Showing papers in "Journal for Research in Mathematics Education in 1995"

Reconstructing Mathematics Pedagogy from a Constructivist Perspective.

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

Splitting, covariation, and their role in the development of exponential functions

Abstract: Exponential and logarithmic functions are typically presented as formulas with which students learn to associate the rules for exponents/logarithms, a particular algebraic form, and routine algorithms. We present a theoretical argument for an approach to exponentials more closely related to students' constructions. This approach is based on a primitive multiplicative operation labeled "splitting" that is not repeated addition. Whereas educators traditionally rely on counting structures to build a number system, we suggest that students need the opportunity to build a number system from splitting structures and their geometric forms. We advocate a "covariation" approach to functions that supports a construction of the exponential function based on an isomorphism between splitting and counting structures.

Children's Concepts of Average and Representativeness.

Abstract: Whenever the need arises to describe a set of data in a succinct way, the issue of mathematical representativeness arises. The goal of this research is to understand the characteristics of fourth through eighth graders' constructions of "average" as a representative number summarizing a data set. Twenty-one students were interviewed, using a series of open-ended problems that called on children to construct their own notion of representativeness. Five basic constructions of representativeness are identified and analyzed. These approaches illustrate the ways in which students are (or are not) developing useful, general definitions for the statistical concept of average. One objective of statistics is to reduce large, unmanageable, and disordered collections of information to summary representations. The need to summarize data is present even among young children. For example, in the surveys conducted by primary-grade students, we see movement from focusing on individual pieces of data ("I have one brother") to highlighting and summarizing the data in some manageable form ("Most of the class members have only one brother or sister"). As soon as there is the need to describe a set of data in a more succinct way, the notion of representativeness arises: What is typical of these data? How can we capture their range and distribution?

Confounding Whole-Number and Fraction Concepts When Building on Informal Knowledge

Abstract: This study examined the development of students' understanding of fractions during instruction with respect to the ways students' prior knowledge of whole numbers influenced the meanings and representations students constructed for fractions as they built on their informal knowledge of fractions. Four third-grade and three fourth-grade students received individualized instruction on addition and subtraction of fractions in a one-to-one setting for 3 weeks. As students attempted to construct meaning for symbolic representations of fractions, they overgeneralized the meanings of symbolic representations for whole numbers to fractions, and they overgeneralized the meanings of symbolic representations for fractions to whole numbers.

Factors Associated with Types of Mathematics Anxiety in College Students.

Abstract: This article investigates the interrelatedness of various types of mathematics anxiety with attitudes toward mathematics, learning preferences, study motives, and strategies. An 80-item version of the Mathematics Anxiety Rating Scale (MARS) was completed by 173 university students enrolled in one of three introductory statistics courses offered by the departments of mathematics, psychology, or sociology. Mathematics attitude scales were also administered in conjunction with the Study Process Questionnaire (SPQ), which depicts students' motivations to use study methods associated with contrasting learning approaches. Factor analysis of the MARS identified six factors labeled as General Evaluation Anxiety, Everyday Numerical Anxiety, Passive Observation Anxiety, Performance Anxiety, Mathematics Test Anxiety, and Problem-Solving Anxiety. The examination of both substantive and peripheral factors supported a multidimensional view of mathematics anxiety. Correlational analysis indicated complex interaction patterns between attitudes toward mathematics and the six MARS factors, depending on the overall level of anxiety experienced. Variation in orientation to learning also proved significantly related to specific types of anxiety, attitudes, and instructional factors. The results confirm the utility of learning-approach theory and instruments for analyzing relationships between cognitive and affective components of mathematics anxiety. Mathematics anxiety has become a euphemism for debilitating test stress, low self-confidence, fear of failure, and negative attitudes toward mathematics learning. Despite substantial inquiry into the dimensions and correlates of mathematics anxiety, several issues remain unresolved. Many researchers conclude that the Mathematics Anxiety Rating Scale (MARS; Suinn, Edie, Nicoletti, & Spinelli, 1972) is a multidimensional construct, without discussing the theoretical meaning(s) of this concept. However, the complexity of mathematics anxiety cannot be limited to factors identified in the MARS. Recent studies also confirm the interrelatedness of mathematics attitudes and anxieties but disregard the importance of cognitive processes in generating proneness to anxiety. This study analyzes mathematics anxiety in relation to learning approaches, cognitive processes, and affective dispositions. The background section summarizes literature that illustrates the empirical and theoretical bases of this analytical framework.

