Focus on Learning Problems in Mathematics

A scholarly journal publishing original research on issues that affect mathematics learning. Content is interdisciplinary and drawn from such fields as mathematics, special education, psychology, and neurology. Coverage includes current research, methods


Vol. 30, No. 1, Winter

Kindergarten and First Graders' Knowledge of the Number Symbols: Production and Recognition
The Hindu-Arabic number system represents amounts of objects by number symbols, without referring to other properties of these objects (e.g., color, size). That is to say that the number symbols, like many other symbols, are linked to the objects that...
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Teacher-Developed Mathematics Performance Assessments in the Context of Reform-Based Professional Development
One of the disappointments associated with the mathematical reform movement is the increasing mismatch between the improvements made in curriculum and instruction and prevalent assessment modes (Firestone & Schorr, 2004; Niss, 1993). Despite early...
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A Complex System Analysis of Practitioners' Discourse about Research
The recent past has seen an increasing frequency of calls for teachers to implement "evidence-based" practices (Davies, 1999; U.S. Department of Education, 2002). At the same time, it has been noted that teachers see research as largely irrelevant...
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From an Operational to a Relational Conception of the Equal Sign: Third Graders' Developing Algebraic Thinking
As algebra has taken a more prominent role in mathematics education, many are advocating introducing children to it in primary school (e.g., Carraher, Schliemann, Brizuela, & Earnest, 2006; Kaput, 2000; National Council of Teacher of Mathematics,...
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Vol. 29, No. 4, Fall

Class within a Class: A Systematic Approach to Teaching High School Mathematics Students with Special Needs
Abstract A general education teacher completed an action research project in order to serve his students with special needs better by truly incorporating a special education teacher in co-teaching. By doing so with a set routine each day, the students...
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A Framework for Identifying Stages of Teacher Change Resulting from Extended Mathematics Professional Development
Abstract With the advent of reform-based curricula and recommendations related to the teaching of mathematics from the National Council of Teachers of Mathematics (NCTM, 2000), teachers need access to ongoing professional development which models...
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Listening as a Vital Element in Synergistic Argumentation during Mathematics Problem Solving
Abstract From research observations of activities during a second-grade mathematics problem-centered learning classroom, synergistic argumentation emerged as a class norm for discussing the students' mathematics. In this paper we analyze the contrast...
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Efficacy Beliefs, Problem Posing, and Mathematics Achievement
Abstract Perceived self-efficacy beliefs have been found to be a strong predictor of mathematical performance while problem posing is considered fundamental in mathematical learning. In this study we examined the relation among efficacy in problem...
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Vol. 29, No. 3, Summer

Preservice Elementary Teachers' Visual Images of Themselves as Mathematics Teachers
Abstract This study investigated preservice elementary teachers' pictorial representations of how they envision their future classroom and describe their own actions as well as those of their students. Drawings were analyzed for the preservice teachers'...
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The Construction of a Classification Schema as a Foundation for Mathematical Understanding in Young Deaf Children
Abstract While it is known that the mathematics achievement levels of deaf children are substantially below that of their hearing peers, it is not known when or in what capacity these delays begin. It is possible that differences in achievement...
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Solving Problems on Functions: Role of the Graphing Calculator
Abstract To study the roles that the graphing calculator plays in solving problems about functions, a small quasi-experimental study was conducted with four pairs of undergraduate students solving problems with and without the graphing calculator....
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Guided Notes: An Interactive Method for Success in Secondary and College Mathematics Classrooms
Abstract This paper reports the results of an action research project that examined the use of interactive guided notes in two sections of freshman level college algebra. This method unifies lecture, in-class guided practice, and cooperative learning...
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Vol. 29, No. 1, Winter

A Preliminary Investigation of Placement and Predictors of Success for Students with Learning Disabilities in University-Required Mathematics Courses
Abstract The number of students with learning disabilities attending institutions of higher education has dramatically increased in the last ten years and will continue to do so. Since federal laws in the United States and other counties require...
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Some Insights about College Students' Interpretations of Histograms
Abstract The interpretation of histograms is a complex process requiring the integration of understanding about how graphs convey information with knowledge about how statistical constructs are displayed graphically. For this study, students in...
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They Know the Math, but the Words Get in the Way
Abstract This study sought to better understand instructional models that could be expected to improve student understanding of graphs of kinematic variables (distance, velocity and acceleration). The effect of using CBL-instruments and cooperative...
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Attitudes toward Mathematics of Precalculus and Calculus Students
Introductory college mathematics courses comprise a large percentage of course offerings in postsecondary institutions, serving over half of all students who ever study mathematics in college (Cohen, 1995). In a report of mathematics classes offered...
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Vol. 28, No. 3-4, Summer

