of concept attainment. Evidence was also sought of any interaction between the strategy and the students' mathematical ability. A significant main effect favoring a rational over a random sequence was found at the formal level; the interaction effect was not significant at either level of attainment. About 6% of the variance in scores at the formal level could be attributed to the treatment manipulation independently of the effects of mathematical ability.
Variables Affecting Word Problem Difficulty in Secondary School Mathematics
Authors: Caldwell Janet H. , Goldin Gerald A.
Published: August 2020

The relative difficulties of concrete versus abstract and factual versus hypothetical verbal problems in mathematics were compared for secondary school students, extending previously reported results for elementary school students. Concrete problems were significantly less difficult than abstract problems (p<.01) at both the junior and senior high school levels, as previously observed at the elementary school level, but the differences became smaller in magnitude with increasing grade level. Factual problems were significantly less difficult than hypothetical problems (p<.01) at both the junior and senior high school levels, in contrast to the elementary school results. There was an interaction between the two experimental factors.
Editorial
Authors: Kilpatrick Jeremy
Published: August 2020

In the final panel discussion at the Psychology of Mathematics Education meeting in Montreal last July, Hermine Sinclair (in remarks delivered in her absence by Carolyn Kieran) posed a paradox that she saw underlying much of the wrangling about constructivism that had occurred at the meeting. Everyone accepts the view that there can be no end to human knowledge, that there will always be something new to be added to what we know. The more difficult view is that our present-day knowledge will change. Learning entails both the conservation and the transformation of our knowledge. The paradox is that although change occurs when we push our ideas until we encounter a conceptual gap or obstacle, which signals that our ideas are in some way false, we can change those ideas only by remaining completely convinced of their truth. Only on looking back can we see how our knowledge changed and how it remained the same.
Integers as Transformations
Authors: Thompson Patrick W. , Dreyfus Tommy
Published: August 2020

To investigate whether elementary school students can construct operations of thought for integers and integer addition that are crucial for understanding elementary algebra, 2 sixth graders were taught for 6 weeks in eleven 40-minute sessions using a computerized microworld that proposed integers as transformations of position, integer addition as composition of transformations, negation as an operator upon integers or integer expressions, and representations of expressions as defined words. By the final session, both students had constructed mental operations for negating arbitrary integers and determining the sign and magnitude of a sum and had constructed a rule of substitution that allowed them to negate integer expressions. One student could negate represented expressions.
Sigal's Ineffective Computer-Based Practice of Arithmetic: A Case Study
Authors: Hativa Nira
Published: August 2020

Sigal, a second-grade girl in Israel, was observed for several months while practicing arithmetic with a structured and controlled computer-assisted instruction (CAI) system. Sigal was average in arithmetic in her class, enjoyed the computer practice, and believed that it helped her in arithmetic. However, observations showed that she consistently failed to solve problems on the computer that she could do successfully with paper and pencil. She was therefore given CAI practice well below her real arithmetic ability. She often oscillated among a few types of exercises, sometimes regressing rather than advancing upward in the hierarchical levels of practice. Several kinds of interventions did not bring about substantial improvement in her work with the computer. Continued observations in the following 2 years showed almost no change in her CAI-related behavior. This paper suggests the reasons for Sigal's problems with the CAI practice and draws implications for CAI design and implementation.
Representation of Addition and Subtraction Word Problems
Authors: Carpenter Thomas P. , Moser James M. , Bebout Harriett C.
Published: August 2020

This study investigated children's ability to write number sentences for simple addition and subtraction word problems. First graders taught to represent problems with open number sentences (e.g., 5 + □ = 8) represented a variety of problems with number sentences that directly modeled the action in the problems. First graders taught to represent all problems with number sentences in standard form (a + b = □, a − b = □) were limited in the problems they could represent. Second graders could represent problems directly with open number sentences or transform them to number sentences in standard form. The results, consistent with research on solutions using modeling and counting strategies, suggest that open number sentences may provide mathematical symbolism that allows young children to build upon informal strategies for representing and solving simple word problems.
Performance On Some Discovery Tasks, Grades 4—7
Authors: Sowder Larry
Published: August 2020

