Search

Fractions remain a challenging area of school mathematics at every stage of education, with impacts that extend far beyond the school years. For this study, researchers engaged in classroom-based design research over a 6-year period to investigate effective strategies for teaching fractions with Canadian students. Participants included 86 teachers (representing 12 collaborative research teams spread across 8 school boards) and over 2000 students from Grades 3-10. Quantitative analyses revealed significant pre-post gains in students' fraction knowledge. Qualitative findings revealed some best practices in fractions instruction, including the importance of focusing on unit fractions and number lines to facilitate student sense-making. These findings lead to a detailed discussion of the benefits of (1) focusing on unit fractions as a central construct that allows students to meaningfully work with fractions and make connections across ideas of increasing complexity; (2) leveraging powerful representations as objects-to-think-with that combine concrete and abstract thinking about fractions; and (3) using a design research methodology in the context of collaborative work with teachers.

The study of linear relationships is foundational for mathematics teaching and learning. However, students' abilities to connect different representations of linear relationships have proven to be challenging. In response, a computer-based instructional sequence was designed to support students' understanding of the connections among representations. In this paper we report on the affordances of this dynamic mode of representation specifically for students with learning disabilities. We outline four results identified by teachers as they implemented the online lessons. [This document was originally published as Beatty, R., & Bruce, C. D. (2011). Supporting students with learning disabilities to explore linear relationships using online learning objects. In B. Ubuz (Ed.), "Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education" (Vol. 2, pp. 129-136). Ankara, Turkey: PME.]

This study examined the effects of peer coaching on mathematics teaching practices and teacher beliefs about their capacity to have an impact on student learning. Twelve teachers in grades 3 and 6 participated in a brief but intensive professional development program over six months. The program focused on effective mathematics teaching strategies and peer coaching opportunities. Data sources included classroom observations, teacher self-assessments, interviews, and field notes. Data were analysed using a two-level qualitative coding strategy with multiple interpreters. Findings showed that teachers implemented key strategies for effective mathematics teaching, especially in facilitating student interaction and improving the quality of tasks assigned. (Contains 2 figures, 1 table, and 4 notes.) [Abstract provided in both English and French.]

This paper finds its origins in a multidisciplinary research group's efforts to assemble a review of research in order to better appreciate how "spatial reasoning" is understood and investigated across academic disciplines. We first collaborated to create a historical map of the development of spatial reasoning across key disciplines over the last century. The map informed the structure of our citation search and oriented an examination of connection across disciplines. Next, we undertook a network analysis that was based on highly cited articles in a broad range of domains. Several "connection gaps"--that is, apparent blockages, one-way flows, and other limitations on communications among disciplines--were identified in our network analysis, and it was apparent that these connection gaps may be frustrating efforts to understand the conceptual complexity and the educational significance of spatial reasoning. While these gaps occur between the academic disciplines that we evaluated, we selected a few examples for closer analysis. To illustrate how this lack of flow can limit development of the field of mathematics education, we selected cases where it is evident that researchers in mathematics education are not incorporating the important work of mathematicians, psychologists, and neuroscientists--and vice versa. Ultimately, we argue, a more pronounced emphasis on transdisciplinary (versus multidisciplinary or interdisciplinary) research might be timely, and perhaps even necessary, in the evolution of educational research.

In an effort to increase our understanding of what mathematics young children are capable of when provided with playful yet purposeful opportunities to learn, Canadian researchers initiated lesson study over 4 years with teams of teachers of young children (ages 4-7). The lesson study process followed a lesson study cycle of goal setting, planning, implementation and reflection, with one major variation: a focus on the design and implementation of exploratory tasks. Exploratory tasks are brief, collaboratively planned activities that focus on precise mathematics content with engaging contexts. The lesson study team featured in this paper presented students with challenging exploratory tasks to investigate length, perimeter and area measurement concepts as they progressed toward the public research lesson. Through the process, lesson study teachers increased their abilities to look and listen for student mathematics thinking that affected their subsequent planning to tasks. Notably, teacher estimations of what young children can do were expanded and raised. Students also made significant gains on pre-post interviews, suggesting that exploratory tasks are a powerful variation of lesson study that increases teacher and student learning due to increases in frequency, emphasis and implementation of quality tasks.

