1 meeting the needs of students with learning disabilities: the role of schema-based instruction...
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Meeting the Needs of Students with Learning Disabilities: The Role of
Schema-Based Instruction
Asha K. JitendraUniversity of Minnesota
Jon StarHarvard University
Paper Presented at the 2008 NCTM Annual Convention, Salt Lake City, UT
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Thanks to …
• Research supported by Institute of Education Sciences (IES) Grant # R305K060075-06)
• 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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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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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 problemChen, 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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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 achieversBaker, Gersten, & Lee., 2002; Jitendra & Xin, 1997; Tuovinen & Sweller, 1999; Xin & Jitendra, 1999
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Math WarsMath Wars
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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 understanding
Rittle-Johnson & Star, 2007; Star & Rittle-Johnson, 2008
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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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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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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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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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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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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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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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Participants• Participants in the larger study - 148 7th 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 = 12.83 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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Note: SBI-SM = schema-based instruction-self-monitoring
Condition
SBI-SM (n = 10) Control (n = 5)
Variable M SD n (%) M SD n (%)
Age (in years) 12.83 0.45 12.82 0.26
Gender:
Male 7 70% 4 80%
Ethnicity
American Indian
Asian
African American
Hispanic
White
1
0
1
2
6
10%
0%
10%
20%
60%
0
1
0
1
3
0%
20%
0%
20%
60%
Free/Subsidized Lunch 2 20% 1 20%
Table 1Student Demographic Characteristics by Condition
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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.
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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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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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Results
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Table 3Student 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.
Condition
SBI (n = 10) Control (n = 5)
Var iable M SD M SD ES
Pretest
Posttest 1
Posttest 1- Pretest Scores
Posttest 2
Posttest 2- Pretest Scores
5.80
10.20
4.40
9.13
3.33
2.15
4.08
3.12
3.94
3.05
8.20
8.00
-0.20
10.60
2.40
2.78
2.45
2.62
3.91
2.00
1.46
0.37
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Figure 2Mathematics Problem-Solving Performance of Students in the SBI-SM Condition
0
10
20
30
40
50
60
70
80
90
100
I1 I2 I3 I4 I5 I6 I7 I8 I9 I10
SBI-SM Group
Percent Correct Score
Pretest Posttest Delayed Posttest
Figure 3Mathematics Problem-Solving Performance by Students in the Control Condition
0
10
20
30
40
50
60
70
80
90
100
C1 C2 C3 C4 C5
Control Group
Percent Correct Score
Pretest Posttest Delayed Posttest
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Summary and Discussion
• 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
SBI-SM led to significant gains in problem-solving skills.
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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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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 Team
CHAPTERSChard, 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 TeamJournal 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 TeamJournal 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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