1. Knowledge of Procedures (familiar)1. 3(h + 2) + 4(h + 2) = 35 2. (x + 1) = 10 Knowledge of Procedures (transfer)3. 3(2x + 3x – 4) + 5(2x + 3x – 4) = 48 4. Knowledge of Flexibility1. Solve the equation 18 = 3(x + 2) in two different ways. 2. Adam’s first step: 2(s + 3(s - 1)) = 18 s + 3(s – 1)= 9 a. What step did Adam use to get from the first line to the second line? Combine like terms Distribute across parentheses Add or Subtract the same quantity on both sides Multiply or Divide by the same quantity on both sides Do you think this is a good way to start this problem? Circle one: (a) Very good way (b) OK, but not a very good way, (c) Not OK Knowledge of Concepts1. Without solving each equation, which of the following equations are NOT equivalent to (will NOT have the same answer as) the equation: x + 25 = 90: (a) x + 25 + 14 = 90 + 14 (b) 2x + 50 = 180 (c) 10x + 25 = 900 (d) all of the equations are equivalent 2. Which of the following is a like term to (could be combined with) 7(j + 4)? (a) 7(j + 10) (b) 7(p + 4) (c) j (d) 2(j + 4) (e) a and d Immediate Introduction to Multiple Procedures Supports Procedural Flexibility in Equation Solving Kelley Durkin 1, Bethany Rittle-Johnson 1, & Jon R. Star 2 1 Vanderbilt University, 2 Harvard University Abstract Knowing multiple procedures and using them adaptively is important for problem-solving. We examined how different methods for developing procedural flexibility affected novices learning equation solving. Students (N = 198) were assigned to one of three conditions that differed in whether students immediately compared procedures, compared procedures after a delay, or did not compare procedures. Students in the immediate compare procedures condition had greater procedural flexibility and accuracy than students in the other conditions. Differences in students’ explanations during the intervention suggest reasons for the benefits of immediate introduction to multiple procedures. Students in the immediate compare procedures condition more frequently compared and evaluated efficiency of procedures than students in other conditions. They also more frequently used efficient procedures during the intervention. Immediate introduction to multiple procedures supports attention to and adaption of efficient procedures, which benefits flexibility. Introduction Procedural fluency, including flexibility in choosing among alternative solution procedures to solve problems, is important for students in problem-solving domains (Baroody & Dowker, 2003; Blöte, Van der Burg, & Klein, 2001; Kilpatrick, Swafford, & Findell, 2001). However, little is known about the best way to improve students’ procedural flexibility. In the current study, we evaluated whether students should be exposed to multiple procedures immediately or after building familiarity with one procedure and whether comparison should be used. We define procedural flexibility as knowing multiple procedures and applying them adaptively to a range of situations (Blöte et al., 2001; Kilpatrick et al., 2001; Rittle-Johnson & Star, 2007). Procedural flexibility is important because people who develop procedural flexibility are more likely to use or adapt existing procedures when faced with unfamiliar transfer problems and to have a greater understanding of domain concepts (e.g., Blöte et al., 2001; Hiebert, et al., 1996). One possible way of improving procedural flexibility is to have students compare multiple procedures. Expert mathematics teachers often have students explicitly compare multiple solution procedures (e.g., Ball, 1993; Lampert, 1990), and comparing procedures seems to help students differentiate important features of the procedures and use multiple procedures when solving problems (Rittle-Johnson & Star, 2007; Schwartz & Bransford, 1998). However, it is not clear when in the learning process students should compare procedures and if comparing procedures is best for all students. For example, in one study with novices there was an aptitude-treatment interaction, such that students who were not familiar with one of the target procedures learned more if they studied the procedures separately, rather than comparing the procedures (Rittle-Johnson, Star, & Durkin, 2009). Past research has suggested that such novices might benefit more by gaining familiarity with one procedure before comparing learning another procedure (Clarke, Ayres, & Sweller, 2005). Target Domain. We investigated the importance of when to introduce multiple procedures with middle-school students learning to solve equations. Many people in mathematics education consider linear equation solving a “basic skill” (National Mathematics Advisory Panel, 2008). Regrettably, students often memorize rules and do not learn flexible and meaningful ways to solve equations (Kieran, 1992). We assessed students’ competence with equations for three critical components of mathematics knowledge: procedural knowledge, procedural flexibility, and conceptual knowledge (Kilpatrick et al., 2001). Current Study. We evaluated the effects of when and how to introduce multiple procedures with middle-school students. We included two different comparison conditions. The first was comparing the same problem solved with two different algebraic solution procedures (Immediate Compare Procedures). The second was comparing different problem types solved with the same solution method at first and then comparing multiple procedures (Delayed Compare Procedures). In the No Compare Procedures condition, students studied the same examples as those in the Delayed Compare Procedures condition, but the examples were presented one at a time on separate pages. Method We examined how learning from immediately comparing procedures (Immediate CP), from comparing procedures after a delay (Delayed CP), and from not comparing procedures (No CP) differed from one another. Participants: 198 (99 female) 8 