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Teaching Ratio and Proportion Problem Solving Using Schema-

based InstructionAsha K. Jitendra,1 Jon Star,2

Kristin Starosta,3 Sheetal Sood,3

Grace Caskie, 3 Jayne Leh, 3 Cheyenne Hughes, 3 Toshi Mack, 3 and Sarah Paskman 3

1University of Minnesota2Harvard University

3Lehigh University

Paper Presented at the 2008 Annual CEC Convention, Boston, MA

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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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Mathematical word problems

• Represent “the most common form of problem solving” (Jonassen, 2003, p. 267) in school mathematics curricula.

• Present difficulties for special education students and low achieving students Cummins, Kintsch, Reusser, & Weimer, 1988; Mayer, Lewis, & Hegarty, 1992; Nathan, Long, & Alibali, 2002; Rittle-Johnson & McMullen, 2004).

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Math WarsMath Wars

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To solve word problems,

• Need to be able to recognize the underlying mathematical structure

• Schemas • 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

Chen, 1999; Sweller, Chandler, Tierney, & Cooper, 1990

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Focus on math structure helps …

• Allows for the organization of problems and identification of strategies based on the underlying mathematical similarity rather than superficial features

• “This is a rate problem”– Rather than “This is a train problem”

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Prior research on SBI has focused on

• Schema priming (Chen, 1999; Quilici & Mayer, 1996; Tookey, 1994),

• Visual representations such as number line diagrams (e.g., Zawaiza & Gerber, 1993) or schematic diagrams (e.g., Fuson and Willis, 1989); Jitendra, Griffin, McGoey, Gardill, Bhat, & Riley, 1998; Xin, Jitendra, & Deatline-Buchman, 2005; Jitendra, Griffin, Haria, Leh, Adams, & Kaduvettoor, 2007; Willis and Fuson, 1988)

• Schema-broadening by focusing on similar problem types (e.g., Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp & Jancek, 2003; Fuchs, Seethaler, Powell, Fuchs, Hamlett, & Fletcher, 2008; )

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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

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Marshall (1990); Mayer (1999); Riley, Greeno, & Heller (1983)

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Teaching proportionality is critical …

• Challenging topic for many students (National

Research Council, 2001) • Current curricula typically do not focus on

developing deep understanding of the mathematical problem structure and flexible solution strategies (NCES, 2003; NRC, 2001).

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Purpose of the study

• To investigate the effectiveness of SBI-SM instruction on students’ ability to solve ratio and proportion problems.

• To evaluate the outcomes 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

• Mean chronological age 153.12 months (range = 137.04 to 174.96; SD = 5.76).

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

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

3% ELLs

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Teacher Participants

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

and Concepts, Course 2

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Study 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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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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Procedure - Both Conditions• Instruction on same topics

• Duration: 40 minutes daily, five days per week across 10 school days

• Classroom teachers delivered all instruction

• Lessons structured as follows: – Students work individually to complete a review problem

and teacher reviews it in a whole class format,

– Teacher introduces the key concepts/skills using a series of examples

– Teacher assigns homework

• Students allowed to use calculators.April 4, 2008

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SBI-SM Condition

• Our intervention unit on ratio and proportion

• Lessons scripted

• Instructional paradigm: Teacher-mediated instruction - guided learning - independent practice, using schematic diagrams and problem checklists (FOPS)

• Teacher and student “think alouds”

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SBI-SM Instructional SequenceLesson 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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Problem Checklist (FOPS)

• Step 1. Find the problem type

• Step 2: Organize the information

• Step 3: Plan to solve the problem

• Step 4: Solve the problem

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Applying SBI-SM to Solve Ratio Problems

Example:

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?