Is There a Role for Concept Mapping in Mathematics? an Introduction to This Special Issue
Contributors to this issue were, together with the editor, all participants in the first International Conference on Concept Mapping held in Pamplona, Spain, in September 2003. As a group of mathematics educators, we met and questioned why comparatively...
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Thde Development of the Concept Mapping Tool and the Evolution of a New Model for Education: Implications for Mathematics Education
Introduction When I began my graduate studies in 1952 at University of Minnesota, the only psychology of learning presented was behavioral psychology, based largely on research with rats, cats and other animals. The only philosophy of knowledge,...
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Concept Maps: An Essential Tool for Teaching and Learning to Learn Science
Abstract This article is meant to offer math teachers a possibility to initiate their own study of concept maps, a powerful heuristic tool designed by Joseph Novak, on the ground that this tool can effectively help university students to face many...
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Developing a More Conceptual Understanding of Matrices & Systems of Linear Equations through Concept Mapping and Vee Diagrams
Abstract The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating...
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Substantive Knowledge and Mindful Use of Logarithms: A Conceptual Analysis for Mathematics Educators
Abstract This study uses concept mapping to investigate the logarithm. Empirical evidence documents the extent to which selected instructors and students command substantive knowledge about logarithms and analyzes their ability to make mindful use...
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Concept Mapping: Important Elements for Intervention
Abstract The aim of this paper is to present the results of an experiment carried out in the Mexican Autonomous Metropolitan University (UAM) based on recent theories of educational research on building up knowledge. The research group was studying...
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Vol. 28, No. 2, Spring

Conjecturing in a Computer Microworld: Zooming out and Zooming In
Teachers and researchers agree that making and testing conjectures is an invaluable part of mathematical learning. This belief is reflected in the NCTM's Reasoning and Proof standard, which states that students at all grade levels should have opportunities...
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CPMP III versus Algebra II
Abstract The research question was: When a non-paper-and-pencil test is used as the source of measurement, do there appear to be differences by students' mathematics curriculum, Algebra II or CPMP III, in the mastery level of problem-solving and...
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Sixth Graders' Oral Retellings of Compare Word Problems
Abstract In this study, we explored the use of oral retellings as a problem solving strategy for compare word problems in mathematics. Twenty-nine students in two sixth-grade mathematics classes, one class labeled as average achieving and the other...
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Vol. 28, No. 1, Winter

Implementation of Multiple Solution Connecting Tasks: Do Students' Attitudes Support Teachers' Reluctance?
Abstract This study focuses on the pitfalls associated with incorporating multiple solution connecting tasks in classroom instruction. We found implementation of these standard-based materials problematic due to teachers' reluctance to use them...
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And Now Put Aside Your Pens and Calculators: On Mental Problem Solving in the High School Mathematics Classroom
The aim of this article is to draw attention to the use of mental mathematics--to promote its use, and to pose several questions connected with it. Unfortunately, the psychological problems associated with the reception of teaching materials, the teaching...
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Students' Preferences When Solving Quadratic Inequalities
Abstract We address the question whether to present students with a single method or with a number of methods for solving quadratic inequalities. Twenty 10th graders were presented with three differently sequenced methods, i.e., the graphic, the...
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Vol. 27, No. 4, Fall

Development of Reasoning about Random Events
Abstract The development of school students' understanding of random events is explored in three related studies based on tasks well known in the research literature. In Study 1,99 students in Grades 3 to 9 were interviewed on three tasks and surveyed...
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Assessing Zimbabwean Children's Mathematics Problem Solving for Cognitively Guided Instruction
Abstract Cognitively Guided Instruction (CGI) has been highly effective in helping elementary school children in America develop number sense and mathematics problem solving ability. This study attempted to determine if children in Zimbabwe, a developing...
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Vol. 27, No. 3, Summer