th graders in Tennessee public schools. Procedure: We paired students within their classrooms and randomly assigned them to a condition. All students completed a pretest in class on Day 1. On Days 2 and 3, students were given a mini-lecture by a researcher followed by time to work on their respective intervention packets. Students studied worked examples of other students’ solutions and answered questions about the solutions. During the intervention, students in all conditions also solved practice problems. On Day 4, students completed a posttest. About one month later, all students completed a retention test. Samples of Intervention Materials Samples of Assessment Items Results 1. Students in the Immediate CP condition had higher procedural accuracy and flexibility. They were also more likely to use more efficient, shortcut methods during the intervention and on subsequent tests. Discussion Overall, students who immediately compared procedures were more likely to attend to the efficiency of procedures and to use more efficient procedures. It appears that having students compare multiple procedures immediately was more beneficial for their procedural flexibility than having them compare procedures after a delay and having them see procedures one at a time. Contrary to what some may think, immediately comparing procedures does not seem to overwhelm novices when they are provided with appropriate scaffolds, and it may help students more carefully focus on strategy efficiency. This focus on efficiency is part of what helps increase procedural flexibility. Delaying the introduction to multiple procedures seems to have harmed learning. One potential reason for this finding is that by delaying comparison of procedures, students do not get as much exposure to alternate, possibly more efficient, procedures. Early introduction to multiple procedures is emphasized in current ideas for best practices in mathematics education in various countries (Becker & Selter, 1996; Klein et al., 1998; National Council of Teachers of Mathematics, 2000), and the current results support this practice. Future research should further investigate under what conditions immediate exposure to multiple procedures may be beneficial. References Ball, D. L. (1993). With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics. The Elementary School Journal, 93, 373-397. Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Erlbaum. Becker, J. P., & Selter, C. (1996). Elementary school practices. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 511-564). Dordrecht, The Netherlands: Kluwer. Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students' flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93(3), 627-638. Clarke, T., Ayres, P., & Sweller, J. (2005). The impact of sequencing and prior knowledge on learning mathematics through spreadsheet applications. Educational Technology Research and Development, 53(3), 15-24. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Human, P., Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12- 21. 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. Kilpatrick, J., Swafford, J. O., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press. Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: Realistic versus gradual program design. Journal for Research in Mathematics Education, 29(4), 443-464. Lampert, M. (1990). When the problem is not the question and the solution is not the answer mathematical knowing and teaching. American Educational Research Journal, 27, 29-63. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council for Teachers of Mathematics. National Mathematics Advisory Panel (2008). Foundations of Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561-574. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836-852. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475-522. Acknowledgements We would like to thank The Children’s Learning Lab at Vanderbilt University. This research was supported by pre-doctorial training grant R305B040110 and by grant R305B040110 provided by the Institute of Education Sciences, U.S. Department of Education. Contact Information Kelley Durkin: kelley.durkin@vanderbilt.edu Bethany Rittle-Johnson: bethany.rittle-johnson@vanderbilt.edu Jon R. Star: Jon_Star@Harvard.edu No Compare Procedures condition Delayed Compare Procedures condition Immediate Compare Procedures condition 2. Immediate CP tended to compare efficiency and evaluate efficiency more frequently than the other two conditions. Explanation CharacteristicSample Explanations Immediate CP Delayed CP No CP 1. Any Comparison a At least one comparison56 7 Compare efficiency b “[I would use] Morgan’s way because it has less steps.” 1670 Compare solution steps b “Alex distributed and Morgan combined like terms” “They both subtracted 4.” 30361 Compare problem features a “If the x + _ numbers are the same, it [Morgan’s way] will work all the time.” 18245 Compare answers c “Both got the same answer”1272 2. Evaluates Efficiency b “James’ way was just faster.” 412519 Problem Features d “Heather's problem has easier number” 81613 Either Evaluation a 474131 3. Justify Mathematically a “Used the right properties at the right times.” 71221 a No CP differs from other two conditions at p <.001; b All three conditions differ from each other at p ≤.001; c All three conditions differ from each other at p <.005; d Delayed CP differs from Immediate CP at p ≤.001 Distribute First Divide Composite Shortcut Multiply Composite Shortcut Combine Composite Shortcut Intervention Day 1 Immediate CP 50.018.2n/a17.2 Delayed CP 83.6***0.3***n/a1.9*** No CP 84.5***1.9***n/a0.0*** Intervention Day 2 Immediate CP 29.612.923.616.9 Delayed CP 49.5*3.0***16.76.7** No CP44.44.6***20.39.2* * Condition differs from Immediate CP at p <.05; ** Condition differs from Immediate CP at p <.01; *** Condition differs from Immediate CP at p ≤.001