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Teaching Ratio and Proportion Problem Solving Using Schema-based Instruction

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 1 University of Minnesota 2 Harvard University

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Teaching Ratio and Proportion Problem Solving Using Schema-based Instruction

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  1. 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

  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 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 Marshall (1990); Mayer (1999); Riley, Greeno, & Heller (1983) April 4, 2008

  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

  13. 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

  14. 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

  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

  17. SBI-SM Instructional Sequence April 4, 2008

  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

  20. 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? April 4, 2008

  21. 2. Organize the information April 4, 2008

  22. 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? April 4, 2008

  23. 2. Organize the information 12 Girls x Children March 27, 2008 AERA 53.026 23

  24. 3. Plan to solve the problem Translate information in the diagram into a math equation Plan how to solve the equation April 4, 2008

  25. 4. Solve the problem Solve the math equation and write the complete answer Check to see if the answer makes sense April 4, 2008

  26. Problem solving strategies A. Cross multiplication April 4, 2008

  27. 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

  28. 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

  29. Additional problem types/schemata April 4, 2008

  30. Control condition • Instructional procedures outlined in the district-adopted mathematics textbook April 4, 2008

  31. 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 April 4, 2008

  32. 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

  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

  34. Results At pretest: SBI-SM and control classes did not differ Scores in each track significantly differed as expected: High > Average > Low No interaction April 4, 2008

  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

  37. Figure 1 Mathematics Problem-Solving Performance by Condition April 4, 2008

  38. Figure 2 Mathematics Problem-Solving Performance by Condition and Students’ Ability Level Status April 4, 2008

  39. 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

  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

  41. Thanks! Asha K. Jitendra (jiten001@umn.edu) Jon R. Star (jon_star@harvard.edu) 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 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

  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

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