Best Practices in Classroom Math Interventions

(Elementary) Jim Wright www.interventioncentral.o

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Workshop PPTs and handout available at

http//www.interventioncentral.org/rtimath

Workshop Agenda RTI Challenges

Core Instruction Tier 1 Intervention Focus of

Inquiry What are the indicators of high-quality

core instruction and classroom (Tier 1)

intervention for math?

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.

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.

Common Core Standards, Curriculum, and Programs

How Do They Interrelate?

School Curriculum. Outlines a uniform sequence

shared across instructors for attaining the

Common Core Standards instructional goals.

Scope-and-sequence charts bring greater detail to

the general curriculum. Curriculum mapping

ensures uniformity of practice across classrooms,

eliminates instructional gaps and redundancy

across grade levels.

An RTI Challenge Limited Research to Support

Evidence-Based Math Interventions

- in contrast to reading, core math programs

that are supported by research, or that have been

constructed according to clear research-based

principles, are not easy to identify. Not only

have exemplary core programs not been identified,

but also there are no tools available that we

know of that will help schools analyze core math

programs to determine their alignment with clear

research-based principles. p. 459

Source Clarke, B., Baker, S., Chard, D.

(2008). Best practices in mathematics assessment

and intervention with elementary students. In A.

Thomas J. Grimes (Eds.), Best practices in

school psychology V (pp. 453-463).

National Mathematics Advisory Panel Report 13

March 2008

Math Advisory Panel Report at http//www.ed.gov/

mathpanel

2008 National Math Advisory Panel Report

Recommendations

- The areas to be studied in mathematics from

pre-kindergarten through eighth grade should be

streamlined and a well-defined set of the most

important topics should be emphasized in the

early grades. Any approach that revisits topics

year after year without bringing them to closure

should be avoided. - Proficiency with whole numbers, fractions, and

certain aspects of geometry and measurement are

the foundations for algebra. Of these, knowledge

of fractions is the most important foundational

skill not developed among American students. - Conceptual understanding, computational and

procedural fluency, and problem solving skills

are equally important and mutually reinforce each

other. Debates regarding the relative importance

of each of these components of mathematics are

misguided. - Students should develop immediate recall of

arithmetic facts to free the working memory for

solving more complex problems.

Source National Math Panel Fact Sheet. (March

2008). Retrieved on March 14, 2008, from

http//www.ed.gov/about/bdscomm/list/mathpanel/rep

ort/final-factsheet.html

The Elements of Mathematical Proficiency What

the Experts Say

Five Strands of Mathematical Proficiency

- Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. - Computing Carrying out mathematical procedures,

such as adding, subtracting, multiplying, and

dividing numbers flexibly, accurately,

efficiently, and appropriately. - Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately.

Source National Research Council. (2002).

Helping children learn mathematics. Mathematics

Learning Study Committee, J. Kilpatrick J.

Swafford, Editors, Center for Education, Division

of Behavioral and Social Sciences and Education.

Washington, DC National Academy Press.

Five Strands of Mathematical Proficiency (Cont.)

- Reasoning Using logic to explain and justify a

solution to a problem or to extend from something

known to something less known. - Engaging Seeing mathematics as sensible, useful,

and doableif you work at itand 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.

Five Strands of Mathematical Proficiency (NRC,

2002)

- Table Activity Evaluate Your Schools Math

Proficiency - As a group, review the National Research Council

Strands of Math Proficiency. - Which strand do you feel that your school /

curriculum does the best job of helping students

to attain proficiency? - Which strand do you feel that your school /

curriculum should put the greatest effort to

figure out how to help students to attain

proficiency? - Be prepared to share your results.

- Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. - Computing Carrying out mathematical procedures,

such as adding, subtracting, multiplying, and

dividing numbers flexibly, accurately,

efficiently, and appropriately. - Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately. - Reasoning Using logic to explain and justify a

solution to a problem or to extend from something

known to something less known. - Engaging Seeing mathematics as sensible, useful,

and doableif you work at itand being willing to

do the work.

