Tracing Ariel’s Algebraic Problem Solving: A Case Study of Cognitive and Language Growth
Creators: name: Sigley, Robert
name: Wilkinson, Louise
Publisher: name: No Publisher Supplied
Year: 2013

description: 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 &amp; Uptegrove 2010; Kieran, 1996; Yerushalmy, 2000; Kaput, Carraher, &amp; 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 &amp; 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. 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