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

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

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

Math Wars 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 April 4, 2008

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

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

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

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

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

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

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

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

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 Control *Referred to in the school as Honors, Academic, Applied, and Essential April 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 April 4, 2008

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

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

SBI-SM Instructional Sequence Lesson 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 April 4, 2008

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

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

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

2. Organize the information April 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? April 4, 2008

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

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

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

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

Additional problem types/schemata April 4, 2008

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

Outcome Measure Mathematical problem-solving (PS) Cronbach’s alpha 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

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

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

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

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

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

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

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

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

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

Asha K. Jitendra (jiten001@umn.edu) Jon R. Star (jon_star@harvard.edu) Thanks! Asha K. Jitendra (jiten001@umn.edu) Jon R. Star (jon_star@harvard.edu) April 4, 2008

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

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

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

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