Students’ use of standard algorithms for solving linear equations

1. Students’ use of standard algorithms for solving linear equations Jon R. Star Michigan State University

2. Acknowledgements • Thanks to graduate students at MSU: Kosze Lee, Beste Gucler, Howard Glasser, Mustafa Demir, and Kuo-Liang Chang • Thanks to Bethany Rittle-Johnson, Vanderbilt, for her collaboration in the design and implementation of this study. • Funds supporting this work provided by small grants from the Michigan State University College of Education. PME-NA 2005

3. Starting definitions • A procedure is a step-by-step plan of action for accomplishing a task • A strategy is a plan of action for accomplishing a task • I use these terms synonymously, as is the norm among many psychologists who study strategy change (e.g., Siegler) PME-NA 2005

4. More definitions • A procedure/strategy can be either: • A heuristic, which is a helpful procedure for arriving at a solution; a rule of thumb • An algorithm, which is a procedure that is deterministic; when one follows the steps in a predetermined order, one is guaranteed to reach the solution PME-NA 2005

5. Standard algorithms • For some problems, a “standard algorithm” (SA) exists • Called “standard” because it is commonly and often explicitly taught as THE way to solve problems within a problem class PME-NA 2005

6. Strategies for 4(x + 5) = 80 • A standard algorithm (SA) 4(x + 5) = 80 4x + 20 = 80 4x = 60 x = 15 • Alternative approach #1 4(x + 5) = 80 x + 5 = 20 x = 15 • Alternative approach #2 4(x + 5) = 80 4x + 20 = 80 4x - 60 = 0 x - 15 = 0 x = 15 PME-NA 2005

7. SA, more generally • Distribute first, to “clear” parentheses • Combine like variable and constant terms on each side • ‘Move’ variable terms to one side and constant terms to the other side • Divide both sides by the coefficient of the variable term PME-NA 2005

8. Reasonably efficient Widely applicable Can be executed often without attending to specifics of the problem Are not always the best or most efficient strategy Over reliance on SA may lead to difficulties on unfamiliar problems Ability to use not always connected with why algorithm is effective; may lead to rote memorization; strategy may be easily forgotten Pros and cons of SA PME-NA 2005

9. Learning standard algorithms • Learning and use of SAs has become a flashpoint issue in US mathematics education • Should they be learned at all? explicitly taught? discovered? • Not a lot of research on students’ learning of SA to help resolve these issues • Particularly on algorithms other than arithmetic PME-NA 2005

10. Not researched in high school? • Key features of elementary school reform instruction are less typical at high school level: • Sharing and comparing of multiple strategies for solving problems • Allowing students to discover their own algorithms, rather than providing direct instruction on a SA • Allowing students to use non-standard algorithms PME-NA 2005

11. Not researched in high school? • Discovery of SAs is presumed to be more difficult, if not highly improbable, in high school • “Are you saying you want my students to ‘discover’ the quadratic formula?!” • As a result, many teachers feel that it is necessary to provide direct instruction on strategies such as the SA • “If I don’t teach students this algorithm, there is no way that they would come up with it on their own.” PME-NA 2005

12. Unquestioned assumptions • Is direct instruction the only way that students will learn the SA? • Can students discover the SA largely on their own? • When some students discover a strategy and others are shown it by direct instruction, is there a difference in how students use the strategy? PME-NA 2005

13. Larger goal: Flexibility • We want students to know the SA but also to be flexible in their knowledge of problem solving strategies, meaning that they: • Know a variety of other strategies (SA and others) that can be used to solve similar problems • Are able to adaptively select the most appropriate strategy (SA and others) for solving a particular problem (Star, 2001, 2002, 2004, 2005) PME-NA 2005

14. Research questions • Do students discover the SA for solving linear equations when allowed to work largely on their own? • Do either of two instructional interventions affect the discovery and use of the SA among algebra equation solvers? • Direct instruction • Alternative ordering task (Star, 2001) • Goal was to see what strategies students develop and how they make sense of, use, and modify these strategies PME-NA 2005

