ERG SES G 05, Mathematics Education Research
One of the most significant areas of mathematics students need to become proficient with are fractions (Lamon, 2012) as they underpin many related areas of mathematics such as real numbers, algebra, measurement, probability and proportional reasoning (Bruce, Chang & Flynn, 2013; Dole, 2010; Hansen et al. 2015; Siemon et.al 2015). Fraction competency is an important determiner of numeracy success at school and beyond (Hilton, Hilton, Dole, & Goos, 2016), yet despite many efforts, contemporary literature has confirmed that “fraction capabilities pose large difficulties for many children and adults, and students' proficiency in them has shown little sign of improvement over the past three decades” (Lortie-Forgues, Tian, & Siegler, 2015, p. 201).
Much of the current research into student’s conceptualisation of fraction ideas has focused on middle and upper primary school students and the difficulties they present from an absence of learning fundamental fraction concepts during their initial years of instruction. In response to these challenges, this paper will present a PhD project on the development of an alternative approach to fraction instruction for the early years of primary school, which is underpinned by the theoretical frameworks of Confrey (2008) and Möhring, Newcombe, Levine & Frick (2016).
Confrey (2008) has developed a learning trajectories approach to fraction instruction which demonstrates that a ‘parallel approach’ to learning fractions is fundamental to rational number knowledge. For example, a parallel approach to fraction instruction explores the meanings of fraction as measure, fraction as operator and fraction as ratio simultaneously throughout the primary years of education, which are not traditionally of focus in many education system’s early years curricula.
The second theoretical framework underpinning this project is based on a spatial reasoning approach. Spatial reasoning is not a single ability or skill, rather it is an overarching term that describes a set of concepts, processes and tools (National Research Council (NRC), 2006). These skills or abilities are underpinned by three core components: “concepts of space, tools of representation and processes of reasoning” (NRC, 2006, p. 3). These core components require the capacity to recognise and perform mental manipulations of visual stimuli; the ability to transform spatial forms of information into other visual arrangements and representations; an awareness of the structural features of spatial information or objects and the critical thinking required to find relationships and solve problems (Arcavi, 2003; Mulligan, 2015). The inclusion of this theoretical framework is supported by Möhring, Newcombe, Levine & Frick, (2016) who illustrated that pre-school children often employ spatial reasoning strategies such as scaling to reason in contexts of proportional continuous measures, which was found to support the development of formal fraction knowledge. Through the identification and application of unit size and magnitude, scaling is strongly connected to the numerical relationships of fractions as measurement and ratio. Thus, capitalising on these intuitive skills young children are familiar and confident with in the early years of schooling in a parallel approach to fraction instruction, is conjectured to enhance children’s formal knowledge of fractions development.
The aims of this study are:
- To draw on educational research to develop an alternative approach to fraction instruction in the early years of primary school.
- To determine the extent to which this alternative approach to fraction instruction impacts student’s fraction knowledge and confidence.
- To explore the potential implications of this alternative approach for developing a new theory of instruction for fractions in the early years of primary school.
This study is situated in two, Year 2 primary classrooms and this presentation will present some preliminary findings for feedback and discussion.
