Foundations of Math Skills & RTI Interventions Jim Wright interventioncentral

1. Foundations of Math Skills& RTI InterventionsJim Wrightwww.interventioncentral.org

2. “Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.” –Anonymous

3. Who is At Risk for Poor Math Performance?: A Proactive Stance “…we use the term mathematics difficulties rather than mathematics disabilities. Children who exhibit mathematics difficulties include those performing in the low average range (e.g., at or below the 35th percentile) as well as those performing well below average…Using higher percentile cutoffs increases the likelihood thatyoung children who go on to have serious math problems will be picked upin the screening.” p. 295 Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

4. Profile of Students with Math Difficulties (Kroesbergen & Van Luit, 2003) [Although the group of students with difficulties in learning math is very heterogeneous], in general, these students have memory deficits leading to difficulties in the acquisition and remembering of math knowledge. Moreover, they often show inadequate use of strategies for solving math tasks, caused by problems with the acquisition and the application of both cognitive and metacognitive strategies. Because of these problems, they also show deficits in generalization and transfer of learned knowledge to new and unknown tasks. Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..

5. The Elements of Mathematical Proficiency: What the Experts Say…

6. 5 Strands of Mathematical Proficiency • Understanding • Computing • Applying • Reasoning • Engagement • 5 Big Ideas in Beginning Reading • Phonemic Awareness • Alphabetic Principle • Fluency with Text • Vocabulary • Comprehension Source: Big ideas in beginning reading. University of Oregon. Retrieved September 23, 2007, from http://reading.uoregon.edu/index.php Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

7. Five Strands of Mathematical Proficiency “ 1. Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean.Understanding refers to a student’s grasp of fundamental mathematical ideas. Students with understanding know more than isolated facts and procedures. They know why a mathematical idea is important and the contexts in which it is useful. Furthermore, they are aware of many connections between mathematical ideas. In fact, the degree of students’ understanding is related to the richness and extent of the connections they have made.” p. 10 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

8. Five Strands of Mathematical Proficiency “ 2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.Computing includes being fluent with procedures for adding, subtracting, multiplying, and dividing mentally or with paper and pencil, and knowing when and how to use these procedures appropriately. Although the word computing implies an arithmetic procedure, … it also refers to being fluent with procedures from other branches of mathematics, such as measurement (measuring lengths), algebra (solving equations), geometry (constructing similar figures), and statistics (graphing data). Being fluent means having the skill to perform the procedure efficiently, accurately, and flexibly.” p. 11 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

9. Five Strands of Mathematical Proficiency “ 3. Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.Applying involves using one’s conceptual and procedural knowledge to solve problems. A concept or procedure is not useful unless students recognize when and where to use it—as well as when and whether it does not apply. … Students …need to be able to pose problems, devise solution strategies, and choose the most useful strategy for solving problems..” p. 13 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

10. Five Strands of Mathematical Proficiency “ 4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known.Reasoning is the glue that holds mathematics together. By thinking about the logical relationships between concepts and situations, students can navigate through the elements of a problem and see how they fit together. One of the best ways for students to improve their reasoning is to explain or justify their solutions to others. …Reasoning interacts strongly with the other strands of mathematical thought, especially when students are solving problems. ” p. 14 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

11. Five Strands of Mathematical Proficiency “ 5. Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.Engaging in mathematical activity is the key to success. Our view of mathematical proficiency goes beyond being able to understand, compute, apply, and reason. It includes engagement with mathematics. Students should have a personal commitment to the idea that mathematics makes sens and that—given reasonable effort—they can learn it and use it both in school and outside school.. ” p. 15-16 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

12. Three General Levels of Math Skill Development (Kroesbergen & Van Luit, 2003) As students move from lower to higher grades, they move through levels of acquisition of math skills, to include: • Number sense • Basic math operations (i.e., addition, subtraction, multiplication, division) • Problem-solving skills: “The solution of both verbal and nonverbal problems through the application of previously acquired information” (Kroesbergen & Van Luit, 2003, p. 98) Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..

13. What is ‘Number Sense’? (Clarke & Shinn, 2004) “… the ability to understand the meaning of numbers and define different relationships among numbers. Children with number sense can recognize the relative size of numbers, use referents for measuring objects and events, and think and work with numbers in a flexible manner that treats numbers as a sensible system.” p. 236 Source: Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248.