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.

Assisting Students Struggling with Mathematics

RtI for Elementary Middle Schools 8

Recommendations

- Recommendation 1. Screen all students to identify

those at risk for potential mathematics

difficulties and provide interventions to

students identified as at risk - Recommendation 2. Instructional materials for

students receiving interventions should focus

intensely on in-depth treatment of whole numbers

in kindergarten through grade 5 and on rational

numbers in grades 4 through 8.

Assisting Students Struggling with Mathematics

RtI for Elementary Middle Schools 8

Recommendations (Cont.)

- Recommendation 3. Instruction during the

intervention should be explicit and systematic.

This includes providing models of proficient

problem solving, verbalization of thought

processes, guided practice, corrective feedback,

and frequent cumulative review - Recommendation 4. Interventions should include

instruction on solving word problems that is

based on common underlying structures.

Assisting Students Struggling with Mathematics

RtI for Elementary Middle Schools 8

Recommendations (Cont.)

- Recommendation 5. Intervention materials should

include opportunities for students to work with

visual representations of mathematical ideas and

interventionists should be proficient in the use

of visual representations of mathematical ideas - Recommendation 6. Interventions at all grade

levels should devote about 10 minutes in each

session to building fluent retrieval of basic

arithmetic facts

Assisting Students Struggling with Mathematics

RtI for Elementary Middle Schools 8

Recommendations (Cont.)

- Recommendation 7. Monitor the progress of

students receiving supplemental instruction and

other students who are at risk - Recommendation 8. Include motivational strategies

in tier 2 and tier 3 interventions.

How Do We Reach Low-Performing Math Students?

Instructional Recommendations

- Important elements of math instruction for

low-performing students - Providing teachers and students with data on

student performance - Using peers as tutors or instructional guides
- Providing clear, specific feedback to parents on

their childrens mathematics success - Using principles of explicit instruction in

teaching math concepts and procedures. p. 51

Source Baker, S., Gersten, R., Lee, D.

(2002).A synthesis of empirical research on

teaching mathematics to low-achieving students.

The Elementary School Journal, 103(1), 51-73..

Activity How Do We Reach Low-Performing

Students? p.5

- Review the handout on p. 5 of your packet and

consider each of the elements found to benefit

low-performing math students. - For each element, brainstorm ways that you could

promote this idea in your math classroom.

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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..

Math Challenge The student can not yet reliably

access an internal number-line of numbers 1-10.

What Does the Research Say?...

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, 234248.

What Are Stages of Number Sense? (Berch, 2005,

p. 336)

- Innate Number Sense. Children appear to possess

hard-wired ability (or neurological foundation

structures) in number sense. Childrens innate

capabilities appear also to be to represent

general amounts, not specific quantities. This

innate number sense seems to be characterized by

skills at estimation (approximate numerical

judgments) and a counting system that can be

described loosely as 1, 2, 3, 4, a lot. - Acquired Number Sense. Young students learn

through indirect and direct instruction to count

specific objects beyond four and to internalize a

number line as a mental representation of those

precise number values.

Source Berch, D. B. (2005). Making sense of

number sense Implications for children with

mathematical disabilities. Journal of Learning

Disabilities, 38, 333-339...

The Basic Number Line is as Familiar as a

Well-Known Place to People Who Have Mastered

Arithmetic Combinations

Internal Number-Line

- As students internalize the number-Line, they

are better able to perform mental arithmetic

(the manipulation of numbers and math operations

in their head).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24 25 26 27 28 29

Math Challenge The student can not yet reliably

access an internal number-line of numbers 1-10.

- Solution Use this strategy
- Building Number Sense Through a Counting Board

Game (Supplemental Packet)

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.

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.

Source Siegler, R. S. (2009). Improving the

numerical understanding of children from

low-income families. Child Development

Perspectives, 3(2), 118-124.

