Using Comparison to Develop Teachers’ Flexibility in Algebra

1. Using Comparison to Develop Teachers’ Flexibility in Algebra Jon R. Star & Courtney PollackHarvard University Christopher Yakes California State University, Chico PME-NA

2. Outline • Background • What is strategic flexibility? • Importance of flexibility • Development of flexibility • Flexibility and teacher professional development • Current study • Goals • Method • Results • Discussion PME-NA

3. What is strategic flexibility? • Knowledge of multiple approaches for solving mathematics problems and the ability to select the most appropriate strategy for a given problem (Star 2005; Star & Seifert, 2006; Star & Rittle-Johnson, 208; see also Verschaffel, Luvel, Torbeyns, & Van Dooren, 2007) PME-NA

4. Importance of flexibility • Flexibility as an outcome is alluded to in recent policy documents • “knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (“Adding It Up,” NRC 2000) • Strong metacognitive component, which is a key finding for improving student learning (How People Learn, NRC, 2000) PME-NA

5. Development of flexibility Three proposed comparison practices: • Present strategies side-by-side rather than sequentially (Rittle-Johnson and Star, 2007) • Engage students in comparison conversations (Silver, Ghousseini, Gosen, Charalambous, & Strawhun, 2005) • Encourage students to generate multiple solution methods to the same problem (Star & Seifert, 2006; Star & Rittle-Johnson, 2008) PME-NA

6. Teacher professional development • Little is known about how to help teachers support student flexibility • Necessary teacher knowledge, beliefs, or skills • PD experiences that help teachers develop the knowledge, beliefs, or skills • Two studies on prospective elementary school teachers address these concerns (Newton, 2008; Berk, Taber, Gorowara, & Poetz, in press) • No previous studies with in-service secondary teachers PME-NA

7. Goals • Design and pilot a professional development activity for in-service secondary math teachers • Investigate impact of the one-day professional development on teachers’ own flexibility and analyze teachers’ self-reports of subsequent classroom practices • Will teachers use comparison? • What are teachers’ perceived benefits and concerns of using comparison? PME-NA

8. Professional development goals • Increase teachers’ awareness of comparison and how to implement comparison in the classroom • Impact teachers’ flexibility by implementing the three comparison practices • Teachers must see value in flexibility before they will regard it as an important instructional outcome • Flexible teachers can choose appropriate problems more amenable to one solution method over another PME-NA

9. Teacher participants • Twenty-four single-subject credentialed mathematics teachers • Passed CSET or completed state-approved credentialing program • 20 taught fewer than 5 years • 18 taught in high schools, 6 in middle schools • All taught algebra or pre-algebra courses • Range of students vary from low-income and underrepresented groups to middle class and relatively affluent students PME-NA

10. Method • Comparison problem solving activities • Groups of 3-4 teachers given two similar math problems and two suggested strategies • Seven sets total • Teacher presentations • Poster of the four combinations of problem and strategy • Model three comparison practices PME-NA

11. Sample Activity PME-NA

12. Sample solutions PME-NA

13. Data collection • Teachers’ impressions of the professional development activity was assessed with a written open-ended survey • Reflection on comparison activity with regard to teaching • Reflection on comparison to teachers’ mathematical ability and understandings • Second reflection four months after PD • Third reflection nine months after PD PME-NA

14. Results • Teacher discussions for comparison activities • Study included seven sets of problems • Highlight results for three • Post-activity survey • Teacher reflections • Comparison in classrooms • Benefits of comparison and concerns with implementation PME-NA

15. Systems of equations • Many teachers already use modified comparison techniques, presenting multiple solution methods • Conversations about methods or method selection were absent • Teacher comments addressed side-by-side comparison and method selection based on original problem form PME-NA

16. Linear inequalities • Many teachers always move the variable to the left of the inequality, and have trouble moving it to the right • Teachers noted moving the variable to the right may eliminate negative coefficients PME-NA

17. Linear equations • Generally, teachers favor slope-intercept form • Point-slope form is useful when students do not fully understand solving for b • Teachers explored fundamental connections • point-slope form and slope formula • point-slope form and the slope-intercept form PME-NA

18. Post-activity survey • General increase in appreciation of comparison in improving students’ flexibility • “If students look at several ways of doing the same problem, they can start to generalize what’s really going on” • “If I were to use this comparison as a review or a recap of the concepts, the students would then be able to engage in fruitful conversation about the various methods” • “Comparison would also be a way for students to check their own work” PME-NA

19. Post-activity survey (cont.) • Challenges to effective comparison implementation • Teachers noted tendencies to do most of the talking and their trouble allowing students to discuss ideas • “I tend to want to lecture and give them my comparisons instead of asking them what they notice.” • Student confusion • “My worry is that some students will be confused if I introduce more than one way to solve a problem on the same day.” PME-NA

20. Post-activity survey (cont.) • Influence on teachers’ thinking • “I learned that in my own thinking and strategic competence that I already have a mental map of comparison strategies… [The discussion] allows students to take ownership of their own learning.” • “I realized that intuitively I choose a method that is best/most efficient/easiest for me when I work on the board, but I have never taken the time to express why or even let the students suggest why.” PME-NA

21. Teacher reflections • Academic year discussions • Online postings during fall • Follow-up discussion in March • Participants reported a number of instances of using comparison in their classrooms • Benefits and concerns of comparison PME-NA

22. Comparison in the classroom • Multiplying binomials • One problem, four different methods • One solution method per quadrant on twice-folded piece of paper • Solve for unknown side in a right triangle using trigonometric functions • Students presented alternate method alongside teacher’s method • Students challenged to consider additional relationships between the sides of a given triangle PME-NA

23. Comparison in the classroom (cont.) • Solving quadratic equations • Using quadratic formula and factoring • Comparison serves as a way to connect formerly unrelated solution methods in the students’ procedural domain • Systems of equations • Use different solution methods to check one another • Having discussion with the class about when and why one should get the same answers using different methods PME-NA

24. Benefits and Concerns • Benefits of comparison • Highlight multiple solution methods • Serve as culmination activity for review • Connect mathematical ideas • Concerns about using comparison • Student explanations could be unclear, especially for other students • Student confusion about what method to use • Student questioning of necessity of learning multiple strategies PME-NA

25. Discussion • Results suggest that comparison-focused professional development gives teachers an adaptable instructional tool • Provides a chance for teachers to examine their own flexibility • Survey results suggest that teachers . . . • Valued flexibility as a valuable instructional goal • Used comparison for a wide range of topics • Expanded their own flexibility PME-NA

26. Discussion (cont.) • The three comparison practices may be a practical way to begin teaching for flexibility in the algebra classroom • Use side-by-side comparisons • Implemented without difficulty, creative variations • Have comparison conversations • Initial excitement, unclear if teachers could facilitate conversations as modeled • Allow students to generate their own solutions • Happened spontaneously, extend to assessments PME-NA

27. Recommendations for future studies • Focus on long-term effects of comparison in the classroom • Direct observations on teacher practices • Quantitative measures to assess teacher flexibility • Study correlation between teachers’ knowledge of multiple strategies and the effectiveness of their use of comparison • Research impact of comparison with and without discussion and student engagement component PME-NA

28. Thank you! PME-NA