Best Practices in Classroom Math Interventions

Best Practices in Classroom Math Interventions (Elementary) Jim Wright www.interventioncentral.org www.interventioncentral.org

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

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

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

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. www.interventioncentral.org 5

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. www.interventioncentral.org

Common Core Standards, ds, Curriculum, u and Programs: How Do They Interrelate? 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. Response to Intervention www.interventioncentral.org 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. 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.

National Mathematics Advisory Panel Report 13 March 2008 www.interventioncentral.org 8

Math Advisory Panel Report at: http://www.ed.gov/mathpanel www.interventioncentral.org 9

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 g 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 www.interventioncentral.org 10

The Elements of Mathematical Proficiency: What the Experts Say 5 Strands of Mathematical Proficiency 1. Understanding 2. Computing 3. Applying 4. Reasoning 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. 5 Big Ideas in Beginning Reading 1. Phonemic Awareness 2. Alphabetic Principle 3. Fluency with Text 4. Vocabulary 5. Engagement 5. Comprehension Source: Big ideas in beginning reading. University of Oregon. Retrieved September 23, 2007, from http://reading.uoregon.edu/index.php www.interventioncentral.org 11

Five Strands of Mathematical Proficiency 1. Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. 2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. p 3. 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. www.interventioncentral.org 12

Five Strands of Mathematical Proficiency (Cont.) 4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. 5. 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. www.interventioncentral.org 13

Five Strands of Mathematical Table Activity: Evaluate Your School s Math Proficiency Proficiency (NRC, 2002) 1. Understanding: Comprehending mathematical concepts, As a group, review the operations, and relations--knowing what mathematical National Research symbols, diagrams, and procedures mean. Council Strands of Math Proficiency. 2. Computing: Carrying out mathematical procedures, such Which strand do you feel as adding, subtracting, multiplying, and dividing numbers that your school / flexibly, accurately, efficiently, i and appropriately. curriculum does the best job of helping students to 3. Applying: Being able to formulate problems attain proficiency? mathematically and to devise strategies for solving them using concepts and procedures appropriately. Which strand do you feel that your school / curriculum should put the 4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. greatest effort to figure out how to help students to attain proficiency? Be prepared to share your results. 5. Engaging: Seeing mathematics ti as sensible, useful, and doable if you work at it and being willing to do the work. www.interventioncentral.org 14

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. www.interventioncentral.org

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 h 8. www.interventioncentral.org 16

Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.) Recommendation 3. Instruction during the intervention ention 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. t www.interventioncentral.org 17

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 ti facts www.interventioncentral.org 18

Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.) Recommendation 7. Monitor the progress of students receiving ing supplemental instruction and other students who are at risk Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions. www.interventioncentral.org 19

How Do We Reach Low-Performing Math Students?: Instructional Recommendations Important elements of math instruction for low-performing students: t 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 lowachieving students. The Elementary School Journal, 103(1), 51-73.. www.interventioncentral.org 20

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. Response to Intervention Activity: How Do We Reach Low-Performing Students? p.5 www.interventioncentral.org 21

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.. www.interventioncentral.org 22

Math Challenge: The student t can not yet reliably access an internal number-line of numbers 1-10. What Does the Research Say?... www.interventioncentral.org 23

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. www.interventioncentral.org 24

What Are Stages of Number Sense? (Berch, 2005, p. 336) 1. 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. 2. 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... www.interventioncentral.org 25

The Basic Number Line is as Familiar as a Well-Known Place to People Who Have Mastered Arithmetic ti Combinations Moravia, NY Number Line: 0-144 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 www.interventioncentral.org 26

Children s Understanding of Counting Rules The development of children s counting ability depends upon the development of: One-to-one correspondence: one and only one word tag, e.g., one, two, is assigned to each counted object. Stable order: the order of the word tags must be invariant across counted sets. Cardinality: the value of the final word tag represents the quantity of items in the counted set. Abstraction: objects of any kind can be collected together and counted. Order irrelevance: items within a given set can be tagged in any sequence. Source: Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4-15. www.interventioncentral.org 27

Math Challenge: The student t can not yet reliably access an internal number-line of numbers 1-10. Solution: Use this strategy: Building Number Sense Through h a Counting Board Game (Supplemental Packet) www.interventioncentral.org 28

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. www.interventioncentral.org 29

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. www.interventioncentral.org 30

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

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: 1. 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. www.interventioncentral.org 32

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: 1. 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. www.interventioncentral.org 33

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: 2. 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. 3. 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 l 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. www.interventioncentral.org 34

