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)

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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 understandingRittle-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

• 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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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 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– 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. 90B. 100C. 900D. 1000E. 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

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 problem-solving 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 (jon_star@harvard.edu)

Asha K. Jitendra (jiten001@umn.edu)

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