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1. Enhancing Mathematical Problem Solving for At-Risk Students Lynn Fuchs Vanderbilt University December 10, 2008

2. Principles of Effective Instruction for At-Risk Learners * Address all valued outcomes (don’t assume transfer will occur). * Incorporate explicit, systematic instruction. * Build instructional design to minimize the learning challenge. * Provide opportunities to verbalize problem-solving strategies. * Incorporate systematic practice and review. * Build motivation for at-risk learners to work hard.

3. In This Presentation, I Illustrate These Principles • With a tutoring program for enhancing at-risk learners’ problem solving, while addressing foundational skills. • We call this tutoring program Pirate Math. • We have developed similar programs at different grade levels to be used for tutoring or for whole-class instruction. • We call our approach to word-problem instruction schema-broadening instruction, which teaches students to recognize problems within “schema” or “problem types.” • Schema-broadening instruction explicitly teaches students: • The underlying structure of each “problem type” • Solution strategies for each problem type • Transfer to recognize novel problems (with irrelevant information, relevant information found in charts, combinations of problem types, etc.) as belonging to the problem types students have learned.

4. Pirate Math Tutoring 48 sessions: 3 per week for 16 weeks 20-30 minutes per session Scripted lessons, which tutors study (not read) Four units Foundational Skills for Word Problems Total Word Problems Difference Word Problems Change Word Problems

5. Pirate Math: Introductory Unit • Teach students: • Efficient counting strategies to answer number combinations • 2-digit procedural calculations • How to solve for X in addition and subtraction equations (a+b=c; x-y=z) • How to check work

6. Remaining Units:Word Problem Lessons Following Unit 1, four activities per session. 1. Flash-card warm up 2. Conceptual/strategic lesson using schema-broadening instruction (in this presentation, we focus on this second activity) 3. Sorting practice on identifying problem types 4. Paper/pencil review

7. 2. LessonPirate Math RUN • Students use “RUN” strategy for every word problem. • Students learn to circle relevant information directly in the text or picture/graph/chart.

8. 2. Lesson3 Problem Types with Transfer • 3 problem types are taught (Total, Difference, and Change at 2nd grade). • For each problem type, transfer features are taught: • Irrelevant information • Money • X in different position • Double-digit calculations • Finding relevant information in graphs and pictures.

9. 2. LessonPirate Math Change • Change problems with a starting amount that increases or decreases (a change) to make it a new amount. • “Sarah had 10 pencils. The she gave 4 pencils to Pamela. How many pencils does Sarah have now?”

10. 2. LessonPirate Math Change Lexie had some comic books in her desk. Then she bought 8 more. Now, she has 12 comic books. How many comic books did Lexie have in her desk to begin with? St = C = E = ST +/- C = E

11. 2. LessonPirate Math Change Lexie had some comic books in her desk. Then she bought 8 more. Now, she has 12 comic books. How many comic books did Lexie have in her desk to begin with? St = X C = + 8 E = 12 X + 8 = 12 X = 4 comic books

12. Monday’s Star Chart Milo Trish David Alicia 0 2 4 6 8 10 Number of Gold Stars 2. Lesson Pirate Math Change Alicia has 3 friends in her math class. The chart shows how many stars Alicia earned on Monday. On Tuesday, Alicia lost 3 stars for talking. How many stars does she have now? St = C = E = ST +/– C = E

13. Monday’s Star Chart Milo Trish David Alicia 0 2 4 6 8 10 Number of Gold Stars 2. Lesson Pirate Math Change Alicia has 3 friends in her math class. The chart shows how many stars Alicia earned on Monday. On Tuesday, Alicia lost 3 stars for talking. How many stars does she have now? St = 8 C = – 3 E = X 8 – 3 = X X = 5 stars

14. 2. LessonPirate Math Total • Total problems have two parts that are combined for a total. • Total amount is the entire or combined amount. • “Sarah has 5 pencils. Pamela has 3 pencils. How many pencils do the girls have in all?”

15. 2. Lesson Pirate Math Difference • Difference problems compare two amounts to find the difference between them. • “Sarah has 7 pencils. Pamela has 12 pencils. How many more pencils does Pamela have than Sarah?”

16. Based on A Series of Field-Based Randomized Control Trials at 2nd and 3rd grades Lynn Fuchs, Sarah Powell, Pamela Seethaler, Paul Cirino, Jack Fletcher, Doug Fuchs, and Carol Hamlett Vanderbilt University and University of Houston Grant #P01046261 National Institute of Child Health and Human Development

17. Efficacy: Fluency with Number Combinations and Procedural Calculations • On number combinations, Pirate Math effects superior improvement compared to control group. Notable, because Pirate Math only allocates an initial lesson and then 4-6 minutes per session on number combinations. • On procedural calculations, Pirate Math effects superior improvement compared to control group and compared to contrasting tutoring conditions. Again, little time spent on procedural calculations.

18. Efficacy: Algebra • On algebra, Pirate Math effects superior outcomes compared to control group and compared to contrasting tutoring conditions. • Algebraic cognition improved even though students were severely deficient in math and young. • Given strong focus on algebra in high schools, given graduation requirements for algebra, and given emphasis in NMAP, introducing algebra earlier in the curriculum may represent a productive innovation.

19. Efficacy: Word Problems • Work on these foundation skills (number combinations, procedural calculations, algebra), combined with schema-broadening instruction, also produced differential growth on word-problem outcomes compared to control group and compared to contrasting tutoring conditions.

20. These Tutoring Protocols Are Transportable Tutoring protocols were comparably effective in Nashville (where tutoring program was developed) and Houston (a distal site).

21. Conclusions For a reasonable amount of tutoring time (48 sessions, each 30 minutes long), Pirate Math enhances word-problem skill, fluency with number combinations, procedural calculation skill, algebraic cognition, and competence with word problems. With a 1-day training and ongoing supervision, non-certified tutors can implement Pirate Math at distal sites with comparable outcomes.

22. Conclusions: Principles of Effective Instruction for At-Risk Learners * Address all valued outcomes (don’t assume transfer will occur). * Incorporate explicit, systematic instruction. * Build instructional design to minimize the learning challenge. * Provide opportunities to verbalize problem-solving strategies. * Incorporate systematic practice and review. * Build motivation for at-risk learners to work hard.

23. For Materials, Contact: Flora Murray flora.murray@vanderbilt.edu Vanderbilt University 228 Peabody College Department of Special Education Nashville, TN 37203 (615) 343-4782

24. Pirate Math: Introductory Unit

25. Pirate Math: Introductory Unit

26. Pirate Math: Introductory Unit

27. Whole-Class Pirate Math • Lesson • Problem review • New content • Partner Work • Challenge Problem