Improving Ratio and Proportion Problem Solving Performance of Seventh Grade

Students Using Schema-Based Instruction

Asha K. Jitendra (University of Minnesota)

Jon R. Star (Harvard University)

Kristin Starosta, Grace Caskie, Jayne Leh, Sheetal Sood, Cheyenne Hughes, and Toshi Mack (Lehigh University)

Poster Presented at the Annual International Academy for Research in Learning Disabilities

June 20, 2008Toronto, Canada

The present study evaluated the effectiveness of schema-based instruction with self-monitoring (SBI-SM). Specifically, SBI-SM emphasizes the role of the mathematical structure of problems and also provides students with a heuristic to aid and self-monitor problem solving. Further, SBI-SM addresses well-articulated problem solving strategies and supports flexible use of the strategies based on the problem situation. One hundred forty eight seventh-grade students and their teachers participated in a 10-day intervention on learning to solve ratio and proportion word problems, with random assignment to SBI-SM or a business-as-usual control. Results indicated that students in SBI-SM treatment classes made greater gains than students in control classes on a problem solving measure, both at posttest and on a delayed posttest administered four months later. However, the two groups’ performance was comparable on a state standardized mathematics achievement test. 2

Abstract

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Study Background• Converging evidence suggests that explicit schema training

using visual representations improves students’ problem solving performance(Fuchs, Seethaler, Powell, Fuchs, Hamlett, Fletcher, 2008; Fuson & Willis, 1989; Griffin & Jitendra, in press; Jitendra, DiPipi, & Perron-Jones, 2001; Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007; Jitendra, Griffin, Haria, Leh, Adams, & Kaduvetoor, 2007; Jitendra, Griffin, McGoey, Gardill, Bhat, & Riley, 1998; Jitendra & Hoff, 1993; Jitendra, Hoff, & Beck, 1999; Lewis, 1989; Willis & Fuson, 1988; Xin, Jitendra, & Deatline-Buchman, 2005; Zawaiza & Gerber, 1993).

• Prior work on schema-based instruction (SBI) by Jitendra and colleagues suggested to us three conclusions:– Focus on students with disabilities and low achieving students

– Emphasis exclusively on word problem solving rather than also addressing the foundational concepts (e.g., ratios, equivalent fractions, rates, fraction and percents) in ratio and proportion problem solving, for example.

– Multiple solution strategies and flexible application of those strategies are not considered

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Our Approach

• Schema-Based Instruction with Self-Monitoring• Translate problem features into a coherent

representation of the problem’s mathematical structure, using schematic diagrams

• Apply a problem-solving heuristic which guides both translation and solution processes

• Focus on multiple solution strategies and flexible application of those strategies

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1. Find the problem type

• Read and retell problem to understand it

The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in the class is 12. How many children are in the class?

• Ask self if this is a ratio problem

• Ask self if problem is similar or different from others that have been seen before

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2. Organize the information

• Underline the ratio or comparison sentence and write ratio value in diagram

• Write compared and base quantities in diagram

• Write an x for what must be solved

The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in the class is 12. How many children are in the class?

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2. Organize the information

12 Girls

x Children

2

5

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3. Plan to solve the problem

• Translate information in the diagram into a math equation

• Plan how to solve the equation

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Problem solving strategies

A. Cross multiplication

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Problem solving strategies

B. Equivalent fractions strategy

“7 times what is 28? Since the answer is 4 (7 * 4 = 28), we multiply 5 by this same number to get x. So 4 * 5 = 20.”

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Problem solving strategies

C. Unit rate strategy

“2 multiplied by what is 24? Since the answer is 12 (2 * 12 = 24), you then multiply 3 * 12 to get x. So 3 * 12 = 36.”

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4. Solve the problem

• Solve the math equation and write the complete answer

• Check to see if the answer makes sense

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Research Questions• What are the differential effects of SBI-SM and

“business as usual” treatment on the acquisition of seventh grade students’ ratio and proportion word problem solving ability?

• Is there a differential effect of the treatment (SBI-SM and business as usual) on the maintenance of problem solving performance four months following the end of intervention

• Do the effects of the treatment transfer to performance on state wide mathematics assessment?

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Participants

• 148 7th grade students (79 girls), in 8 classrooms, in one urban public middle school

• 54% Caucasian, 22% Hispanic, 22% African American

• 42% Free/reduced lunch• 15% receiving special education services

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Teachers

• 6 teachers (3 female)• (All 7th grade teachers in the school)• 8.6 years experience (range 2 to 28 years)• Text: Glencoe Mathematics: Applications and

Concepts, Course 2• Intervention replaced normal instruction on

ratio and proportion

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Design• Pretest-intervention-posttest-delayed

posttest with random assignment to condition by class

• Four “tracks” - Advanced, High, Average, Low*# classes High Average Low

SBI-SM 1 2 1

Control 1 2 1*Referred to in the school as Honors, Academic, Applied, and Essential

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Instruction• 10 scripted lessons taught over 10 days

Lesson Content

1 Ratios

2 Equivalent ratios; Simplifying ratios

3 & 4 Ratio word problem solving

5 Rates

6 & 7 Proportion word problem solving

8 & 9 Scale drawing word problem solving

10 Fractions and percents

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Professional development• SBI-SM teachers received one full day of PD

immediately prior to unit and were also provided with on-going support during the study– Understanding ratio and proportion problems– Introduction to the SBI-SM approach– Detailed examination of lessons

• Control teachers received 1/2 day PD– Implementing standard curriculum on

ratio/proportion

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Treatment fidelity

• Treatment fidelity checked for all lessons

• Mean treatment fidelity across lessons for intervention teachers was 79.78% (range = 60% to 99%)

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Outcome measure• Mathematical problem-solving (PS) test

– 18 items from TIMSS, NAEP, and state assessments

– Cronbach’s alpha (0.73, 0.78, and 0.83 for the pretest, posttest, and delayed posttest)

• Mathematics subtest of the Pennsylvania System of School Assessment (PSSA).– Cronbach’s alpha > 0.90

Results

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ES = 0.45 ES = 0.56

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Results

• SBI-SM and control classes did not differ

• Scores in each track significantly differed as expected:

• High > Average > Low

• No interaction

On the PSSA posttest:

Conclusion• SBI-SM led to significant gains in problem-

solving skills• The benefits of SBI-SM persisted four months

after the intervention • The effects for SBI-SM were not mediated by

ability level, suggesting that it may benefit a wide range of students

• The SBI-SM treatment did not show an advantage over the control treatment on the statewide mathematics test (possibly due to the short-term intervention) 23

Thanks!

Asha K. Jitendra (jiten001@umn.edu)

Jon R. Star (jon_star@harvard.edu)

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