Workshop PPTs and handout available at: interventioncentral/rtimath

1. Best Practices in Classroom Math Interventions (Elementary)Jim Wrightwww.interventioncentral.org

2. Workshop PPTs and handout available at: http://www.interventioncentral.org/rtimath

3. Defining Research-Based Principles of Effective Math Instruction & Intervention Finding Effective, Research-Based Math Interventions Understanding the Student With ‘Math Difficulties’ Screening and Progress-Monitoring for Students With Math Difficulties Finding Web Resources to Support Math Assessment & Interventions Workshop Agenda: RTI Challenges…

4. Core Instruction & Tier 1 InterventionFocus of Inquiry: What are the indicators of high-quality core instruction and classroom (Tier 1) intervention for math?

5. “ ” “Tier I of an RTI model involves quality core instruction in general education and benchmark assessments to screen students and monitor progress in learning.” p. 9 “ ” “It is no accident that high-quality intervention is listed first [in the RTI model], because success in tiers 2 and 3 is quite predicated on an effective tier 1. “ p. 65 Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York.

6. Common Core State Standards Initiative http://www.corestandards.org/View the set of Common Core Standards for English Language Arts (including writing) and mathematics being adopted by states across America.

7. Commercial Instructional and Intervention Programs. Provide materials for teaching the curriculum. Schools often piece together materials from multiple programs to help students to master the curriculum. It should be noted that specific programs can change, while the underlying curriculum remains unchanged. Common Core Standards, Curriculum, and Programs: How Do They Interrelate? School Curriculum. Outlines a uniform sequence shared across instructors for attaining the Common Core Standards’ instructional goals. Scope-and-sequence charts bring greater detail to the general curriculum. Curriculum mapping ensures uniformity of practice across classrooms, eliminates instructional gaps and redundancy across grade levels. Common Core Standards. Provide external instructional goals that guide the development and mapping of the school’s curriculum. However, the sequence in which the standards are taught is up to the district and school.

8. An RTI Challenge: Limited Research to Support Evidence-Based Math Interventions “… in contrast to reading, core math programs that are supported by research, or that have been constructed according to clear research-based principles, are not easy to identify. Not only have exemplary core programs not been identified, but also there are no tools available that we know of that will help schools analyze core math programs to determine their alignment with clear research-based principles.” p. 459 Source: Clarke, B., Baker, S., & Chard, D. (2008). Best practices in mathematics assessment and intervention with elementary students. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 453-463).

9. National Mathematics Advisory Panel Report13 March 2008

10. Math Advisory Panel Report at:http://www.ed.gov/mathpanel

11. 2008 National Math Advisory Panel Report: Recommendations • “The areas to be studied in mathematics from pre-kindergarten through eighth grade should be streamlined and a well-defined set of the most important topics should be emphasized in the early grades. Any approach that revisits topics year after year without bringing them to closure should be avoided.” • “Proficiency with whole numbers, fractions, and certain aspects of geometry and measurement are the foundations for algebra. Of these, knowledge of fractions is the most important foundational skill not developed among American students.” • “Conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided.” • “Students should develop immediate recall of arithmetic facts to free the “working memory” for solving more complex problems.” Source: National Math Panel Fact Sheet. (March 2008). Retrieved on March 14, 2008, from http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-factsheet.html

12. 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: 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. Source: Big ideas in beginning reading. University of Oregon. Retrieved September 23, 2007, from http://reading.uoregon.edu/index.php The Elements of Mathematical Proficiency: What the Experts Say…

13. Five Strands of Mathematical Proficiency • Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. • Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. • Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. 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.

14. Five Strands of Mathematical Proficiency (Cont.) • Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. • Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work. 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.

15. Table Activity: Evaluate Your School’s Math Proficiency… • As a group, review the National Research Council ‘Strands of Math Proficiency’. • Which strand do you feel that your school / curriculum does the best job of helping students to attain proficiency? • Which strand do you feel that your school / curriculum should put the greatest effort to figure out how to help students to attain proficiency? • Be prepared to share your results. Five Strands of Mathematical Proficiency(NRC, 2002) • Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. • Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. • Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. • Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. • Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.

