Assessing Fractional Conceptual Knowledge Reveals Algebraic Thinking

Author(s):

Catherine Pearn(presenting / submitting)Max Stephens

Conference:

ECER 2015

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 04, Rational Numbers

Paper Session

Time:

2015-09-09

09:00-10:30

Room:

659.Oktatóterem [C]

Chair:

Svein Arne Sikko

Contribution

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. 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 the critical foundations for algebra learning. According to Wu (2001) the ability to efficiently manipulate fractions is "vital to a dynamic understanding of algebra" (p. 17). Teaching, especially in the primary and middle years, needs to be informed by a clear awareness of what these links are before introducing students to formal algebraic notation.

Kieren (1980) suggested that difficulties experienced by children solving rational number tasks arise because rational number ideas are sophisticated and different from natural number ideas and that children have to develop the appropriate images, actions, and language to precede the formal work with fractions, decimals, and rational algebraic forms. 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) argue that the key to learning algebra meaningfully is to help students: "to see the continuities among whole numbers, fractions and algebra" (p. 411). They suggest that students should develop and use computational procedures using relational thinking to integrate their learning of whole numbers and 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 with fractions to solve problems. Effective reciprocal or reverse thinking requires multiplicative as opposed to additive thinking. Reverse thinking is required when a given number of objects is identified as a fraction of an unknown whole, which students are then required to find.

The overarching question of this study is to identify those aspects of fractional competence and fractional reasoning that predict algebraic readiness in the middle years. The research design has been informed by Siegler et al.’s analysis (2012) of longitudinal data from the United States and United Kingdom, and Lee and Hackenburg's small scale research (2013). Siegler et al.’s (2012) analysis of longitudinal data 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 (Lee, 2012; Lee & Hackenburg, 2013) conducted research with 18 middle school and high school students. Their research showed that fractional knowledge appeared to be closely related to establishing algebra knowledge in the domains of writing and solving linear equations and concluded: “Teaching fraction and equation writing together can create synergy in developing students’ fractional knowledge and algebra ideas" (p. 9). Their research used both a Fraction based interview and an Algebra based interview. The two interview protocols were designed so that the reasoning involved in the Fraction based interview provided a foundation for solving problems in the Algebra interview. In both interviews students were asked to draw a picture as part of the solution. For the Fraction tasks they were also asked to calculate the answer whereas in the Algebra tasks they were asked to also write the appropriate equations.

Method

Sixty-seven Year 6 students from an eastern suburban metropolitan school in Melbourne, Australia, were tested using a paper and pencil Fraction Screening Test (Pearn & Stephens, 2014) which took approximately 30 minutes. This paper focuses on items from the test that demonstrate clear links between students’ solutions to fractional tasks involving reverse processes and algebraic thinking. Unlike the Lee and Hackenburg study (2013) that used both a Fraction Interview and a separate Algebra interview, our study is based on analysis of students’ performances in a single paper and pencil test of fractional thinking. The Fraction Screening Test (Pearn & Stephens, 2014) was divided into three parts. Part A included 12 tasks, 11 tasks had been trialled previously (Pearn & Stephens, 2007). Part A tasks included routine fraction tasks such as equivalent fractions, ordering fractions and recognising simple representations. Part A also included a simple reverse thinking task showing a collection of four lollies and saying: “This is one-half of the lollies I started with. How many lollies did I start with?” This task was correctly answered by the majority of students and was one of the easiest questions in Part A. Part B included five number line tasks with four tasks trialled previously (Pearn & Stephens, 2007). One number line task, involving reverse thinking, gave a number line showing where both zero and the number one-third were placed. Students were asked to: “Put a cross (x) where you think the number 1 would be on the number line.” Fewer students were successful with this task and many put a mark at the end of the number line. Part C included three questions that required students to use reverse thinking using less familiar fractions. For example, students were told that 12 CDs were four-sevenths of the total and asked to find the total number of CDs. We added the Part C items to offer students opportunities to use more explicit algebraic thinking that was not needed to solve the previous tasks in Parts A and B. Each question was allocated marks up three. For example, in a task that was given three marks, no response scored zero, one mark was given if there was some evidence of a diagram that the student did not take further (starting point). Two marks were given for a correct answer without explanation and three marks were given for a fully elaborated explanation.

Expected Outcomes

Analysis of three Part C items from the Fraction Screening Test, that involved reverse thinking, identified five distinct groups. This paper will focus on the results from the 19 students who correctly and completely solved all three questions. These students, confident reverse thinkers, were 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, both nationally and internationally, 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 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 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 show that several efficient and successful multiplicative methods can be used 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 solution of algebraic equations. These findings from this study will be further investigated in a much larger study with students from Years 5 to 8.

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 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. (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 http://www.icme12.org/upload/UpFile2/TSG/0766.pdf 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. (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. 601–610. Adelaide: 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. Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10-17.

Author Information

Catherine Pearn (presenting / submitting)

The University of Melbourne

Science and Mathematics Education

Melbourne

Max Stephens

The University of Melbourne, Australia

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