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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

Math Wars

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 problems 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. Robinsons class is

25. 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. Robinsons class is

25. 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. Robinsons class is

25. The number of girls in the class is 12. How

many children are in the class?

April 4, 2008

2. Organize the information

12 Girls

x Children

March 27, 2008

AERA 53.026

23

3. Plan to solve the problem

- Translate information in the diagram into a math

equation - Plan how to solve the equation

April 4, 2008

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

- A. Cross multiplication

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
- Cronbachs alpha
- 0.73 for the pretest
- 0.78 for the posttest
- 0.83 for the delayed posttest

April 4, 2008

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? - 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 gt Average gt 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 gt Average gt 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 gt Average gt Low
- No interaction

April 4, 2008

Figure 1 Mathematics Problem-Solving Performance

by Condition

April 4, 2008

Figure 2 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

Thanks!

- Asha K. Jitendra (jiten001_at_umn.edu)
- Jon R. Star (jon_star_at_harvard.edu)

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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