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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, 2008
Toronto, Canada
Abstract
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-asusual 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
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mathematics achievement test.
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
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not considered
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
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?
• Write compared and base quantities in
diagram
• Write an x for what must be solved
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2. Organize the information
12 Girls
2
5
x Children
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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
5
Ratio word problem solving
Rates
6&7
Proportion word problem solving
8&9
Scale drawing word problem
solving
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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
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Results
ES = 0.45
ES = 0.56
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Results
On the PSSA posttest:
• SBI-SM and control classes did not
differ
• Scores in each track significantly
differed as expected:
• High > Average > Low
• No interaction
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Conclusion
• SBI-SM led to significant gains in problemsolving 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)
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Thanks!
Asha K. Jitendra ([email protected])
Jon R. Star ([email protected])
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