Motivation and Ability as Factors in Mathematics Experience and Achievement.

Abstract: This study examined relationships among interest, achievement motivation, mathematical ability, the quality of experience when doing mathematics, and mathematics achievement. One hundred eight freshmen and sophomores (41 males, 67 females) completed interest ratings, an achievement motivation questionnaire, and the Preliminary Scholastic Aptitude Test. These assessments were followed by 1 week of experience sampling. Mathematics grades were available from the year before the study started, from the same year, and from the following 3 years. In addition, a measure of the students' course level in mathematics was included. The results showed that quality of experience when doing mathematics was mainly related to interest. Grades and course level were most strongly predicted by level of ability. Interest was found to contribute significantly to the prediction of grades for the second year and to the prediction of course level. Quality of experience was significantly correlated with grades but not course level.

A study of intrinsic motivation in the mathematics classroom: a personal constructs approach

Abstract: This article examines the relationship between teachers' and students' personal constructs with regard to intrinsic motivation in the mathematics classroom. The research focused on (a) the ways in which teachers attempted to build student motivation into their lessons and (b) the belief systems of teachers as compared to those of their students. In a repertory grid task, students and teachers were asked to distinguish what they believed makes mathematics motivating. Results revealed that the individuals studied,whether teachers or students, were similar in their constructs systems. Both students and teachers tended to stress the interrelationship between arousal and control levels in determining the intrinsic motivation of mathematics activities. Despite these similarities, the extent to which teachers can anticipate the motivation of their students may depend more on teachers' personal conceptions of intrinsic motivation than their beliefs about their students. Most of the teachers studied had little notion of the motivational beliefs of their students. Results are examined in relation to the need to inform teachers regarding the dynamics of student motivation and to pay particular attention to the individual differences in students' motivational beliefs. In general, results indicate that when teachers are able to predict their students' beliefs, they are better able to fine-tune their instruction to turn kids on to mathematics. Research has shown that when children are motivated intrinsically' to perform an academic activity, they spend more time engaged in the activity, learn better, and enjoy the activity more than when they are motivated extrinsically (Lepper, 1988). Clearly, getting children to engage in learning "for its own sake" is a primary goal of educators: not only will children learn better at the immediate task, they will also tend to seek out similar activities in the future. Thus, designing intrinsically motivating activities is of paramount importance in developing lifelong learners. Why do children prefer intrinsically motivating activities? Is it because there is something "intrinsic" to the activity that induces them to engage, or could it be that there are certain characteristics of the activity that fit the individual's notion of motivation? The distinction between these two cases may seem trivial at first. Does it really matter to the educator if the motivation stems from the activity or from the child? In the case of mathematics, it seems that it does matter.

Toward a Working Model of Constructivist Teaching: A Reaction to Simon.

Abstract: We start this discussion of the article by Martin A. Simon with two major conjectures that we will try to substantiate. The first is that there is a kind of teaching that can legitimately be called "constructivist teaching." The second is that Simon's model of teaching his prospective elementary school teachers, if modified, would fit our understanding of constructivist teaching, even though his basic premise is that constructivism does not tell us how to teach mathematics (Simon, 1995). The issue concerning whether constructivism tells us how to teach mathematics resides in how constructivism is understood. One way to understand constructivism is in terms of basic tenets like "knowledge is not passively received but is actively built up by the cognizing subject," and "the function of cognition is adaptive and serves in the organization of the experiential world rather than in the discovery of ontological reality" (von Glasersfeld, 1989, p. 162). These basic tenets are orienting, but as indicated by Simon, they do not stipulate a particular model of teaching mathematics. Neither do they tell us how to do family therapy or how to provide psychiatric counseling. People who engage in these types of human activities, however, can use the basic tenets of constructivism in building models of the realities of those with whom they interact. If a teacher formulates a model of children's mathematical knowledge, including its construction, we claim that the model is an important part of the teacher's understanding of constructivism. Similarly, if the teacher formulates a model of how she makes sense of children's mathematical knowledge, including its construction, this would be a constructivist model of teaching. It, too, would be an important part of the teacher' s meaning for constructivism. Regarding the teacher as a learner in the activity of teaching is essential in our understanding of constructivist teaching, and it is a basic aspect of Simon's model of teaching as well.