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

Math Challenge: The student has not yet acquired math facts. What Does the Research Say?... www.interventioncentral.org 36

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 ti 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. www.interventioncentral.org 37

Big Ideas: The Four Stages of Learning Can Be Summed Up in the Instructional ti 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. www.interventioncentral.org 38

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 t 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. www.interventioncentral.org 39

Students Who Understand Mathematical Concepts Can Discover Their Own Shortcuts t 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. www.interventioncentral.org 40

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. www.interventioncentral.org 41

Math Short-Cuts: Subtraction (Supplemental Packet) When zero is subtracted from the original number, the answer is the original i 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. www.interventioncentral.org 42

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 li 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. www.interventioncentral.org 43

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. www.interventioncentral.org 44

Math Challenge: The student has not yet acquired math facts. Solution: Use these strategies: Strategic Number Counting Instruction (Supplemental Packet) Incremental Rehearsal Cover-Copy-CompareC C Peer Tutoring in Math Computation with Constant Time Delay www.interventioncentral.org 45

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. www.interventioncentral.org 46

Strategic Number Counting Instruction MATERIALS: Number-line Number combination (math fact) flash cards for basic addition and subtraction ti 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. www.interventioncentral.org 47

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= 0 1 2 3 4 5 6 7 8 9 10 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. www.interventioncentral.org 48

Strategic Number Counting Instruction PREPARATION: The tutor trains the student to use these two counting strategies for addition and subtraction: SUBTRACTION: With access to a number-line, the student t is taught to refer to the first number appearing in the subtraction problem (the minuend) as 'the number you start with' and to refer to the number appearing after the minus (subtrahend) as 'the minus number'. The student starts at the minus number on the number-line and counts up to the starting number while keeping a running tally of numbers counted up on his or her fingers. The final tally of digits separating the minus number and starting number is the answer to the subtraction problem. E..g., 6 2 = 0 1 2 3 4 5 6 7 8 9 10 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. www.interventioncentral.org 49

Strategic Number Counting Instruction INTERVENTION STEPS: For each tutoring session, the tutor follows these steps: 1. Create Flashcards. The tutor creates addition and/or subtraction flashcards of problems that the student is to practice. Each flashcard displays the numerals and operation sign that make up the problem but leaves the answer blank. 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. www.interventioncentral.org 50

Strategic Number Counting Instruction INTERVENTION STEPS: For each tutoring session, the tutor follows these steps: 2. Review Count-Up Strategies. At the opening of the session, the tutor asks the student to name the two methods for answering a math fact. The correct student response is 'Know it or count up.' The tutor next has the student describe how to count up an addition problem and how to count up a subtraction problem. Then the tutor gives the student two sample addition problems and two subtraction problems and directs the student to solve each, using the appropriate count-up strategy. 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. www.interventioncentral.org 51

Strategic Number Counting Instruction INTERVENTION STEPS: For each tutoring t session, the tutor t follows these steps: 3. Complete Flashcard Warm-Up. The tutor reviews addition/subtraction flashcards with the student for three minutes. Before beginning, the tutor reminds the student that, when shown a flashcard, the student should try to 'pull the answer from your head' but that if the student does not know the answer, he or she should use the appropriate countup strategy. The tutor then reviews the flashcards with the student. Whenever the student makes an error, the tutor directs the student to use the correct count-up strategy to solve. NOTE: If the student cycles through all cards in the stack before the three-minute period has elapsed, the tutor shuffles the cards and begins again. At the end of the three minutes, the tutor counts up the number of cards reviewed and records the total correct responses and errors. 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. www.interventioncentral.org 52

Strategic Number Counting Instruction INTERVENTION STEPS: For each tutoring t session, the tutor t follows these steps: 4. Repeat Flashcard Review. The tutor shuffles the math-fact flashcards, encourages the student to try to beat his or her previous score, and again reviews the flashcards with the student for three minutes. As before, whenever the student makes an error, the tutor directs the student to use the appropriate count-up strategy. Also, if the student completes all cards in the stack with time remaining, the tutor shuffles the stack and continues presenting cards until the time is elapsed. At the end of the three minutes, the tutor once again counts up the number of cards reviewed and records the total correct responses and errors. 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. www.interventioncentral.org 53

Strategic Number Counting Instruction INTERVENTION STEPS: For each tutoring t session, the tutor t follows these steps: 5. Provide Performance Feedback. The tutor gives the student feedback about whether (and by how much) the student's performance on the second flashcard trial exceeded the first. The tutor also provides praise if the student beat the previous score or encouragement if the student failed to beat the previous score. 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. www.interventioncentral.org 54

