Many students entering Algebra I hold persistent mathematical misconceptions about equation-solving that hinder their ability to succeed in the course (e.g., Baroody & Ginsburg, 1993). Unfortunately, these misconceptions persist even after targeted classroom instruction (Authors, 2007), indicating the need for researchers to identify more effective teaching methods that help students to correct their misconceptions and, importantly, to disseminate this knowledge among practicing educators in ways to efficiently meet the needs of Algebra students.

 Theoretical Perspectives

            An instructional intervention to correct misconceptions beyond what has already been attempted in traditional classroom instruction is explored in this study by pairing the worked example problems with self- explanations.  Emphasizing worked example problems helps to extinguish erroneous strategies in ways that procedural practice of problems cannot.  Overlapping waves theory (Siegler, 1996) maintains that individuals have a variety of strategies for solving any given type of problem, and choose among those possible strategies in each instance.  By exposing students to worked example problems, students will be able to identify the wrong strategy and choose the correct one more quickly.  Additionally, coupling worked examples with self-explanation – generating explanations about instructional material – can improve students’ explicit knowledge of correct concepts and procedures by generating new knowledge to fill gaps and replace faulty knowledge (Chi, 2000).

         Two outstanding questions are whether correctly or incorrectly solved examples are most beneficial, and whether the optimal type of instruction varies for students with different levels of background knowledge (e.g., Kalyuga et al., 2003). In particular, incorrect examples may improve learning by providing negative feedback and causing cognitive conflict (e.g., Siegler, 2002), which may be particularly beneficial for low-ability students.  On the other hand, one might argue that being exposed to incorrect examples could further confuse these students.

Methods

The purpose of the present study is to examine the effects of self-explaining correct versus incorrect examples for learning in Algebra.. The research questions are as follows:  

1)      Are students’ conceptual and procedural knowledge impacted differently by correct versus incorrect examples?

2)      Do students with different levels of background knowledge benefit differently from the presence of correct versus incorrect examples?

         Sixty-four participants were recruited from eighth-grade Algebra I classrooms that used the Cognitive Tutor software in an inner-ring suburban school in the mid-western United States. Students were randomly assigned to one of three conditions: Incorrect examples only (N=21), correct and incorrect examples (‘Both’; N=21), and Correct examples only (N=22).  Each condition incorporated 12 experimental problems interspersed between the typical problem-solving activities within the Solving Two-Step Equations unit of the Algebra 1 Cognitive Tutor. In each of the experimental problems, students were asked to explain what was done in the example and why the strategy was either correct or incorrect. 

Data Sources

          Each participant took a pretest immediately before beginning the intervention and a posttest immediately following completion of the intervention.  The pretest and posttest measured both student conceptual and procedural knowledge of Algebra.  There were two measures of conceptual knowledge: An encoding task, which measured a student’s ability to reconstruct an equation after viewing it for 6 seconds, and a conceptual features test, which probed students’ knowledge of terms and variables, negatives, and the equal sign.  Two measures of procedural knowledge were administered: Equation-solving and flexibility (the ability to choose and use the most effective strategy when solving an equation). 

Results and Conclusions   

           A significant main effect of condition was found for the conceptual features task (students in the Both condition (M = 72% correct) scored higher than those in the Correct only condition (56% correct)); this effect was strongest for knowledge of terms and of the equals sign. A marginal but moderate effect of condition was also found for flexibility, with students in the Both condition (M = 38%) scoring higher than those in the Incorrect condition (M = 25%)) No effects of condition were found for the encoding task or equation solving.  Initial results also suggest an interaction between the condition and students’ pretest conceptual scores such that the Incorrect condition was most beneficial for low-ability students, the Both condition was most beneficial for average students, and high-ability students benefited equally from the Incorrect and the Both condition. No group benefited most from the Correct only condition.  

            In conclusion, students learn more about the conceptual features of an equation after having examined both correct and incorrect examples. The improved conceptual knowledge yielded in the Both condition may be responsible for the increased procedural flexibility seen in that condition; surprisingly, this group outperformed even the Correct condition, which purposefully exposed students to unconventionally solved problems. In general, the Both condition helps to intervene with misconceptions more than the Correct and Incorrect conditions, though the Incorrect only condition may be even more beneficial for low performers.

Educational Importance

          With misconceptions of solving equations being a significant hindrance to students’ ability to successfully complete Algebra, this study identifies a plan for intervention that significantly reduces misconceptions in solving two-step equations for students in the general population.  Overall, the Both condition was better than the Correct condition, showing that it is important for students to receive, examine, and answer questions about incorrect examples. Far from confusing students, receiving incorrect examples may be crucial for developing the knowledge of conceptual features and has huge implications for classroom equation solving practices. 

          Results from this study can lead to the development of individualized instructional programs that provide students with the optimal form of instruction based on their level of prior knowledge.  In small groups and beyond, we will talk about further ways this research can be converted into classroom practice.

References

Authors (2007).

Baroody, A. & Ginsburg, H. (1993). The effects of instructions on children’s understanding of the equals sign. The Elementary School Journal, 84, 199-212.

Chi, M.T.H. (2000) Self-explaining expository texts: The dual processes of generating inferences and repairing mental models.  In Glaser, R. (Ed.) Advances in Insturctional Psychology, Mahwah, NJ: Lawrence Erlbaum Associates, pp. 161-238.

Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The Expertise Reversal Effect. Educational Psychologist, 38, 23-31.

Siegler, R.S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press.

Siegler, R.S. (2002). Microgenetic studies of self-explanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31-58). New York: Cambridge University.