A cognitive analysis of U.S. and Chinese students' mathematical performance on tasks involving computation, simple problem solving, and complex problem solving

Abstract: The mathematical performance of 250 U.S. sixth-grade students from both private and public schools and 425 Chinese sixth graders from both key and common schools was examined on multiple-choice tasks assessing computation and simple problem solving and on open-ended tasks assessing complex problem solving. Chinese students performed significantly better than U.S. students on both computation and simple problem solving. The results were about the same for the two samples on complex problem solving. Moreover, when subsets of U.S. and Chinese students were matched on their computational performance, the U.S. students scored significantly higher than comparable Chinese students on the measures of both simple and complex problem solving. U.S. and Chinese students had similar overall performance on complex problem solving, but a detailed cognitive analysis of students' written responses revealed not only many similarities in the solutions but also many subtle differences. For example, the types of strategies employed and the types of errors made by the Chinese students were similar to those for the U.S. students, although the Chinese students' solutions tended to be more elegant. Also, U.S. students tended to use visual representation more frequently than Chinese students, who tended to use symbolic representation (e.g., algebraic equations) more frequently. The results of this study suggest not only the complexity of examining mathematical performance differences, but also the inadequacy of using a limited range of tasks to measure mathematical performance in cross-national studies. One of the main contributions of this study is its use of a variety of mathematical tasks to capture the thinking and reasoning of U.S. and Chinese students. Another contribution is the scheme used to analyze student performance, a scheme based not solely on the percentage correct or incorrect, but rather on a detailed analysis of students' strategies, representations, and errors. This range of tasks and the associated methodology supported the discovery of findings of similarities and differences between U.S. and Chinese students that have not been reported previously.

The frequency of arithmetic facts in elementary texts: addition and multiplication in grades 1-6

Abstract: We tabulated the frequency with which simple addition and multiplication facts occur in elementary school arithmetic texts for grades 1-6. The results indicated a strong "small-fact bias" in both addition and multiplication. "Large" facts, with operands larger than 5, occurred up to half as frequently as those with operands in the 2-5 range. As was also found in an earlier tabulation for grades K-3, facts with operands of 0 and 1 occurred relatively infrequently; the ostensible exceptions to this pattern, high frequencies for combinations like 1 + 2 and 1 x 3, were caused by the small-fact bias in multicolumn problems. The small-fact bias in the presentation of basic arithmetic, at least to the degree observed here, probably works against a basic pedagogical goal, mastery of simple arithmetic. It may also provide a partial explanation of the widely reported problem size or problem difficulty effect, that children's and adults' responses to larger basic facts are both slower and more error prone than their solutions to smaller facts. Theoretical and practical implications of the small-fact bias are discussed briefly. It is a common observation that elementary school students have difficulty learning and mastering their large number facts, combinations with relatively large, single-digit operands such as 7 + 6 and 8 x 9. This difficulty is apparent in casual classroom observation, in testing situations, and certainly in later arithmetic instruction when multicolumn addition and multiplication are taught. In the laboratory, the difficulty is most clearly observed when subjects are tested in a reaction time (RT) task, asked either to produce the answer to a given fact or to judge an answer as true or false. The effect is referred to as the problem size or problem difficulty effect: As the operands in a basic fact problem increase, so do the individual's RTs and errors. It is by far the most commonly obtained result in the literature on children's and adults' mental arithmetic performance. It characterizes both RT performance and error rates across the entire developmental span, from kindergartners and

The tensions and contradictions of the school mathematics tradition

Abstract: The ubiquitousness and robustness of traditional practices in the teaching of school mathematics raise two fundamental and interrelated questions: Will the fate of the current reform effort in mathematics education be any different from that of previous reform efforts? Why have these traditional practices been so constant and durable? Certainly these are questions that cannot be answered in a single study. However, in an effort to develop a research basis that could aid in thinking about these questions, I conducted an ethnographic case study of a beginning high school mathematics teacher's acculturation into the school mathematics tradition (i.e., the beliefs and practices that characterize the traditional approach to school mathematics). The analysis presented here focuses on the tensions and contradictions that underlie the beliefs and practices of the school mathematics tradition and on the ways that teachers cope with these tensions and contradictions. It also indicates how the explanations and strategies that teachers employ to cope with these tensions and contradictions can actually help to sustain the tradition.

