The interactive constitution of mathematical meaning in one second grade classroom: an illustrative example

The manner in which a horizontal addition and subtraction number sentence activity was constituted in one second grade classroom is analyzed for the purpose of discussing and illustrating how mathematical meaning is interactively constituted in the classroom. In particular, the teacher's emphasis on different solutions contributed to students' development of increasingly sophisticated concepts of ten. In turn, students' solutions contributed to the teacher's development of an increasingly sophisticated understanding of the children's mathematical activity and their concepts of ten.     

Asking mathematical questions mathematically

It is proposed that the style and format of the questions used by lecturers and tutors profoundly influence students' conceptions of what mathematics is about and how it is conducted. By looking at reasons for asking questions, and becoming aware of different types of questions which mathematicians typically ask themselves, we can enrich students' experience of mathematics. Drawing on recent work by Watson and Mason stimulated by the ideas of Zygfryd Dyrszlag, the paper proposes that mathematical themes, powers, heuristics and activities generate a mathematical discourse which is not always represented in the questions students are asked, and that the real purpose of questions is to provoke students into construal, into constructing their own stories which constitute meaning and understanding, and which equip them for the future. The use of questions of whatever type depends on both scaffolding and fading their use with and in front of students, so that students internalize the questions into their own learning and doing of mathematics. The framework directed—prompted—spontaneous is proposed as an alternative to scaffolding—fading for informing interactions with students.     

From formal standards to everyday practice of mathematics learning

This paper uses the example of six Japanese teachers and their mathematics lessons to illustrate how clear, high standards for mathematics instruction are combined with teachers' holistic concern for students. We draw upon data from the Third International Math and Science Study Case Study Project in Japan that was designed to elucidate the context behind the high achievement of Japanese students. Using everyday examples of classroom practice, we illustrate both flexibility in teachers' approach to teaching and adherence to Monbusho's (Ministry of Education, Science, Sports, and Culture)Course of Study. Our purpose is to emphasize how flexibility and attention to individual needs by Japanese teachers combine with quality mathematics instruction based on the detailed Japanese curricula. Six teachers' characteristics and lessons (two teachers at each educational level—elementary, junior high, and high school) are described in order to show the variety of teachers who exist in Japan. These teachers use their understanding of theCourse of Study and are supported by their school environment to enhance their students' conceptual understanding of the fundamentals of mathematics. Characteristics of their teaching include: 1) involving the whole class in learning. 2) using extremely focused curriculum guidelines that expect mastery of concepts at each grade level, 3) thoroughly covering mathematics units in an organized and in-depth manner, 4) leading classes as facilitators or guides more often than as lecturers, and 5) focusing on problem solving with the primary goal of developing students' ability to reason, especially to reason inductively. The examples in this paper show how these methods develop in individal classrooms.     

Middle Grades Students' Algebraic Understanding in a Reform Curriculum

The National Council of Teachers of Mathematics' Curriculum and Evaluation Standards in 1989 was pivotal in mathematics reform. The National Science Foundation funded several curriculum projects to address the vision described in the Standards. This study investigates students' learning in one of these Standards‐based curricula, the Connected Mathematics Project (CMP). The authors of CMP believe that the teaching and learning of algebra is an ongoing activity woven through the entire curriculum, rather than being parceled into a single grade level. The content of the study investigates students' ability to symbolically generalize functions. The data regards the solutions of four performance tasks dealing with three different types of relationships—linear, quadratic, and exponential situations—completed by five pairs of eighth‐grade students. The major finding claims that middle to high achieving students who had 3 years in the CMP curriculum demonstrated achievement in five strands of mathematical proficiency of a significant piece of algebra.     

Computer Use and Visualization in Students' Construction of Reflection and Rotation Concepts

The effects of a dynamic instructional environment (based on use of The Geometer's Sketchpad, N. Jackiw, 1991 , in a computer lab) and visualization on eighth-grade students' (N= 241) construction of the concepts of reflection and rotation were investigated. Also investigated were the effects of the environment on students' two- and three-dimensional visualization. After controlling for initial differences, it was concluded that students experiencing the dynamic environment significantly outperformed students experiencing a traditional environment on content measures of the concepts of reflection and rotation, as well as on measures of two-dimensional visualization. The students' environment did not significantly affect their three-dimensional visualization.     

