24 SES 16, Proportional Reasoning
The links between fractional knowledge and readiness for algebra have been highlighted by many researchers such as Wu (2001); Jacobs, Franke, Carpenter, Levi, and Battey, (2007); Empson, Levi, and Carpenter, (2011) and Siegler and colleagues (2012).
This paper builds on previous research by the authors (see for example, Pearn & Stephens, 2015; 2016) that investigated the links between fractional competence and algebraic thinking. The initial stage of the research focussed on middle years students’ written responses to the Fraction Screening Test (Pearn & Stephens, 2015) and a test of algebraic reasoning, the Algebraic Thinking Questionnaire (Pearn & Stephens, 2016). The key research question is: 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? 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.
This paper focuses on the final stage of the research. An interview protocol was developed based on a group of reverse fraction tasks, similar to those in the Fraction Screening Test but with progressive levels of abstraction. The interview was designed to investigate whether students who used multiplicative thinking in their written responses to the three reverse fraction tasks from the Fraction Screening Test could be advanced to more robust generalisation. The interview was also designed to investigate whether students who had relied on the use of diagrams or a mix of additive and multiplicative strategies could adopt more consistent multiplicative and generalisable strategies.
The interview protocol was explicitly guided by the research of Marton, Runesson and Tsui (2004) that shows how numbers can be varied in order to foster a generalisable pattern. We wondered whether stronger evidence of algebraic thinking would be obtained if students were consistently able to use the same methods in similar reverse fraction questions where the quantities were changed but the fractions remained the same. In other words, were students able to treat different given fractions as ‘quasi-variables’ (Fuji & Stephens, 2001) applying similar operations to given quantities regardless of the particular fraction used.
The interview incorporated one set of three questions using the same three fractions as in the Fraction Screening Test but with different quantities. In a second set of three questions the first part of the question used one of the given fractions with a new quantity; and the second part of the question started with: “If I gave you any number of counters which is also a (given fraction) of the number I started with, what would you need to do to find the number of counters I started with?” If students could satisfactorily complete these six interview questions, they were asked to consider what they would do if they were given any number of counters which represented any fraction. Responses to these seven interview questions were analysed to answer the following questions:
- Can students who used multiplicative thinking in their written solutions to the three reverse fraction tasks in the Fraction Screening Test be moved forward to more robust generalisation as they complete the interview?
Can students who correctly used visual or a mix of multiplicative or additive strategies adopt more consistent multiplicative and generalizable strategies during the interview?
At the end of the 2016 school year 46 students from Years 5 and 6 from an inner-city Melbourne primary school completed two paper and pencil tests: The Fraction Screening Test (Pearn & Stephens, 2015) and the Algebraic Thinking Questionnaire (Pearn & Stephens, 2016). The students’ responses to the three reverse fraction tasks from the Fraction Screening Test were analysed to determine the dominant strategies students were using across the three tasks. The overall responses were classified according to six categories: Not Clear, Diagram Dependent, Additive/Subtractive, Mixed, Multiplicative, and Advanced Multiplicative. The term Not Clear refers to students whose written responses were incomplete or who did not attempt any or all three reverse fraction tasks from the Fraction Screening Test. Diagram Dependent refers to students who partitioned either the given diagrams, or those diagrams they created themselves, before using additive or subtractive strategies. Additive/Subtractive refers to students who used additive or subtractive methods without explicit partitioning of the given diagram, or creating a new diagram, to find the whole. These students could find the quantity of objects needed to represent the unit fraction and then added or subtracted the appropriate number of objects needed to make the whole. Mixed refers to students who used multiplicative strategies to solve at least one task while still using additive/subtractive strategies to solve at least one other task. Multiplicative refers to students who only used multiplicative reasoning to successfully solve at least two questions. These students usually found the quantity represented by the unit fraction and then scaled up or down to find the whole. Advanced multiplicative describes students who used either algebraic notation or a one-step method to find the whole. Seventeen students were chosen from the 32 students who successfully solved at least two of the three reverse fraction tasks from the initial Fraction Screening Test. There were nine Year 5 and eight Year 6 students interviewed. These students represented the range of strategies described above. Each interview took approximately 15 minutes. Interviewers encouraged students to verbalise their thinking before recording their responses. An Interview Scoring Framework was developed which included five levels where Level 1 was: “Completed some or all of questions using known fraction and given quantity” and Level 4 was “Completed all questions using consistent multiplicative methods and used suitable algebraic notation to give a multiplicative response to question with unknown quantity and unknown quantity.”
Analysis of the written solution methods used by students to solve the initial three reverse fraction tasks revealed that 12 students did not respond or gave incorrect responses, four were dependent on the diagrams, nine used additive/subtractive strategies, 14 used both multiplicative and additive/subtractive strategies, five consistently used multiplicative strategies while two students divided by the reciprocal of the given fraction. All students who used multiplicative, or a mix of multiplicative and additive/subtractive methods, to solve the three reverse fraction tasks were able to deal with variations in both fractions and corresponding quantities and to generalise their solutions to the interview tasks. The interview allowed students to treat variations in the given fractions as ‘quasi-variables’, and recognised that the same multiplicative operations applied regardless of the fraction. Careful scaffolding of interview tasks assisted students who had appeared to rely previously on additive methods to use multiplicative and generalizable methods to solve questions with either an unknown quantity or with both an unknown fraction and an unknown quantity. Analysis of the interview results has revealed that students, who relied on visual or additive methods, were unable to generalise or use a multiplicative approach and would experience difficulty transitioning to algebra. These findings highlight specific aspects of fractional operations that have implications both nationally and internationally. Students must understand fraction equivalence and realise that to determine the number of objects representing the whole group, the same operations need to be applied to both the number of objects, and the fraction that those objects represent. Students must develop efficient and successful multiplicative methods rather than rely on diagrams or additive methods, which may work only with simple fractions. All three aspects are essential for the solution of algebraic equations. This paper builds on research presented in 2015 and 2016.
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 Fuji, T. & Stephens, M. (2001). Fostering an understanding of algebraic generalisation through numerical expressions: The role of quasi-variables. In K. Stacey, H. Chick & M. Kendal (Eds.), Proceedings of the 12th ICMI Study Conference: The future of the teaching and learning of algebra. Pp 259-264. Melbourne: University of Melbourne. 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. Marton, F., Runesson, U., and Tsui, A. B. M. (2004). The space of learning. In F. Marton, & A. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3-40) Mahwah, N. J .Erlbaum Associates. Pearn, C. & Stephens (2016). Competence with fractions in fifth or sixth grade as a unique predictor of algebraic thinking? In B. White, M. Chinnappan & S.Trenholm (Eds.), Opening up mathematics education research. Proceedings of the 39th Annual Conference of MERGA, pp.519-526. Adelaide: MERGA. 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., Davis-Kean, P., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. http://pss.sagepub.com/content/early/2012/06/13/0956797612440101 Accessed: 20 February 2017 Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10-17. 211
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