Mathematical Beliefs Research Articles

Inconsistency between a Beginning Elementary School Teacher's Mathematics Beliefs and Teaching Practice

Inconsistency between a Beginning Elementary School Teacher's Mathematics Beliefs and Teaching Practice

Deepening the Analysis: Longitudinal Assessment of a Problem-Centered Mathematics Program

Longitudinal analyses of the mathematical achievement and beliefs of 3 groups of elementary pupils are presented. The groups consist of those students who had received 2 years of problem-centered mathematics instruction, those who had received 1 year, and those who had received textbook instruction. Comparisons are made for the groups using a standardized norm-referenced achievement test from first through fourth grade. Next, student comparisons are made using instruments developed to measure conceptual understanding of arithmetic and beliefs and motivation for learning mathematics. The results of the analyses indicate that after 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than do those in textbook instruction. In addition, these differences remain after problem-centered students return to classes using textbook instruction. Comparisons of pupils in problem-centered classes for only 1 year reveal that after returning to textbook instruction, these students' mathematical achievement and beliefs are more similar to the textbook group. Also included are exploratory analyses of the pedagogical beliefs held by teachers before and after teaching in problem-centered classes, and those held by teachers in textbook classes. The results of the student and teacher analyses are interpreted in light of research on children's construction of nonstandard algorithms and the nature of classroom environments.

The Effect of Year-Long Instruction in Mathematical Problem Solving on Middle-School Students' Attitudes, Beliefs, and Abilities

Three classes of middle-school students who received 1 year of problem-solving instruction were compared with 3 classes of students who were taught mathematics in a more traditional manner. At the end of the school year, all the students completed a questionnaire that explored their mathematical beliefs. In addition, 3 students of varying ability levels per class were interviewed and asked to solve 4 nonroutine problems. Compared with the students who had received traditional mathematics instruction, the students who had received problem-solving instruction displayed greater perseverance in solving problems, more positive attitudes about the usefulness of mathematics, and more sophisticated definitions of mathematical understanding. A limitation of the problem-solving instruction is that the students tended to equate problem solving with the problem-solving skills they had learned, seeing them as “rules” to solve all problems. The implications of these findings for the reform of mathematics instruction are discussed.

The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond

The ArgumentThe belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs.

Sociomathematical Norms, Argumentation, and Autonomy in Mathematics

This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated. For the past several years, we have been engaged in a research and development project at the elementary school level that has both pragmatic and theoretical goals. On one hand, we wish to support teachers as they establish classroom environments that facilitate students' mathematical conceptual development. On the other hand, we wish to investigate children's mathematical learning in the classroom. The latter involves developing perspectives that are useful for interpreting and attempting to make sense of the complexity of classroom life. The purpose of this paper is to set forth a way of interpreting classroom life that aims to account for how students develop specific mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics, that is, how they come to develop a mathematical disposition (National Council of Teachers of Mathematics, 1991). To that end, we focus on classroom norms that we call sociomathematical norms. These norms are distinct from general classroom social norms in that they are specific to the mathematical aspects of students' activity. As a means of introducing and elaborating the theoretical discussion in this paper, we present episodes from a classroom that we have studied extensively. The episodes have been

Sociomathematical Norms, Argumentation, and Autonomy in Mathematics

This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.

A Problem‐Solving Based Mathematics Course and Elementary Teachers' Beliefs

This study examined the impact a problem‐solving based mathematics content course for preservice elementary education teachers (PSTs) had on challenging the beliefs they held with respect to mathematics and themselves as doers of mathematics. Nine PSTs were interviewed to gain insight into changes that occurred to their belief systems and what aspect(s) of the course were instrumental in producing those changes. Surveys to measure how strongly PSTs subscribed to five mathematical beliefs were administered to 137 PSTs who were enrolled in the course. Significant positive changes (p < .01) were observed for four of the five beliefs. When changes were studied by achievement level, students with final grades of A or B showed statistically significant changes (p < .005) in three of the five beliefs. Students who were interviewed consistently reported increased confidence in their mathematical abilities as a result of the course.