Design of a logo environment for elementary geometry

One rationale for Logo programming is that students will learn geometry for utilizing concepts that aid them in understanding and directing the Logo turtle's movements. Research on this claim has yielded mixed results; however, an analysis of these research findings provides significant guidance in the teaching and learning of geometry with Logo. This article (a) reviews the body of research that led to the design of a new Logo environments tailored for elementary students' learning of geometry; (b) describes this environment, Geo-Logo, and its connections to this research corpus; and (c) presents empirical data from the first field test of the environment, which generally support the efficacy of the design.     

Fuzzy Logic,Neural Networks,Genetic Algorithms: Views of Three Artificial Intelligence Concepts Used in Modeling Scientific Systems

Students' conceptions of three major artificial intelligence concepts used in the modeling of systems in science, fuzzy logic, neural networks, and genetic algorithms were investigated before and after a higher education science course. Students initially explored their prior ideas related to the three concepts through active tasks. Then, laboratories, project work, use of computer modeling of scientific systems, and cooperative group work were used to help students construct key characteristics of each concept. Finally, they applied each concept in contexts different from that in which it had been previously studied. In postcourse interviews using a set of scenarios for each of the major course concepts, 49% of students' applications included key characteristics of the concepts studied versus an application of 5% in precourse interviews. Students' post interview applications were inconsistent even though they were more frequent, indicating a state of transition in their conceptual change. Applications were most consistent when used with scenarios deemed very familiar to the students, indicating the effects of context in conceptual change.     

High School Students' Goals for Working Together in Mathematics Class: Mediating the Practical Rationality of Studenting

In this article I explore high school students' perspectives on working together in a mathematics class in which they spent a significant amount of time solving problems in small groups. The data included viewing session interviews with eight students in the class, where each student watched video clips of their own participation, explaining and justifying their behaviors. Analysis of data involved an investigation of students' goals for working together, which were found to vary along multiple dimensions. The dimensions that emerged from these data were mathematical versus nonmathematical goals, individual versus group goals, and personal versus normative goals. I present cases of four individual students to illustrate these dimensions. Such goals are important for illuminating how students' practical rationality is mediated by their personal goals for working together; additionally, these goal dimensions can be used as tools for considering challenges involved with using small group collaboration in high school classes where students' goals may be diverse.     

Taiwanese Gifted Students' Views of Nature of Science

This study examined the conceptions of nature of science (NOS) possessed by a group of gifted seventh‐grade students from Taiwan. The students were engaged in a 1‐week science camp with emphasis on scientific inquiry and NOS. A Chinese version of a NOS questionnaire was developed, specifically addressing the context of Chinese culture, to assess students' views on the development of scientific knowledge. Pretest results indicated that the majority of participants had a basic understanding of the tentative, subjective, empirical, and socially and culturally embedded aspects of NOS. Some conflicting views and misconceptions held by the participants are discussed. There were no significant changes in students' views of NOS after instruction, possibly due to time limitations and a ceiling effect. The relationship between students' cultural values and development of NOS conceptions and the impact of NOS knowledge on students' science learning are worth further investigation.     

On problematic aspects in learning trigonometry

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.

The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.     

Analogies and Reconstruction of Probability Knowledge

The effectiveness of utilizing analogies to effect conceptual change in students' alternative probability concepts was investigated. Forty-one senior high school mathematics students were engaged in a knowledge reconstruction process regarding their beliefs about common everyday probability situations, such as sports events or lotteries. The students were given situations similar to those shown in previous research to reveal alternative mathematical conceptions. They were also given analogous researcher-generated anchoring situations that had been pretested and found to elicit mathematically acceptable responses. The cognitive dissonance produced by the conflicting responses motivated students to reconstruct their knowledge. The results of the investigation showed that analogies can be effective in producing a desired conceptual change in high school students' probability concepts.     

Table Bases as Unions of Proper Closed Subsets

If the distinguished basis of a table algebra is an irredundant union of n proper closed subsets, and if the positive structure constants of the quotient table algebra (rescaled to be standard) modulo the intersection of these closed subsets are all at least 1, then it is proved that the order of this quotient algebra is bounded above by a function of n. This generalizes a result of B. H. Neumann for finite groups, applies directly to association schemes, and also yields the following result: if G is a finite group, ?? is the set of minimal members (with respect to containment) of the set of kernels of irreducible characters of G, and N = ∏ _K∈?? K, then |N| is bounded above by a function of |??|. Table algebras where the table basis is a union of three or four proper closed subsets are characterized as well.     