Strategic t Number Counting Instruction Score Sheet www.interventioncentral.org 55

RTI at Tier 1: The Teacher as First Responder Focus of Inquiry: What does Tier 1 intervention look like for the general-education classroom teacher who is supporting struggling students? www.interventioncentral.org 56

RTI Pyramid of Interventions Tier 3 Tier 2 Response to Intervention Tier 3: Intensive interventions. Students who are nonresponders to Tiers 1 & 2 are referred to the RTI Team for more intensive interventions. Tier 2 Individualized interventions. Subset of students receive interventions targeting specific needs. Tier 1 Tier 1: Universal interventions. Available to all students in a classroom or school. Can consist of whole-group or individual strategies or supports. www.interventioncentral.org 57

Tier 1 Core Instruction Tier I core instruction: Is universal available to all students. Can be delivered within classrooms or throughout the school. Is an ongoing process of developing strong classroom instructional practices to reach the largest number of struggling learners. All children have access to Tier 1 instruction/interventions. Teachers have the capability to use those strategies without requiring outside assistance. Tier 1 instruction encompasses: The school s core curriculum. All published or teacher-made materials used to deliver that curriculum. Teacher use of whole-group teaching & management strategies. Tier I instruction addresses this question: Are strong classroom instructional strategies sufficient to help the student to achieve academic success? www.interventioncentral.org 58

Tier 1 intervention: Response to Intervention Tier I (Classroom) Intervention Targets red flag students t who are not successful with core instruction alone. Uses evidence-based strategies to address student academic or behavioral concerns. Must be feasible to implement given the resources available in the classroom. Tier I intervention addresses the question: Does the student make adequate progress when the instructor uses specific academic or behavioral strategies matched to the presenting concern? www.interventioncentral.org 59

The Key Role of Classroom Teachers as Interventionists ti i t in RTI: 6 Steps 1. The teacher defines the student academic or behavioral problem clearly. 2. The teacher decides on the best explanation for why the problem is occurring. 3. The teacher selects research-based interventions. 4. The teacher documents the student s Tier 1 intervention plan. 5. The teacher monitors the student s s response (progress) to the intervention plan. 6. The teacher knows what the next steps are when a student fails to make adequate progress with Tier 1 interventions alone. www.interventioncentral.org 60

Supplemental Packet www.interventioncentral.org 61

RTI Interventions: What If There is No Commercial Intervention Package or Program Available? Although commercially prepared p programs and manuals and materials are inviting, they are not necessary. A recent review of research suggests that interventions are research based and likely to be successful, if they are correctly targeted and provide explicit instruction in the skill, an appropriate level of challenge, sufficient opportunities to respond to and practice the skill, and immediate feedback on performance Thus, these [elements] e e could be used as criteria with which to judge potential interventions. p. 88 Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York. www.interventioncentral.org 62

Motivation Deficit 1: The student is unmotivated because he or she cannot do the assigned work. Profile of a Student with This Motivation Problem: The student lacks essential skills required to do the task. Handout pp.2-3 www.interventioncentral.org 63

Motivation Deficit 1: Cannot Do the Work Profile of a Student with This Motivation Problem (Cont.): Areas of deficit might include: Basic academic skills. Basic skills have straightforward criteria for correct performance (e.g., the student defines vocabulary words or decodes text or computes math facts ) and comprise the building-blocks blocks of more complex academic tasks (Rupley, Blair, & Nichols, 2009). Cognitive strategies. Students employ specific cognitive strategies as guiding procedures to complete more complex academic tasks such as reading comprehension or writing (Rosenshine, 1995). Academic-enabling enabling skills. Skills that are academic enablers (DiPerna, 2006) are not tied to specific academic knowledge but rather aid student learning across a wide range of settings and tasks (e.g., organizing work materials, time management). www.interventioncentral.org 64

Motivation Deficit 1: Cannot Do the Work (Cont.) What the Research Says: When a student lacks the capability to complete an academic task because of limited or missing basic skills, cognitive strategies, or academicenabling skills, that student is still in the acquisition stage of learning (Haring et al., 1978). That student cannot be expected to be motivated or to be successful as a learner unless he or she is first explicitly taught these weak or absent essential skills (Daly, Witt, Martens & Dool, 1997). www.interventioncentral.org 65

