Booth
Contents
 1 Improving skill at solving equations through better encoding of algebraic concepts
Improving skill at solving equations through better encoding of algebraic concepts
Julie Booth, Robert Siegler, Ken Koedinger & Bethany RittleJohnson
 PI: Julie Booth
 Key faculty: Ken Koedinger, Robert Siegler, Bethany RittleJohnson
 Studies: 4 complete
Exp 2  Data available in DataShop  Dataset: Corrective Self Explanation  2006 (CTAT)

Exp 3  Data available in DataShop  Dataset: Self Explanation Riverview Fall 2007 (CTAT) Dataset: Self Explanation CWCTC Winter 2008 (CL)

Abstract
This project examines the effectiveness of corrective selfexplanation, or explanation of incorrect worked examples, for improving students' knowledge components for solving algebraic equations. Students in classrooms which are using the Algebra Cognitive Tutor curriculum will complete such exercises during their otherwise typical experience solving equations with the Tutor to determine whether coordination of the two instructional methods increases robust learning; as a control, other students in the classroom will receive typical selfexplanation exercises (explanation of correct worked examples) or no selfexplanation exercises amid their tutor problems (Ecological control group). Robust learning will be assessed using measures of longterm retention, transfer, and accelerated future learning.
Glossary
 Corrective SelfExplanations: Selfexplanations of incorrect worked examples; explaining how and why they are incorrect
 Incorrect worked examples: Examples that include errors
Research Question
Students tend to learn overgeneralized knowledge components and apply them when attempting to solve algebra problems with incorrect features. How can we help them to learn correct knowledge components? And when is combining exercises designed to improve conceptual knowledge with tutored procedural exercises effective for improving robust learning of algebraic problemsolving?
Background and Significance
Errors are inevitable when individuals are first learning any skill; solving algebraic equations is no exception. Students often use incorrect knowledge components when learning Algebra (Lerch, 2004; Sebrechts, Enright, Bennett, & Martin, 1996), and use of incorrect knowledge components has been attributed to misunderstandings or gaps in students’ conceptual knowledge of Algebra (Anderson, 1989; Van Lehn & Jones, 1993). Results from Experiment 1 in Booth et al.’s current PSLC project confirm this hypothesis; a lack of knowledge about certain features in the problems (e.g., negatives, equals sign, like terms) was associated with use of related incorrect knowledge components on the problemsolving task. For example, students who do not see negatives as integral parts of terms in algebraic equations, or who believe that negatives can enter and exit equations without consequence tend to apply knowledge components with incorrect (or incomplete) features when solving equations, such as behaving in accordance with “to remove a term from an equation, subtract it from both sides” rather than with a knowledge component with feature validity that specifies “positive term” in the predicate, Thus, improving students’ knowledge of the conceptual features that underlie Algebra may be necessary for robust learning to occur.
Siegler’s overlapping waves theory suggests that there are two important steps that are necessary to improve knowledge (Siegler, 1996): 1) Weaken the incorrect knowledge component, and 2)Construct and strengthen correct knowledge component. One way to accomplish this is through selfexplanation, or generating explanations about instructional material. Chi and her colleagues (e.g., Chi, 2000; Roy & Chi, 2005) have shown that selfexplanation is useful for a variety of purposes, including generating new knowledge to fill gaps and repairing faulty knowledge. Siegler (2002) suggests that one particular type of selfexplanation may be especially useful for repairing faulty knowledge: explaining why the procedures used in incorrect worked examples are wrong. This selfexplanation of incorrect worked examples (why they’re wrong) can weaken students’ overgeneralized knowledge components by helping them to understand both that the knowledge components are incorrect and what relevant features make them incorrect.
Examples of corresponding misconceptions and incorrect knowledge components:
Dependent Variables
Normal posttest. Near transfer—immediate posttest in which isomorphic problems to instruction are included for students to solve. (e.g., 3x + 10 = 20, 4x/3 + 4 = 16, 2/(5x) = 14)
Robust Learning Measures:
Longterm retention. Embedded assessment within instruction by the Cognitive Tutor. We will collect log data from the review portion of the next equationsolving Tutor unit to determine whether correct knowledge components are applied.
Transfer. Problems included on the posttest in two forms: 1)Procedural format with more difficult problems/problems with additional features (e.g., 2x  7 = 5x + 9, 4/(6x) – 7 = 32). 2) Conceptual format assessing knowledge of features (e.g., State whether each of the following is the same as 3 – 4x: a. 3 + 4x b. 3 + (4x) c. 4x – 3 d. 4x + 3)
Accelerated future learning. We will collect log data during tutor instruction in the next equationsolving Tutor unit when treatment is no longer in place to determine whether the slope of the learning curve is greater for students who received the corrective selfexplanation treatment.