Despite the large number of studies devoted to discovery learning during the last decade (see, for example, Shulman & Keislar, 1966), little statistically supported data have appeared about how much information students require in order to form generahzations. No doubt many teachers, through classroom experience, know that pupils of different intelligence levels and at different grade levels are indeed capable of discovering, but there has also seemed to be a lack of quantitative informati0n about how performance differs among intelligence levels or among grade levels. The study reported here examined the performance of boys and girls from three intelligence levels at grades 4, 5, 6, and 7 on some numerical discovery tasks. In addition, a linear model was posited to see how well easily-obtained data would account for variability in performance.
The Relationship of Grade Placement to Programming Aptitude and Fortran Programming Achievement
Authors: Alspaugh John W.
Published: August 2020

This investigation to assess the influence of grade placement upon programming aptitude and FORTRAN programming achievement employed 2 groups of subjects equated on their high school Ohio Psychological Test scores. 1 group consisted of high school juniors and seniors, and the other group contained college juniors and seniors. A t-test revealed a significant difference in the IBM Programmer Aptitude Test scores for the 2 groups. By giving the high school subjects twice as much instruction time as the college students the anticipated difference in programming achievement was considerably reduced.
International Association for The Evaluation of Educational Achievement (Iea)—The Mathematics Study
Authors: Postlethwaite T. N.
Published: August 2020

In 1960, educational research institutes from ten countries decided to carry out a joint survey of mathematics achievement in their respective countries. This association of research centers called themselves the International Project for the Evaluation of Educational Achievement (IEA). The countries were Belgium, England, Finland, France, Israel, Japan, the Netherlands, Scotl and, Sweden, and the United States. In late 1962 and early 1963 Research centers from Australia and the Federal Republic of Germany e ntered the project on an associate basis.
Some Considerations of the Findings
Authors: Becker Jerry P. , Mckellar Marie T.
Published: August 2020

The results of the massive International Study of Achievement in Mathematics: A Comparison of Twelve Countries were reported by the mass media in the United States in March 1967. In general, the reports evoked concern and alarm among interested people across the country. References were made to the “hardly flattering” performance of American students; the media commented that Japan did the best overall job of teaching mathematics to its youngsters and that the United States was among the worst in this respect. In reporting on the study, the New York Times (March 1967) criticized poor mathematics teaching in America and lack of student interest in mathematics. The article also referred to a lack of interest on the part of the mathematical community in communicating with the American people.
A Theory of Mathematical Knowledge: Can Rules Account for Creative Behavior?
Authors: Scandura Joseph M.
Published: August 2020

This paper describes some elements of an emerging theory of mathematical knowledge. It is proposed that the mathematical behavior any given (idealized) S is potentially capable of at any given stage of learning can be accounted for precisely in terms of a finite set of rules. In particular, it is hypothesized that rule characterizations of this sort can account for creative, as well as more routine, behavior. The main argument centers on the notion of allowing rules to operate on other rules (i.e., to act as higher order rules) and thus allowing rule sets to “grow.” (Newly generated rules are assumed to be available from then on.) Attention is first centered on the behaviors involved in knowing mathematical systems (semantic knowledge). Then, the same ideas are extended to logical interrelationships between properties of mathematical systems, e.g., proving theorems.
Negative Instances and Mathematical Concept Formation: A Preliminary Study
Authors: Shumway Richard J.
Published: August 2020

Are there measurable differences in general mathematics achievement, inductive reasoning, syllogistic reasoning, perceptual speed, and tendency to overgeneralize between groups learning mathematical concepts through the use of positive instances (C groups), and groups learning mathematical concepts through the use of both positive and negative instances (E groups)? 84 eighth grade mathematics students in 4 classes served as subjects for 65 days. Analysis of variance and covariance of pre- and posttest means yielded significant differences (p<.05) indicating that the E groups tended to overgeneralize less frequently than the C groups. No other differences in means were significant (p<.05). It appears that the use of negative instances may be a means for controlling the common error of overgeneralization.
Conceptual Bases of Arithmetic Errors: The Case of Decimal Fractions
Authors: Resnick Lauren B. , Nesher Pearla , Leonard Francois , Magone Marla , Omanson Susan , Peled Irit
Published: August 2020

