1. PROBLEM SOLVING

2. Problem Solving Big Ideas  Make problems meaningful  Use C-S-A strategy  Teach strategies for basic problem solving (schema or number family)  For complex problem solving…  Teach problem solving processes/strategies  Use self-regulation strategies  Teach problem solving as a regular strand in your math instruction

3. Importance of Problem Solving  Important Daily Living Activity  Budgeting, spending, saving, investing  Scheduling, traveling  Daily living activities (cooking, watching TV, taking meds, phone service, temperature, etc.)  Application of mathematical knowledge to solve real problems

4. Math Problem-Solving  Mathematical problem solving is a complex cognitive activity involving a number of processes and strategies.  Problem solving has two stages:  problem representation and  problem execution.

5. Challenge of Teaching Problem Solving  Requires an thorough understanding of mathematical concepts and their application to authentic contexts.  Integrates conceptual, procedural, and declarative knowledge.  Requires higher order/divergent thinking  Requires high quality instruction over an extended period of time

6. Math Problem Solving  Many students with LD do not easily acquire the skills and strategies needed to “read the problem” and “decide what to do” to solve it.  Need explicit instruction in mathematical problem solving skills and strategies to solve problems in their math textbooks and in their daily lives.

7. Instructional Sequence  Solve simple word problems using concrete/pictorial models  Use key words to write number sentences (equations)  Use graphic organizers (schema based, number families, tables) to teach basic problem types  Change; action; temporal sequence  Group; classification  Comparison; compare; difference  Multiplication/division; vary; ratio; rate  Multi-step problems  Complex problem solving (anchored instruction; functional applications) Concrete Abstract Represen- tational

8. Addition and Subtraction Introducing the Concept  What are the preskills needed needed before starting problem solving?  What vocabulary should one teach in advance?

9. Introducing the Concept (K and 1st grade)  Use simple action word problems that can be acted out or drawn  Teach students to write simple number sentences.  Teach key verbs for getting more (+) and getting less (-)

10. Introducing the Concept Phrase by phrase translation with key words  Tammy has 4 apples. She buys 2 more. She end up with how many?  Isaac has 5 apples. He gives 3 away. How many does he end up with? 4 + 2 =  5 - 3 =                    

11. Limitation of key word strategy  Does it work for these problems?  Traci had 5 marbles. She finds some more. She now has 8 marbles. How many did she find?  Joel has 6 marbles. He gives some away. He now has 4. How many did he give away?  April’s fish tank has 10 fish. 7 are guppies. How many are not guppies?  Matt is 6 feet tall. He is 1 foot taller than Carrie. How tall is Carrie?

12. Story Problems Types  Addition/Subtraction  Change (Action or Temporal Sequence)  Classification/Group  Comparison  Multiplication/Division  Key words  Without key words  Division with remainders  Multiplicative compare  Ratio/rate problems  Larger numbers  Extraneous information  Complex grammar (indirectly stated, complex sentence structure)  Difficult vocabulary  Multi-step problems Task Scaffolding (Task Difficulty) Story Structure

13. What Type of Problem? Change/action; Class/group, Compare, Multip/Div)  Harry had a party and invited 6 boys and some girls. There were 10 kids at the party. How many were girls?  Ron is 10 inches taller than Malfoy. Ron is 60 inches tall. How tall is Malfoy?  The Firebolt racing broom can travel 2 miles per second. How many miles can it travel in 30 seconds?  The cook at Gryffindor put out a plate of 12 cookies. Hermione ate 3. How many cookies were left?

14. Revise task difficulty - how could you rewrite problems  For students who have learned addition and subtraction with renaming of three digit numbers?  With a distractor (numerical quantity)?  With more difficult vocabulary or grammatical structure?  To make it into a multistep problem?  To make it more meaningful to the students?

15. Scheme-based strategy instruction (Jitendra)  Teaches students to represent quantitative relationships graphically to solve problems  Elements:  Problem Identification  Problem Representation  Problem Solution

16. Schema Based Strategy for Change Problems Ending Amount Beginning Amount Change + or -

17. Schema Based Strategy for Change Problems  Hank had 7 CDs. He bought 2 more. How many CDs does he have now? ? 7 +2 Ending Amount Beginning Amount Change

18. Schema Based Strategy for Change Problems  Jillanna had 9 CDs in her collection. She lost some. She now has 6 CDs. How many did she lose? 6 9 -? Ending Amount Beginning Amount Change

19. Classification or Group Problems  How do students know if it’s a classification problem?  Language skill - recognizing categories or groups of things  Preskill: Differentiating which group is the biggest group or class.  Examples:  boys, girls, or students?  fish, sharks, or salmon?

20. Schema Based Strategy for Group Problems  Smaller group amounts  Larger group amount Gregg has 5 pencils and 3 pens. How many writing tools does he have altogether?

21. Schema Based Strategy for Group Problems  Smaller group amounts  Larger group amount 53 ? Greg has 5 pencils and 3 pens. How many writing tools does he have altogether?

22. Comparison or Difference Problems  How do you know if it’s a comparison problem?  Tells about 2 things and asks about the difference between them  Preskill: Which person or thing in the problem is the big number?