Learning not to think in a textbook-based mathematics class

This study coordinates anthropological and cognitive perspectives on one child's learning of the standard addition algorithm. Changes in the child's mathematical beliefs and constructions were analyzed as he moved from a experimental 2nd-grade mathematics class characterized by inquiry mathematics to a textbook-based third grade. Piagetian clinical interviews focusing on this mathematical understanding before, during, and after 8 weeks of instruction in third grade were coordinated with an interactional analysis of a typical textbook lesson selected from a larger microethnographic study of his classroom mathematical community. The resulting description of the social norms and mathematical practices of this community provide a background against which the child's mathematical development is examined. The analysis shows that he had abandoned his self-generated computational algorithms in favor of less understood conventional procedures.

Preservice Elementary Teachers and Self-Reflection: How Innovation in Mathematics Teacher Preparation Challenges Mathematics Beliefs

Preservice Elementary Teachers and Self-Reflection: How Innovation in Mathematics Teacher Preparation Challenges Mathematics Beliefs

One Preservice Secondary Teacher's Understanding of Function: The Impact of a Course Integrating Mathematical Content and Pedagogy

This study examines the evolving knowledge and beliefs of a preservice secondary mathematics teacher as she participated in a mathematics education course that emphasized mathematical and pedagogical connections and applications of the function concept. Her conceptions were revealed during a 10-week period through interviews, observations, and written work. The teacher's initial understanding of functions as computational activities (e.g., function machines, point plotting, vertical line test) was consistent with her larger view of mathematics as a collection of “concrete” procedures. Although her understanding of function grew substantially during the study, her anticipated approach to teaching, which was dominated by her narrow view of mathematics, was less significantly affected by course activities.

Students' Beliefs about Learning School Mathematics

In this study we considered the mathematical beliefs and achievement of 62 students in grades 1-6 from 1 school where teachers were participating in a project to improve mathematics teaching. Students, most of whom were from lower and lower-middle socioeconomic families, were interviewed about whether they liked school and mathematics, whether their parents helped them with schoolwork, whether they were confident of their abilities in mathematics, whether they felt mathematics was useful, and whether they felt all children had the ability to learn mathematics. Students were also tested for computation and problem-solving ability. Trends that emerged for this sample included (a) students in the upper-elementary grades were able to articulate their beliefs more clearly than students in the lower grades, (b) students who enjoyed mathematics were also confident of their ability in mathematics, (c) students with moderate achievement were just as confident of their abilities as high achievers, (d) most students...

Debates About Realism Transposed to a New Key

This book is concerned with a family of current and recent debates which may be loosely gathered under the title of realisms versus anti-realisms. Wright's first aim is to establish a set of theses which should be by all parties to the various debates-though they have not been so in fact. Setting aside these agreed theses, Wright proceeds to sketch some of the cruces at which, he thinks, realists and anti-realists should properly part company. Some of Wright's cruces are familiar, but some are radical reworkings of familiar cruces. In short, Wright proposes to generate a new agenda for debates about realism. Finally, Wright conjures up a spectre from Wittgenstein's rule following considerations which threatens to produce an outright victory for anti-realism, and Wright tries, tentatively, to contain this threat. The anti-realisms most discussed include theories of moral discourse (locus classicus John Mackie 1977) and of mathematical discourse (locus classicus Hartry Field 1980). Error theorists have held that our ordinary moral and mathematical assertions are apt for being or false, in a sense of true which is not merely disquotational but supports the idea that statements correspond to the moral or modal facts. But they claim that such statements are in fact false, because the world does not contain what would be required to make them true, and hence we are in massive and systematic error in our moral and mathematical beliefs. Other anti-realist theories in view are expressive, non-cognitive, projective accounts of moral or modal discourses, which claim that moral or modal utterances are not properly regarded as being apt for truth or falsity, if truth is not merely disquotational but supports the idea of correspondence to moral or modal facts. Also in the picture is the debate between a Dummettian realist, who claims that our grasp of truth conditions is such that truth is not constrained by possible evidence, and his opponent, the Dummettian anti-realist. The list of anti-realisms is left open-ended. Wright argues that all these various anti-realists should agree, pace expressivists, non-cognitivists, etc., that the target discourse, moral discourse or modal discourse for example, is apt for truth or falsity, where the concept of truth concerned is not merely disquotational but expresses a genuine property which