23March 27, 2008 AERA 53.026 23

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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Control condition

• Instructional procedures outlined in the district-adopted mathematics textbook

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Outcome Measure

• Mathematical problem-solving (PS)– 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

Figure 1. 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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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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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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Figure 1Mathematics Problem-Solving Performance by Condition

0

10

20

30

40

50

60

70

80

90

100

Pretest Posttest Delayed Posttest

Percent Correct Score

SBI-SM Control

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10

20

30

40

50

60

70

80

90

100

Pretest Posttest Delayed Posttest

Percent Correct Score

SBI-SM Academic Control Academic SBI-SM Applied Control Applied SBI-SM Essential Control Essential

Figure 2Mathematics Problem-Solving Performance by Condition and Students’ Ability Level Status

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Summary and Discussion

• A low moderate effect size (0.45) at Time 1 • A strong moderate effect (0.56) at Time 2

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

SBI-SM led to significant gains in problem-solving skills.

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Discussion

• Three issues undermined the potential impact of SBI-SM

– One high ability control classroom teacher deviated from the textbook presentation

– One intervention teacher experienced classroom management difficulties

– Variation in implementation fidelity

• Intervention was time-based (10 days) rather than criterion-based (mastery of content)

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Thanks!

Asha K. Jitendra (jiten001@umn.edu)

Jon R. Star (jon_star@harvard.edu)

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SBI References from our Research Team

BOOKS AND CURRICULAR MATERIALS• Jitendra, A. K. (2007). Solving math word problems:

Teaching students with learning disabilities using schema-based instruction. Austin, TX: Pro-Ed.

• Montague, M., & Jitendra, A. K. (Eds.) (2006). Teaching mathematics to middle school students with learning difficulties. New York: The Guilford Press.

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SBI References from our Research TeamCHAPTERS

Chard, D. J., Ketterlin-Geller, L. R., & Jitendra, A. K. (in press). Systems of instruction and assessment to improve mathematics achievement for students with disabilities: The potential and promise of RTI. In E. L. Grigorenko (Ed.), Educating individuals with disabilities: IDEIA 2004 and beyond. New York, N.Y.: Springer.

Xin, Y. P., & Jitendra, A. K. (2006). Teaching problem solving skills to middle school students with mathematics difficulties: Schema-based strategy instruction. In M. Montague & A. K. Jitendra (Eds.), Teaching mathematics to middle school students with learning difficulties (pp. 51-71). New York: Guilford Press.

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SBI References from our Research Team

Journal Articles• Griffin, C. C. & Jitendra, A. K. (in press). Word problem solving

instruction in inclusive third grade mathematics classrooms. Journal of Educational Research.

• Jitendra, A. K., Griffin, C., Deatline-Buchman, A., & Sczesniak, E. (2007). Mathematical word problem solving in third grade classrooms. Journal of Educational Research, 100(5), 283-302.

• Jitendra, A. K., Griffin, C., Haria, P., Leh, J., Adams, A., & Kaduvetoor, A. (2007). A comparison of single and multiple strategy instruction on third grade students’ mathematical problem solving. Journal of Educational Psychology, 99, 115-127.

• Xin, Y. P., Jitendra, A. K., & Deatline-Buchman, A. (2005). Effects of mathematical word problem solving instruction on students with learning problems. Journal of Special Education, 39(3), 181-192.

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SBI References from our Research Team

Journal Articles• Jitendra, A. K. (2005). How design experiments can inform teaching and

learning: Teacher-researchers as collaborators in educational research. Learning Disabilities Research & Practice, 20(4), 213-217.

• Jitendra, A. K., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of word problem-solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. Journal of Special Education, 36(1), 23-38.

• Jitendra, A. K., Hoff, K., & Beck, M. (1999). Teaching middle school students with learning disabilities to solve multistep word problems using a schema-based approach. Remedial and Special Education, 20(1), 50-64.

• Jitendra, A. K., Griffin, C., McGoey, K., Gardill, C, Bhat, P., & Riley, T. (1998). Effects of mathematical word problem solving by students at risk or with mild disabilities. Journal of Educational Research, 91(6), 345-356.

• Jitendra, A. K., & Hoff, K. (1996). The effects of schema-based instruction on mathematical word problem solving performance of students with learning disabilities. Journal of Learning Disabilities, 29(4), 422-431.

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