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Verify the Presence of This Motivation Problem: The teacher collects information (e.g., through observations of the student engaging in academic tasks; interviews with the student; examination of work products, quizzes, or tests) demonstrating that the student lacks basic skills, cognitive strategies, or academic-enabling enabling skills essential to the academic task. www.interventioncentral.org 66

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Fix This Motivation Problem: Students who are not motivated because they lack essential skills need to be taught those skills. Direct-Instruction Format. Students learning new material, concepts, or skills benefit from a direct instruction approach. (Burns, VanDerHeyden & Boice, 2008; Rosenshine, 1995; Rupley, Blair, & Nichols, 2009). www.interventioncentral.org 67

www.interventioncentral.org

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Fix This Motivation Problem: When following a direct-instruction instruction format, the teacher: ensures that the lesson content is appropriately matched to students abilities. opens the lesson with a brief review of concepts or material that were previously presented. states the goals of the current day s lesson. breaks new material into small, manageable increments, or steps. www.interventioncentral.org 69

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Fix This Motivation Problem: When following a direct-instruction instruction format, the teacher: throughout the lesson, provides adequate explanations and detailed instructions for all concepts and materials being taught. NOTE: Verbal explanations can include talk-alouds (e.g., the teacher describes and explains each step of a cognitive strategy) and think-alouds (e.g., the teacher applies a cognitive strategy to a particular problem or task and verbalizes the steps in applying the strategy). regularly checks for student understanding by posing frequent questions and eliciting group responses. www.interventioncentral.org 70

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Fix This Motivation Problem: When following a direct-instruction instruction format, the teacher: verifies that students are experiencing sufficient success in the lesson content to shape their learning in the desired direction and to maintain student motivation and engagement. provides timely and regular performance feedback and corrections throughout the lesson as needed to guide student learning. www.interventioncentral.org 71

Motivation Deficit 1: Cannot Do the Work (Cont.) How to Fix This Motivation Problem: When following a direct-instruction instruction format, the teacher: allows students the chance to engage in practice activities distributed throughout the lesson (e.g., through teacher demonstration; then group practice with teacher supervision and feedback; then independent, individual student practice). ensures that students have adequate support (e.g., clear and explicit instructions; teacher monitoring) to be successful during independent seatwork practice activities. www.interventioncentral.org 72

www.interventioncentral.org

Activity: Good Instruction is Research-Based Review the elements of effective direct instruction that appear on page 3 of your handout. Discuss how you can use this checklist to verify that small-group or 1:1 remedial math instruction developed d and delivered d by the teacher can be considered research-based. www.interventioncentral.org 74

Cover-Copy-Compare: Math Computational Fluency-Building Intervention The student is given sheet with correctly completed math problems in left column and index card. For each problem, the student: studies the model covers the model with index card copies the problem from memory solves the problem uncovers the correctly completed model to check answer Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420. www.interventioncentral.org 75

Cover-Copy-Compare: Math Computational Fluency-Building Intervention Here is one way to create CCC math worksheets, using the math worksheet generator on www.interventioncentral.org: 1. From any math operations page, select the computation target. 2. Then click the Cover-Copy-Compare button. A scaffolded version of the CCC worksheet will be created that provides the student with both a completed model and a partially completed model. Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420. www.interventioncentral.org 76

Cover-Copy-Compare: Math Computational Fluency-Building Intervention Here is another way to create CCC math worksheets, using the math worksheet generator on www.interventioncentral.org: ti t 1. From any math operations page, select a computation skill for the CCC worksheet. 2. Next, set the Number of Columns setting to 1. 3. Then set the Number of Rows setting to the number of CCC problems that you would like the student to complete. 4. Click the Single-Skill Skill Computation Probe button. 5. Print off only the answer key and use it as your student s CCC worksheet. Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420. www.interventioncentral.org 77

Peer Tutoring in Math Computation with Constant Time Delay pp. 20-26 www.interventioncentral.org

Peer Tutoring in Math Computation with Constant Time Delay DESCRIPTION: This intervention employs students as reciprocal peer tutors to target acquisition of basic math facts (math computation) using constant time delay (Menesses & Gresham, 2009; Telecsan, Slaton, & Stevens, 1999). Each tutoring session is brief and includes its own progress-monitoring component--making this a convenient and time-efficient math intervention for busy classrooms. www.interventioncentral.org 79

Peer Tutoring in Math Computation with Constant Time Delay MATERIALS: Student Packet: A work folder is created for each tutor pair. The folder contains: 10 math fact cards with equations written on the front and correct answer appearing on the back. NOTE: The set of cards is replenished and updated regularly as tutoring pairs master their math facts. Progress-monitoring form for each student. Pencils. www.interventioncentral.org 80