This study examines children's efforts to make sense of new mathematics instruction. The study documents major categories of errors that appear consistently as children learn decimal fractions. It then establishes the conceptual sources of these errors. Whole number errors derive from children's applying rules for interpreting multidigit integers. Fraction errors derive from children's efforts to interpret decimals as fractions. Different curriculum sequences influence the probability that these classes of errors will appear. It is suggested that errors are a natural concomitant of students' attempts to integrate new material that they are taught with already established knowledge. Since errorful rules cannot be avoided in instruction, educators are encouraged to use them as useful diagnostic tools to detect the nature of children's understanding of a mathematics topic.
Cross-National Achievement with Differential Retention Rates
Authors: Miller M. David , Linn Robert L.
Published: August 2020

In the First International Mathematics Study, Husén (1967b) showed that cross-national achievement differences were related to the proportion of students remaining, or retained, in the educational system. In addition, in countries retaining a high proportion of students in school, the more able students fared better than in countries retaining few students. For data gathered in 1981–1982 for the Second International Mathematics Study, the retention rates of countries accounted for some achievement differences, but achievement differences for the more able students across countries were not related to the rate at which countries retained students. Instead, variables such as opportunity to learn were needed to explain these differences.
Proof Frames of Preservice Elementary Teachers
Authors: Martin W. Gary , Harel Guershon
Published: August 2020

This study asked 101 preservice elementary teachers enrolled in a sophomore-level mathematics course to judge the mathematical correctness of inductive and deductive verifications of either a familiar or an unfamiliar statement. For each statement, more than half the students accepted an inductive argument as a valid mathematical proof. More than 60% accepted a correct deductive argument as a valid mathematical proof; 38% and 52% accepted an incorrect deductive argument as being mathematically correct for the familiar and unfamiliar statements, respectively. Over a third of the students simultaneously accepted an inductive and a correct deductive argument as being mathematically valid.
Math Lessons: A Contrast of Novice and Expert Competence
Authors: Leinhardt Gaea
Published: August 2020

Three important elements needed for constructing expert mathematics lessons are identified and described: rich agendas, consistent but flexible lesson structures, and explanations that meet the goals of clarifying concepts and procedures and having students learn and understand them. Each element is discussed in terms of its hypothetical characteristics and then tested by comparing two novice and four expert elementary mathematics teachers. Information about the teachers' agendas was gathered from preclass interviews; data on the structure of their lessons and on the characteristics of the explanations they gave came from lesson videotapes and related interviews. The novice-expert contrast highlighted the nature of the competencies expert teachers possessed and suggested some areas of instruction for future teachers.
Review: Consensus, More or Less
Authors: Clements Douglas H.
Published: August 2020

These books are the result of two impressive research programs in the area of early counting and number concepts, conducted by researchers who have dedicated a large number of years to this endeavor. Steffe and Cobb used a constructivist, case-study paradigm to investigate the development of ftrst and second graders' counting structures, including the meaning they gave to their counting and the strategies they used to find sums and differences. Fuson generally employed crosssectional and interview approaches to study changes in the number word sequence, correspondence errors, and concepts of cardinality of children from 2 to 8 years of age. Both projects reaffirm that work in this area lies at the cutting edge of research in both mathematics education and educational/cognitive psychology.
Children's Solution Strategies for Equivalent Set Multiplication and Division Word Problems
Authors: Kouba Vicky L.
Published: August 2020

One hundred twenty-eight children in Grades 1, 2, and 3 were given two multiplication and four division word problems that differed in semantic structure. The children's solution strategies were classified by degree of abstraction into direct representation, double counting, transitional counting, additive or subtractive, or recalled number fact. Strategies were also classified by use of physical objects: use as representations of individual elements, use as tallies or repeated references, or no use. Analyses of the solution strategies indicated the existence of a two-step intuitive model for multiplication, two common intuitive models for partitive division and measurement division each of which was related to an intuitive model for subtraction, and a third model for partitive division.
The Influence of Instructional Aids on Mathematics Achievement
Authors: Raphael Dennis , Wahlstrom Merlin
Published: August 2020