14. What Are Stages of ‘Number Sense’? (Berch, 2005, p. 336) • Innate Number Sense. Children appear to possess ‘hard-wired’ ability (neurological ‘foundation structures’) to acquire number sense. Children’s innate capabilities appear also to be to ‘represent general amounts’, not specific quantities. This innate number sense seems to be characterized by skills at estimation (‘approximate numerical judgments’) and a counting system that can be described loosely as ‘1, 2, 3, 4, … a lot’. • Acquired Number Sense. Young students learn through indirect and direct instruction to count specific objects beyond four and to internalize a number line as a mental representation of those precise number values. Source: Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333-339...

15. Number Line: 0-144 Moravia, NY 0 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100101 102 103 104 105 106 107 108 109 110111 112 113 114 115 116 117 118 119 120121 122 123 124 125 126 127 128 129 130131 132 133 134 135 136 137 138 139 140141 142 143 144 The Basic Number Line is as Familiar as a Well-Known Place to People Who Have Mastered Arithmetic Combinations

16. 3 X 7 = 21 28 ÷ 4 = 7 9 – 7 = 2 2 + 4 = 6 Internal Numberline As students internalize the numberline, they are better able to perform ‘mental arithmetic’ (the manipulation of numbers and math operations in their head). 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1920 21 22 23 24 25 26 27 28 29

17. Mental Arithmetic: A Demonstration 332 x 420 = ? Directions: As you watch this video of a person using mental arithmetic to solve a computation problem, note the strategies and ‘shortcuts’ that he employs to make the task more manageable.

18. 7. Add Intermediate Products: Chunk into Smaller Computation Tasks 6. Use Mnemonic Strategy to Remember Intermediate Product 5. Continue with Next ‘Chunk’ of Problem: Math Shortcut 2. Break Problem into Manageable Chunks 4. Use Mnemonic Strategy to Remember Intermediate Product 3. Apply Math Shortcut: Add Zeros in One’s Place for Each Multiple of Ten Solving for… 132,800 x 6,640 132,800 + 6000 = 138,800 132,800 + 640 = 139,440 332 x 20 332 x 10 = 3320 3320 x 2 = 6640 332 x 4 1,328 332 X 420 6,640 ’66 is a famous national road’ & ’40 is speed limit in front of house’ 132,800 ‘1=3-2’ & ‘800 is a toll-free number’ 1,328 x 100 132,800 1. Estimate Answer 300 x 400 120,000 \Mental Arithmetic Demonstration: What Tools Were Used?

19. Math Computation: Building FluencyJim Wrightwww.interventioncentral.org

20. "Arithmetic is being able to count up to twenty without taking off your shoes." –Anonymous

21. Benefits of Automaticity of ‘Arithmetic Combinations’ (: (Gersten, Jordan, & Flojo, 2005) • There is a strong correlation between poor retrieval of arithmetic combinations (‘math facts’) and global math delays • Automatic recall of arithmetic combinations frees up student ‘cognitive capacity’ to allow for understanding of higher-level problem-solving • By internalizing numbers as mental constructs, students can manipulate those numbers in their head, allowing for the intuitive understanding of arithmetic properties, such as associative property and commutative property Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

22. Associative Property • “within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed” • Example: • (2+3)+5=10 • 2+(3+5)=10 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Associative

23. Commutative Property • “the ability to change the order of something without changing the end result.” • Example: • 2+3+5=10 • 2+5+3=10 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Commutative

24. Least efficient strategy: Count out and group 3 objects; count out and group 8 objects; count all objects: =11 + More efficient strategy: Begin at the number 3 and ‘count up’ 8 more digits (often using fingers for counting): 3 + 8 More efficient strategy: Begin at the number 8 (larger number) and ‘count up’ 3 more digits: 8+ 3 Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in memory and automatically retrieved: Answer = 11 How much is 3 + 8?: Strategies to Solve… Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

25. Math Skills: Importance of Fluency in Basic Math Operations “[A key step in math education is] to learn the four basic mathematical operations (i.e., addition, subtraction, multiplication, and division). Knowledge of these operations and a capacity to perform mental arithmetic play an important role in the development of children’s later math skills. Most children with math learning difficulties are unable to master the four basic operations before leaving elementary school and, thus, need special attention to acquire the skills. A … category of interventions is therefore aimed at the acquisition and automatization of basic math skills.” Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114.