Building Number Sense Through a Counting Board

Game

- INTERVENTION STEPS A counting-board game

session lasts 12 to 15 minutes, with each game

within the session lasting 2-4 minutes. Here are

the steps - Introduce the Rules of the Game. The student is

told that he or she will attempt to beat another

player (either another student or the

interventionist). The student is then given a

penny or other small object to serve as a game

piece. The student is told that players takes

turns spinning the spinner (or, alternatively,

tossing the block) to learn how many spaces they

can move on the Great Number Line Race! board. - Each player then advances the game piece, moving

it forward through the numbered boxes of the

game-board to match the number "1" or "2"

selected in the spin or block toss.

Source Siegler, R. S. (2009). Improving the

numerical understanding of children from

low-income families. Child Development

Perspectives, 3(2), 118-124.

Building Number Sense Through a Counting Board

Game

- INTERVENTION STEPS A counting-board game

session lasts 12 to 15 minutes, with each game

within the session lasting 2-4 minutes. Here are

the steps - Introduce the Rules of the Game (cont.). When

advancing the game piece, the player must call

out the number of each numbered box as he or she

passes over it. For example, if the player has a

game piece on box 7 and spins a "2", that player

advances the game piece two spaces, while calling

out "8" and "9" (the names of the numbered boxes

that the game piece moves across during that

turn).

Source Siegler, R. S. (2009). Improving the

numerical understanding of children from

low-income families. Child Development

Perspectives, 3(2), 118-124.

Building Number Sense Through a Counting Board

Game

- INTERVENTION STEPS A counting-board game

session lasts 12 to 15 minutes, with each game

within the session lasting 2-4 minutes. Here are

the steps - Record Game Outcomes. At the conclusion of each

game, the interventionist records the winner

using the form found on the Great Number Line

Race! form. The session continues with additional

games being played for a total of 12-15 minutes. - Continue the Intervention Up to an Hour of

Cumulative Play. The counting-board game

continues until the student has accrued a total

of at least one hour of play across multiple

days. (The amount of cumulative play can be

calculated by adding up the daily time spent in

the game as recorded on the Great Number Line

Race! form.)

Source Siegler, R. S. (2009). Improving the

numerical understanding of children from

low-income families. Child Development

Perspectives, 3(2), 118-124.

Source Siegler, R. S. (2009). Improving the

numerical understanding of children from

low-income families. Child Development

Perspectives, 3(2), 118-124.

Math Challenge The student has not yet acquired

math facts.

What Does the Research Say?...

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 childrens

later math skills. Most children with math

learning difficulties are unable to master the

four basic operations before leaving elementary

school and, thus, need special attention to

acquire the skills. A category of interventions

is therefore aimed at the acquisition and

automatization of basic math skills.

Source Kroesbergen, E., Van Luit, J. E. H.

(2003). Mathematics interventions for children

with special educational needs. Remedial and

Special Education, 24, 97-114.

Big Ideas The Four Stages of Learning Can Be

Summed Up in the Instructional Hierarchy

(Supplemental Packet) (Haring et al., 1978)

- Student learning can be thought of as a

multi-stage process. The universal stages of

learning include - Acquisition The student is just acquiring the

skill. - Fluency The student can perform the skill but

must make that skill automatic. - Generalization The student must perform the

skill across situations or settings. - Adaptation The student confronts novel task

demands that require that the student adapt a

current skill to meet new requirements.

Source Haring, N.G., Lovitt, T.C., Eaton, M.D.,

Hansen, C.L. (1978). The fourth R Research in

the classroom. Columbus, OH Charles E. Merrill

Publishing Co.

Math Shortcuts Cognitive Energy- and Time-Savers

- Recently, some researchershave argued that

children can derive answers quickly and with

minimal cognitive effort by employing calculation

principles or shortcuts, such as using a known

number combination to derive an answer (2 2

4, so 2 3 5), relations among operations (6

4 10, so 10 -4 6) and so forth. This

approach to instruction is consonant with

recommendations by the National Research Council

(2001). Instruction along these lines may be much

more productive than rote drill without linkage

to counting strategy use. p. 301

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Students Who Understand Mathematical Concepts

Can Discover Their Own Shortcuts

- Students who learn with understanding have less

to learn because they see common patterns in

superficially different situations. If they

understand the general principle that the order

in which two numbers are multiplied doesnt

matter3 x 5 is the same as 5 x 3, for

examplethey 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.

