Using Comparison to Develop Teachers’ Flexibility in Algebra

Jon R. Star & Courtney PollackHarvard University

Christopher YakesCalifornia State University, Chico

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Outline• Background

– What is strategic flexibility?– Importance of flexibility– Development of flexibility– Flexibility and teacher professional development

• Current study– Goals– Method– Results – Discussion

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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)

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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)

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Development of flexibilityThree 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)

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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

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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?

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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

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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

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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

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Sample Activity

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2. Linear Inequalities: Find the solution sets for the inequalities by (S1) moving the variable to the right-hand side of the inequality and (S2) moving the variable to the left-hand side of the inequality: (P1) 1053 x (P2) 23 xx

Sample solutions

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(S1P2) (S2P2) (with mistake)

i. 23xx

ii. xxxx 23

iii. 220 x

iv. 22220 x

v. x22

vi. x1

i. 23 xx

ii. xxxx 3233

iii. 22 x

iv. 2222 x

v. 1x

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

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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

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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

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1. Systems of Equations: Solve the following systems by (S1) substitution, and (S2) linear combinations:

(P1) 852234

yxyx

(P2) 82523

yxxy

• 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

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2. Linear Inequalities: Find the solution sets for the inequalities by (S1) moving the variable to the right-hand side of the inequality and (S2) moving the variable to the left-hand side of the inequality: (P1) 1053 x (P2) 23 xx

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

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5. Finding Linear Equations: Determine the equation of the line passing through the two points by (S1) using the slope-intercept form of the linear equation, and (S2) using the point-slope form of the linear equation:

(P1) (0, 4) and (5, –2) (P2) (–2, –2) and (6,1)

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”

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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.”

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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.”

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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

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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

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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

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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

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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

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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

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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

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Thank you!

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