Teachers' Interpretations of "CGI" after Four Years: Meanings and Practices.

Abstract: Twenty primary teachers were interviewed who, three or four years earlier, had participated in in-service workshops on Cognitively Guided Instruction (CGI). Three patterns of CGI use seemed related to the meanings teachers constructed for CGI itself. Teachers who reported developing their use of CGI until it formed the mainstay of their mathematics teaching saw CGI conceptually. They also reported learning mainly through their interactions with students and other teachers and developing beliefs about the conceptual nature of mathematics, the constructivist nature of learning, and the students' central role in that learning. Teachers who reported never having used CGI more than supplementally saw CGI as a group of procedures and espoused more traditional beliefs in these areas. Teachers who reported using CGI more at first, but less currently, showed a marked incongruity between their espoused beliefs and reported practices. The authors ask whether additional reseacher support, collegial interaction, or perhaps prescriptiveness in the intervention might have helped teachers in this third group enact their conceptually based beliefs.

Cultural Tools and Mathematical Learning: A Case Study.

Abstract: This study investigates the role that four second graders' use of the hundreds board played in supporting their conceptual development over a 10-week period. Particular attention is given to the transition from counting by ones to counting by tens and ones. The analysis indicates that the children's use of the hundreds board did not support the construction of increasingly sophisticated concepts of ten. However, children's use of the hundreds board did appear to support their ability to reflect on their mathematical activity once they had made this conceptual advance. The constructivist perspective exemplified in the analysis is contrasted with a sociocultural perspective on mathematical development. The differing roles attributed to cultural tools are clarified, and potentially complementary aspects of the two perspectives are discussed. The study reported in this article investigates second graders' use of the hundreds board in a classroom where instruction was broadly compatible with recent reform recommendations (e.g., National Council of Teachers of Mathematics, 1989, 1991). The children's use of this particular instructional device was analyzed for two purposes. The first is pragmatic and concerns the role that the hundreds board might play in supporting children's construction of increasingly sophisticated numerical concepts. The second purpose is theoretical and concerns the differences between constructivist and sociocultural accounts of conceptual development. Below, I discuss each of these issues in turn, thereby elaborating the rationale for the study.

Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8.

Abstract: This study assessed attitude, computational preferences, and mental computational performance of 176, 187, 186, and 206 Japanese students in grades 2, 4, 6, and 8, respectively. A sample of students in grades 4 and 8 scoring in the upper and middle quintiles on the mental computation test was interviewed to identify strategies used to mentally compute. All data were collected during the last month of the school year. A wide range of performance on mental computation was found with respect to all types of numbers (whole numbers, decimals, and fractions) and operations at every grade level; the mode of presentation (visual or oral) significantly affected performance levels, with visual items generally producing higher performance; and the range of strategies (initial and alternative) used to do mental computation was narrow, with the most popular approach reflecting a mental version of a learned "paper/pencil" algorithm.

Learning to Solve Addition and Subtraction Word Problems Through a Structure-Plus-Writing Approach.

Abstract: The aim of this study was to design and field-test instruction intended to help students construct knowledge about addition and subtraction story problems and determine if this knowledge would transfer to actually solving problems. The study tests two related hypotheses: (a) structure-pluswriting instruction will result in improved word-problem solving, and (b) this improvement will be more enduring than that resulting from a more traditional heuristic and practice-based approach. To test these hypotheses, 401 third-grade and fourth-grade students from 21 classrooms in six schools participated in a study in which the problem solving of children taught by a structure-plus-writing approach was compared to that of (a) a control group receiving no explicit instruction in arithmetic word-problem solving, and (b) a group receiving instruction based largely on practice and explicit heuristics. Both hypotheses were supported by the results. The structure-plus-writing group outperformed the group receiving practice and explicit heuristics instruction. Moreover, the structure-plus-writing group not only maintained this superiority but actually widened the gap, as shown by retention test results 10 weeks after treatment. The effectiveness of instruction based on a structured approach to authoring arithmetic word problems was strongly supported.

