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Using Schema-based Instruction to
Improve Seventh Grade Students’
Learning of Ratio and Proportion
Jon R. Star (Harvard University)
Asha K. Jitendra (University of Minnesota)
Kristin Starosta, Grace Caskie, Jayne Leh, Sheetal Sood,
Cheyenne Hughes, and Toshi Mack (Lehigh University)
Thanks to...
• Research supported by Institute of Education
Sciences (IES) Grant # R305K060075-06
• All participating teachers and students
(Shawnee Middle School, Easton, PA)
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Solving word problems in math
• Is very hard for students
• Yet plays a critical role in our instructional goals
in mathematics
• Something that low achieving students
particularly struggle with
Cummins, Kintsch, Reusser, & Weimer, 1988; Mayer, Lewis, & Hegarty, 1992;
Nathan, Long, & Alibali, 2002; Rittle-Johnson & McMullen, 2004
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To solve word problems,
• Need to be able to recognize underlying
mathematical structure
• Allows for the organization of problems and
identification of strategies based on underlying
mathematical similarity rather than superficial
features
• “This is a rate problem”
– Rather than “This is a bicycle problem”
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Schemata
• Domain or context specific knowledge structures
that organize knowledge and help the learner
categorize various problem types to determine
the most appropriate actions needed to solve
the problem
Sweller, Chandler, Tierney, & Cooper, 1990; Chen, 1999
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Develop schema knowledge?
• Math education: A student-centered, guided
discovery approach is particularly important for
low achievers (NCTM)
• Special education: Direct instruction and
problem-solving practice are particularly
important for low achievers
Baker, Gersten, & Lee., 2002; Jitendra & Xin, 1997; Tuovinen & Sweller, 1999;
Xin & Jitendra, 1999
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Our approach
• Collaboration between special education
researcher (Jitendra) and math education
researcher (Star)
• Direct instruction
• However, “improved” in two ways by connecting
with mathematics education literature:
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Exposure to multiple strategies
• Weakness of some direct instruction models is
focus on a single or very narrow range of
strategies and problem types
• Can lead to rote memorization
• Rather, focus on and comparison of multiple
problem types and strategies linked to flexibility
and conceptual understanding
Rittle-Johnson & Star, 2007; Star & Rittle-Johnson, 2008
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Focus on structure
• Avoid key word strategies present in some direct
instruction curricula
– in all means total, left means subtraction, etc.
• Avoid procedures that are disconnected from
underlying mathematical structure
– cross multiplication
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SBI-SM
• 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
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An example problem
• 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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1. Find the problem type
• Read and retell problem to understand it
• Ask self if this is a ratio problem
• Ask self if problem is similar or different from
others that have been seen before
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
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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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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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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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Additional problem types/schemata
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Our questions
• Does the SBI-SM approach improve students’
success on ratio and proportion word problems,
as compared to “business as usual” instruction?
• Is SBI-SM more or less effective for students of
varying levels of academic achievement?
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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% AfrAm
• 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, to be taught over 10 days
Lesson
Content
1
Ratios
2
Equivalent ratios; Simplifying ratios
3&4
5
Rates
6&7
Proportion word problem solving
8&9
Scale drawing word problem solving
10
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Ratio word problem solving
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
– 18 items from TIMSS, NAEP, and state assessments
• Cronbach’s alpha
– 0.73 for the pretest
– 0.78 for the posttest
– 0.83 for the delayed posttest
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Sample PS test item
• If there are 300 calories in 100g of a certain
food, how many calories are there in a 30g
portion of this food?
A. 90
B. 100
C. 900
D. 1000
E. 9000
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Results
• At pretest:
• 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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Results
• At posttest:
• Significant main effect for treatment: SBI-SM
scored higher than control classes
– Low medium effect size of 0.45
• Significant main effect for track as expected
– High > Average > Low
• No interaction
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Results
• At delayed posttest:
• Significant main effect for treatment: SBI-SM
scored higher than control classes
– Medium effect size of 0.56
• Significant main effect for track as expected
– High > Average > Low
• No interaction
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Results
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In sum...
• SBI-SM led to significant gains in problemsolving skills
• Developing deep understanding of the
mathematical problem structure and fostering
flexible solution strategies helped students in the
SBI-SM group improve their problem solving
performance
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Thanks!
Jon R. Star ([email protected])
Asha K. Jitendra ([email protected])
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