Peer Tutoring in Math Computation with Constant Time Delay PREPARATION: To prepare for the tutoring program, the teacher selects students to participate and trains them to serve as tutors. Select Student Participants. Students being considered for the reciprocal peer tutor program should at minimum meet these criteria (Telecsan, Slaton, & Stevens, 1999, Menesses & Gresham, 2009): Is able and willing to follow directions; Shows generally appropriate classroom behavior; Can attend to a lesson or learning activity for at least 20 minutes. www.interventioncentral.org 81

Peer Tutoring in Math Computation with Constant Time Delay Sl Select Student Std t Participants tii t (Cont.). Students t being considered d for the reciprocal peer tutor program should at minimum meet these criteria (Telecsan, Slaton, & Stevens, 1999, Menesses & Gresham, 2009): Is able to name all numbers from 0 to 18 (if tutoring in addition or subtraction math facts) and name all numbers from 0 to 81 (if tutoring in multiplication or division math facts). Can correctly read aloud a sampling of 10 math-facts (equation plus answer) that will be used in the tutoring sessions. (NOTE: The student does not need to have memorized or otherwise mastered these math facts to participate just be able to read them aloud from cards without errors). [To document a deficit in math computation] When given a two-minute math computation probe to complete independently, computes fewer than 20 correct digits (Grades 1-3) or fewer than 40 correct digits (Grades 4 and up) (Deno & Mirkin, 1977). www.interventioncentral.org 82

Peer Tutoring in Math Computation: ti Teacher Nomination Form www.interventioncentral.org 83

Peer Tutoring in Math Computation with Constant Time Delay Tutoring Activity. Each tutoring session last for 3 minutes. The tutor: Presents Cards. The tutor presents each card to the tutee for 3 seconds. Provides Tutor Feedback. [When the tutee responds correctly] The tutor acknowledges the correct answer and presents the next card. [When the tutee does not respond within 3 seconds or responds incorrectly] The tutor states the correct answer and has the tutee repeat the correct answer. The tutor then presents the next card. Provides Praise. The tutor praises the tutee immediately following correct answers. Shuffles Cards. When the tutor and tutee have reviewed all of the math-fact carts, the tutor shuffles them before again presenting cards. www.interventioncentral.org 84

Peer Tutoring in Math Computation with Constant Time Delay Progress-Monitoring Activity. The tutor concludes each 3-minute tutoring session by assessing the number of math facts mastered by the tutee. The tutor follows this sequence: Presents Cards. The tutor presents each card to the tutee for 3 seconds. Remains Silent. The tutor does not provide performance feedback or praise to the tutee, or otherwise talk during the assessment phase. Sorts Cards. Based on the tutee s responses, the tutor sorts the math-fact cards into correct and incorrect piles. Counts Cards and Records Totals. The tutor counts the number of cards in the correct and incorrect piles and records the totals on the tutee s progress-monitoring chart. www.interventioncentral.org 85

Peer Tutoring in Math Computation with Constant Time Delay Tutoring Integrity Checks. As the student pairs complete the tutoring activities, the supervising adult monitors the integrity with which the intervention is carried out. At the conclusion of the tutoring session, the adult gives feedback to the student pairs, praising successful implementation and providing corrective feedback to students as needed. NOTE: Teachers can use the attached form Peer Tutoring in Math Computation with Constant Time Delay: Integrity Checklist to conduct integrity checks of the intervention and student progressmonitoring components of the math peer tutoring. www.interventioncentral.org 86

Peer Tutoring in Math Computation: Intervention Integrity Sheet: (Part 1: Tutoring Activity) it www.interventioncentral.org 87

Peer Tutoring in Math Computation: Intervention Integrity Sheet (Part 2: Progress- Monitoring) i www.interventioncentral.org 88

Peer Tutoring in Math Computation: Score Sheet www.interventioncentral.org 89

Activity: Making Use of Student Power in Classroom Interventions Consider the Peer Tutoring in Math Computation With Constant Time Delay intervention on p. 20 Discuss ways that your school/grade level/classroom l/ l can enlist students t to take an active guiding role in math interventions. www.interventioncentral.org 90

Math Challenge: The student t has acquired math computation skills but is not yet fluent. What Does the Research Say?... www.interventioncentral.org 91

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 p and commutative property p 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. www.interventioncentral.org 92

Associative Property Response to Intervention 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 www.interventioncentral.org 93

Commutative Property Response to Intervention the ability to change the order of something without t 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 www.interventioncentral.org 94