Students' use of part-whole and direct comparison strategies for comparing partitioned rectangles

Abstract: Thirty-six students, twelve each at the fourth-, sixth-, and eighth-grade levels, were asked to solve 21 comparison-of-area tasks during a clinical interview. The areas were partitioned and shaded so that the tasks could be solved by using rational number knowledge. The types of strategies students used to compare the areas were identified and classified into categories. Initially, fractional terms and symbols were not introduced into the tasks in order to compare the types of strategies students used without and with symbols introduced. Most students tended to ignore the part-whole relationships inherent in the tasks and used a Direct Comparison strategy when they compared the areas. The use of a Part-Whole strategy increased with the introduction of fractional terms and symbols, especially at the eighth-grade level.

Numeracy as Cultural Practice: An Examination of Numbers in Magazines for Children, Teenagers, and Adults.

Abstract: Many have argued for the importance of numeracy, yet little is known about the opportunities for numeracy presented to people in their daily lives. In this study, we analyzed and compared the characteristics of rational numbers in magazines written for children, teenagers, and adults. Our analysis indicates that difficult mathematical concepts that appear in the media, such as fractions, percents, and averages, are much more prevalent in adults' magazines than in those written for children and teenagers. Adults are often presented with rational numbers that are related to each other. Numbers in teenagers' texts do not appear to form a transition to those found in adults' texts, despite the fact that through formal schooling teenagers have covered all the mathematical concepts that are frequently found in adults' texts. Implications for preparing students for the numeracy demands of everyday life are discussed.

Elaborating models of mathematics teaching: A response to Steffe and D'Ambrosio

Abstract: In my article, I framed a developing model of mathematics teaching. As such, the model presented was neither complete nor elaborated in detail. The article was designed to generate discussion that can contribute to the further development of this and other models. Steffe and D'Ambrosio's response contributes in important ways to this discussion. They indicate an acceptance of many of the components of the model and elaborate several of them.

The Relevance of Practice: A Response to Orton.

Abstract: In this response to Robert Orton, I address each of the major points he raises and attempt to clarify the discrepancies in our positions. I give particular attention to his approach of reframing issues in terms of the categories of traditional academic philosophy. In adopting this stance, Orton (a) presents a highly idiosyncratic interpretation of the Cartesian dualism, (b) creates a gulf between theory and practice, and (c) implies that the social and cultural aspects of mathematical activity can be dismissed. I discuss each of these points and further develop my position by outlining Putnam's (1987) pragmatic realism, clarifying why Rorty's (1979) work might be of interest to mathematics educators, and revisiting Bereiter's (1985) learning paradox. I then conclude by exploring the relationship between the resulting nondualist approach and John Dewey's philosophy and pedagogy.

A critical look at practice in mathematics and mathematics education

Abstract: Math Worlds is divided into an introductory chapter by Restivo and three major sections on the philosophy, politics, and sociology of mathematics. The first section offers a refreshing alternative to the usual discussions of Platonism, constructivism/intuitionism, and formalism that have dominated past discourse in mathematical philosophy. An anti-Platonist feeling runs through the book as a whole, but as Restivo points out in his introductory chapter, Platonism has not been vanquished by appeals to take mathematics as practiced more seriously. All four chapters in this first section take the question of Platonism as a jumping-off point and bring a variety of intriguing epistemological standpoints to the debate. In general, I found the philosophical discussions in the first section both accessible and challenging. As a philosophically unsophisticated reader, I was sometimes overwhelmed by numerous references to unfamiliar works. At the same time, I found the arguments so compelling that I was motivated to find and peruse many of the works cited. By this measure, the book satisfied one of my (admittedly idiosyncratic) criteria for excellence: it left me interested and intrigued, wanting more background on the arguments and viewpoints presented. The second section deals more with the politics of mathematics education: the place of mathematics and mathematics education in contributing to the good of society.