This study is positioned in an interpretivist paradigm. This paradigm is underpinned by the belief that interpretations of knowledge are located in a particular context and time and are open to re-interpretation and negotiation through dialogue to build a more sophisticated understanding of the social world (Cohen & Crabtree, 2006). Thus, this paradigm research is conducted with a critical focus on dialogue between participants, the classroom teacher and the PhD supervisors to fully understand and interpret the phenomenon of the student’s learning experiences. Design based research (DBR) is an appropriate methodology for this paradigm as it allows the exploration of how an alternative approach to teaching fractions in the early years of primary school can attempt to explain the intuitive ideas and misconceptions of students, and how these can be capitalized upon to develop formal fraction knowledge. DBR is an appropriate methodology as it is employed by researchers to bridge theory and practice in education by developing a local instruction theory that can be elaborated and refined iteratively during the intended design research (Gravemeijer & Cobb, 2006). DBR allows the researcher to put in harm’s way (Cobb et al. 2003) a rationale and hypothesis for generating a theory (Kelly, 2004) through the trialling, refining and application of an alternative approach to teaching and learning fractions. Thus, the learning experiences designed will be analysed, refined and the skills and abilities participants exhibit and under what conditions, will be closely considered to lead towards the generation of a theory for teaching and learning of fractions in the early years. Based upon a literature review, a four-week unit of work (with pre and post-tests) will be developed and implemented by the researcher to the first participating class. Examples of tasks and student responses will be presented in this presentation. Qualitative data will be collected to interpret the conditions in which this proposed conjecture for fraction instruction may or may not be successful. This includes student work samples, observations and discussions with the primary classroom teachers and PhD supervisors, which will inform the continual reflection and refinement of the learning design during the unit implementation. Post test data will be utilized to reflect on the unit of work and further refine it, and then the researcher will re-implement with another class of students of the same age.
This PhD study is designed to determine the efficacy of an alternative approach to fraction instruction in the early years of schooling. This alternative approach is based on the conjecture that children predominantly access the fundamental meanings of fractions in Confrey’s parallel approach by utilising their intuitive spatial abilities. If this is so, this study will provide evidence to rethink current curricular sequences in many education systems, including the role and prevalence of spatial reasoning in contexts other than geometry and shape; as well as provide evidence-based pedagogical advice to help mitigate the challenges many children face in developing conceptual fraction knowledge.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational studies in mathematics, 52(3), 215-241. Bruce, C., Chang, D., Flynn, T., & Yearley, S. (2013). Foundations to learning and teaching fractions: Addition and subtraction. Retrieved from http://www.edugains.ca/resourcesDP/Resources/PlanningSupports/FINALFoundationstoLearningandTeachingFractions.pdf Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9-13. Cohen, D., & Crabtree, B. (2006). Qualitative research guidelines project. Retrieved from http://www.qualres.org/HomeInte-3516.html Confrey, J. (2008). Learning Trajectories and Rational Number Reasoning [PowerPoint slides]. Retrieved from https://www.human.cornell.edu/sites/default/files/HD/nsfalw/Confrey-NSF.pdf Dole, S. (2010). Making connections to the big ideas in mathematic: Promoting proportional reasoning. 2009 - 2018 ACER Research Conferences. 5. Retrieved from https://research.acer.edu.au/research_conference/RC2010/17august/5 Cobb, P., & Gravemeijer, K. (2006). Design research from a learning design perspective. In Educational design research (pp. 29-63). Routledge. Hansen, N., Jordan, N. C., Fernandez, E., Siegler, R. S., Fuchs, L., Gersten, R., & Micklos, D. (2015). General and math-specific predictors of sixth-graders’ knowledge of fractions. Cognitive Development, 35, 34-49. Hilton, A., Hilton, G., Dole, S., & Goos, M. (2016). Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers. Educational Studies in Mathematics, 92(2), 193-219. Kelly, A. (2004). Design research in education: Yes, but is it methodological?. The journal of the learning sciences, 13(1), 115-128. Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Routledge. Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201-221. Möhring, W., Newcombe, N. S., Levine, S. C., & Frick, A. (2016). Spatial proportional reasoning is associated with formal knowledge about fractions. Journal of Cognition and Development, 17(1), 67-84. Mulligan, J. (2015). Looking within and beyond the geometry curriculum: connecting spatial reasoning to mathematics learning. ZDM, 47(3), 511-517. National Research Council, (2006). Learning to think spatially. Washington D.C.: Retrieved from https://www.nap.edu/read/11019/chapter/1 Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2015). Teaching mathematics: Foundations to middle years. (2nd ed). South Melbourne: Oxford.
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