Digging Deeper in the Tiered System of Support For Mathematics Laura Colligan Academic Consultant, Ingham ISD

 Universal Screener: MCOMP, Now What?  6 Areas of Demands and Difficulties for Students  Identify research based strategies to use with student emphasis on fluency  Individual reflect-and-write Massachusetts Department of Elementary and Secondary Education 2

Participants will…  Be able to dig deeper into MCOMP assessments to inform instruction  Identify 6 Areas of Demands and Difficulties for students in mathematics  Identify research based strategies to use with students emphasis on fluency Learning Targets

 Task: Take 8 minute assessment Score assessment  Small group discussion about MCOMP Topics to ponder:  Strengths  Common skills across probes  What could we take back to our schools

Massachusetts Department of Elementary and Secondary Education = 14 ÷ 7 = 12 x 2 = = 6 x 5 = 9 ÷ 9 = = 17 x 2 = 8 ÷ 4 = 4 x 3 = = 9 ÷ 2 = 8 x 7 = = 6 x 2 = = = = Directions: Solve the following basic facts. You have 1 minute to complete this quiz. Please remember that the + symbol means multiply, the - symbol means divide, the ÷ symbol means add, and the x symbol means subtract.

 How did it feel to be in the place of the quiz taker?  How might this experience translate into ways in which students with disabilities respond to typical classroom learning experiences? Massachusetts Department of Elementary and Secondary Education8

 Chosen because they have an impact on mathematics learning. Massachusetts Department of Elementary and Secondary Education9 Memory Conceptual Understanding Attention OrganizationLanguage Visual/Spatial Understanding

 Difficulties storing and retrieving facts  Math facts  Students’ lack fluency and accuracy  Working memory  Impacts work on multi- step problems  Other theories: difficulties with language of number words or difficulties with visual representations, e.g. number lines  Difficulty holding information in mind while solving a problem  May be related to difficulties inhibiting correct answers Sources: Gersten et. al., 2008; Mazzocco, The impact of Memory on learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 Lack of focus on details  Lack of routines to follow  Too much text on a page  Finding key words or phrases to solve problems  Focus on only one aspect of a problem Source: Allsopp et al., Impact of Attention for learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 Aligning columns and rows for computation  Problem solving  Ordering of numbers and symbols  Constant movement of manipulatives  Creating graphs  Matching tables with patterns Source: Allsopp et al., Impact of Organization for learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 Reading Text  Math Vocabulary  Writing explanations  Sharing ideas in groups  Listening to instruction  Writing math stories Source: Allsopp et al., Impact of Language for learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 Number sense  Problem solving  Moving from concrete to abstract, i.e. equations  Making generalizations  Applying strategies to new situations  Reflecting on thinking— metacognition Source: Allsopp et al., Impact of Conceptual Understanding for learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 Reading tables  Diagrams  Visual examples  Trouble following graphs  May not line up numbers correctly  Following patterns from drawings Source: Allsopp et al., Impact of Visual/Spatial Understanding for learning mathematics includes: Massachusetts Department of Elementary and Secondary Education

 What are the essential barriers that students with these difficulties experience?  What experiences have you had with this area of demand with students or with teachers? Massachusetts Department of Elementary and Secondary Education16

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 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 19 Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38,

 “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 20 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from

 “the ability to change the order of something without changing the end result.”  Example: ◦ 2+3+5=10 ◦ 2+5+3=10 21 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from

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

“[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.” 23 Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24,

The three essential elements of effective student learning include: 1. 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. 2. 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. 3. 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. 24 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 ). Pacific Grove, CA:Brooks/Cole.

1. The student is given a math computation worksheet of a specific problem type, along with an answer key [Academic Opportunity to Respond]. 2. 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. 3. 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] 4. 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] 5. The student records the day’s score of TOTAL number of correct digits on his or her personal performance chart. 6. The student receives praise or a reward if he or she exceeds the most recently posted number of correct digits. 25 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 ). Pacific Grove, CA:Brooks/Cole.

26 Worksheets created using Math Worksheet Generator. Available online at:

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 27 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,

“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 =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 Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38,

 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 x 1 9 x 29 x 39 x 49 x 59 x 69 x 79 x 89 x 99 x 10 Source: Russell, D. (n.d.). Math facts to learn the facts. Retrieved November 9, 2007, from

“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 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.