26. Big Ideas: Learn Unit (Heward, 1996) The three essential elements of effective student learning include: • Academic Opportunity to Respond. The student is presented with a meaningful opportunity to respond to an academic task. A question posed by the teacher, a math word problem, and a spelling item on an educational computer ‘Word Gobbler’ game could all be considered academic opportunities to respond. • Active Student Response. The student answers the item, solves the problem presented, or completes the academic task. Answering the teacher’s question, computing the answer to a math word problem (and showing all work), and typing in the correct spelling of an item when playing an educational computer game are all examples of active student responding. • Performance Feedback. The student receives timely feedback about whether his or her response is correct—often with praise and encouragement. A teacher exclaiming ‘Right! Good job!’ when a student gives an response in class, a student using an answer key to check her answer to a math word problem, and a computer message that says ‘Congratulations! You get 2 points for correctly spelling this word!” are all examples of performance feedback. Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

27. Math Intervention: Tier I or II: Elementary & Secondary: Self-Administered Arithmetic Combination Drills With Performance Self-Monitoring & Incentives • The student is given a math computation worksheet of a specific problem type, along with an answer key [Academic Opportunity to Respond]. • The student consults his or her performance chart and notes previous performance. The student is encouraged to try to ‘beat’ his or her most recent score. • The student is given a pre-selected amount of time (e.g., 5 minutes) to complete as many problems as possible. The student sets a timer and works on the computation sheet until the timer rings. [Active Student Responding] • The student checks his or her work, giving credit for each correct digit (digit of correct value appearing in the correct place-position in the answer). [Performance Feedback] • The student records the day’s score of TOTAL number of correct digits on his or her personal performance chart. • The student receives praise or a reward if he or she exceeds the most recently posted number of correct digits. Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

28. Self-Administered Arithmetic Combination Drills:Examples of Student Worksheet and Answer Key Worksheets created using Math Worksheet Generator. Available online at:http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php

29. Reward Given Reward Given Reward Given Reward Given No Reward No Reward No Reward Self-Administered Arithmetic Combination Drills…

30. How to… Use PPT Group Timers in the Classroom

31. Math Shortcuts: Cognitive Energy- and Time-Savers “Recently, some researchers…have argued that children can derive answers quickly and with minimal cognitive effort by employing calculation principles or “shortcuts,” such as using a known number combination toderive an answer (2 + 2 = 4, so 2 + 3 =5), relations among operations (6 + 4 =10, so 10 −4 = 6), n+ 1, commutativity, and so forth. This approach to instruction is consonant with recommendations by the National Research Council (2001). Instruction along these linesmay be much more productive than rote drill without linkage to counting strategy use.”p. 301 Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

32. 9 x 1 9 x 2 9 x 3 9 x 4 9 x 5 9 x 6 9 x 7 9 x 8 9 x 9 9 x 10 Math Multiplication Shortcut: ‘The 9 Times Quickie’ • The student uses fingers as markers to find the product of single-digit multiplication arithmetic combinations with 9. • Fingers to the left of the lowered finger stands for the ’10’s place value. • Fingers to the right stand for the ‘1’s place value. Source: Russell, D. (n.d.). Math facts to learn the facts. Retrieved November 9, 2007, from http://math.about.com/bltricks.htm

33. Students Who ‘Understand’ Mathematical Concepts Can Discover Their Own ‘Shortcuts’ “Students who learn with understanding have less to learn because they see common patterns in superfically different sicuations. If they understand the general principle that the order in which two numbers are multiplied doesn’t matter—3 x 5 is the same as 5 x 3, for example—they have about half as many ‘number facts’ to learn.” p. 10 Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

34. Application of Math Shortcuts to Intervention Plans • Students who struggle with math may find computational ‘shortcuts’ to be motivating. • Teaching and modeling of shortcuts provides students with strategies to make computation less ‘cognitively demanding’.