The ability to mentally rotate objects in space has been singled out by cognitive scientists as a central metric of spatial reasoning (see Jansen, Schmelter, Quaiser-Pohl, Neuburger, & Heil, 2013; Shepard & Metzler, 1971 for example). However, this is a particularly undeveloped area of current mathematics curricula, especially in North America. In this article we discuss what we mean by mental rotation, why it is important, and how it can be developed with young children in classrooms. We feature results from one team of teacher-researchers in Canada engaged in Lesson Study to develop enhanced theoretical understandings as well as practical applications in a geometry program that incorporates 2D and 3D mental rotations. Children in the Lesson Study classrooms (ages 4-8 years) demonstrated large gains in their mental rotation skills during 4 months of Lesson Study intervention in the Math for Young Children research program. The results of this study suggest that young children from a wide range of ability levels can engage in, and benefit from, classroom-based mental rotation activities. The study contributes to bridging a gap between cognitive science and mathematics education literature.

This paper outlines the new opportunities that that will be changing the landscape of geometry education at the primary school level. These include: the research on spatial reasoning and its connection to school mathematics in general and school geometry in particular; the function of drawing in the construction of geometric meaning; the role of digital technologies; the importance of transformational geometry in the curriculum (including symmetry as well as the isometries); and, the possibility of extending primary school geometry from its typical emphasis on vocabulary (naming and sorting shapes by properties) to working on the composing/decomposing, classifying, comparing and mentally manipulating both two- and three-dimensional figures. We discuss these opportunities in the context of historical developments in the nature and relevance of school geometry. The aim is to motivate and connect the set of papers in this special issue.

Teaching and learning fraction concepts provides challenges in primary schools all over the world. In this article, Shelley Yearley and Catherine Bruce describe a fractions-based research project conducted in Ontario, Canada.

Recent research demonstrates that in many countries gender differences in mathematics achievement have virtually disappeared. Expectancy-value theory and social cognition theory both predict that if gender differences in achievement have declined there should be a similar decline in gender differences in self-beliefs. Extant literature is equivocal: there are studies indicating that the male over female advantage in self-efficacy and beliefs about math learning is as strong as ever and there are studies reporting an absence of gender differences in belief. Using data from 996 grades 7-10 Canadian students, we found that gender differences in beliefs continued, even though gender differences in achievement were near zero. Gender differences, favoring males, were larger for self-beliefs (math self-efficacy and fear of failure) and weaker for functional and dysfunctional beliefs about math learning. There were also gender differences in the structure of a model linking beliefs about math, beliefs about self and achievement.

Researchers examine the outcomes of professional collaborative inquiry in mathematics on teacher efficacy in a three-year study of teacher professional learning in Canada. The study applies a mixed methods approach involving over 200 teachers and 1000 students as well as case study sites in English and French. The collaborative inquiry-based professional learning program (called CIL-M) focuses on teacher collaboration, mathematics knowledge for teaching, and student mathematical thinking. The program, which was refined annually based on research report recommendations, was found to increase teacher efficacy, student achievement and positive student beliefs.

The authors extend findings from qualitative research on the effects of action research by reporting two linked quantitative studies (N = 80 and 105). They found that teachers who participated in collaborative action research experienced statistically significant improvements in attitudes to educational research and teacher efficacy. The pre-post changes were robust across conditions of teacher gender, career stage, and qualifications. Teachers benefited more if they (i) recognized the importance of the data analysis and process reflection stages of action research; (ii) participated in action research that was rigorous and/or led to changes in their conceptual understanding; (iii) worked in schools that fostered professional learning; and (iv) had participated in research activities prior to these action research studies. (Contains 2 notes, 7 tables and 3 figures.)

