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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. Jitendra

University of Minnesota

Jon Star

Harvard University

Paper Presented at the 2008 NCTM Annual Convention, Salt Lake City, UT

- 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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- 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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- 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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- 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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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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Math Wars

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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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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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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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- 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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- 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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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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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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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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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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12 Girls

x Children

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Translate information in the diagram into a math equation

Plan how to solve the equation

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Solve the math equation and write the complete answer

Check to see if the answer makes sense

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A. Cross multiplication

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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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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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- 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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- 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 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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Table 1

Student Demographic Characteristics by Condition

Note: SBI-SM = schema-based instruction-self-monitoring

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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

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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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- 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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- 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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Instructional procedures outlined in the district-adopted mathematics textbook

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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

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- 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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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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Figure 2

Mathematics Problem-Solving Performance of Students in the SBI-SM Condition

Figure 3

Mathematics Problem-Solving Performance by Students in the Control Condition

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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- 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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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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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.

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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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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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