16. What Works Clearinghouse Practice Guide: Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools http://ies.ed.gov/ncee/wwc/This publication provides 8 recommendations for effective core instruction in mathematics for K-8.

17. Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations • Recommendation 1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk • Recommendation 2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8.

18. Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.) • Recommendation 3. Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review • Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

19. Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.) • Recommendation 5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas • Recommendation 6. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts

20. Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.) • Recommendation 7. Monitor the progress of students receiving supplemental instruction and other students who are at risk • Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions.

21. How Do We Reach Low-Performing Math Students?: Instructional Recommendations Important elements of math instruction for low-performing students: • “Providing teachers and students with data on student performance” • “Using peers as tutors or instructional guides” • “Providing clear, specific feedback to parents on their children’s mathematics success” • “Using principles of explicit instruction in teaching math concepts and procedures.” p. 51 Source:Baker, S., Gersten, R., & Lee, D. (2002).A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103(1), 51-73..

22. Activity: How Do We Reach Low-Performing Students? p.5 • Review the handout on p. 5 of your packet and consider each of the elements found to benefit low-performing math students. • For each element, brainstorm ways that you could promote this idea in your math classroom.

24. 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..

25. Math Challenge: The student can not yet reliably access an internalnumber-line of numbers 1-10. What Does the Research Say?...

26. 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.

27. What Are Stages of ‘Number Sense’? (Berch, 2005, p. 336) • Innate Number Sense. Children appear to possess ‘hard-wired’ ability (or neurological ‘foundation structures’) in 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...

28. 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

29. 3 X 7 = 21 28 ÷ 4 = 7 9 – 7 = 2 2 + 4 = 6 Internal Number-Line As students internalize the number-Line, 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

30. Math Challenge: The student can not yet reliably access an internalnumber-line of numbers 1-10. • Solution: Use this strategy: • Building Number Sense Through a Counting Board Game (Supplemental Packet)

31. Building Number Sense Through a Counting Board Game DESCRIPTION: The student plays a number-based board game to build skills related to 'number sense', including number identification, counting, estimation skills, and ability to visualize and access specific number values using an internal number-line (Siegler, 2009). Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

32. Building Number Sense Through a Counting Board Game MATERIALS: • Great Number Line Race! form • Spinner divided into two equal regions marked "1" and "2" respectively. (NOTE: If a spinner is not available, the interventionist can purchase a small blank wooden block from a crafts store and mark three of the sides of the block with the number "1" and three sides with the number "2".) Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

33. Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

34. Building Number Sense Through a Counting Board Game INTERVENTION STEPS: A counting-board game session lasts 12 to 15 minutes, with each game within the session lasting 2-4 minutes. Here are the steps: • Introduce the Rules of the Game. The student is told that he or she will attempt to beat another player (either another student or the interventionist). The student is then given a penny or other small object to serve as a game piece. The student is told that players takes turns spinning the spinner (or, alternatively, tossing the block) to learn how many spaces they can move on the Great Number Line Race! board. Each player then advances the game piece, moving it forward through the numbered boxes of the game-board to match the number "1" or "2" selected in the spin or block toss. Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

35. Building Number Sense Through a Counting Board Game INTERVENTION STEPS: A counting-board game session lasts 12 to 15 minutes, with each game within the session lasting 2-4 minutes. Here are the steps: • Introduce the Rules of the Game (cont.).When advancing the game piece, the player must call out the number of each numbered box as he or she passes over it. For example, if the player has a game piece on box 7 and spins a "2", that player advances the game piece two spaces, while calling out "8" and "9" (the names of the numbered boxes that the game piece moves across during that turn). Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

36. Building Number Sense Through a Counting Board Game INTERVENTION STEPS: A counting-board game session lasts 12 to 15 minutes, with each game within the session lasting 2-4 minutes. Here are the steps: • Record Game Outcomes. At the conclusion of each game, the interventionist records the winner using the form found on the Great Number Line Race! form. The session continues with additional games being played for a total of 12-15 minutes. • Continue the Intervention Up to an Hour of Cumulative Play. The counting-board game continues until the student has accrued a total of at least one hour of play across multiple days. (The amount of cumulative play can be calculated by adding up the daily time spent in the game as recorded on the Great Number Line Race! form.) Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

37. Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.

