Developing Procedural Flexibility: When Should Multiple Solution Procedures Be Introduced? Bethany Rittle-Johnson Jon Star Kelley Durkin

Procedural Flexibility Defined Procedural flexibility: person’s ability to flexibly choose among alternative solution procedures to solve problems Know more than one way to solve a problem Know when it is appropriate to use a particular procedure, including strengths and relative efficiencies of procedures Use most appropriate procedures – (Beishuizen, van Putten, & van Mulken, 1997; Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006) Common for people to know procedures that they do not initially use (e.g., utilization deficiency; Miller & Seier, 1994), so important to assess both knowledge and use. 2AERA 2010

Why Procedural Flexibility Matters Related to greater expertise in mathematics. Students with greater procedural flexibility: – Have better transfer – Have greater knowledge of domain concepts (Blöte, et al., 2001; Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Hiebert & Wearne, 1996; Resnick, 1980; Rittle-Johnson & Star, 2007). 3AERA 2010

Pathways to Procedural Flexibility: When and How How does procedural flexibility develop? – Comparing multiple solution methods is very helpful (Rittle-Johnson & Star, 2007; Star & Rittle-Johnson, 2009). When in the learning process should multiple procedures be introduced? – Should multiple methods be introduced from the very beginning or only after students have some mastery of one method? – If knowledge of one method is desirable, how should competence with that method be supported? What role should comparison play? 4AERA 2010

When To Introduce Multiple Procedures In Mathematics Instruction? Your intuitions: Which is better No Delay – Introduce multiple method from beginning After a Delay - Learn one procedure first 5AERA 2010

When to Introduce Multiple Procedures Early introduction to multiple procedures is emphasized in current ideas for best practices in mathematics education in some countries (Becker & Selter, 1996; Klein, Beishuizen, & Treffers, 1998; National Council of Teachers of Mathematics, 2000). Prior research has not experimentally compared the effectiveness of early vs. delayed introduction of multiple procedures. AERA 20106

After A Delay Learn one procedure first – Avoid overwhelming students’ limited working- memory capacity (Sweller, van Merrienboer, & Paas, 1998) – Adhere to instructional design principle that students need sequenced, building up of multiple ideas, not simultaneous presentation of ideas (Carnine, 1997). – Recognize teacher concerns about “possibly confusing students with multiple solutions.” (Silver, Ghousseini, Gosen, Charalambous, & Strawhun, 2005, p. 292). 7AERA 2010

No Delay Introduce multiple method from beginning – Immediate and explicit attention to multiple procedures was associated with greater flexibility than when exposure to multiple procedures was delayed in two year-long classroom studies (Blote et al 2001; Klein et al 1998) 8AERA 2010

Current Study Experimentally evaluated the impact of early introduction to multiple procedures. Explore ways that comparison can support learning of the procedures. – Compare two different procedures for solving the same problem – Compare two different problems solved with the same procedure (Rittle-Johnson & Star, 2009) AERA 20109

Current Study Three conditions differed in – Whether multiple methods were introduced immediately or after practice with a single method – Whether comparison of examples was supported. AERA CompareNo Compare No Delay1. No-Delay-Comparex Delay2. Delay-Compare3. Delay-No-Compare

Target Domain: Equation Solving Method #1: 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 Method #2: 3(x + 1) = 15 x + 1 = 5 x = 4 Tested these conditions with 8 th graders learning to solve multi- step equations Equation solving is an area of weakness for students, but is necessary for advancing in mathematics (Blume & Heckman, 1997; Schmidt et al., 1999). A focal point for instruction in 7 th & 8 th grade (NCTM, 2006) Students tend to memorize one method, even though there are multiple methods that vary in efficiency (e.g., # of solution steps) ◦ Example: Solving 3(x + 1) = 15 11AERA 2010