Rigorous Mathematical Thinking: Mediated Learning and Psychological Tools
Abstract The proposed rigorous mathematical thinking paradigm is based on two theories of learning: Vygotsky's sociocultural theory, with particular emphasis on his concept of psychological tools as mediators of cognitive processes, and Feuerstein's...
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In-Service Elementary Mathematics Teachers' Views of Errors in the Classroom
Educators generally agree that familiarity with students' conceptions and ways of thinking about mathematical topics are essential for teachers (e.g., Australian Education Council, 1991; Even & Tirosh, 2002; Fennema, Carpenter, Franke, Levi, Jacobs,...
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Exploring Mathematical Exploration: How Two College Students Formulated and Solved Their Own Mathematical Problems
Preparation of this paper is supported, in part, by a grant from the University of Delaware Research Foundation and a grant from the University of North Carolina at Charlotte. However, any opinions expressed herein are those of the authors. The first...
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Vol. 27, No. 1-2, Spring

Culture, Communication, and Mathematics Learning: An Introduction
History is replete with politically and ideologically motivated resistance to change in teaching mathematics. In 415 AD, St. Cyril, Christian Patriarch of Alexandria ordered the assassination of the Greco-Egyptian mathematician Hypatia. She was taken...
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Social Constructivism in Practice: Case Study of an Elementary School's Mathematics Program
Abstract This research investigated implications for the implementation of social constructivist epistemology on teaching/learning of mathematics in a K-4 public school with particular focus on African American fourth grade students. In addition,...
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Culture, Social Interaction, and Mathematics Learning
Abstract The research report investigates connections between five 3rd garde students (one African American boy, one African American girl, one Caucasian girl, one Asian boy, and one Middle Eastern boy) belief, attitudes, and practices about mathematics...
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Passion for Teaching/learning Mathematics: A Story of Two Fourth Grade African American Students
Abstract The intent of this study was to investigate the experiences and reflections of two African American children, their teachers, and their parents about mathematics learning and what these experiences imply for educators as they attempt to...
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The Impact of a State Mathematics Test on the Structure and Culture of a K-4 School
Abstract This study explored the impact of state mathematics testing on the structure and the culture of a K-4 school. We selected a school that was teaching students to the state mathematics test. Students at the school were doing well on the test....
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Vol. 26, No. 4, Fall

Learning Disabilities in Mathematics: A Review of the Issues and Children's Performance across Mathematical Tests
Abstract In recent years, the research literature on mathematics disabilities (MD) has shown relative growth. The characteristics of children with MD have thus been investigated from different perspectives. The purpose of this paper was to review...
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Middle School Mathematics Classroom Practices and Achievement: A TIMSS-R Analysis
Abstract Recent debate in mathematics education has focused attention on the value of teaching mathematics reform-oriented strategies and/or to take a more "back-to-basics" approach. Data from the TIMSS-R Population 2 (13 year-olds) International...
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On Two Types of Learning (in Mathematics) and Implications for Teaching
Abstract A preliminary identification is made of two types of understanding--nominal and explanatory. The distinction has a bearing on teaching mathematics. To help illustrate the ideas, a detailed outline for an introduction to the dot product...
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Are Visual Perceptual Skills Related to Mathematics Ability in Second through Sixth Grade Children?
Abstract Controversy exists regarding the relation between visual perceptual ability and mathematics. The purpose of this study was to investigate whether or not poor mathematics achievement was related to performance on tests of visual discrimination...
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Vol. 26, No. 3, Summer

K-8 Teacher Candidates' Use of Mathematical Representations and the Development of Their Pedagogical Content Knowledge as Exhibited in Their Lesson Planning
Abstract A study was carried out involving thirty-one K-8 teacher candidates enrolled in an elementary mathematics methods course to investigate and document their thinking as they plan for mathematics instruction. The teacher candidates submitted...
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Pre-Service Elementary Teachers' Understanding of Geometric Similarity: The Effect of Dynamic Geometry Software
Since the use of technology in mathematics classrooms has increased dramatically during the past few decades, critical issues such as the role of technological advances in the learning and doing of mathematics need to be addressed (Balacheff &...
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Repeating Pattern or Number Pattern: The Distinction Is Blurred
"Life itself is a creator of patterns." (Piaget, 1950, p. 167) Introduction Many argue that patterns are the cornerstone of mathematics. They are the foundation that the whole of the subject is built on. From the earliest tally systems to the development...
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Inaccurate Mental Addition and Subtraction: Causes and Compensation
Abstract This paper reports on a study of seven Year 3 students' diminished performance in mental computation, and compares their mental architecture. Although all students were identified as being inaccurate, three students used some variety of...
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Vol. 26, No. 2, Spring