Data from the Second International Mathematics Study were analyzed using dual scaling or correspondence analysis to identify dimensions on which 103 Ontario Grade 8 mathematics teachers varied in reported use of instructional aids. These dimensions were then related to teacher and student characteristics and student achievement in mathematics. Teachers reporting greater use of a variety of instructional aids in the teaching of geometry and of ratio, proportion, and percent also reported greater course coverage. Experienced teachers reported more use of instructional aids. Student achievement in geometry was related to teaching experience and occasional use of a variety of instructional aids. Student achievement in ratio, proportion, and percent was related to teaching experience but was associated with extensive rather than occasional use of aids. The effect of instructional aids did not remain after the influence of topic coverage was removed. The use of instructional aids in the teaching of measurement was unrelated to virtually all teacher and student characteristics as well as student achievement.
Editorial
Authors: Carpenter Thomas P.
Published: August 2020

One score and ten years ago our forebears brought forth on this continent a new curriculum, conceived in academia and dedicated to the proposition that all students can learn mathematics with understanding.
Van Hiele Levels and Achievement in Writing Geometry Proofs
Authors: Senk Sharon L.
Published: August 2020

This study investigated relations between van Hiele levels, achievement in writing geometry proofs, and achievement in standard geometry content. Two hundred forty-one secondary school students who were enrolled in full-year geometry classes were tested in the fall for van Hiele level of thinking and entering knowledge of geometry, and in the spring for van Hiele level, standardized geometry achievement, and proof-writing achievement. Proof-writing achievement correlated significantly, .5 with fall van Hiele level, .6 with spring van Hiele level and with entering knowledge of geometry, and .7 with standardized geometry achievement in the spring. Proof-writing achievement also varied significantly with van Hiele level when either entering knowledge of geometry or geometry achievement in the spring was used as a covariate. The predictive validity of the van Hiele model was supported. However, the hypothesis that only students at Levels 4 or 5 can write proofs was not supported.
Solutions of Simple Word Problems by Nigerian Children: Language and Schooling Factors
Authors: Adetula Lawal O.
Published: August 2020

The problem-solving strategies used by schooled and unschooled Nigerian children to solve simple addition and subtraction word problems were studied using clinical interviews to find the effect of schooling on advanced strategies, and to assess the influence of the children's native language. Forty-eight schooled children from Grades 1 through 4 (ages 6 1/2 through 9 1/2 years) and 47 unschooled children from ages 7 through 14 years solved a broad range of addition and subtraction word problems. The schooled children answered more problems correctly and used more advanced strategies when the problems were presented in their native language than when the problems were presented in English. The performance of the schooled and the unschooled children was similar when 2-4 years allowance for age difference was made for the unschooled children.
NCTM Standards for School Mathematics: Visions for Implementation
Authors: Crosswhite F. Joe , Dossey John A. , Frye Shirley M.
Published: August 2020

We can expect that the release of NCTM's Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, Commission on Standards for School Mathematics, 1989) (Standards) will become one of those memorable moments that mark the history of our Council. We may remember this event in any of several ways.
Teachers' Beliefs about Small-Group Instruction in Elementary School Mathematics
Authors: Good Thomas L. , Grouws Douglas A. , Mason DeWayne A.
Published: August 2020

This study examines teachers' beliefs about, and practices in, small-group teaching of mathematics. A sample of 1509 teachers from 126 elementary schools in 10 districts in three states completed the survey. Roughly 80% of the teachers reported using the whole-class format as the basic organization for mathematics lessons. About 13% of the teachers reported using two or more groups and that while they worked with one group, students assigned to other groups worked alone. Only 5% of the teachers reported using two or more groups in which students were encouraged to work cooperatively. The paper discusses the various beliefs teachers report in deciding whether or not to group for instruction and describes how teachers use group instruction in mathematics classes. Grade-level differences are also explored.
Spatial Visualization and Gender Differences in High School Geometry
Authors: Battista Michael T.
Published: August 2020

The balance between visual-spatial and verbal-logical thought may determine “mathematical casts of mind” that influence how an individual processes mathematical information. Thus, to investigate the role that spatial thinking plays in learning, problem solving, and gender differences in high school geometry, spatial thought was examined along with its counterpart verbal-logical thought. The results suggest that whereas males and females differed in spatial visualization and in their performance in high school geometry, they did not differ in logical reasoning ability or in their use of geometric problem-solving strategies. There was evidence of gender differences in profiles of those mental abilities that are important for geometry performance and of a teacher-by-gender interaction on geometry achievement.
Outcomes and Processes on Estimation Test Items in Different Formats
Authors: Schoen Harold L. , Blume Glendon , Hoover H. D.
Published: August 2020