12 Method Participants: 198 8th grade students in TN classes with limited algebra instruction. Design: – Pretest - Intervention – Posttest – 1-month-Retention Test – Randomly assigned to condition No-delay-compare, n = 67 Delay-compare, n = 62 Delay-no-compare, n = 69 – Intervention Occurred in partner work during 2 math classes (about 80 min each) Studied worked examples and answered prompts with partner Solved practice problems on own 12AERA 2010

No-Delay - Compare 13AERA 2010

Delay - Compare Day 1: Compared two similar equations solved using the same procedure Day 2: Compared same equation solved in two different ways (identical to no-delay-compare condition) 14AERA 2010

Delay – No-Compare 15AERA 2010

Conditions ConditionDay 1Day 2 No-Delay-CompareLearn multiple procedures by comparing two ways to solve same problem Delay-CompareLearn 1 procedure by comparing different problems solved same way Learn multiple procedures by comparing two ways to solve same problem Delay – No CompareLearn 1 procedure by studying examples one at a time Learn new procedures by studying examples one at a time; review original procedure too 16AERA 2010

17 Procedural knowledge assessment Equation Solving – Intervention: ¼ (x + 8) = 5 – Posttest Familiar: ½ (x + 1) = 10 – Posttest Novel: 17AERA 2010

18 Procedural flexibility Use of more efficient solution methods on procedural knowledge assessment (i.e., method with fewer solution steps) Knowledge of multiple methods – Solve each equation in two different ways. – Evaluate methods: Looking at the problem shown above, do you think that this first step is a good way to start this problem? Circle your answer below and explain your reasoning. (a) Very good way(b) Ok, but not a very good way(c) Not OK to do 18AERA 2010

19 Conceptual knowledge assessment 19AERA 2010

No Delay Increases Flexible Use Main effect of condition (p’s ≤.01); No-Delay-Compare more flexible use than delay conditions 20AERA 2010

No Delay Increases Flexible Knowledge Main effect of condition (p =.09 at post, p =.02 at retention); No-Delay-Compare more flexible knowledge than Delay-Compare 21AERA 2010

No Delay Improves Procedural Accuracy at Retention Test No effect of condition at posttest, but main effect of condition after 1 month delay (p =.03) No-Delay-Compare has greater procedural knowledge than Delay-Compare 22AERA 2010

Conceptual Knowledge Same Regardless No effect on conceptual knowledge. Note: This was a new measure, with alpha = AERA 2010

Relations Between Knowledge Measures Both measures of flexibility related to greater procedural and conceptual knowledge Flexibility mediated the effect of condition on procedural knowledge at retention. AERA

Intervention Data Why did no-delay-compare help flexibility and long-term accuracy? – No Delay increased use of more efficient procedures on the practice problems. – No Delay focused students’ attention on the efficiency of different methods when studying the worked examples. 25AERA 2010

Summary Immediate introduction and comparison of multiple procedures focused students’ attention on the efficiency of different methods and increased adoption of more efficient methods. The pay off was greater procedural flexibility in the short-term, as well as greater procedural accuracy in the long-term. 26AERA 2010

Conclusions Contrary to concerns about overwhelming novices with multiple procedures, these findings support early introduction of students to multiple procedures. – Converges with classroom-based research on 2 nd graders learning multi-digit arithmetic procedures (Blöte, Van der Burg & Klein, 2001) – Teachers should be reassured that it is a good idea to teach multiple procedures even when teaching a lesson for the first time (Silver et al., 2005) However, this may not be true when procedures are in different representational formats (such as graphs and tables). Additional research is needed. 27AERA 2010

What’s Next? Support teachers’ use of comparison throughout Algebra I Identifying types of comparisons and the strengths of each 28 AERA 2010

29 Acknowledgements for Funded by a grant from the Institute for Education Sciences, US Department of Education Thanks to research assistants: – Kristen Tremblay, Holly Harris, Anna Krueger, Vivien Haupt, Chrissy Tanner, and Meredith Murray 29AERA 2010