Session Information
24 SES 10, Arithmetics and Number Sense. Proportions, Fractions and Multiplication
Paper Session
Contribution
The National Mathematics Advisory Panel (NMAP, 2008) stated that the conceptual understanding of fractions and fluency in using procedures to solve fraction problems are central goals of students’ mathematical development and are critical foundations for algebra learning. Many researchers believe that much of the basis for algebraic thought requires a clear understanding of rational number concepts (Kieren, 1980; Lamon, 1999; Wu, 2001) and the ability to manipulate common fractions. Stephens and Pearn (2003) identified Year 8 proficient fractional thinkers as students who demonstrated a capacity to represent fractions in various ways, and to use reverse thinking to solve fraction problems. Reverse thinking is required when the given number of objects is identified as a fraction of an unknown whole, which students are then required to find. Effective reverse thinking requires multiplicative as opposed to additive thinking.
Kieran (2004) described algebraic thinking as: analysing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modelling, justifying, proving, and predicting. Empson, Levi and Carpenter (2010) suggest that students should develop and use computational procedures using relational thinking to integrate their learning of whole numbers and fractions. Three distinct aspects of algebraic thinking identified by Jacobs, Franke, Carpenter, Levi, and Battey (2007) and by Stephens and Ribeiro (2012) are important for this study. They are students’ understanding of equivalence, transformation using equivalence, and the use of generalisable methods.
The overarching aim of this study is to identify aspects of fractional competence and fractional reasoning that predict algebraic readiness in the middle years. The research design has been informed by Siegler and colleagues’ analysis (2012) of longitudinal data from the United States and United Kingdom, and Lee and Hackenburg's small scale research (2013). Siegler and colleagues’ analysis of longitudinal data (2012) showed that competence with fractions and division in fifth or sixth grade is a uniquely accurate predictor of their attainment in algebra and overall mathematics performance five or six years later when other factors were controlled. They controlled for factors such as whole number arithmetic, intelligence, working memory, and family background. Lee and Hackenburg’s research (2013) showed that fractional knowledge appeared to be closely related to establishing algebra knowledge in the domains of writing and solving linear equations.
An international group of 15 mathematicians, mathematics educators, and mathematics teachers was asked to analyse two sets of student responses to fraction tasks requiring reciprocal or reverse thinking to determine which responses were ‘anticipating algebra’ or algebraic readiness. One set of student work samples highlighted successful multiplicative responses while the other contained correct responses using pictorial or additive methods. While there was no uniform agreement about what anticipated algebra in the student work samples many experts drew attention to features that demonstrated a shift away from arithmetical thinking. The mathematics teachers stated that students using multiplicative strategies used methods that were algebraic through the use of equivalence and generalisability. This study seeks to develop a stronger conceptual framework for the terms “anticipating algebra” or “algebraic readiness”.
The key research question is: How does middle years students’ fractional competence and reasoning show evidence of non-symbolic algebraic thinking and its progression towards more traditional algebraic thinking as experienced in lower secondary classes i.e. “anticipating algebra”. For this study algebraic thinking is defined in terms of students’ capacity to identify an equivalence relationship between a given collection of objects and the fraction this collection represents of an unknown whole, and then to operate multiplicatively on both in order to find the whole. This definition has been derived from an investigation of the literature and in consultation with current mathematics experts.
Method
Expected Outcomes
References
Empson, S. B., Levi, L., and Carpenter, T. P. (2010). The algebraic nature of fractions: developing relational thinking in elementary school. In J. Cai and E. Knuth (Eds), Early Algebraization: Cognitive, Curricular and Instructional Perspectives. New York: Springer Jacobs, V., Franke, M., Carpenter, T., Levi, L. & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education 38(3), 258–288. Kieran, C. (2004): Algebraic thinking in the early grades: What is it? – In: The Mathematics Educator (Singapore) 8(No. 1), p. 139-151 Kieren, T. E. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent Research on Number Learning (pp. 125-149). Columbus: Ohio State University. (ERIC Document Reproduction Service No. ED 212 463). Lamon, S. J. (1999). Teaching Fractions and Ratios for Understanding: Essential Knowledge and Instructional Strategies for Teachers. Mahwah, NJ: Lawrence Erlbaum Associates. Lee, M.Y. & Hackenburg, A. (2013). Relationships between fractional knowledge and algebraic reasoning: The case of Willa. International Journal of Science and Mathematics Education. 12(4) 975-1000 National Mathematics Advisory Panel [NMAP] (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education. http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf Pearn, C. & Stephens, M. (2015). Strategies for solving fraction tasks and their link to algebraic thinking. In M. Marshman, V. Geiger, & A. Bennison (Eds.) Mathematics Education in the Margins. Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia, pp. 493 – 500. Sunshine Coast: MERGA Siegler, R., Duncan, G. Pamela E. Davis-Kean, P. Kathryn Duckworth, K Amy Claessens, A. Mimi Engel, M. Maria Ines Susperreguy, M. & Meichu Chen, M. (2012). Early Predictors of High School Mathematics Achievement. Retrieved from http://pss.sagepub.com/content/early/2012/06/13/0956797612440101 Stephens, M., & Pearn, C. (2003). Probing whole number dominance with fractions. In L. Bragg, C.Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics Education Research: Innovation, Networking, Opportunity. Proceedings of the Twenty-sixth Annual Conference of the Mathematics Education Research Group of Australasia, pp. 650-657. Sydney: MERGA. Stephens, M. & Ribeiro, A. (2012). Working towards Algebra: The importance of relational thinking. Revista Latinoamericano de Investigacion en Matematica Educativa, 15(3), 373 - 402 Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10-17.
Search the ECER Programme
- Search for keywords and phrases in "Text Search"
- Restrict in which part of the abstracts to search in "Where to search"
- Search for authors and in the respective field.
- For planning your conference attendance you may want to use the conference app, which will be issued some weeks before the conference
- If you are a session chair, best look up your chairing duties in the conference system (Conftool) or the app.