Studies incorporating technology into the teaching of trigonometry, although sparse, have demonstrated positive effects on student achievement. The optimal sequence for integrating technology with teacher-led mathematics instruction has not been determined. Our research investigated whether technology has a greater impact on student achievement and attitudes if it is implemented before or after whole class teaching. The curriculum context of the study was a set of learning objects (CLIPS: Trig) designed to support student learning of transformations of trigonometric functions. The software includes functional features identified in prior research: it relieves students of the tedium of creating graphs by hand; sliders give students control of the simulations within program parameters; there are easy transitions between algebraic and graphic representations; the environment is dynamic; animation and visualization are included with graphing functions. Twenty Canadian classrooms (N = 489 grade 11-12 students, aged 17-18 years) were randomly assigned to two instructional sequences: CLIPS: Trig followed by whole-class teaching (CLIPS early treatment) and whole-class teaching followed by CLIPS: Trig (CLIPS late treatment). We found that in the pre-test to post-test comparisons, students who experienced CLIPS: Trig after whole-class teaching of core concepts learned more than students who began the unit with technology-supported simulations. However, there were no statistically significant differences in the pre-test to delayed post-comparisons. Beginning the trigonometry unit with CLIPS: Trig enhanced the impact of whole-class teaching, while beginning with whole-class teaching enriched students' technology experience. The findings suggest that a tight integration of whole-class and technology-assisted instruction is preferable.

The aim of this study was to identify when and how the interactive whiteboard (IWB) functioned as a productive tool that impacted student learning in mathematics. Using video data, field notes, and interview transcripts from 1 school year in two optimal case study classrooms, we were able to examine the unique opportunities afforded by the size of the IWB screen, the manipulation of virtual objects onscreen, and related communication using gestures. We: (i) established criteria for defining "significant learning moments"; (ii) assessed these significant learning moments to determine how the interactive whiteboard was supporting the learning; and (iii) isolated the use of gesture during IWB use to magnify the grain size of our analysis and understanding. The data fell into three types of IWB use: productive (89%), reproductive (2%), and problematic (9%). The study recommends that in order to best support student learning, professional development for teachers should emphasize direct and active "student" use of the IWB to engage students in inquiry of mathematics. (Contains 4 tables.)

In 2008-09, the Elementary Teachers' Federation of Ontario (ETFO) sponsored a professional learning initiative entitled, Teachers Learning Together: The Math Journey. The context for teachers was one of team-based, collaborative action research (CAR) dealing with a mathematics topic of their own choosing. University researcher teams in provincial regions were also enlisted to facilitate the CAR activities and to conduct case study research and reporting. Each teacher team designated a Lead Teacher (LT) for the duration of the project. This paper highlights the case study research findings pertaining to perceptions of role and responsibility of the LTs, and focuses on four emergent roles, namely that of manager, motivator, model, and mediator. Lead Teachers are shown to be essential to the collaborative action research process. Notwithstanding, recommendations regarding the actual LT designation, and a clearer communication of LT role and responsibility are also made.

The purpose of this paper is to report on the nature of collaboration in a multi-year, large-scale collaborative action research project in which a teachers' federation (in Ontario, Canada), university researchers and teachers partnered to investigate teacher-selected topics for inquiry. Over two years, 14 case studies were generated involving six university researchers and 61 teacher researchers. As teacher researchers collected their topic-specific data, university researchers worked in parallel on case studies at the 14 sites in an attempt to gain insights into the types and value of collaboration in collaborative action research. University researchers gathered 20-30 points of data for each case-study site (including teacher researcher final reports as well as journal entries, focus group interviews, field notes, video footage and questionnaires). Through comprehensive cross-case analysis, researchers were able to examine common traits as well as relationships between the nature of the teams, the nature of the projects and their relative success, to highlight conditions that led to robust collaborations. Through pattern matching, hypothesised events were compared with observations to test the claim that a triad of relationships between the team, the team lead and the researchers functioned as a positive collaborative force that propelled teacher team success in a generative ethos, even when faced with substantial challenges. (Contains 2 figures, 2 tables, and 1 note.)

