1 Teaching Ratio and Proportion Problem Solving Using Schema- based Instruction Asha K. Jitendra, 1...
-
Upload
joseph-blair -
Category
Documents
-
view
215 -
download
0
Transcript of 1 Teaching Ratio and Proportion Problem Solving Using Schema- based Instruction Asha K. Jitendra, 1...
1
Teaching Ratio and Proportion Problem Solving Using Schema-
based InstructionAsha 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 Minnesota2Harvard University
3Lehigh University
Paper Presented at the 2008 Annual CEC Convention, Boston, MA
2
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
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).
April 4, 2008
4
Math WarsMath Wars
April 4, 2008
5
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
6
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
7
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
8
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
April 4, 2008
Marshall (1990); Mayer (1999); Riley, Greeno, & Heller (1983)
9
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
10
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
11
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
12
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
13April 4, 2008
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
14April 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
15
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
16
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
17April 4, 2008
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
18
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
19
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
20April 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?
21April 4, 2008
2. Organize the information
22April 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?
23March 27, 2008 AERA 53.026 23
2. Organize the information
12 Girls
x Children
2
5
24April 4, 2008
3. Plan to solve the problem
• Translate information in the diagram into a math equation
• Plan how to solve the equation
25April 4, 2008
4. Solve the problem
• Solve the math equation and write the complete answer
• Check to see if the answer makes sense
26April 4, 2008
Problem solving strategies
A. Cross multiplication
27April 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.”
28April 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.”
29April 4, 2008
Additional problem types/schemata
30
Control condition
• Instructional procedures outlined in the district-adopted mathematics textbook
April 4, 2008
31April 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
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
April 4, 2008
33
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
34April 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
35
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
36
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
Figure 1Mathematics Problem-Solving Performance by Condition
0
10
20
30
40
50
60
70
80
90
100
Pretest Posttest Delayed Posttest
Percent Correct Score
SBI-SM Control
April 4, 2008
0
10
20
30
40
50
60
70
80
90
100
Pretest Posttest Delayed Posttest
Percent Correct Score
SBI-SM Academic Control Academic SBI-SM Applied Control Applied SBI-SM Essential Control Essential
Figure 2Mathematics Problem-Solving Performance by Condition and Students’ Ability Level Status
April 4, 2008
39
Summary and Discussion
• 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
SBI-SM led to significant gains in problem-solving skills.
April 4, 2008
40
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
42
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
43
SBI References from our Research TeamCHAPTERS
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
44
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
45
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