Using Comparison to Develop Teachers’ Flexibility in Algebra Jon R. Star & Courtney Pollack Harvard University Christopher Yakes California State University, Chico 9/29/091PME-NA

Outline Background –What is strategic flexibility? –Importance of flexibility –Development of flexibility –Flexibility and teacher professional development Current study –Goals –Method –Results –Discussion 9/29/092PME-NA

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) 9/29/09PME-NA3

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) 9/29/09PME-NA4

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) 9/29/09PME-NA5

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 9/29/09PME-NA6

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? 9/29/09PME-NA7

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 9/29/09PME-NA8

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 9/29/09PME-NA9

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 9/29/09PME-NA10

Sample Activity 9/29/09PME-NA11

Sample solutions 9/29/09PME-NA12

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 9/29/09PME-NA13

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 9/29/09PME-NA14

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 9/29/09PME-NA15

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 Linear inequalities 9/29/09PME-NA16

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 9/29/09PME-NA17

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” 9/29/09PME-NA18

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.” 9/29/09PME-NA19

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.” 9/29/09PME-NA20

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 9/29/09PME-NA21

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 9/29/09PME-NA22

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 9/29/09PME-NA23

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 9/29/09PME-NA24

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 9/29/09PME-NA25

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 9/29/09PME-NA26

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 9/29/09PME-NA27

Thank you! 9/29/09PME-NA28