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The “Insertion” Error in Solving Linear Equations Kosze Lee and Jon R. Star, Michigan State University Introduction This proposed research investigates a particular phenomenon that occurred during a study of students’ flexibility in solving linear equations (Star, 2004). Method 153 6th graders participated in five hours (over five days) of algebra problem solving. In the first hour, the students were given a pretest and a brief lesson on four different steps that could be used to solve algebraic equations (adding to both sides, multiplying on both sides, distributing, and combining like terms). Students then spent three hours solving a series of unfamiliar linear equations with minimal facilitation. 23 students (randomly selected from all participants) were interviewed while working individually with a tutor/interviewer. On the last day of the project, students completed a post- test. Results Analyses of students’ work made apparent an interesting type of error, named “insertion”, in 12 (7.8%) students’ of which three (given the pseudonyms as Adam, Bryan and Cindy) were interviewed. The insertion error was evident when 3x = 6x + 6 became 6x 3x = 6x 6x + 6. Similarly, 2(x + 5) = 4(x + 5) became 2 – 2(x + 5) = 4 – 2(x + 5). In another case, 2(x + 1) = 10 became 2(x + 1 – 1) = 10 – 1. Interestingly, this type of errors has not previously been reported nor classified in the literature on linear equation solving (e.g., Matz, 1980; Payne & Squibb, 1990). Out of the many proposed classifications of students’ rule-based errors in computational or algebraic problems (Matz, 1980; Payne & Squibb, 1990; Sleeman, 1984), Ben-Zeev’s (1998) classification is the most relevant here. Its context of solving unfamiliar problems is very similar to the context of the present research. In this framework, the errors are classified into two major types: critic-related failures and inductive failures. Contact Information Jon R. Star, jonstar@msu.edu; Kosze Lee leeko@msu.edu. College of Education, Michigan State University, East Lansing, Michigan, 48824. This poster can be downloaded at www.msu.edu/~jonstar. References Ben-Zeev, T. (1998). Rational errors and the mathematical mind. Review of General Psychology, 2(4), 366-383. Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3(1), 93-166. Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14(3), 445- 481. Star, J. R. (2004). The development of flexible procedural knowledge of equation solving. Paper presented at the American Educational Research Association, San Diego, CA. Critic-related failures are due to the students’ failure to signal a violation of a rule while inductive failures are due to student’s over- generalization or over-specialization of conceptual interpretations or surface-structural features of worked examples. Here the latter, “syntactic induction”, is useful in our analysis. The interview transcripts of three students suggest that they have over-generalized the procedure of subtracting the same term on both sides in order to eliminate a term of a linear equation. As a result, two erroneous procedures are created – one that violates the subtraction law by inserting “TERM –” to both sides, and the other that violates the distributive law by inserting “ – TERM” in between a coefficient p and its associated term (x + n) or inside the parenthesis. The former is seen in Adam’s transcript (Table 1) while Cindy (Table 2) and Bryan (Figure 2) makes the latter error. However, they stopped making these errors after they were made aware of the violation of such rules. Conclusions The data analysis thus proposes to include the following into Ben-Zeev’s classification: 1) another critic-based failure whereby prior rules can be suppressed by the desired effect of a new procedure, and 2) errors which are generated by the confluence of over-generalized rules and critic-based failures even though this may be rare in the case of the “insertion” error. Adam’s work that exhibits “insertion” error for the second time (“-3x” is a later correction) Bryan’s work that exhibits both types of “insertion” errors that violate the distributive law Figure 1Figure 2

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