38. Math Challenge: The student has not yet acquired math facts. What Does the Research Say?...

39. 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.

40. Big Ideas: The Four Stages of Learning Can Be Summed Up in the ‘Instructional Hierarchy’ (Supplemental Packet)(Haring et al., 1978) Student learning can be thought of as a multi-stage process. The universal stages of learning include: • Acquisition: The student is just acquiring the skill. • Fluency: The student can perform the skill but must make that skill ‘automatic’. • Generalization: The student must perform the skill across situations or settings. • Adaptation: The student confronts novel task demands that require that the student adapt a current skill to meet new requirements. Source: Haring, N.G., Lovitt, T.C., Eaton, M.D., & Hansen, C.L. (1978). The fourth R: Research in the classroom. Columbus, OH: Charles E. Merrill Publishing Co.

41. 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 to derive an answer (2 + 2 = 4, so 2 + 3 =5), relations among operations (6 + 4 =10, so 10 −4 = 6) … and so forth. This approach to instruction is consonant with recommendations by the National Research Council (2001). Instruction along these lines may 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.

42. 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 superficially different situations. 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.

43. Math Short-Cuts: Addition (Supplemental Packet) • The order of the numbers in an addition problem does not affect the answer. • When zero is added to the original number, the answer is the original number. • When 1 is added to the original number, the answer is the next larger number. Source: Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40.

44. Math Short-Cuts: Subtraction (Supplemental Packet) • When zero is subtracted from the original number, the answer is the original number. • When 1 is subtracted from the original number, the answer is the next smaller number. • When the original number has the same number subtracted from it, the answer is zero. Source: Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40.

45. Math Short-Cuts: Multiplication (Supplemental Packet) • When a number is multiplied by zero, the answer is zero. • When a number is multiplied by 1, the answer is the original number. • When a number is multiplied by 2, the answer is equal to the number being added to itself. • The order of the numbers in a multiplication problem does not affect the answer. Source: Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40.

46. Math Short-Cuts: Division (Supplemental Packet) • When zero is divided by any number, the answer is zero. • When a number is divided by 1, the answer is the original number. • When a number is divided by itself, the answer is 1. Source: Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40.

47. Math Challenge: The student has not yet acquired math facts. • Solution: Use these strategies: • Strategic Number Counting Instruction (Supplemental Packet) • Incremental Rehearsal • Cover-Copy-Compare • Peer Tutoring in Math Computation with Constant Time Delay

48. Strategic Number Counting Instruction DESCRIPTION: The student is taught explicit number counting strategies for basic addition and subtraction. Those skills are then practiced with a tutor (adapted from Fuchs et al., 2009). Source: Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2009). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences 20(2), 89-100.

49. Strategic Number Counting Instruction MATERIALS: • Number-line (attached) • Number combination (math fact) flash cards for basic addition and subtraction • Strategic Number Counting Instruction Score Sheet Source: Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2009). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences 20(2), 89-100.

50. 0 1 2 3 4 5 6 7 8 9 10 Strategic Number Counting Instruction PREPARATION: The tutor trains the student to use these two counting strategies for addition and subtraction: • ADDITION: The student is given a copy of the number-line. When presented with a two-addend addition problem, the student is taught to start with the larger of the two addends and to 'count up' by the amount of the smaller addend to arrive at the answer to the problem. E..g., 3 + 5= ___ Source: Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2009). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences 20(2), 89-100.