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## Enhancing Mathematical Problem Solving for At-Risk Students

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**Enhancing Mathematical Problem Solving for At-Risk Students**Lynn Fuchs Vanderbilt University December 10, 2008**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.**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.**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**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**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**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.**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.**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?”**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**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**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**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**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?”**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?”**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**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.**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.**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.**These Tutoring Protocols Are Transportable**Tutoring protocols were comparably effective in Nashville (where tutoring program was developed) and Houston (a distal site).**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.**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.**For Materials, Contact:**Flora Murray flora.murray@vanderbilt.edu Vanderbilt University 228 Peabody College Department of Special Education Nashville, TN 37203 (615) 343-4782**Whole-Class Pirate Math**• Lesson • Problem review • New content • Partner Work • Challenge Problem