35. Math Computation: Motivate With ‘Errorless Learning’ Worksheets In this version of an ‘errorless learning’ approach, the student is directed to complete math facts as quickly as possible. If the student comes to a number problem that he or she cannot solve, the student is encouraged to locate the problem and its correct answer in the key at the top of the page and write it in. Such speed drills build computational fluency while promoting students’ ability to visualize and to use a mental number line. TIP: Consider turning this activity into a ‘speed drill’. The student is given a kitchen timer and instructed to set the timer for a predetermined span of time (e.g., 2 minutes) for each drill. The student completes as many problems as possible before the timer rings. The student then graphs the number of problems correctly computed each day on a time-series graph, attempting to better his or her previous score. Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282

36. ‘Errorless Learning’ Worksheet Sample Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282

37. Math Computation: Two Ideas to Jump-Start Active Academic Responding Here are two ideas to accomplish increased academic responding on math tasks. • Break longer assignments into shorter assignments with performance feedback given after each shorter ‘chunk’ (e.g., break a 20-minute math computation worksheet task into 3 seven-minute assignments). Breaking longer assignments into briefer segments also allows the teacher to praise struggling students more frequently for work completion and effort, providing an additional ‘natural’ reinforcer. • Allow students to respond to easier practice items orally rather than in written form to speed up the rate of correct responses. Source: Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing opportunities for responding and influencing students to choose to respond. Psychology in the Schools, 42, 389-403.

38. Math Computation: Problem Interspersal Technique • The teacher first identifies the range of ‘challenging’ problem-types (number problems appropriately matched to the student’s current instructional level) that are to appear on the worksheet. • Then the teacher creates a series of ‘easy’ problems that the students can complete very quickly (e.g., adding or subtracting two 1-digit numbers). The teacher next prepares a series of student math computation worksheets with ‘easy’ computation problems interspersed at a fixed rate among the ‘challenging’ problems. • If the student is expected to complete the worksheet independently, ‘challenging’ and ‘easy’ problems should be interspersed at a 1:1 ratio (that is, every ‘challenging’ problem in the worksheet is preceded and/or followed by an ‘easy’ problem). • If the student is to have the problems read aloud and then asked to solve the problems mentally and write down only the answer, the items should appear on the worksheet at a ratio of 3 ‘challenging’ problems for every ‘easy’ one (that is, every 3 ‘challenging’ problems are preceded and/or followed by an ‘easy’ one). Source: Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal ratios on fifth-grade students’ mathematics accuracy. School Psychology Review, 34, 543-555..

39. How to… Create an Interspersal-Problems Worksheet

40. Additional Math InterventionsJim Wrightwww.interventioncentral.org

41. Math Instruction: Unlock the Thoughts of Reluctant Students Through Class Journaling Students can effectively clarify their knowledge of math concepts and problem-solving strategies through regular use of class ‘math journals’. • At the start of the year, the teacher introduces the journaling weekly assignment in which students respond to teacher questions. • At first, the teacher presents ‘safe’ questions that tap into the students’ opinions and attitudes about mathematics (e.g., ‘How important do you think it is nowadays for cashiers in fast-food restaurants to be able to calculate in their head the amount of change to give a customer?”). As students become comfortable with the journaling activity, the teacher starts to pose questions about the students’ own mathematical thinking relating to specific assignments. Students are encouraged to use numerals, mathematical symbols, and diagrams in their journal entries to enhance their explanations. • The teacher provides brief written comments on individual student entries, as well as periodic oral feedback and encouragement to the entire class. • Teachers will find that journal entries are a concrete method for monitoring student understanding of more abstract math concepts. To promote the quality of journal entries, the teacher might also assign them an effort grade that will be calculated into quarterly math report card grades. Source: Baxter, J. A., Woodward, J., & Olson, D. (2005). Writing in mathematics: An alternative form of communication for academically low-achieving students. Learning Disabilities Research & Practice, 20(2), 119–135.

42. 5 x 3 =__ 5 x 5 =__ 2 x 6 =__ 3 x 8 =__ 9 x 2 =__ 4 x 7 =__ 7 x 6 =__ 9 x 7 =__ 3 x 6 =__ 8 x 4 =__ 3 x 5 =__ 4 x 5 =__ 3 x 2 =__ 6 x 5 =__ 8 x 2 =__ Math Review: Incremental Rehearsal of ‘Math Facts’ Step 1: The tutor writes down on a series of index cards the math facts that the student needs to learn. The problems are written without the answers.

