1. Prompts to Self-Explain Why examples are (in-)correct Focus on Procedures 58% of explanations were procedure- based Self-explanation is thought to facilitate construction of inference rules that are used in the formation of general principles (Chi et al., 1989) Focus on Goal of Procedure Highlighted by correct & incorrect examples and their implied goals; the correct goal being to make both sides have the same value Explanation supports an increase in awareness of the problem’s goal- structure (Crowley & Siegler, 1999) Goal Highlights Underlying Principle Focus on getting both sides to be equal, regardless of equation structure Highlights main principle of mathematical equivalence: the equal sign indicates that both sides are equivalent Increased Conceptual Knowledge A fuller understanding of a problem’s goal structure & underlying principle would allow for greater conceptual understanding via: Implicit knowledge of allowable equation structures Explicit knowledge that “=“ means “the same” Increased Transfer Performance Greater conceptual understanding of the problem’s goal structure and guiding principle may allow for less incorrect strategy use, and more flexible strategy use and greater success on transfer items 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 Summary 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 Children ʼ s 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. 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 predoctorial training 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. The Effect of Self-Explanation on Conceptual and Procedural Knowledge Katherine L. McEldoon, Kelley Durkin & Bethany Rittle-Johnson Vanderbilt University Effect of Condition Current Focus Method Mathematical Equivalence Katherine L. McEldoon K.McEldoon@Vanderbilt.edu Psychology & Human Development, Peabody College, Vanderbilt University, Nashville, TN How does self-explaining enact its effect on conceptual and procedural knowledge? 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 References Content of Explanations Contact Acknowledgments How Self-Explanation May Increase Conceptual & Procedural Knowledge 75 2 nd through 4 th graders with less than 80% correct at pretest on conceptual and procedural knowledge of mathematical equivalence Conceptual KnowledgeProcedural Transfer 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) 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 Overall Research Question 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? Assessment Components 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 Conceptual Knowledge Explicit- Equal Sign Knowledge What does the equal sign mean? Implicit- Equation Structure Knowledge 3 + 5 = 5 + 3 True or False Pretes t One on One Intervention Immediate Posttest Delayed Retention Test Three Conditions 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? Students who mention equivalent sides have marginally higher conceptual knowledge scores at retention test Structure Judgment 7 + 6 = 6 + 6 + 1 True False Don’t Know ** ** Control Solve 6 problems Self-Explain Solve 6 problems & explain Additional-Practice Solve 12 problems Matched for amount of practice Matched for amount of time on task Explanation Type 9 + 3 + 5 = __ + 9 Equals (6%) Recognizes that both sides of equation are equal or displays understanding of the equals sign. Procedure (58%) Talks about a procedure for solving the equation – mentions an operation or number(s), even if incorrect. Why do you think 8 is the right answer? Because if you add 9+3=12, then 12+5=17, its, and um there's an equal sign there so you have to make the other side equal to it. 8+9=17. Because 9+3=12, and 12+5=17, and 17+, I mean 17- 9 is 8. Why do you think 26 is a wrong answer? 26 is not right because 9+26 doesn't equal 17 like 9+3+5 does equal 17. Because 9+3+5= um 17, and then um 17+9=26 and that's not really right. Explaining increased conceptual knowledge, especially implicit conceptual knowledge of allowable equation structures Structure Encoding 5 + 4 + 8 = 5 + __ Reconstruct from memory after a 5s delay 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 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. m * * * ** Self-ExplainContro l Additional Practice Cognitive Science Society Conference, July 2011, Boston MA