 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’. 31

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. 32 Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80,

33 Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80,

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. 34 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,

 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). 35 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,

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37 2 x 6 =__ 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. 3 x 8 =__9 x 2 =__4 x 7 =__7 x 6 =__5 x 5 =__5 x 3 =__3 x 6 =__8 x 4 =__3 x 5 =__4 x 5 =__3 x 2 =__6 x 5 =__8 x 2 =__9 x 7 =__

38 2 x 6 =__ 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. 3 x 8 =__4 x 7 =__7 x 6 =__5 x 3 =__3 x 6 =__8 x 4 =__4 x 5 =__3 x 2 =__6 x 5 =__9 x 7 =__9 x 2 =__3 x 5 =__8 x 2 =__5 x 5 =__ ‘KNOWN’ Facts‘UNKNOWN’ Facts

39 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 =__ 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. 3 x 8 =__ 4 x 5 =__3 x 8 =__

Instructional Strategy (Hattie Effect Size) Effect Size for Special Education Students Effect Size for Low- Achieving Students 1. Visual and graphic descriptions of problems 0.50 (moderate)N/A 2. Systematic and explicit instruction (0.59)1.19 (large) 0.58 (moderate to large) 3. Student think-alouds (0.69)0.98 (large)N/A 4. Use of structured peer-assisted learning activities involving heterogeneous ability groupings (0.72) 0.42 (moderate)0.62 (large) 5. Formative assessment data provided to teachers (0.90) 0.32 (small to moderate)0.51 (moderate) 6. Formative assessment data provided directly to students(0.90) 0.33 (small to moderate)0.57(moderate)

 Method of delivery (‘Who or what delivers the treatment?’) Examples include teachers, paraprofessionals, parents, volunteers, computers.  Treatment component (‘What makes the intervention effective?’) Examples include activation of prior knowledge to help the student to make meaningful connections between ‘known’ and new material; guide practice (e.g., Paired Reading) to increase reading fluency; periodic review of material to aid student retention. As an example of a research-based commercial program, Read Naturally ‘combines teacher modeling, repeated reading and progress monitoring to remediate fluency problems’. 42

 Interventions. An academic intervention is a strategy used to teach a new skill, build fluency in a skill, or encourage a child to apply an existing skill to new situations or settings. An intervention is said to be research-based when it has been demonstrated to be effective in one or more articles published in peer–reviewed scientific journals. Interventions might be based on commercial programs such as Read Naturally. The school may also develop and implement an intervention that is based on guidelines provided in research articles—such as Paired Reading (Topping, 1987). 43

 Modifications. A modification changes the expectations of what a student is expected to know or do—typically by lowering the academic expectations against which the student is to be evaluated. Examples of modifications are reducing the number of multiple-choice items in a test from five to four or shortening a spelling list. Under RTI, modifications are generally not included in a student’s intervention plan, because the working assumption is that the student can be successful in the curriculum with appropriate interventions and accommodations alone. 44

 Accommodations. An accommodation is intended to help the student to fully access the general-education curriculum without changing the instructional content. An accommodation for students who are slow readers, for example, may include having them supplement their silent reading of a novel by listening to the book on tape. An accommodation is intended to remove barriers to learning while still expecting that students will master the same instructional content as their typical peers. Informal accommodations may be used at the classroom level or be incorporated into a more intensive, individualized intervention plan. 45

 PALs Strategy  Rocket Math  Origo Education Program  Websites and online resources

 On your candy bar personality/feedback sheet give feedback about session and needs or wants for future professional development. Thanks for coming today! Laura Colligan Academic Consultant, Ingham ISD

Milk Chocolate You're an all American who loves baseball, Mom & apple pie. You're a cheerleader for your program, level-headed, a good PR person and a great fundraiser. You're also kind, thoughtful, and always remember everyone's birthday. You are nurturing, dependable, loyal, and help others to "shine". Others often turn to you for help. Krackel You're creative, optimistic, always see the cup as half full. You're messy (messy desk or classroom) but organized (eventually find a missing item or believe you will). You like to be a hands-on person. You're a little off-beat, ditzy, funny, friendly and outgoing person who is always will to help. You like the surprising things in life, the "krackel". You like situations that allow flexibility, change and growth. Mr. Goodbar You're analytical and logical. You gather data first before giving an opinion, play the devil's advocate at meetings, tend to see all the possibilities and drive people crazy by sharing all the "what ifs". You hate deadlines and put off starting things; you're a procrastinator. You like to be the expert but in your own time frame. You can analyze things to death. You like there to be rules that everyone follows. You like a lot of structure and hate surprises. Special Dark You're a patient, thoughtful individualist and problem-solver. You like to see a project through from start to finish. You're a good grant writer and work well with difficult people. You are reflective and insightful and have little patience for incompetence or liars. You set high standards for yourself and others. You are dependable, resourceful, and loyal. Candy Bar Personality Test What’s your favorite bar say about you?