The Role of Technology in Students' Conceptual Constructions in a Sample Case of Problem Solving
It is well recognized that an important component in mathematical instruction is to provide students with an opportunity to develop and use diverse representational systems in order to solve a variety of mathematical tasks (National Council of Teachers...
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The Effect of a Problem Centered Approach to Mathematics on Low-Achieving Sixth Graders
ABSTRACT: Twenty-six sixth grade low achievers in the rural south experienced a problem centered mathematics curriculum for nine weeks. Potentially meaningful tasks were utilized in the class, which was divided into small collaborative groups of...
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Preservice Teachers' Number Sense
Many calls for reform in school mathematics emphasize number sense (Australian Education Council, 1991; Cockroft, 1982; Japanese Ministry of Education, 1989; National Council of Teachers of Mathematics, 1989; 2000; Swedish Ministry of Education and...
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A Review of Literature on Mathematics Instruction for Elementary Students with Learning Disabilities
Introduction The majority of students with learning disabilities (LD) are taught in the general education classroom today because of the trend toward inclusive services in special education. However, there is limited literature on teaching math...
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Vol. 26, No. 1, Winter

The Importance of the Statement in Addition and Subtraction Word Problems
1. Introduction The research carried out on simple addition and subtraction word problems (solved by x + y = z or x - y = z) has been very extensive. Ample bibliographical details on this subject may be found in research surveys by Fuson (1992)...
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Beginning Secondary Mathematics Teachers: A Snapshot across One State
Introduction The National Council of Teachers of Mathematics (NCTM) has called for reform in the teaching of mathematics (1989; 2001). Beginning secondary mathematics teachers are being educated in mathematics methods courses to teach by NCTM-oriented...
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The Role of Playing Games in Developing Algebraic Reasoning, Spatial Sense, and Problem-Solving
Introduction and Literature Review Purpose and Relevance Using games in the classroom to facilitate learning has been common practice among teachers for many years. Blum and Yocum (1996) supported instructional game-playing in the classroom because...
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Girls' Experiences in Learning School Mathematics
During the past few decades, gender issues in mathematics education emerged as a controversial topic. Researchers found that female students, at both elementary and secondary levels, perform less well than their male counterparts on standardized tests...
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Vol. 25, No. 4, Fall

Issues and Challenges in Teacher Education
Introduction and Background From its origin, ACME has been about maximizing mathematics learning, focusing both on student learning and what teachers do to help or hinder student's mathematics learning. Thus, the preparation of teachers has been...
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Basing Instruction on Theory and Research: What Is the Impact of an Extreme Case?
Introduction In the last decade, it has been widely recommended that mathematics teaching consider and address students' correct and incorrect ideas concerning the subject matter (NCTM, 1991; 2000). Teachers are expected, among other things, to...
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Language, Arithmetic and Young Children's Interpretations
Throughout the last two decades the language of mathematics has been acknowledged as being important to the development of mathematics understanding. It is believed that one of the main reasons children have difficulties with mathematics is the learning...
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Using Metacognitive Skills to Improve 3rd Graders' Math Problem Solving
For the past twenty years, problem solving has been touted as a primary focus for mathematics instruction at all grade levels (National Council of Supervisors of Mathematics, 1978; National Council of Teachers of Mathematics, 1980, 1989, 2000; Mathematical...
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Vol. 25, No. 3, Summer

Profiling Coursework Patterns in Mathematics: Grades 8 to 12
Interest in student coursework, especially in mathematics and science, has been rapidly increasing over the past decade, because the emerging global economy requires new generations to become more knowledgeable in mathematics and science and more...
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The Persuasiveness of Computer-Based Simulations on Students' Probabilistic Misconceptions
People of all ages have been found to have misconceptions and lack sound intuition in situations of uncertainty (Kahneman, Slovic, & Tversky, 1982; Bar-Hillel & Falk, 1982; Shaughnessy, 1981; Shaughnessy, 1992). Further, many people hold...
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Get It Right and Get It Fast! Building Automaticity to Strengthen Mathematical Proficiency
Introduction The important role mathematical fluency, or automaticity, plays in daily life is perhaps best appreciated when the fluency is absent. Have you ever spent time in a checkout line as the shopper in front of you slowly counted out the...
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Vol. 25, No. 2, Spring