43. 5 x 5 =__ 3 x 8 =__ 8 x 2 =__ 9 x 2 =__ 3 x 5 =__ 9 x 7 =__ 6 x 5 =__ 3 x 2 =__ 8 x 4 =__ 3 x 6 =__ 5 x 3 =__ 7 x 6 =__ 4 x 7 =__ 2 x 6 =__ 4 x 5 =__ Math Review: Incremental Rehearsal of ‘Math Facts’ ‘KNOWN’ Facts ‘UNKNOWN’ Facts Step 2: The tutor reviews the ‘math fact’ cards with the student. Any card that the student can answer within 2 seconds is sorted into the ‘KNOWN’ pile. Any card that the student cannot answer within two seconds—or answers incorrectly—is sorted into the ‘UNKNOWN’ pile.

44. Step 3: The tutor is now ready to follow a nine-step incremental-rehearsal sequence: First, the tutor presents the student with a single index card containing an ‘unknown’ math fact. The tutor reads the problem aloud, gives the answer, then prompts the student to read off the same unknown problem and provide the correct answer. Step 3: Next the tutor takes a math fact from the ‘known’ pile and pairs it with the unknown problem. When shown each of the two problems, the student is asked to read off the problem and answer it. Step 3: The tutor then repeats the sequence--adding yet another known problem to the growing deck of index cards being reviewed and each time prompting the student to answer the whole series of math facts—until the review deck contains a total of one ‘unknown’ math fact and nine ‘known’ math facts (a ratio of 90 percent ‘known’ to 10 percent ‘unknown’ material ) 3 x 8 =__ 3 x 8 =__ 3 x 8 =__ 2 x 6 =__ 4 x 7 =__ 5 x 3 =__ 3 x 6 =__ 8 x 4 =__ 3 x 2 =__ 6 x 5 =__ 4 x 5 =__ 4 x 5 =__ Math Review: Incremental Rehearsal of ‘Math Facts’

45. Step 4: At this point, the last ‘known’ math fact that had been added to the student’s review deck is discarded (placed back into the original pile of ‘known’ problems) and the previously ‘unknown’ math fact is now treated as the first ‘known’ math fact in new student review deck for future drills. Step 4: The student is then presented with a new ‘unknown’ math fact to answer--and the review sequence is once again repeated each time until the ‘unknown’ math fact is grouped with nine ‘known’ math facts—and on and on. Daily review sessions are discontinued either when time runs out or when the student answers an ‘unknown’ math fact incorrectly three times. 3 x 8 =__ 9 x 2 =__ 2 x 6 =__ 8 x 4 =__ 3 x 2 =__ 6 x 5 =__ 4 x 5 =__ 2 x 6 =__ 4 x 7 =__ 5 x 3 =__ 3 x 6 =__ 8 x 4 =__ 3 x 2 =__ 6 x 5 =__ 4 x 5 =__ 3 x 6 =__ 5 x 3 =__ 3 x 8 =__ Math Review: Incremental Rehearsal of ‘Math Facts’

46. Applied Math: Helping Students to Make Sense of ‘Story Problems’Jim Wrightwww.interventioncentral.org

47. ‘Advanced Math’ Quotes from Yogi Berra— • “Ninety percent of the game is half mental." • “Pair up in threes." • “You give 100 percent in the first half of the game, and if that isn't enough in the second half you give what's left.”

48. Applied Math Problems: Rationale • Applied math problems (also known as ‘story’ or ‘word’ problems) are traditional tools for having students apply math concepts and operations to ‘real-world’ settings.

49. Sample Applied Problems • Once upon a time, there were three little pigs - ages 2, 4, and 6. Are their ages even or odd? • Every day this past summer, Peter rode his bike to and from work. Each round trip was 13 kilometers. His friend Marsha rode her bike18 kilometers' each day, but just for exercise. How much further did Marsha ride her bike than Peter in one week? • Suzy is ten years older than Billy, and next year she will be twice as old as Billy. How old are they now?

50. Applied Math Problems: Some Required Competencies For students to achieve success with applied problems, they must be able to: • Comprehend the text of written problems. • Understand specialized math vocabulary (e.g., ‘quotient’). • Understand specialized use of ‘common’ vocabulary (e.g., ‘product’). • Be able to translate verbal cues into a numeric equation. • Ignore irrelevant information included in the problem. • Interpret math graphics that may accompany the problem. • Apply a plan to problem-solving. • Check their work.