The performance of students in grades five through eight on test items in formats designed to measure various aspects of computational estimation was examined. Paper-pencil test data for 1376 students and interviews with 20 students were analyzed. Performance differed by item format, types of numbers and operations in the items, and grade level of students. Students tended to round numbers to the nearest leading power of 10 even when the items required other estimation processes.
Using a Base-Ten Blocks Learning/Teaching Approach for First- and Second-Grade Place-Value and Multidigit Addition and Subtraction
Authors: Fuson Karen C. , Briars Diane J.
Published: August 2020

A learning/teaching approach used base-ten blocks to embody the English named-value system of number words and digit cards to embody the positional base-ten system of numeration. Steps in addition and subtraction of four-digit numbers were motivated by the size of the blocks and then were carried out with the blocks; each step was immediately recorded with base-ten numerals. Children practiced multidigit problems of from five to eight places after they could successfully add or subtract smaller problems without using the blocks. In Study 1 six of the eight classes of first and second graders (N=169) demonstrated meaningful multidigit addition and place-value concepts up to at least four-digit numbers; average-achieving first graders showed more limited understanding. Three classes of second graders (N=75) completed the initial subtraction learning and demonstrated meaningful subtraction concepts. In Study 2 most second graders in 42 participating classes (N=783) in a large urban school district learned at least four-digit addition, and many children in the 35 classes (N=707) completing subtraction work learned at least four-digit subtraction.
A Part-Part-Whole Curriculum for Teaching Number in the Kindergarten
Authors: Fischer Florence E.
Published: August 2020

The set-subset relationship is an important mathematical construct that underlies many mathematical concepts developed by young children. Forty-two kindergarten children who received number concept instruction using a part-part-whole curriculum that stressed set-subset relationships developed a more mature concept of number, were more successful in solving addition and subtraction word problems, and developed greater understanding of place value in the base-10 numeration system than a comparable group of 44 kindergarten children who received standard instruction on number concepts.
Evaluating a Test of van Hiele Levels: A Response to Crowley and Wilson
Authors: Usiskin Zalman , Senk Sharon
Published: August 2020

Test instruments are an important element of almost every study in mathematics education, and the test that is used obviously affects the results of the study. Yet often a test is assumed both valid and reliable, and neither its content nor its psychometric properties are given scrutiny. The analyses Crowley and Wilson have done with the Van Hiele Geometry Test are welcome.
Research on Mathematics Education Reported in 1989
Authors: Suydam Marilyn N. , Crocker Deborah A.
Published: August 2020

As in previous years, this 20th annual listing of mathematics education research to appear in the JRME contains research summaries, articles, and djssertations, listed alphabetically by author within each category. For each reference, the grade or age level is indicated (when the researcher has provided this information or it can be inferred). An index is included at the end to help readers locate studies on certain, generally broad, topics.
The Effects of Logo on Children's Conceptualizations of Angle and Polygons
Authors: Clements Douglas H. , Battista Michael T.
Published: August 2020

It has been claimed that in Logo programming children learn mathematics by using concepts that aid them in understanding and directing the turtle's movements. The fundamental basis for this claim is that appropriate Logo environments can help children elaborate on, and become cognizant of, the mathematics implicit in certain kinds of intuitive thinking. To investigate this claim, 12 fourth graders were interviewed three times, at the beginning, middle, and end of 40 sessions of Logo graphics programming experience. The six Logo children, but not the comparison children, progressed from their original intuitive notions to more mathematically sophisticated and elaborate ideas of angle, angle size, and rotation. In addition, more Logo children explicitly mentioned geometric properties of shapes, indicating that they were beginning to think of the shapes in terms of their properties instead of as visual gestalts. Thus, there was support for the hypothesis that Logo experiences, especially those enriched with appropriate activities and discussions, can help children become cognizant of their mathematical intuitions and move to higher levels of geometric thinking.
Review: Apples and Oranges Again
Authors: Kilpatrick Jeremy
Published: August 2020