15. Method • 130 6th graders (82 girls, 48 boys) • 5 one-hour classes in one week (Mon - Fri) • Class size 8 to 15 students; students worked individually • Pre-test (Mon), post-test (Fri); three problem-solving sessions (Tues, Wed, Thurs) • Domain was linear equation solving • 3(x + 1) = 12 • 2(x + 3) + 4(x + 3) = 24 • 9(x + 2) + 3(x + 2) + 3 = 18x + 9 PME-NA 2005

16. Prior knowledge & instruction • Students had no prior knowledge of symbolic approaches for solving equations • Minimal instruction and feedback provided • 30 minute benchmark lesson • Combine like terms, add to both sides, multiply to both sides, distribute • How to use each step individually • No strategic guidance provided during study • No worked examples PME-NA 2005

17. Alternative ordering task • During problem solving, some students were asked to re-solve a previously completed problem, but using a different ordering of steps (Star, 2001) • Random assignment to condition by class • Control group solved new but isomorphic problem • 2(x + 1) = 10 • 3(x + 2) = 15 PME-NA 2005

18. Direct instruction • At start of 2nd problem solving class (Wed), 3 worked examples presented to direct instruction classes • “This is the way I solve this equation.” • Each problem solved with using a different method; one was the SA • Total time was 8 minutes of supplemental instruction • Random assignment to condition by class PME-NA 2005

19. Analysis • Students’ written work was analyzed for use of SA • Booklet problems (Tues, Wed, Thurs sessions) - total of 31 equations attempted • Post-test problems - total of 9 equations attempted • Three “markers” of SA: • Distribute first • Combine like terms before moving • Divide as a final step PME-NA 2005

20. Results. • About 2/3 of students did not discover SA • Of those who did, a small number started using SA very early PME-NA 2005

21. Results.. PME-NA 2005

22. Results... • Those who discovered and used SA performed better on the post-test than those who did not use SA (p < .01) PME-NA 2005

23. Results... PME-NA 2005

24. Results.... • Direct instruction on SA did not increase chances that a student would use SA on post-test PME-NA 2005

25. Results..... PME-NA 2005

26. Results...... • Stated somewhat differently (and not including 12 Early Users): • 25% (16 of 65) of students in the Direct Instruction condition used SA on the post-test • 30% (16 of 53) of students in the Discovery condition used SA on the post-test PME-NA 2005

27. Results....... • The alternative ordering task made it less likely that a student would use the SA on the post-test (p < .05) • Alternative ordering task made it more likely that students would use other, more efficient or innovative strategies than the SA on the post-test PME-NA 2005

28. Results........ PME-NA 2005

29. Summary of results. • Do students discover the SA for solving linear equations when allowed to work largely on their own? • Most did not • Only about one-fourth of students learned the SA on their own • Is one-fourth high or low? PME-NA 2005

30. Summary of results.. • Do either of two instructional interventions affect the discovery and use of the SA among algebra equation solvers? • There was no difference in the rate of SA use between the direct instruction and discovery conditions • The alternative ordering condition made it less likely that students used the SA on the post-test PME-NA 2005

31. Implications for SA learning • Neither a short period of direct instruction (viewing of worked examples) nor pure discovery was particularly effective in promoting development of the SA • Is learning the SA a goal of algebra instruction? If so, how should it best be taught? PME-NA 2005

32. Implications for flexibility • Flexibility aided by activities such as the alternative ordering task, where students generate and compare multiple strategies for solving procedural problems • Direct instruction did not improve chances of discovering the SA, so activities such as the alternative ordering task appear to be a win-win proposition PME-NA 2005

33. This presentation and other related papers can be downloaded at: www.msu.edu/~jonstar Jon R. Star Michigan State University jonstar@msu.edu