# Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking

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5 shown, arguing that each pair of dots represents 1 and that the whole can be found by subtracting two dots. This solution, as is Emily s solution to Question (see Figure 2), involves similar two-step thinking as those students who first divide the 14 counters by 7 to find how many counters are represented by 1 and then to multiply (scale up) by to find a whole. These two questions, even with a diagram provided for Question 7, were more difficult than Question 5. Analysis In this section, we focus on the 19 students (Group A) who gave completely correct responses and adequate explanations to all three questions. Some explanations were briefly written leaving some thinking unstated and raising a question of whether these students may have been using a routine. Each of the 19 students was asked to provide a short written elaboration of their initial explanation to one question selected by the researchers. In looking at their initial responses and their subsequent elaborations our goal was to identify those features that could be confidently taken to indicate evidence of algebraic thinking. Our focus was to identify instances of student thinking that could be clearly classified as algebraic; namely, understanding of equivalence, transformation using equivalence, and use of generalisable methods. Students in Group A offered the best chance to show this. Confident Reverse Thinkers Responses of Group A students show that confident reverse thinkers are able to step back from a visual representation, and to relate the fraction to the numerical quantity it represents. These students know how to scale down and scale up fractions and the quantities they represent to obtain a measure for the whole. Scaling down and scaling up is a reliable two-step procedure for finding the whole. It may even be compacted into onestep. These students are not dependent on using additive strategies which may be appropriate for simple fraction problems like the one-half task in Part A. From the 19 fully correct responses four different types of responses were evident: Response Type 1. Eleven students employed equivalent operations using fractions and whole number quantities in parallel. See for example Emily s response to Question in Figure 2 where she wrote 4 4 = 1 7 = 7 = 1 on one line and 12 4 = 3, 3 7 = 21 on the one underneath tracking both fractional and whole number computation in parallel. Figure 2. Emily s response to Question 497

6 Emily s response can be directly compared to a two-step solution for x = 12. Like 7 some other students, Emily uses an equal sign idiosyncratically to connect her steps as in the first line of her response. However, Emily clearly understands the need for equivalent operations to relate the two lines of her solution. Other students write equivalence relationships involving fractions and whole numbers together. For example, in Question some students wrote: 7 4 = 12, 7 1 = 3, 3 7 = 21 Sometimes a two-step reverse operation is compacted into one step as Kenneth s response to Question 5 as shown in Figure 3. 4 Figure 3. Kenneth s response to Question 5 Kenneth s response mirrors very closely the kind of transformational thinking needed to solve the algebraic equation 2 3 x = 10 x = Response Type 2. Six students left the fraction unstated and operated directly on the whole number quantity. While scaling up the fraction is left invisible, this transformation clearly guides the operations on the associated whole numbers using equivalence: For example, one two-step response to Question was 12 4 = 3, 3 7 = 21; or by another student on the same question: 12 7 = 84, 84 4 = 21 or in one compacted step by another student for Question 7 was These strategies explicitly show the kind of generalisable algebraic thinking needed to solve the equation 7 x = 14 Response Type 3. Symbolic representation using an unknown was used by one student only. Figure 4 shows Julie s response to Question : 12 is 7 4 of x, meaning that x = which is now Figure 4. Julie s response to Question 7 498

7 Julie s response shows a clear understanding of equivalence and transformation. It is also generalizable unlike Response Type 4 which relies on written descriptions involving continued adding. This was used by one student for Question who stated: 4 of Kay s 7 CD collection is 12. That means that 1 is 3. I started adding 3 onto 12 until it reached That number is 21. Multi-step responses like this correctly establishing that one-seventh is equivalent to 3 then rely on additive strategies to achieve the whole. This is a more limiting strategy than shown in the preceding Response Types which demonstrate reciprocal thinking. Mixed Methods Julie, who used a pro-numeral expression for Question, used the second and also generalisable method to solve other questions (e.g to solve Question 5). While 3 some Group A students tended to use either the first or second method consistently, most used a mix of methods. We wondered, for example, if the student who wrote 14 7 = 2 = 12 might be using a routine, but this student later explained that 14 was split into seven numerator groups. Adding, I could have taken one group away. A similar range of methods, excluding symbolic representation, was evident among students in Groups B and C. However, among students in Groups C and D additive processes became more evident, like this Type 4 explanation from a student in Group C for Question 5: I had to halve 10 because 2 is 10, halve 2 to get 1, and so I did this to get 5. I 3 just added it (5) on after (to get 15). Among students in Group D explanations begin to show less evidence of multiplicative (reverse) thinking: Started with 10 to get 15 ; or Every 5 is 1 ; or Because there are 5 3 in each row and 10 is 2 of 15 ; or 3 1 = 5, 2 = 10, 1 = 15. There is clear evidence of 3 3 equivalence but these additive strategies have less algebraic potential compared to the efficient multiplicative (reverse) strategies shown by those using Response Types 1, 2, and 3. Algebraic thinking, as we have defined it, requires more than use of equivalence. It needs to be reflected in confident and appropriate transformations of the fractional entities involved. Conclusion and Implications Confident reverse thinkers are able to scale down and scale up (or scale up and then scale down) based on the meaning of the particular fractional relationship. This is exactly what is required in solving for x in corresponding algebraic representations. Their working shows that scaling down and scaling up of fractional quantities must be accompanied by equivalent changes in the quantities represented by a particular fraction. These methods and their resulting mathematical relationships are indicative of algebraic thinking, by which students demonstrate that they can manipulate the fractional and numerical quantities independently of any diagram or visual representation. The algebraic significance of these findings is that they draw attention to three quite specific aspects of fractional operations that are not sufficiently emphasised in earlier studies. The first is being able to transform (operate on) a given fraction in order to return it to a whole, regardless of whether the fraction is expressed in proper or improper form. The second is students understanding of equivalence, meaning that the operations that are 499

8 required to restore a fraction to a whole need to be applied to the corresponding numerical quantities represented by the fraction. The third is to utilise efficient and generalisable multiplicative methods to achieve this goal; in contrast to other methods, usually additive, which may work only with simple fractions. All three aspects are essential for the subsequent solution of algebraic equations. Teachers especially need to help students identify and use these efficient and generalisable strategies. References Harel, G. & Confrey, J. (1994). The development of multiplicative reasoning in the learning of mathematics. New York: SUNY Press. 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), Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics 12(3), Kieren, T. E. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent Research on Number Learning (pp ). Columbus: Ohio State University. (ERIC Document Reproduction Service No. ED ). 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. (2012). Fractional Knowledge and equation writing: The cases of Peter and Willa. Paper presented at the 12th International Congress on Mathematical Education 8 15th July, 2012, Seoul, Korea. Last accessed 19th March 2014 from 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), National Mathematics Advisory Panel [NMAP] (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education. Pearn, C. & Stephens, M. (2014). Fractional knowledge as a signpost to algebraic readiness. Presentation at the annual meeting of the Mathematical Association of Victoria, Maths Rocks at La Trobe University, December 4-5. Pearn, C. & Stephens, M. (2007). Whole number knowledge and number lines help develop fraction concepts. In J. Watson & K. Beswick (Eds), Mathematics: Essential research, essential practice. Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, Vol. 2, pp Adelaide: 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. Retrieved from 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 Sydney: MERGA. Stephens, M. & Ribeiro, A. (2012). Working towards Algebra: The importance of relational thinking. Revista Latinoamericano de Investigacion en Matematica Educativa, 15(3), Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2),

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