In 1964, samples of students from some 5000 secondary schools in 12 countries took mathematics tests and filled out opinion and informational questionnaires in the first effort by a group now known as the International Association for Educational Achievement (IEA) to compare students' subject maner knowledge crossnationally and relate that knowledge to characteristics of schools, teachers, and educational systems. The release of the report of that study (Husén, 1967), which later became known as the First International Mathematics Study (FIMS), touched off a furor in the United States as American mathematics educators rushed to explain (or explain away) the relatively poor performance of the American students. The study itself was more a study in comparative education that used mathematics as an indicator of what had been learned in school than a study in mathematics education with clear implications for curricula and instruction. The press release and subsequent accounts in the media, however, portrayed the study as an international competition with winners and losers (Japan won, the United States lost). The bulk of a special issue of this journal on the FIMS (Wilson & Peaker, 1971) was given over to critiques by Americans of the IEA's assumptions, instruments, analyses, reporting, and interpretations.
Logo Programming and Geometric Understanding: An In-Depth Study
Authors: Olive John
Published: August 2020

The Logo work of 30 ninth-grade students was analyzed from three theoretical perspectives: the van Hiele levels of thinking, the SOLO taxonomy, and Skemp's model of mathematical understanding. Using computer dribble files of the students' Logo work, summary analyses across four tasks examined the Logo programming aspects and the geometric aspects of the tasks. Relationships among the three theoretical models and between Logo programming and geometric understanding emerged from the summary analyses. SOLO learning cycles in which students achieve relating level responses to a task are more likely to result in relational understanding of the task when students are able to approach the task with at least a transition towards descriptive level thought. Success in Logo programming appears to be necessary but not sufficient for success with the geometric aspects of the tasks. The results suggest the possibility of constructing an integrated model of teaching and learning based on the three theoretical perspectives.
Algebra Students' Knowledge of Equivalence of Equations
Authors: Steinberg Ruth M. , Sleeman D. H. , Ktorza David
Published: August 2020

This study examined algebra students' understanding of equivalent equations. Ninety-eight eighth- and ninth-grade students were presented with pairs of equations and asked to judge whether the equations in each pair were equivalent. The students were also asked to justify their answers. The responses fell into three major groups: computed solutions, judgments based on transformations of equations, and incorrect solutions. Most of the students consistently gave justifications from one group. Children who gave reasons based on transformations had a significantly higher rate of correct responses than the children who calculated solutions. In the incorrect group, the success rate was much lower than in the other two groups. Most students in the transformations group were able to transfer their knowledge to transformations not used in solving equations.
Children's Learning in a Computer Microworld for Transformation Geometry
Authors: Edwards Laurie D.
Published: August 2020

Twelve middle school students working in pairs used a computer microworld to explore an introductory curriculum in transformation geometry. The microworld linked a symbolic representation (a set of simple Logo commands) with a visual display that showed the effects of each transformation. Worksheets were designed with the objective of encouraging the students to find and express mathematical patterns in the domain. The students were successful in constructing an accurate working understanding of the transformations. There was a tendency for symbolic overgeneralization in some activities, but the students were able to use visual feedback from the microworld and discussions with their partners to correct their own errors.
Review: Justice in Mathematics Education: Reaching beyond Equity
Authors: Becker Joanne Rossi
Published: August 2020

Although not a definitive volume on the current state of our knowledge concerning gender and mathematics. this book nevertheless collects in one place a discussion of some of the critical variables that relate to inequity in mathematics education for females. The book is edited by two of the leading researchers in this area of endeavor, each well known within and outside of her native country. This collaboration between Fennema and Leder provides some parallel research from their respective countries, the United States and Australia, thus allowing some cross-cultural comparisons. Most of the volume. however. reports about research conducted in the U.S. by former students of Fennema.
Number Sense as Situated Knowing in a Conceptual Domain
Authors: Greeno James G.
Published: August 2020

This paper presents a preliminary attempt to characterize number sense theoretically as a form of cognitive expertise. I propose a way to view conceptual domains using a metaphor of an environment in which one can know how to find resources and use them to make things. I discuss the domain of numbers and quantities as an example of a conceptual environment, and I interpret number sense as a set of capabilities for constructing and reasoning within mental models. This perspective provides reasons that support considering various aspects of number sense as features of students' general condition of knowing in the domain of numbers and quantities, rather than skills that should be given specific instruction. Some current trends in research about cognition and learning support this view of knowing in conceptual domains, including number sense.
Review: Constructivism in the Classroom
Authors: Clement John
Published: August 2020

