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Teaching Ratio and Proportion Problem Solving Using Schema-based Instruction

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### Teaching Ratio and Proportion Problem Solving Using Schema-based Instruction

Asha 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 Minnesota

2Harvard University

3Lehigh University

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

Thanks to …

- Research supported by Institute of Education Sciences (IES) Grant # R305K060075-06)
- All participating teachers and students (Shawnee Middle School, Easton, PA)

April 4, 2008

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).

April 4, 2008

April 4, 2008

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

April 4, 2008

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”

April 4, 2008

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; )

April 4, 2008

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

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

April 4, 2008

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).

April 4, 2008

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.

April 4, 2008

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

April 4, 2008

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

April 4, 2008

Study Design

Pretest-intervention-posttest-delayed posttest with random assignment to condition by class

Four “tracks” - Advanced, High, Average, Low*

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

April 4, 2008

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

April 4, 2008

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

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”

April 4, 2008

SBI-SM Instructional Sequence

April 4, 2008

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

April 4, 2008

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?

April 4, 2008

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?

April 4, 2008

2. Organize the information

April 4, 2008

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?

April 4, 2008

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

April 4, 2008

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.”

April 4, 2008

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.”

April 4, 2008

Additional problem types/schemata

April 4, 2008

Control condition

- Instructional procedures outlined in the district-adopted mathematics textbook

April 4, 2008

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

April 4, 2008

If there are 300 calories in 100g of a certain food, how

many calories are there in a 30g portion of this food?

90

100

900

1000

9000

April 4, 2008

Treatment Fidelity

- Treatment fidelity checked for all lessons.
- Mean treatment fidelity across lessons for intervention teachers was 79.78% (range = 60% to 99%).

April 4, 2008

Results

At pretest:

SBI-SM and control classes did not differ

Scores in each track significantly differed as expected:

High > Average > Low

No interaction

April 4, 2008

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

April 4, 2008

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

April 4, 2008

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

April 4, 2008

Summary and Discussion

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

- 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

April 4, 2008

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)

April 4, 2008

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.

April 4, 2008

SBI References from our Research Team

CHAPTERS

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.

April 4, 2008

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.

April 4, 2008

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.

April 4, 2008

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