1. Exploring the Development of Flexibility in Struggling Algebra Students Kristie J. Newton (Temple University) Jon R. Star (Harvard University) Katie Lynch (Harvard University) 9/24/091PME-NA

2. Strategy Flexibility Flexibility includes knowledge of multiple strategies as well as the adaptive use of them (Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006). Students seem to gain knowledge of multiple strategies for solving problems before they began to use them regularly (e.g., Blöte et al., 2001; Star & Rittle-Johnson, 2008). Star and colleagues have demonstrated that comparing worked examples can be effective in promoting flexibility with linear equations while developing procedural and conceptual knowledge. 9/24/092PME-NA

3. Gaps in the Literature Prior research has been done within controlled environments, using brief interventions with middle school students. Few studies have examined how flexibility might develop for students who struggle with mathematics. Most studies have relied exclusively on written assessments to understand flexibility. The current study included interviews in order to understand why the students chose particular methods. 9/24/093PME-NA

4. Research Questions How do knowledge and use of multiple methods change during an algebra course focused on promoting flexibility? How does prior instruction in algebra impact students’ flexible use of solution methods? How does prior knowledge of mathematics in general impact students’ flexible use of solution methods in algebra? 9/24/094PME-NA

5. Method 9/24/095PME-NA

6. Participants Two boys and four girls, all enrolled in the same high school. Xavier, Ricardo, and Nicole were all new to the school and were entering the ninth grade. Annemarie and Naomi were entering tenth grade. Yvonne was entering eleventh grade. All six students had taken a first course in algebra prior to the summer course. 9/24/096PME-NA

7. Measures Algebra exam ◦ 55 items, served as pretest and posttest ◦ adapted from exam designed by experienced algebra teacher ◦ given to eight experts Intermediate assessments ◦ homework, quizzes, tests ◦ prompted students to solve problems in more than one way Interviews ◦ Pre/post interview asked students to solve problems in more than one way, to evaluate two methods of solving, and to share their views on learning multiple ways to solve problems. ◦ Intermediate interview asked about solution methods from Test I and about students’ views on learning multiple ways to solve problems. 9/24/097PME-NA

8. Procedure 14 instructional days plus one day for final exam Met Monday – Friday for two hours each afternoon Three units ◦ Unit I – solving/graphing linear equations, solving systems ◦ Unit II – simplifying exponential and radical expressions ◦ Unit III – factoring, solving/graphing quadratic equations A quiz and a test were administered for each of these units, and homework was assigned nearly every evening. 9/24/098PME-NA

9. A typical lesson Students compared and contrasted worked examples of problems relevant to the day’s topic. Students discussed which methods they preferred, why they preferred them, and under what circumstances. Worked examples were designed to draw students’ attention to crucial features of the equations. 9/24/099PME-NA

10. Example problem types and strategies Solving linear equations ◦ For problems of the type a(x + b) = c, students were shown distributing as a first step and “clearing” a as a first step. Simplifying radical expressions ◦ Students were shown multiple strategies, including taking the square root first and simplifying the radicand first. 9/24/0910PME-NA Divide first Distribute first

11. Results 9/24/0911PME-NA

12. Knowledge and Use of Multiple Methods Students gained knowledge of multiple strategies but limited their use of alternate strategies in some cases. They tended to use alternate strategies: ◦ when their first strategy posed difficulty ◦ when all strategies for a problem type were equally familiar to them Familiarity, efficiency, and form of the problem were primary reasons for strategy choices 9/24/0912PME-NA

13. Solving Quadratic Equations (Xavier, post-interview) Efficiency: Xavier preferred not to use the quadratic formula because, for the other two methods, “there are less steps and it’s just, there’s a clear way to get the answer.” Form: He stated that the first equation has a “perfect square” and the second one was “pretty easy to factor, it’s already in a factorable form.” 9/24/0913PME-NA

14. Impact of Prior Instruction on Flexibility Prior instruction in algebra negatively influenced flexibility when students began the course with considerable fluency with a particular strategy. The most prominent example of this trend concerned linear equations of the type a(x + b) = c where a was a whole number. 9/24/0914PME-NA

15. 7(n + 2) = 49 (Yvonne, intermediate interview) Let’s look at how you solved some of the problems. This one here? I distributed the 7 to the n + 2. Distributed. Okay. Why did you choose that as your first step? Um, just because it was what came to mind first and it just seemed easiest. Do you know another way to do it? You could divide the 49 by 7 and just get n + 2 = 7, and then you would get n = 5. And which way do you like better? The first way. Just the way I did it. I mean, I guess it doesn’t matter, but it was like instinct, sort of. 9/24/0915PME-NA

16. Impact of Prior Knowledge on Flexibility Weak knowledge of algebra in some students interfered with their ability to attend to and implement efficient methods for certain problem types. The presence of fractions in a problem was often a key factor in leading students to change strategies. Students seemed grateful to have a “backup” method. In general, concern with accuracy was often a driving force for deciding how to solve problems. 9/24/0916PME-NA

17. Simplifying with exponents Xavier, Test 2 #2 9/24/0917PME-NA

18. Distribute for a(x + b) = c, unless a is a fraction For a = 6/5: “Because this one was totally confusing. To distribute was really confusing…It was getting confusing multiplying 6/5 by this. It was easier just to get rid of it” (Ricardo, intermediate interview). For a = 1/2: “Distributing fractions kind of scare me. Distributing them, I’m not good with fractions so I was afraid that if I distributed it, I would distribute the fraction wrong” (Annemarie, intermediate interview). 9/24/0918PME-NA

19. What are some advantages to learning more than one way to solve math problems? “Well it’s an advantage because in case you get stuck doing one way, you can always have a backup.” (Nicole, post-interview). “There were, like with the radicals, how you showed me two methods. My teacher taught me, my old teacher taught me one method, and it was a method that I really didn’t understand. And when you showed me the other method I was like, wow, this is easier. And I could understand it more and I always hated radicals, but now I am starting to get them.” (Annemarie, post-interview) 9/24/0919PME-NA

20. Discussion Students seemed grateful to have a “backup” method. Accuracy seemed to be the primary force driving their strategy decisions. Efficiency and form were secondary forces – prevalent when all choices were relatively equal in familiarity and difficulty. There was some sense that searching for and using efficient approaches became a norm of the class. Future studies should examine the impact of flexibility on attitudes and beliefs about mathematics. 9/24/0920PME-NA

21. Thanks! Questions and discussion? 9/24/0921PME-NA