The International Congress on Mathematical Education, which convenes every four years to discuss issues in mathematics education research and practice, last met in 1988. This book is the product of the subgroup on early childhood education within the Congress, which took as its charge “to identify the issues, problems, and opportunities presented by constructivism for mathematics education in early childhood and to make recommendations for the work in this area over the next four years.” Thus the book not only contains reports on previous research, but also recommendations for future research and practice in early childhood mathematics education.
Meta-Analysis of Problem-Solving Studies: A Critical Response
Authors: Goldin Gerald A.
Published: August 2020

In the past decade, meta-analysis has been applied with increasing frequency to the synthesis of educational research findings. My initial skepticism of its use has. if anything. grown greater during this period. Some of the difficulties with the methods described in the early texts (Glass. McGaw, & Smith, 1981; Hunter, Schmidt, & Jackson, 1982) were, I would begin by acknowledging. effectively addressed through careful delineation of the assumptions underlying the statistical procedures and greater attention to the limits of their applicability (Hedges & Olkin, 1985). Meta-analysis can in fact provide answers to certain kinds of carefully formulated questions, when great caution is exercised (e.g., see Wachter, 1988, and Iyengar, 199 1, for interesting and critical perspectives). Nevertheless, a substantial fraction of meta-analytic studies in education violate the conditions that would be necessary for acceptance of their conclus ions. When this occurs, the method lends an aura of quantitative certainty to questionable generalizations.
A Process Model of Mathematics Achievement and Attitude
Authors: Reynolds Arthur J. , Walberg Herbert J.
Published: August 2020

A structural model of mathematics achievement and attitude was tested with a national probability sample of 3,116 young adolescents from the Longitudinal Study of American Youth using structural modeling. A three-wave longitudinal design incorporated data from students, teachers, and parents to construct a prespecified theoretical model of mathematics achievement and mathematics attitude. A revised model provided a better fit than the original specification and than regression models used in 23 previous studies. Supported in cross-validation, the model revealed a complexity of direct and indirect effects not apparent in previous studies. Prior achievement and home environment influenced subsequent achievement most powerfully; motivation, exposure to extramural reading media, peer environment, and instructional exposure also had significant influences on achievement. Previous attitude had the most powerful influence on subsequent attitude, although the direct effects of instructional quality and the indirect effects of motivation and home environment were also notable. Appropriate teacher use of instructional time, thorough textbook coverage, and daily introduction of new material, although educationally alterable, are themselves influenced by previous student achievement. Similarly, instructional practices are significant alterable influences on mathematics attitudes, but such practices are themselves influenced by students' initial attitudes.
Mental Strategies and Materials or Models for Addition and Subtraction up to 100 in Dutch Second Grades
Authors: Beishuizen Meindert
Published: August 2020

Dutch mathematics programs emphasize mental addition and subtraction in the lower grades. For two-digit numbers up to 100, instruction focuses on “counting by tens from any number” (N10), a strategy that is difficult to learn. Therefore, many children prefer as an easier alternative “decomposition” in tens (1010) and units. Instead of the use of arithmetic blocks (BL), the hundredsquare (HU) was introduced in the 1980s because of a (supposed) better modeling function for teaching N10. In a field study with several schools, (a) we compared the strategies N10 and 1010 on procedural effectiveness and error types, and (b) we assessed the influence of the support conditions BL versus HU on the acquisition of mental strategies (we had also a control condition NO with no extra materials or models). Results confirmed the greater effectiveness of N10 but also the preference of many weaker children for 1010. Support for BL or HU had differential effects on mental strategies. Differences are discussed in terms of cognitive psychology: the role of declarative knowledge and the relation between conceptual and procedural knowledge. New Dutch proposals for the 1990s emphasize teaching both strategies N10 and 1010 to enhance the flexibility of students' mental arithmetic.
Editorial
Authors: Lester Frank K.
Published: August 2020

“Editors certainly are fussy people!” moaned an author who phoned to discuss revisions to her recently accepted manuscript. “Putting a manuscript into publishable form requires attention to so many details!”
Inconsistencies in Students' Reasoning About Probability
Authors: Konold Clifford , Pollatsek Alexander , Well Arnold , Lohmeier Jill , Lipson Abigail
Published: August 2020

