Results of Collaborative Action Research on Fractions (2011-2012)
Knowledge Network on Applied Educatio Research (KNAER) Project
Dr Catherine D. Bruce (Trent uuniversity) and research team: Tara Flynn, Rich McPherson, Shelley Yearley
During the 2011-2012 school year, the Ontario Ministry of Education Curriculum and Assessment Policy Branch attained a grant from Knowledge Network for Applied Education Research (KNAER) to build and extend understanding of effective teaching and learning of fractions. Following a review of the research literature and the development of documents that supported teachers with linking this research to their practice more easily, the professional learning series began in three school boards, Kawartha Pine Ridge DSB, Ottawa Carleton DSB, and Simcoe County DSB. These three boards were selected based on readiness factors that included explicit and precise long-term professional learning plans focused on mathematics, as well as board-level mathematics leadership capacity. The teams engaged in collaborative action research where researchers and teachers investigated areas of mutual interest (see www.tmerc.ca/digitalpapers/). The teams inquired into learning fractions in grades 4 through 7 with a particular emphasis on representing, comparing and ordering fractions. The project maintained a focus on fractions throughout the year by co-developing and organizing fractions lessons in “bundles” that were implemented (including co-teaching and carefully observing students) in a punctuated schedule throughout one term, rather than as a single unit of study.
Participants had a wide range of experience (from first year teachers to teachers with over 20 years of experience, as well as special education and French Immersion teachers, instructional and division leads) in a variety of contexts (rural and urban schools, stable student populations, highly transient student populations, range of socio-economic backgrounds, English Language Learners, and students with special needs). After identifying student strengths and needs, the teams investigated which representations of fraction ideas were most helpful and least helpful to students, based on context, and facility of the representation in building robust knowledge and understanding of fractions. Team members also aimed to refine their content and pedagogical knowledge for teaching fractions, a critical factor for student success.
The collaborative action research process focused on the precise mathematics content of understanding fractions and ways of thinking about fractions led to gains for both teachers and students. The learning process is documented in detail in the Fractions Digital Paper (go to www.edugains.ca/ → Math →Paying Attention to Math).
Throughout the collaborative action research project, the teacher teams grappled with a number of dilemmas that extended beyond the fractions content. These dilemmas can be categorized using Windschitl’s (2002) framework. A small sample of these dilemmas are presented in the table below.
- Why are fractions important? Do students even need to know about them?
- Should I modify my teaching to include the use of a variety of manipulatives? Which manipulatives?
- How can I structure my lessons so that I have time to hear and understand the explanations and reasoning of my students?
- What tasks support student conceptual understanding of fractions?
- How can I transition students from a more traditional math experience where the focus has been on getting the right answer to a community of learners engaged in math talk with a focus on reasoning and proving?
- How can I balance the need to meet the reporting requirements with this type of cyclical learning focused on fractions?
Through the collaborative action research process, teacher team members supported each other in identifying/generating and testing strategies to address the above dilemmas. Teachers tested out a range of tasks and teaching strategies in collaboration, and subsequently met to reflect on and share their observations. This process was repeated in cycles, leading to teacher-reported increases in teacher efficacy. The sources of efficacy information included:
- mastery experiences (where the teacher observed that instructional shifts increased student success which in turn increased the teacher’s willingness to take risks with challenging but effective strategies in the classroom);
- vicarious experiences (where the teacher observed a colleague who was similar to themselves having success in a lesson);
- social and verbal persuasion (where the teacher had opportunities to compare his/her perspectives of an experience with that of a respected colleague);
- physiological and emotional cues (where the teacher articulated feelings of confidence and positive shifts in their mathematics teaching).
Key Understandings Identified by Teachers and Researchers about the Teaching and Learning of Fractions
- Students need to understand that the numerator and denominator together represent a single quantity or number.
- Students benefit rom the use of benchmarks (0, 1, 1⁄2, or 3⁄4) to compare fractions as well as seeing fractions alongside other number systems, such as decimals and percentages
- There are multiple meanings of fractions depending on context:
Educators in the project inquired into how different representations might help or hinder student thinking. They
noticed an overuse of part-whole models, especially circle area representations, which introduced errors when
dealing with numbers into which circles are not easily partitioned (such as sevenths or twelfths).
- Linear measure
- Part-part (e.g. a ratio)
- Fraction as quotient
- Fraction as an operator
- The use of a number line for representing, comparing and ordering fractions was examined across multiple classrooms and grades. The number line showed strong potential for representing fractions with greater accuracy and for helping students develop conceptual understanding as well as proportional reasoning.