From "Easy" to "Difficult" or Vice Versa: The Case of Infinite Sets
It is commonly agreed that students' ways of thinking should be taken into account when planning instruction and that teachers should choose or construct sequences of instruction for use and discussion in class. The NCTM document, "Principles and...
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The Development of Children's Understanding of Equality and Inequality Relationships in Numerical Symbolic Contexts
A fundamental concept in arithmetic and algebra is the equivalence relationship represented by the '=' symbol. In order to understand equations such as 7+8 = 7+2+6 or 2x+2 = 3x-6, children have to understand that the term 'equals' as represented...
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Patterns of Reasoning about Mathematical Models: A Case Study of High School Mathematics Teachers
This case study illustrates the results of instruction in a mathematical modeling course for high school mathematics teachers. In this course the teachers were introduced to the process of mathematical modeling. They explored various problems and...
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Vol. 25, No. 1, Winter

Different Representations in Instruction of Vertical Angles
It is widely accepted that teachers should be aware of and familiar with students' mathematical ideas and conceptions, and it is believed that such awareness and knowledge should play a significant role in planning and carrying out instruction (e.g.,...
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Instructional Decisions: Helping Students Build Links between Representations
Instructional decisions about the nature and extent of teacher input in mathematics classrooms typically lie on a continuum somewhere between teachers directly telling students how to carry out mathematical procedures and free exploration situations...
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Learning Mathematics in Community Accommodating Learning Styles in a Second-Grade Problem Centered Classroom
A typical elementary classroom in the United States consists of anywhere between 20 and 30 students of varying abilities and learning styles with the teacher responsible for providing educational opportunities for students in all major subject areas....
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A Survey of Paper Cutting, Folding and Tearing in Mathematics Textbooks for Prospective Elementary School Teachers
Origami or the art of paper folding receives substantial endorsement from current reform initiatives in mathematics education. Particularly, at the elementary school level, the National Council of Teachers of Mathematics in its Principles and Standards...
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Vol. 24, No. 2, Spring

Trends in the Degree of Importance Assigned to the NCTM's Standards by Elementary Preservice teachers.((National Council of Teachers of Mathematics)(Statistical Data Included)
Introduction The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) was issued to reflect what should be of value and to promote reform in mathematics education. The Professional...
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The Competition between Numbers and Structure: Why Expressions with Identical Algebraic Structures Trigger Different Interpretations
The common perception of school algebra as generalized arithmetic implies that algebraic structural rules are perceived as rules that draw their legitimization and meaning from rules that are valid in the world of numbers (Buxton, 1984; Davis, 1985;...
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Aspects of Word-Problem Context That Influence Children's Problem-Solving Performance
Students tend to deem word problems one of the most distasteful and anxiety-inducing tasks in the mathematics classroom. They--at least those in the United States--perform poorly on solving word problems (e.g., Kouba, Brown, Carpenter, Lindquist,...
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EPA2000: Assessing Off-Line Metacognition in Mathematical Problem Solving
Introduction Research from different theoretical approaches has provided information regarding processes that are important for young children to solve mathematical problems adequately (Donlan, 1998; Koriat, 1995; Lucangeli & Cornoldi, 1997;...
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Vol. 24, No. 1, Winter

Challenges to Reform: An Overview of the Study
In 1997, the Department of Education of a mid-western state contracted with a Canadian university to coordinate research on school change initiatives. By spring of 1997, twelve schools from a sample of 500 nominated schools were selected for study....
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The Challenges of Instructional Leadership School Renewal
Two major publications addressed the issue of reforming American mathematics education in 1989. These reports proposed wide-ranging, radical changes in mathematics learning and teaching. Everybody Counts: A Report to the Nation on the Future of...
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Authentic Assessment: A School's Interpretation
In the previous paper we discussed the challenges of instructional leadership for reforming mathematics education in a K-4 school. In this paper we describe how the school principals and teachers-leaders developed authentic assessment consistent...
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Exploring the Evolving Nature of Three Elementary Preservice Teachers' Beliefs and Practices: Three Parallel Case Studies
The two preceding papers reported on the complexity of reforming mathematics education in one Midwestern state K-A elementary school. This paper focuses on teacher preparation for mathematics teaching and follows three preservice elementary teachers...
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