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doi:10.7282/T3N0186C

While research has shown that understanding the concept of a function is essential for success in other areas of mathematics (Carlson, 1998; Rasmussen, 2000) students continue to struggle learning the concept (Vinner and Dreyfus, 1989). Research has revealed that young children, who are engaged in problem-solving activities designed to elicit justifications for their solutions, develop an understanding of fundamental algebraic ideas such as function (Maher, Powell & Uptegrove 2010; Kieran, 1996; Yerushalmy, 2000; Kaput, Carraher, & Blanton, 2008). Davis (1985) advocated the introduction of algebra to elementary school students as young as grade 3. He argued that the idea of function can be built intuitively by students as they engage in explorations of problems requiring identification of increasingly more challenging patterns; further Davis claimed that students can build the conceptual idea before formal notation is introduced. Davis (1985) offered sets of tasks for student exploration, and he video/audio recorder problem children’s problem solving as they constructed solutions that can be expressed with linear, quadratic and exponential functions (Mayansky 2007; Giordano, 2008). Extending this work, Bellisio and Maher (1998) studied students who provided verbal expressions of algebraic function prior learning to write the rules in symbolic form.
This analytic extends this earlier work by examining how one student, Ariel, builds an understanding of the linear function concept and represents his understanding of the basic algebra ideas underlying the construction. One focus was to see if students could provide a general solution to the problem. A second focus is on use of the mathematics register, the specialized kind language used in mathematics teaching and learning that is characterized by precision and linguistics in both oral and written language (Silliman & Wilkinson, 2015). This analytic presents a task that requires students to determine how many light green Cuisenaire rods are needed to build a ladder with different number of rungs. The shortest ladder has only one rung and can be built with 5 light green Cuisenaire rods. A 2-rung ladder would be modeled using 8 light green rods. The problem was presented as follows: The Ladders Problem: Build a rod model to represent a 3-rung ladder. How many rods did you use? How many rods would you need to build a ladder with 10 rungs? How could you represent the number of rods needed if you were to build a ladder with any number of rungs? Justify your solution. This analytic reveals how Ariel first approaches the problem using an arithmetically proportional approach to build a recursive composite function that depends on whether the numbers of rods are even or odd. When he revisits the problem 18-months later his approach changes. He develops a function table, uses first differences, and constructs a general solution to the problem. His gradual adoption of the mathematics register is exemplified in his oral explanation of the meaning of his symbolic notation. This analytic highlights that early, informal open-ended problem solving tasks provide students opportunities to construct their knowledge. These problem solving tasks are explorations at the heart of developing mathematical understanding---not as simple follow-up activities to procedural instruction. One implication of this work is that teachers include both time and tasks for students to explore, examine, revisit, and connect ideas and concepts through investigations. In so doing students have authentic opportunities to build strong intuitions of the problem conditions. Students’ engagement in activities, such as The Ladders Problem, provide them with the foundation for gaining insights and deeper understandings of mathematics. Ariel used such an opportunity and built his algebra knowledge. His success is revealed in the elegance of his solution, the understanding of his earlier work, and his confidence in offering clear justifications.

REFERENCES
Bellisio, C. and Maher, C. A. (1998). What kind of notations do children build to express algebraic thinking? Proceedings of the 20th Annual Conference of the North American Group for the Psychology of Mathematics Education. Raleigh, NC, 159-165: PME-NA.

Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. Research in Collegiate Mathematics Education III, Conference Board of the Mathematical Sciences, Issues in Mathematics Education, 7, 114-163.

Confrey, J., & Smith, E. (1995). Splitting, co-variation, and their role in the development of exponential function. Journal for Research in Mathematics Education, 26(1), 66-86.

Davis, R. B. (1985). ICME-5 Report: Algebraic thinking in the early grades. Journal of Mathematical Behavior, 4, 195-208.

Davis, R. B. (1964) Discovery in mathematics: A text for teachers. Menlo Park, CA: Addison-Wesley.

Giordano, Patricia (2008). Learning the concept of function: Guess my rule activities with Dr. Robert B. Davis. Dissertation, Rutgers. New Brunswick,

Kaput, J., Carraher, D. & Blanton, M. (2008) (Eds.), Algebra in the Early Grades. Mahwah, NJ, Erlbaum.

Kieran, C. (1996). The changing face of school algebra. In 8th International Congress on Mathematical Education, Selected Lectures (pp. 271-286). S.A.E.M. THALES.

Maher, C. A., Powell, A. B. & Uptegrove, E. (Eds.), (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms. New York: Springer Publishers.

Mayansky, Ella. (2007). An analysis of the pedagogy of Robert B. Davis: Young children working on the tower of Hanoi problem. Dissertation, Rutgers. New Brunswick, NJ.

Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An evolving analytical model for understanding the development of mathematical thinking using videotape data. Journal of Mathematical Behavior, 22(4), 405-435.

Rasmussen, C. L. (2000). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. Journal of Mathematical Behavior, 20(1), 55-87.

Schliemann, A., Carraher, D., Brizuela, B., Earnest, D., Goodrow, A., Lara-Roth, S., & Peled, I. (2003). Algebra in elementary school. In N. A. Pateman, B. J. Doherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 127-134). Honolulu, USA: PME.

Silliman, E. & Wilkinson, L. C. (2015). Challenges of the academic language register for students with language learning disabilities. In Bahr, R., & Silliman, E. (eds.) Routledge Handbook of Communication Disorders. New York & Oxford: Routledge Taylor Francis.

Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 136-150).

Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-234). Albany, NY: SUNY Press.

Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.

Yerushalmy, M. (2000). Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra. Educational Studies in Mathematics, 43(2), 125-147.

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