Subjects were asked to select from among four possible sequences the “most likely” to result from flipping a coin five times. Contrary to the results of Kahneman and Tversky (1972), the majority of subjects (72%) correctly answered that the sequences are equally likely to occur. This result suggests, as does performance on similar NAEP items, that most secondary school and college-age students view successive outcomes of a random process as independent. However, in a follow-up question, subjects were also asked to select the “least likely” result. Only half the subjects who had answered correctly responded again that the sequences were equally likely; the others selected one of the sequences as least likely. This result was replicated in a second study in which 20 subjects were interviewed as they solved the same problems. One account of these logically inconsistent responses is that subjects reason about the two questions from different perspectives. When asked to select the most likely outcome, some believe they are being asked to predict what actually will happen, and give the answer “equally likely” to indicate that all of the sequences are possible. This reasoning has been described by Konold (1989) as an “outcome approach” to uncertainty. This prediction scheme does not fit questions worded in terms of the least likely result, and thus some subjects select an incompatible answer based on “representativeness” (Kahneman & Tversky, 1972). These results suggest that the percentage of secondary school students who understand the concept of independence is much lower than the latest NAEP results would lead us to believe and, more generally, point to the difficulty of assessing conceptual understanding with multiple-choice items.
Models of Problem Solving: A Study of Kindergarten Children's Problem-Solving Processes
Authors: Carpenter Thomas P. , Ansell Ellen , Franke Megan L. , Fennema Elizabeth , Weisbeck Linda
Published: August 2020

Seventy kindergarten children who had spent the year solving a variety of basic word problems were individually interviewed as they solved addition, subtraction, multiplication, division, multistep, and nonroutine word problems. Thirty-two children used a valid strategy for all nine problems and 44 correctly answered seven or more problems. Only 5 children were not able to answer any problems correctly. The results suggest that children can solve a wide range of problems, including problems involving multiplication and division situations, much earlier than generally has been presumed. With only a few exceptions, children's strategies could be characterized as representing or modeling the action or relationships described in the problems. The conception of problem solving as modeling could provide a unifying framework for thinking about problem solving in the primary grades. Modeling offers a parsimonious and coherent way of thinking about children's mathematical problem solving that is relatively straightforward and is accessible to teachers and students alike.
Review: School Mathematics and Multiple Realities: New Paradigms for Assimilation
Authors: Ferrini–Mundy Joan
Published: August 2020

The mathematics education community owes thanks to Bob Davis and Carolyn Maher for their attempt to promote a discussion, through the conference that preceded this book and the volume itself, about how wide-ranging societal issues interact with mathematics education reform. Rutgers's Dean of the Graduate School of Education Loujse Cherry Wilkinson introduces the volume as a critical analysis of the mathematics education reform movement in the United States. The title, and the DavisMaher introduction, suggest a different agenda for this volume, one which auends seriously to the “world of reality.” They argue that schools are not designed to serve the needs of today's children. who will live and work in the 21st century, and claim that the volume chapters suggest a theoretical foundation for the creation of new approaches that incorporate broader concerns about chools and society. Mixed missions notwithstanding, the collected papers form a readable, interesting set that give some new perspectives on mathematics reform and provide a sampler of the issues that are important to individual projects and research groups.
The Participation, Beliefs, and Development of Arithmetic Meaning of a Third-Grade Student in Mathematics Class Discussions
Authors: Lo Jane-Jane , Wheatley Grayson H. , Smith Adele C.
Published: August 2020

In order to study learning in mathematics class discussion, we examined the participation, beliefs, and development of arithmetic meaning of four grade 3 students over one school year. In this paper, we report on class discussion from the perspective of one of these four students, Brad. Enjoying the intellectual challenges from mathematics discussions, Brad took an active role in making sense of other students' explanations and assisting mathematics communication among his classmates. In so doing, he created his learning opportunities. The analysis of Brad's arithmetical development provided additional insights into Brad's participation in mathematics class discussion. On the basis of this study, and especially the knowledge gained from Brad, we suggest that a fruitful mathematics class discussion necessitates the negotiation of social norms and mathematics meanings. Each class must work out its own patterns of interaction. The process of negotiation is dialectic: the negotiation of social norms makes possible the negotiation of mathematics meaning, and the negotiation of mathematics meaning makes possible the negotiation of social norms. Students learn how to discuss mathematics as a group and how to make sense of mathematical ideas as individuals by having the opportunities to discuss and make sense of different explanations.