The use of computers in teaching mathematics—Beliefs and misbeliefs

Three case studies illustrate how mathematics teaching in the Swedish school system has moved away from routine manipulation to problem solving. A numbers game based on factors and primes requires mathematical thinking to gain an advantage. When tested with 134 student teachers, most of them failed to see how mathematics was involved in the game and tended to undervalue it for children's learning. Almost half the students objected to the way the computer ‘cheated’, although that was based on a mathematical device also. Another task involving derivatives demonstrated that students good at mathematics had difficulties in interpreting their meaning outside the routine and manipulative area. Similarly, when presented with a cubic equation, 85% of students considered the use of the computer to be an inaccurate method compared with the analytical solution. The students and the researcher agreed the following criteria for ability in mathematics: mathematicians should be able to observe and formulate problems, model problems mathematically, reformulate problems for the tools at hand, select methods and tools, and decide when a problem was solved.

Reliability in Mathematical Physics

In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematical physics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.

Compassionate Math

The current emphasis on nonroutine problem-solving techniques causes mathematics teachers to confront anxiety in their students more than ever. Using methods from Transactional Analysis, caring teachers can expand their affective role in helping students with mathematics anxiety by showing them how to get past their resentments and limiting beliefs in mathematics.

What Is Empiricism?

What Is Empiricism?

What Myths About Mathematics are Held and Conveyed by Teachers?

As teachers of mathematics, we want to help our students to develop a solid understanding of mathematical concepts and to become good problem solvers who are confident of their ability to do mathematics. Research on students' beliefs about mathematics reveals a set of mathematical beliefs, “math myths,” that can be counterproductive to achieving these goals. This article is a report on math myths believed by a group of preservice teachers. However. as you read, you might want to ask yourself these questions: Which of these myths do I believe? Which of them could my teaching methods convey to my students?

Explorations of Students' Mathematical Beliefs and Behavior

Explorations of Students' Mathematical Beliefs and Behavior

Explorations of Students' Mathematical Beliefs and Behavior

A questionnaire with 70 closed and 11 open questions was administered to 230 mathematics students enrolled in Grades 10 through 12, the majority of whom were enrolled in the traditional year-long 10th-grade course in plane geometry. Sections of the questionnaire dealt with the students' attributions of success or failure; their comparative perceptions of mathematics, English, and social studies; their view of mathematics as a discipline; and their attitude toward mathematics. The data, which are closely tied to a series of classroom and protocol studies, suggest the resolution of contradictory patterns of data in other attitude surveys, where students simultaneously claim that “mathematics is mostly memorizing” but that mathematics is a creative and useful discipline in which they learn to think.

Teacher's Pedagogical Content Beliefs in Mathematics

This study examined relationships among first-grade teachers' pedagogical content beliefs, teachers' pedagogical content knowledge, and students' achievement in mathematics. Teachers (N = 39) completed structured questionnaires and interviews on their beliefs and knowledge about instruction, children's learning, and the mathematics content in addition and subtraction. Results indicated significant positive relationships among teachers' beliefs, teachers' knowledge, and students' problem-solving achievement. Compared to teachers with a less cognitively based perspective (LCB teachers), teachers with a more cognitively based perspective (CB teachers) made extensive use of word problems in introducing and teaching addition and subtraction. They also spent time developing children's counting strategies before teaching number facts. CB teachers had greater knowledge of word problem types and greater knowledge of their children's problem-solving strategies than did LCB teachers. Furthermore, CB teachers obtained this latter knowledge by observing their children in problem situations rather than by relying on tests or formal assessments. Children with CB teachers scored higher on word problem-solving achievement than did children with LCB teachers, but children from both types of classes did equally well on addition/subtraction number facts.