S tudents who self- explained had: S uperior conceptual knowledge

1. Effect of Condition Assessment Components Overall Research Question Current Focus Content of Explanations Acknowledgments Mathematical Equivalence Procedural Transfer Conceptual Knowledge Method Contact References Summary The Effect of Self-Explanation on Conceptual and Procedural Knowledge Katherine L. McEldoon, Kelley Durkin & Bethany Rittle-Johnson Vanderbilt University Knowledge of equivalence is typically assessed through(e.g., Rittle-Johnson, Matthews, Taylor & McEldoon, 2011;Behr, Erlwanger, & Nichols, 1980; Falkner, Levi, & Carpenter, 1999 , McNeil, 2007; Rittle-Johnson & Alibali, 1999) • Students who self-explained had: • Superior conceptual knowledge • Small improvements in procedural transfer • The proposed pathway for how prompts to self-explain enact their effect is supported through: • Content of Explanations: Explainers who mention equivalence have higher conceptual knowledge scores than those who don’t. • Conceptual Knowledge: Explaining increases conceptual knowledge, especially implicit knowledge of allowable equation structures, as knowing why these structures are allowable entails the correct relational understanding • Procedural Transfer: This increased conceptual knowledge supports performance on transfer items, since successful completion relies on knowing the goal or underlying principle of the problem, in order to modify the problem solving strategy appropriately. • - Prompts to explain why an answer is correct or not focuses the learner on the underlying principle, via a consideration of procedures, goal and problem structure, resulting in increased conceptual knowledge • - Overall, self-explanation may benefit conceptual and procedural knowledge when prompts focus the learner on the underlying principle by making the problem solving goal more salient • Prompting students to self-explain benefitted conceptual knowledge when compared to students who received the same amount of practice problems or the same amount of instructional time. This benefit was maintained over a two-week delay. Specifically, prompts to self-explain: • Increased conceptual knowledge • Increased in implicit conceptual knowledge of allowable equation structures How does self-explaining enact its effect on conceptual and procedural knowledge? • Conceptual Knowledge • Explicit- Equal Sign Knowledge • What does the equal sign mean? • Implicit- Equation Structure Knowledge • 3 + 5 = 5 + 3 True or False • Procedural Knowledge • Learning Items- Same as those practiced during the intervention • 7 + 6 + 4 = 7 + __ • Transfer Items- Different from those practiced during the intervention • 8 + __ = 8 + 6 + 4 6 - 4 + 3 = __ + 3 * * Is there a unique benefit of self-explaining on conceptual and procedural knowledge of mathematical equivalence compared to amount of practice and time on task controls? * * How Self-Explanation May Increase Conceptual & Procedural Knowledge Control Self-Explain Additional Practice • Mathematical equivalence is the principle that two sides of an equation represent the same value • Foundational for algebra (Falkner, Levi, & Carpenter, 1999) • 3 + 5 + 6 = __ + 6 • Operational View: View “=“ as a command to carry out arithmetic operations • 3 + 5 + 6 = __ + 6, most get 14 or 20 • Relational View: View “=“ as meaning two sides of an equation have the same value 75 2nd through 4th graders with less than 80% correct at pretest on conceptual and procedural knowledge of mathematical equivalence Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching(92), 13-15. Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). How Students Study and Use Examples in Learning to Solve Problems. Cognitive Science, 13, 145-182. Crowley, K., & Siegler, R. S. (1999). Explanation and Generalization in Young Childrenʼs Strategy Learning. Child Development, 70(2), 304-316. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232-236. Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Simon & Schuster. McNeil, N. M. (2007). U-shaped development in math: 7-year-olds outperform 9-year-olds on equivalence problems. Developmental Psychology, 43(3), 687-695. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189. Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103(1), 85-104. One on One Intervention Immediate Posttest Delayed Retention Test Pretest Students who mention equivalent sideshave marginally higher conceptual knowledge scores at retention test Explaining increased conceptual knowledge, especially implicit conceptual knowledge of allowable equation structures Three Conditions • Structure Judgment • 7 + 6 = 6 + 6 + 1 • True False Don’t Know • Structure Encoding • 5 + 4 + 8 = 5 + __ • Reconstruct from memory after a 5s delay Matched for amount of practice Control Solve 6 problems m Self-Explain Solve 6 problems & explain * * * Matched for amount of time on task Additional-Practice Solve 12 problems Procedural Instruction All students were taught a correct procedure: Add up all numbers on left, subtract number of right, place value in blank Intervention Problems 6 + 3 + 4 = 6 + __ 3 + 4 + 8 = __ + 8 Self-Explanation Prompts 6 + 3 + 4 = 6 + _19_ Why is 19 a wrong answer? 6 + 3 + 4 = 6 + _7_ Why is 7 the right answer? Katherine L. McEldoon K.McEldoon@Vanderbilt.edu Psychology & Human Development, Peabody College, Vanderbilt University, Nashville, TN When also asked “How do you know?”, self-explainers noted that both sides have the same sum or same value, or that inverse is true significantly more often. • Explainers show small improvements in transfer performance • __ + 2 = 6 + 4 • Higher mean performance on transfer items • Significantly less incorrect strategy use • Significantly less use of a prevalent incorrect operational-understanding based strategy We would like to thank Dr. Marci DeCaro, Laura McLean, and Kristen Trembley for their help and guidance. The first author is supported by a predoctorialtraining grant provided by the Institute of Education Sciences, U.S. Department of Education, through Vanderbilt’s Experimental Education Research Training (ExpERT II) grant (David S. Cordray, Director; grant number R305B080025). The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education. This work was also supported by an NSF CAREER Grant (#DRL0746565) awarded to Dr. Bethany Rittle-Johnson. * * Cognitive Science Society Conference, July 2011, Boston MA