1. Using Schema-based Instruction to Improve Seventh Grade Students’ Learning of Ratio and Proportion Jon R. Star (Harvard University) Asha K. Jitendra (University of Minnesota) Kristin Starosta, Grace Caskie, Jayne Leh, Sheetal Sood, Cheyenne Hughes, and Toshi Mack (Lehigh University)

2. March 27, 2008AERA 53.0262 Thanks to... Research supported by Institute of Education Sciences (IES) Grant # R305K060075-06 All participating teachers and students (Shawnee Middle School, Easton, PA)

3. March 27, 2008AERA 53.0263 Solving word problems in math Is very hard for students Yet plays a critical role in our instructional goals in mathematics Something that low achieving students particularly struggle with Cummins, Kintsch, Reusser, & Weimer, 1988; Mayer, Lewis, & Hegarty, 1992; Nathan, Long, & Alibali, 2002; Rittle-Johnson & McMullen, 2004

4. March 27, 2008AERA 53.0264 To solve word problems, Need to be able to recognize underlying mathematical structure Allows for the organization of problems and identification of strategies based on underlying mathematical similarity rather than superficial features “This is a rate problem” –Rather than “This is a bicycle problem”

5. March 27, 2008AERA 53.0265 Schemata 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 Sweller, Chandler, Tierney, & Cooper, 1990; Chen, 1999

6. March 27, 2008AERA 53.0266 Develop schema knowledge? 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

7. March 27, 2008AERA 53.0267 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:

8. March 27, 2008AERA 53.0268 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

9. March 27, 2008AERA 53.0269 Focus on structure Avoid key word strategies present in some direct instruction curricula –in all means total, left means subtraction, etc. Avoid procedures that are disconnected from underlying mathematical structure –cross multiplication

10. March 27, 2008AERA 53.02610 SBI-SM S chema- B ased I nstruction with S elf- M onitoring 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

11. March 27, 2008AERA 53.02611 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?

12. March 27, 2008AERA 53.02612 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?

13. March 27, 2008AERA 53.02613 2. Organize the information

14. March 27, 2008AERA 53.02614 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?

15. March 27, 2008AERA 53.02615 2. Organize the information 12 Girls x Children

16. March 27, 2008AERA 53.02616 3. Plan to solve the problem Translate information in the diagram into a math equation Plan how to solve the equation

17. March 27, 2008AERA 53.02617 4. Solve the problem Solve the math equation and write the complete answer Check to see if the answer makes sense

18. March 27, 2008AERA 53.02618 Problem solving strategies A. Cross multiplication

19. March 27, 2008AERA 53.02619 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.”

20. March 27, 2008AERA 53.02620 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.”

21. March 27, 2008AERA 53.02621 Additional problem types/schemata

22. March 27, 2008AERA 53.02622 Our questions Does the SBI-SM approach improve students’ success on ratio and proportion word problems, as compared to “business as usual” instruction? Is SBI-SM more or less effective for students of varying levels of academic achievement?

23. March 27, 2008AERA 53.02623 Participants 148 7th grade students (79 girls), in 8 classrooms, in one urban public middle school 54% Caucasian, 22% Hispanic, 22% AfrAm 42% Free/reduced lunch 15% receiving special education services

24. March 27, 2008AERA 53.02624 Teachers 6 teachers (3 female) (All 7th grade teachers in the school) 8.6 years experience (range 2 to 28 years) Text: Glencoe Mathematics: Applications and Concepts, Course 2 Intervention replaced normal instruction on ratio and proportion

25. March 27, 2008AERA 53.02625 Design Pretest-intervention-posttest-delayed posttest with random assignment to condition by class Four “tracks” - Advanced, High, Average, Low* # classes HighAverageLow SBI-SM121 Control121 *Referred to in the school as Honors, Academic, Applied, and Essential

26. March 27, 2008AERA 53.02626 Instruction 10 scripted lessons, to be taught over 10 days LessonContent 1Ratios 2Equivalent ratios; Simplifying ratios 3 & 4Ratio word problem solving 5Rates 6 & 7Proportion word problem solving 8 & 9Scale drawing word problem solving 10Fractions and percents

27. March 27, 2008AERA 53.02627 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

28. March 27, 2008AERA 53.02628 Treatment fidelity Treatment fidelity checked for all lessons Mean treatment fidelity across lessons for intervention teachers was 79.78% (range = 60% to 99%)

29. March 27, 2008AERA 53.02629 Outcome measure Mathematical problem-solving –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

30. March 27, 2008AERA 53.02630 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. 90 B. 100 C. 900 D. 1000 E. 9000

31. March 27, 2008AERA 53.02631 Results At pretest: SBI-SM and control classes did not differ Scores in each track significantly differed as expected: High > Average > Low No interaction

32. 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 March 27, 2008AERA 53.02632

33. 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 March 27, 2008AERA 53.02633

34. Results March 27, 2008AERA 53.02634

35. In sum... SBI-SM led to significant gains in problem- solving skills 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 March 27, 2008AERA 53.02635

36. Thanks! Jon R. Star (jon_star@harvard.edu) Asha K. Jitendra (jiten001@umn.edu) March 27, 2008AERA 53.02636