digging deeper in the tiered system of support for mathematics laura colligan academic consultant,...

MTSS Implementers Professional Development

Friday, March 7, 2014Digging Deeper in the Tiered System of Support

For MathematicsLaura Colligan

Academic Consultant, Ingham [email protected]

517.244.1258

mailto:[email protected]

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

Agenda

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 assessmentScore assessment

Small group discussion about MCOMPTopics to ponder:

StrengthsCommon skills across probesWhat could we take back to our schools

MCOMP, NOW WHAT

Compare Reading Foundation and Mathematical Proficiency Skills

Massachusetts Department of Elementary and Secondary

Education 7

The Quiz

8 + 2 =

14 ÷ 7 =

12 x 2 =

10 - 2 =

6 x 5 =

9 ÷ 9 =

10 - 5 =

17 x 2 =

8 ÷ 4 =

4 x 3 =

15 - 3 =

9 ÷ 2 =

8 x 7 =

14 - 7 =

6 x 2 =

8 + 5 =

9 - 1 =

8 - 4 =

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?


Education 8

Reactions to the Quiz

Chosen because they have an impact on mathematics learning.


Education 9

6 Areas of Demands and Difficulties for Students: What’s Involved in Learning Mathematics?

Memory

Conceptual Understanding

Attention

Organization

Language

Visual/Spatial Understandin

g

Memory

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, 2007

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The impact of Memory on learning mathematics includes:

Massachusetts Department of Elementary and Secondary Education

Attention

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., 2003

11

Impact of Attention for learning mathematics includes:


Organization

Aligning columns and rows for computation

Problem solving

Ordering of numbers and symbols

Constant movement of manipulatives

Creating graphs Matching tables with

patterns


12

Impact of Organization for learning mathematics includes:


Language

Reading Text

Math Vocabulary

Writing explanations

Sharing ideas in groups

Listening to instruction

Writing math stories


13

Impact of Language for learning mathematics includes:


Conceptual Understanding

Number sense

Problem solving

Moving from concrete to abstract, i.e. equations

Making generalizations

Applying strategies to new situations

Reflecting on thinking— metacognition


14

Impact of Conceptual Understanding for learning mathematics includes:


Visual/Spatial Understanding

Reading tables Diagrams Visual

examples

Trouble following graphs May not line up numbers

correctly Following patterns from

drawings


15

Impact of Visual/Spatial Understanding for learning mathematics includes:


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?


Education 16

Discuss

Math Computation: Building FluencyJim Wrightwww.interventioncentral.org

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

–Anonymous

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

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Benefits of Automaticity of ‘Arithmetic Combinations’ (Gersten, Jordan, & Flojo, 2005)

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.

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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): 3 + 8More efficient strategy: Begin at the number 8 (larger number) and ‘count up’ 3 more digits: 8 + 3Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in memory and automatically retrieved: Answer = 11


“[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.”

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Math Skills: Importance of Fluency in Basic Math Operations

Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114.

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.

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Big Ideas: Learn Unit (Heward, 1996)

Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

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.

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Math Intervention: Tier I or II: Elementary & Secondary: Self-Administered Arithmetic Combination Drills With Performance Self-Monitoring & Incentives

Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

26

Self-Administered Arithmetic Combination Drills:Examples of Student Worksheet and Answer Key

Worksheets created using Math Worksheet Generator. Available online at:http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php

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

Cover-Copy-Compare: Math Computational Fluency-Building Intervention

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.

“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

28

Math Shortcuts: Cognitive Energy- and Time-Savers


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.

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Math Multiplication Shortcut: ‘The 9 Times Quickie’

9 x 19 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 http://math.about.com/bltricks.htm

“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

30

Students Who ‘Understand’ Mathematical Concepts Can

Discover Their Own ‘Shortcuts’

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’.

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Application of Math Shortcuts to Intervention Plans

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

Math Computation: Motivate With ‘Errorless Learning’ Worksheets

Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282

33

‘Errorless Learning’ Worksheet Sample

Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282

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

Math Computation: Two Ideas to Jump-Start Active Academic Responding

Source: Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing opportunities for responding and influencing students to choose to respond. Psychology in the Schools, 42, 389-403.

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

Math Computation: Problem Interspersed Technique

Source: Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal ratios on fifth-grade students’ mathematics accuracy. School Psychology Review, 34, 543-555..

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How to… Create an Interspersal-Problems Worksheet

37

Math Review: Incremental Rehearsal of ‘Math Facts’

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)

Effect Sizes for Instructional Variables for Special Education Students and Other Low-Achieving Students http://www.nctm.org/news/content.aspx?id=8452

Measuring the ‘Intervention Footprint’: Issues of Planning, Documentation, & Follow-ThroughJim Wrightwww.interventioncentral.org

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’.

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Essential Elements of Any Academic or Behavioral Intervention (‘Treatment’) Strategy:

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).

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Interventions, Accommodations & Modifications: Sorting Them Out

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.

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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 MathOrigo Education ProgramWebsites and online resources

Researched Based Strategies

On your candy bar personality/feedback sheet give feedback about session and needs or wants for future professional development.

Thanks for coming today!

Laura ColliganAcademic Consultant, Ingham [email protected]

Wrap UP:

mailto:[email protected]

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 TestWhat’s your favorite bar say about

you?

digging deeper in the tiered system of support for mathematics laura colligan academic consultant,...

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