This paper reports on the impact of a classroom-embedded professional learning (PL) program for mathematics teaching in two contrasting districts in Canada, and investigates the relationship between teacher efficacy and student achievement. Before the PL, District A had lower teacher efficacy and student achievement than District B, but after the PL, this situation was reversed. Qualitative analysis revealed that the two districts reported learning very different things from the PL opportunity. The complexities of context, prior learning experiences, goal setting, and persistence of participants all factored into what and how teachers learned. (Contains 4 figures and 3 tables.)

Previous research has examined factors influencing teacher decisions to integrate technology using between-teacher designs. This study used a within-teacher design to compare students who were assigned multi-media learning objects for learning fractions with students taught by the same teachers who were not assigned to the technology. There were two conditions: (1) teachers were asked to limit the number of assigned students to 25% of their class (N = 375 grade 7-10 students) and (2) teachers could assign as many students as they wanted (N = 149 grade 7 students). In the constrained decision setting, students assigned to the technology were more likely than students not assigned to score lower on a fractions achievement test, have dysfunctional attitudes towards mathematics learning, have low self-efficacy, exert low effort, and be male. In the unconstrained decision setting, 70% of students were assigned the technology and the only statistically significant predictor was prior achievement. Teachers' criteria were congruent with research identifying correlates of mathematics achievement and comfort with technology.

Students have difficulty learning fractions, and problems in understanding fractions persist into adulthood, with moderate to severe consequences for everyday and occupational decision-making. Remediation of student misconceptions is hampered by deficiencies in teachers' knowledge of the discipline and pedagogical content knowledge. We theorized that a technology resource could provide the sequencing and scaffolding that teachers might have difficulty providing. Five sets of learning objects, called CLIPS (Critical Learning Instructional Paths Supports), were developed to provide remediation on fraction concepts. In this article, we describe one stage in a research program to develop, implement and evaluate CLIPS. Two studies were conducted. In Study 1, 14 grade 7-10 classrooms were randomly assigned, within schools, to early and late treatment conditions. A pre-post, delayed treatment design found that CLIPS had no effect on achievement for the Early Treatment group due to unforeseen implementation problems. These hardware and software issues were mitigated in the late treatment in which CLIPS contributed to student achievement (Cohen's d = 0.30). Study 2 was a pre-post, single group replication involving 18 grade 7 classrooms. The independent variable was the number of CLIPS completed. Completion of all five CLIPS contributed to higher student achievement: Cohen's d = 0.53, compared to students who completed none (d = 0.00) or 1-4 CLIPS (d = 0.02). The two studies indicate that a research-based set of learning objects is effective when the full program is implemented. Incomplete sequences deprive students of instruction in one or more constructs linked to other key ideas in the conceptual map and reduce the amount of practice required to remediate student misconceptions. (Contains 5 tables and 1 figure.)

Self-assessment is a powerful technique for improving achievement. In this article we outline a theory of teacher change that links self-assessment by teachers to their professional growth. This theory provides avenues for peers and change agents to influence teacher practice. We apply the theory to change in mathematics teaching and report an explanatory case study in which use of the self-assessment tool, in combination with other elements, contributed to change in the instructional practice of a grade 8 mathematics teacher. Provision of a self-assessment tool contributed to teacher growth by: (1) influencing the teacher's definition of excellence in teaching and increasing his ability to recognize mastery experiences; (2) helping the teacher select improvement goals by providing him with clear standards of teaching, opportunities to find gaps between desired and actual practices, and a menu of options for action; (3) facilitating communication with the teacher's peer; and (4) increasing the influence of external change agents on teacher practice. The study argues that providing a self-assessment tool is a constructive strategy for improving the effectiveness of in-service provided it is bundled with other professional growth strategies: peer coaching, observation by external change agents, and focused input on teaching strategies.

There has been a strong call for increased clarity and transparency of method in qualitative research. Although qualitative data analysis has been detailed, data management has not been made as transparent in the literature. How do data collection and analysis interact in practical terms? What constitutes sufficient data? And can research be both planful and emergent? In this paper, the author highlights several methodological strategies for addressing data management challenges in a grounded theory study of preservice mathematics teachers. [ABSTRACT FROM AUTHOR]