Transitioning from Rods to the Number Line: Assimilating Prior Representations to Construct New Meaning

According to Davis (1984), children construct ideas by building on prior experiences. When students learn concepts in a meaningful way they construct "powerful representations" which can be used to solve new problems. The development of more advanced concepts is facilitated as students apply their previous knowledge and representations to understand new ideas. In a model which Davis terms an assimilation paradigm, learners assimilate the knowledge they have constructed from previous experiences into new frames which they construct as they build meaning from new experiences. Often learners use several assimilation paradigms in making sense of a new experience. According to Davis and Maher (1990), new frames are constructed and knowledge is assimilated when learners recognize new experiences as related to previous experiences for which they have already built mental representations. The mental representations are then "either validated, modified or rejected" (Steencken & Maher, 2003, p. 114) as the learner seeks to assimilate them with the new problem structure. This analytic will explore the way in which students assimilate previous knowledge and "powerful representations" which they have built from working on fraction tasks with Cuisenaire rods and their knowledge about the properties of rulers into new frames they construct to solve a task involving the placement of fractions on the number line. The videos examined in this analytic are culled from a study of fourth grade students from Colts Neck, a suburban New Jersey district (Maher, Martino, & Davis, 1994). The session, facilitated by researcher Carolyn Maher, took place in November of 1993, after students explored fraction ideas using materials such as Cuisenaire rods, string, and meter sticks for approximately two months. The students examined in this analytic have constructed "powerful representations" of fractions by modeling solutions to tasks involving the concepts of fraction parts, fraction equivalence, fraction comparison, and operations with fractions (Maher et al. 1994; Shmeelk, 2010; Shmeelk & Alston, 2010). After a class discussion regarding the comparative size of the fractions ¼, 1/5, and 1/9 in which students recognized that as the denominator grows larger the size of the fraction grows smaller, the researcher challenged the class to place the numbers ½, 1/3, ¼, 1/5, and 1/10 on a number line (Maher et al. 1994). This analytic explores the discussion and models of these students as they transition from working with models of fractions using Cuisenaire rods, considering fractions as operators, to placing fractions on the number line, considering fractions as numbers, and demonstrates how they build on their previous experiences and mental representations to construct new representations appropriate for the task. This transition begins in a natural way as students begin by sketching fractional parts in a representational structure which mimics the structure of Cuisenaire rods. Their choice of representation suggests that students are using the representations they have constructed as an assimilation paradigm with which to approach the new problem. Students’ first construction of a number line is almost identical in size and fraction placement to their representation of Cuisenaire rods. The assimilation of their old representations to the frames they are constructing to solve the new task allows the students to build on their previous experiences working with Cuisenaire rods to fashion the new concept of fraction placement on a number line. As the discussion progresses, students debate whether or not each representation of the interval 1/3 can be marked on the number line as 1/3. Alan argues that doing so would be valid, drawing on a model of rods to justify his claim. Andrew disagrees, arguing that the representation of numbers on a number line must proceed in ascending order. This disagreement highlights an important difference between students’ previous conceptions of fraction parts represented by rods and the representation of numbers on the number line. In an attempt to assist the students in assimilating their previous knowledge into the new frames they are constructing regarding the number line, the researcher draws an analogy between a ruler, which maintains the additive quality of measurement, and the number line. In this way, the researcher enables students to use more than one assimilation paradigm, the representation of rods and the representation of a ruler, in order to construct a new representation for fractions on the number line. The events depicted by this analytic highlight the benefit to students when they are able to draw on assimilation paradigms when transitioning to the number line. According to Flores, Samson, and Yanik (2006) and Alston, Davis, Maher, and Martino (1994), students find it difficult to relate number line placement to fraction lengths. Additionally, a common misconception among students is the notion that unit fractions are placed at equal distances along the number line. This analytic demonstrates that students are aided in making the transition to the number line when they have built powerful mental representations which can serve as assimilation paradigms for constructing meaning around the number line. References Alston, A. S., Davis, R. B., Maher, C. A., & Martino, A. M. (1994). Children’s use of alternative structures. In J. P. da Ponte and J. F. Matos (Eds.), Proceedings of the 18th Annual Conference of the International Group for the Psychology of Mathematics Education, (2), 248-255. Lisboa, Portugal: University of Lisboa. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Greenwood Publishing Group. Davis, R. B. & Maher, C. A. (1990). What do we do when we do mathematics? In R. B. Davis, Maher, C. A., & Noddings, N. (Eds.), Constructivist Views on the Teaching and Learning of Mathematics: Journal for Research in Mathematics Education Monograph, 4, 65-78. Reston, VA: National Council of Teachers of Mathematics. Flores, A., Samson, J., & Yanik, H.B. (2006). Quotient and measurement interpretations of rational numbers. Teaching Children Mathematics, 13(1), 34-39. Maher, C. A., Martino, A. M., & Davis, R. B. (1994). Children’s different ways of thinking about fractions. In Proc. 18th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp. 208-215). Schmeelk, S. E. (2010). Tracing students’ growing understanding of rational numbers (Doctoral dissertation, Rutgers University-Graduate School of Education). Schmeelk, S., & Alston, A. (2010). From Equivalence to Rational Numbers: The Case of Meredith. In Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematical Education (Vol. VI, pp. 720–727). Steencken, E. P., & Maher, C. A. (2003). Tracing fourth graders’ learning of fractions: early episodes from a year-long teaching experiment. The Journal of Mathematical Behavior, 22(2), 113-132