1. Meeting the Needs of Students with Learning Disabilities: The Role of Schema-Based Instruction
Asha K. Jitendra
University of Minnesota
Jon Star
Harvard University
Paper Presented at the 2008 NCTM Annual Convention, Salt Lake City, UT

2. Thanks to …
Research supported by Institute of Education Sciences (IES) Grant # R305K )
Project Collaborators: Kristin Starosta, Grace Caskie, Jayne Leh, Sheetal Sood, Cheyenne Hughes, Toshi Mack, and Sarah Paskman (Lehigh University)
All participating teachers and students (Shawnee Middle School, Easton, PA)
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3. 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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4. 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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5. 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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6. Bridging the gap …
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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7. Math Wars
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8. 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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9. 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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10. Focus on structure Avoid key word strategies
in all means total, left means subtraction, etc.
Avoid procedures that are disconnected from underlying mathematical structure
cross multiplication
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11. Theoretical framework for SBI …
Draws on Cognitively Guided Instruction (CGI)
categorization of problems as the basis for instruction (Carpenter, Fennema, Franke, Levi, Empson, 1999)
understanding students’ mathematical thinking in proportional reasoning situations (Weinberg, 2002).
Differs from CGI by including teacher-led discussions using schematic diagrams to develop students’ multiplicative reasoning (Kent, Arnosky, & McMonagle, 2002).
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12. 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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13. SBI-SM: 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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14. 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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15. 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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16. 2. Organize the information
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17. 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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18. 2. Organize the information
12 Girls
x Children
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19. 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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20. 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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21. Problem solving strategies
A. Cross multiplication
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22. 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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23. 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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24. Additional problem types/schemata
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25. Proportionality is critical …
Challenging topic for 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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26. Goal of the study
To investigate the effectiveness of SBI-SM instruction on solving ratio and proportion problems as compared to “business as usual” instruction.
Specifically, what are the outcomes for special education students
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27. Participants
Participants in the larger study th graders from 8 classrooms in one urban public middle school
The total number of special education students was 15 (10%).
Mean chronological age of special education students = years (range = SD = .39 years)
60% Caucasian, 20% Hispanic, 7% African American, and 7% American Indian and Asian
Approximately 20% of students received free or subsidized lunch
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28. Student Demographic Characteristics by Condition
Table 1
Student Demographic Characteristics by Condition
Note: SBI-SM = schema-based instruction-self-monitoring
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29. 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
Control
*Referred to in the school as Honors, Academic, Applied, and Essential
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30. 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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31. 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.
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32. 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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33. SBI-SM Instructional Sequence
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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34. Control Condition
Instructional procedures outlined in the district-adopted mathematics textbook
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35. Outcome Measure Mathematical problem-solving (PS) Cronbach’s alpha
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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36. 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
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37. Results
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38. Student Problem Solving Performance by Time and Condition
Table 3
Student Problem Solving Performance by Time and Condition
Note: Scores ranged from 0 to 18 on the problem solving test; SBI-SM = schema-based instruction- self-monitoring.
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39. Figure 2
Mathematics Problem-Solving Performance of Students in the SBI-SM Condition

40. Figure 3
Mathematics Problem-Solving Performance by Students in the Control Condition

41. Summary and Discussion
SBI-SM led to significant gains in problem-solving skills.
A large effect size (1.46) at Time 1 and a low moderate effect (0.37) at Time 2 in favor of the treatment group.
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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42. Discussion Two issues undermined the potential impact of SBI-SM
One intervention teacher experienced classroom management difficulties.
Variation in treatment implementation fidelity
Intervention was time-based (10 days) rather than criterion-based (mastery of content)
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43. Asha K. Jitendra (jiten001@umn.edu) Jon R. Star (jon_star@harvard.edu)
Thanks!
Asha K. Jitendra
Jon R. Star
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44. 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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45. 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 ). New York: Guilford Press.
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46. 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),
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,
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),
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47. 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),
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),
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),
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),
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),
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