Small-group searches for mathematical proofs and individual reconstructions of mathematical concepts

This is a study of mathematics students working in small groups. Our research methodology allows us to examine how individual ideas develop in a social context. The research perspective used in this study is based on a co-constructive view of learning. Groups of three or four undergraduate mathematics majors, with prior experience writing mathematical proofs together, were asked to prove three statements. Computer software, such as Geometers Sketchpad, was available. Group work sessions were videotaped. Later, individuals viewed segments of the group video and were asked to reflect on group activities. Students in some groups did not share a common conception of proof, which seemed to hamper their collaboration. We observed interactions that fit with the co-constructive theory, with bidirectional interactions that shaped both group and individual conceptions of the tasks. These changes in understanding may result from parallel and successive internalization and externalization of ideas by individuals in a social context.     

Investigating Elementary Teachers' Conceptions of Animal Classification

Numerous studies have been conducted regarding alternative conceptions about animal diversity and classification, many of which have used a cross‐age approach to investigate how students' conceptions change over time. None of these studies, however, have investigated teachers' conceptions of animal classification. This study was intended to augment the findings of past research by exploring the conceptions that elementary teachers possess about animal classification. Using interviews and written items, we documented teachers' conceptions about animal classification and compared them with student conceptions identified in previous research studies. Many of the teachers' conceptions observed in this study were similar to students' conceptions in that they were often too limited or too general compared with scientifically accepted conceptions. Also, the teachers in this study frequently used “non‐defining” characteristics, such as locomotion and habitat, to classify animals. As a result, several misclassifications were observed in the teachers' responses to the written items. Notably, the results of our study demonstrate that teachers often have the same alternative conceptions about animal classification as students. We explore some possible explanations for these alternative conceptions and discuss the instructional implications of the findings.     

Associativity in multiary quasigroups: the way of biased expansions

A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an additional structure similar to the combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative structure. Some but not all biased expansions are constructed from groups (group expansions); these include all biased expansions of complete graphs (with at least four nodes), which correspond to Dowling’s lattices of a group and encode an iterated group operation. We show that any biased expansion of a 3-connected graph (with at least four nodes) is a group expansion, and that all 2-connected biased expansions are constructed by the identification of edges from group expansions and irreducible multiary quasigroups. If a 2-connected biased expansion covers every base edge at most three times, or if every four-node minor that contains a fixed edge is a group expansion, then the whole biased expansion is a group expansion. We deduce that if a multiary quasigroup has a factorization graph that is 3-connected, or if every ternary principal retract is an iterated group isotope, it is isotopic to an iterated group. We mention applications of generalizing Dowling geometries and of transversal designs of high strength.     

Teachers' Conceptions of Integrated Mathematics Curricula

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state‐mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions—integration by strands, integration by topics, interdisciplinary integration, and contextual integration—refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways.     

The Contribution of Conceptual Change Texts Accompanied by Concept Mapping to Students' Understanding of the Human Circulatory System

The present study was conducted to investigate the contribution of conceptual change texts accompanied by concept mapping instruction to 10^th— grade students' understanding of the human circulatory system. To determine misconceptions concerning the human circulatory system, 10 eleventh-grade students were interviewed. In the light of the findings obtained from student interviews and related literature, the Human Circulatory System Concepts Test was developed. The data were obtained from 26 students in the experimental group taught with the conceptual change texts accompanied by concept mapping, and 23 students in the control group taught with the traditional instruction. Besides treatment, previous learning in biology and science process skills were other independent variables involved in this study. Multiple Regression Correlation analysis revealed that science process skill, the treatment, and previous learning in biology each made a statistically significant contribution to the variation in students' understanding of the human circulatory system. It was found that the conceptual change texts accompanied by concept mapping instruction produced a positive effect on students' understanding of concepts. The mean scores of experimental and control groups showed that students in the experimental group performed better with respect to the human circulatory system. Item analysis was carried out to determine and compare the proportion of correct responses and misconceptions of students in both groups. The average percent of correct responses of the experimental group was 59.8%, and that of the control group was 51.6% after treatment.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.