Independent Variables
Two types of selfexplanation exercises:
1) Typical selfexplanation (explanation of correct worked examples)
2) Corrective selfexplanation (explanation of incorrect worked examples)
The design is a 2 x 2 factorial with two levels of typical selfexplanation (yes or no) and two levels of corrective selfexplanation (yes and no). The result is that within any participating classroom, one fourth of students received typical selfexplanation, one fourth received corrective selfexplanation, one fourth received both, and one fourth received neither (the current tutor asis, the Ecological control group.
Hypothesis
Selfexplanation of incorrect worked examples (why they’re wrong) combined with procedural practice can lead to robust learning through two processes: 1) Weaken lowfeature validity knowledge components (know that they’re wrong and why they’re wrong) 2) Facilitate construction of highfeature validity knowledge components. See Corrective selfexplanation (relevant instructional principle)
Findings
Findings for Experiments 1a and 1b:
 Pretest misconceptions about features are related to use of specific incorrect knowledge components to solve problems Concepts related to specific buggy procedures
 Students with incorrect or missing equality or terms features are more likely to make errors of those types (p’s < .05 and .01); nonsignificant trend in the same direction for negative feature
 Having knowledge of the features of negativity and equality predicts correctness on procedural problems (p’s < .05 and .01)
 Students do improve from pretest to posttest, but roughly 1/3 of students still lack each of the features at posttest
 Pretest conceptual knowledge predicts students' pretest to posttest gain in procedural knowledge after using the Algebra Tutor as is
 Improvement in conceptual knowledge of the equals sign feature leads to more learning on the procedural problems than if that knowledge was not improved.
Findings from Experiments 2 and 3:
 Students who received any kind of selfexplanation exercises show greater learning of procedural knowledge components for solving algebraic equations compared with students who did not get any type of selfexplanation exercises (Ecological control group). (Booth, 2009; Booth, Koedinger, & Siegler, 2008)
 Improved the number of problems solved by 10%, while control group improved by 5%. No significant differnence, but experimental group students did at least as well as the control students, even though they had less practice solving problems.
 Improved their percent of conceptual questions answered correctly by 7% (control group decreased by 1%; p < .05)
 Recieving Corrective selfexplanation exercises leads to greater improvement in released items from standardized achievement tests.
 Corrective selfexplanation may affect students differently based on the amount and quality of their knowledge components prior to beginning the treatment.
 Students with low and mediumlevel conceptual knowledge at pretest tend to perform better with corrective selfexplanation than typical selfexplanation
 Students with high conceptual knowledge at pretest perform better with typical selfexplanation than corrective selfexplanation.
Explanation
Students who receive corrective selfexplanation exercises are expected to gain improved explicit conceptual knowledge about the features in problems that make certain knowledge components inappropriate. This expected to lead to greater robust learning compared with the other conditions that gain only implicit knowledge or explicit knowledge but do not have this additional knowledge about when it is appropriate to apply their knowledge components.
Descendents
None
Presentations/Publications
 Booth, J.L. (2009). Improving Algebra Learning in Real World Classrooms with Worked Examples and SelfExplanation. Paper presented at the Presidential Symposium entitled The New Learning Sciences at the annual meeting of the Eastern Psychological Association, Pittsburgh, PA, March 58, 2009.
 Booth, J.L., & Koedinger, K.R. (2008). Key misconceptions in algebraic problem solving. In B.C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Cognitive Science Society (pp. 571576). Austin, TX: Cognitive Science Society.
 Booth, J.L., Koedinger, K.R., & Siegler, R.S. (2008, July). Using selfexplanation to improve algebra learning. Poster presented at the 30th annual meeting of the Cognitive Science Society, Washington, D.C.
 Booth, J.L., Koedinger, K.R., & Siegler, R.S. (2007a). The effect of prior conceptual knowledge on procedural performance and learning in algebra. Poster presented at the 29th Annual Cognitive Science Society conference in Nashville, TN. Abstract
 Booth, J.L., Koedinger, K., & Siegler, R.S. (2007b). The effect of corrective and typical selfexplanation on algebraic problem solving. Poster presented at the Science of Learning Centers Awardee’s Meeting in Washington, DC, October, 2007.
 Presentation to the PSLC Advisory Board, Fall 2006. Link to Powerpoint slides
References
Anderson, J.R. (1989). The analogical origins of errors in problem solving. In D. Klahr & K. Kotovsky (Eds). Complex information processing: The impact of Herbert A. Simon. (pp. 343371). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc.
Chi, M.T.H. (2000) Selfexplaining 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. 161238.
Lerch, C. M. (2004). Control decisions and personal beliefs: Their effect on solving mathematical problems. Journal of Mathematical Behavior, 23, 2136.
Roy, M. & Chi, M.T.H. (2005). Selfexplanation in a multimedia context. In R. Mayer (Ed.), Cambridge Handbook of Multimedia Learning (pp. 271286). Cambridge Press.
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 selfexplanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 3158). New York: Cambridge University.
Van Lehn, K., & Jones, R.M. (1993). What mediates the selfexplanation effect? Knowledge gaps, schemas, or analogies? In M. Polson (Ed.) Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society (pp. 10341039).