23. Schema Based Strategy for Compare Problems Bigger amount Smaller amount Difference Justin is 1 foot taller than Jessica. Justin is 6 feet tall. How tall is Jessica? =

24. Schema Based Strategy for Compare Problems 6 Smaller amount 1 Justin is 1 foot taller than Jessica. Justin is 6 feet tall. How tall is Jessica? =

25. DI Number Family Strategy  Based on number family concept  Students apply to all types of problems and  Determine which number is the total (or big number)  Which numbers are parts of the total (or small numbers) 63 236 485 6 + 3 = 9 485 - 236 = 249

26. DI Number Family - Change  Debbie had 7 CDs. She bought 2 more. How many CDs does she have now?  Wendy had 7 CDs. She bought some more. She now has 9. How many CDs did she buy? 7 2 7 9 Missing total: Solve with addition Missing part: Solve with subtraction

27. DI Number Family Classification Problems  Megan has 5 roses and 4 tulips. How many flowers does she have?  There are 9 students. 5 are girls. How many are boys? ? 54 Missing total: Solve with addition 9 5? Missing part: Solve with subtraction

28. DI Number Family Comparison Problems  Amanda got 2 more math problems correct than Michelle. Michelle got 8 problems correct. How many problems did Amanda get correct?  Justin is 1 foot taller than Jessica. Justin is 6 feet tall. How tall is Jessica? ? 28 Missing total: Solve with addition 6 1? Missing part: Solve with subtraction

29. Schema Based Strategy for Multiplication / Division Problems Ratio or Rate The Firebolt racing broom can travel 2 miles per second. How many miles can it travel in 30 seconds? Compared Referent

30. Schema Based Strategy for Multiplication / Division Problems __?_miles_ 30 sec. The Firebolt racing broom can travel 2 miles per second. How many miles can it travel in 30 seconds? 2 miles second

31. Solve this Problem  There are 57 students in the two 3 rd grade classes. 27 are in Mr. Lee’s class. Sixteen of these students are boys. There are 14 boys in Ms. Kay’s class. How many girls are there in the 3 rd grade?

32. Organizing Data in Tables Total BoysGirls 16 14 27 57 Mr. Lee’s class Ms. Kay’s class

33. Organizing Data in Tables Total BoysGirls 16 14 27 57 Mr. Lee’s class Ms. Kay’s class

34. Organizing Data in Tables Total BoysGirls 16 14 27 57 Mr. Lee’s class Ms. Kay’s class 30

35. Organizing Data in Tables Total BoysGirls 16 14 27 57 Mr. Lee’s class Ms. Kay’s class 30

36. Organizing Data in Tables Total BoysGirls 16 1416 11 30 27 57 27 30 Mr. Lee’s class Ms. Kay’s class

37. Diagnosis and Remediation  Describe the 5 types of error patterns and general remediation for word problems. 1. Fact error 2. Computation error 3. Decoding error 4. Vocabulary error 5. Translation error

38. Teaching Complex Problem Solving Simple Problem Solving Convergent Thinking Divergent Thinking Complex Problem Solving Phrase by phrase translation Number family strategies Schema based strategies Simple tables Cognitive processes Self-regulation strategies Problem solving strategies > add “think alouds”

39. Difficulties for Students with Disabilities…  Reading and processing information  Distinguishing relevant information in problems  Low motivation & self-efficacy to learn due to repeated failure  Reasoning & problem solving skills  Self-monitoring and self-regulation  Arithmetic and computational deficits Maccini & Gagnon, 2000

40. Example  The Art Club is having a cookie sale. Each box of cookies costs $2.00. The first day Jennifer sold 6 boxes, Carlos sold 7, and Alex sold 3. How much did the Art Club make the first day of cookie sales?

41. Example  The 24 students in Mrs. Smith’s class formed a line. Every third student was wearing red. Every fifth student was wearing blue. What was the fifteenth student in line wearing?

43. Solve this problem with a partner the way you think a student might solve it The water taxi in the Boston Harbor can carry 8 passengers to the city from the airport. Mrs. Allen and her 34 students need to ride the water taxi into Boston on their field trip. How many trips will it take to get everyone to the city?

44. Generic Problem Solving Strategy Read the Problem Analyze Problem Select a strategy Calculate solution Check for Accuracy Selecting a strategy depends largely on student prior knowledge  Draw a picture  Make a chart  Organize a list  Guess and check  Use a formula  Work backward

45. Instructional Approaches  Sequence and segment  Repetition and practice  Directed questioning  Both process and content  Controlled difficulty  Strategy cues  Verbal Rehearsal  Process modeling/thinking aloud  Peer Coaching

46. What to Teach: Conceptual Understanding  What is the problem asking?  What kinds of math calculations are needed to solve the problem? (add, multiply, etc.)  What math concepts are needed to solve the problem? (odd/even numbers; mean/median; graphing coordinates)  What numbers or information is needed from the problem?  What information is missing?  What will the answer look like?

47. What to Teach: Processes and Strategies  What strategy should be used? (Teach one at a time)  Draw a picture  Make a chart (or table)  Look for Patterns  Guess and check  Organize a list  Work backwards  Is the information organized and labeled?  Are the steps to the solution all shown?

48. What to Teach: Verification or defense  Look over the problem again.  Show that the strategy will solve the problem  or recheck with a calculator  or rework problem again a different way

49. What to Teach: Communication  Explain what strategy you chose and why or check that your steps are clear and organized  Check that drawings, tables, charts, lists, etc. are labeled.

50. What to Teach: Accuracy  Check that the final answer is complete, justifiable and clearly identified. It matches what the problem is asking.  This isn't a step you will specifically teach in the context of problem solving. However, when you are ready to practice the entire process with sample problems, include this step in evaluation process.

51. Think, Plan, Solve, Look Back

52. Procedural Strategies - SIGNS  SIGNS (Watanabe, 1991)  Survey question  Identify key words and labels  Graphically draw the problem  Note type of operation  Solve and check the problem

53. DRAW (Mercer & Miller, 1992)  Discover the sign  Read the Problem  Answer or DRAW a conceptual representation of the problem  Write and answer and check

54. STAR Strategy (Maccini & Hughes, 2000)  Search the word problem  Translate the words into an equation in picture form  Answer the problem  Review the solution