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.

Math Short-Cuts Subtraction (Supplemental Packet)

- When zero is subtracted from the original number,

the answer is the original number. - When 1 is subtracted from the original number,

the answer is the next smaller number. - When the original number has the same number

subtracted from it, the answer is zero.

Source Miller, S.P., Strawser, S., Mercer,

C.D. (1996). Promoting strategic math performance

among students with learning disabilities. LD

Forum, 21(2), 34-40.

Math Short-Cuts Multiplication (Supplemental

Packet)

- When a number is multiplied by zero, the answer

is zero. - When a number is multiplied by 1, the answer is

the original number. - When a number is multiplied by 2, the answer is

equal to the number being added to itself. - The order of the numbers in a multiplication

problem does not affect the answer.

Source Miller, S.P., Strawser, S., Mercer,

C.D. (1996). Promoting strategic math performance

among students with learning disabilities. LD

Forum, 21(2), 34-40.

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.

Math Challenge The student has not yet acquired

math facts.

- Solution Use these strategies
- Strategic Number Counting Instruction

(Supplemental Packet) - Incremental Rehearsal
- Cover-Copy-Compare
- Peer Tutoring in Math Computation with

Constant Time Delay

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.

Strategic Number Counting Instruction

- MATERIALS
- Number-line (attached)
- Number combination (math fact) flash cards for

basic addition and subtraction - Strategic Number Counting Instruction Score Sheet

Source Fuchs, L. S., Powell, S. R., Seethaler,

P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D.,

Hamlett, C. L. (2009). The effects of strategic

counting instruction, with and without deliberate

practice, on number combination skill among

students with mathematics difficulties. Learning

and Individual Differences 20(2), 89-100.

Strategic Number Counting Instruction

- PREPARATION The tutor trains the student to use

these two counting strategies for addition and

subtraction - ADDITION The student is given a copy of the

number-line. When presented with a two-addend

addition problem, the student is taught to start

with the larger of the two addends and to 'count

up' by the amount of the smaller addend to arrive

at the answer to the problem. E..g., 3 5 ___

Source Fuchs, L. S., Powell, S. R., Seethaler,

P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D.,

Hamlett, C. L. (2009). The effects of strategic

counting instruction, with and without deliberate

practice, on number combination skill among

students with mathematics difficulties. Learning

and Individual Differences 20(2), 89-100.

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 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 ___

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.

Strategic Number Counting Instruction

- INTERVENTION STEPS For each tutoring session,

the tutor follows these steps - 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.

Strategic Number Counting Instruction

- INTERVENTION STEPS For each tutoring session,

the tutor follows these steps - 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.

Strategic Number Counting Instruction

- INTERVENTION STEPS For each tutoring session,

the tutor follows these steps - 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

count-up 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.

Strategic Number Counting Instruction

- INTERVENTION STEPS For each tutoring session,

the tutor follows these steps - 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.

Strategic Number Counting Instruction

- INTERVENTION STEPS For each tutoring session,

the tutor follows these steps - 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.

Strategic Number Counting Instruction Score Sheet

Acquisition Stage Math Review Incremental

Rehearsal of Math Facts

Step 1 The tutor writes down on a series of

index cards the math facts that the student needs

to learn. The problems are written without the

answers.

Math Review Incremental Rehearsal of Math Facts

KNOWN Facts

UNKNOWN Facts

Step 2 The tutor reviews the math fact cards

with the student. Any card that the student can

answer within 2 seconds is sorted into the

KNOWN pile. Any card that the student cannot

answer within two secondsor answers

incorrectlyis sorted into the UNKNOWN pile.

Math Review Incremental Rehearsal of Math Facts

Math Review Incremental Rehearsal of Math Facts

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.

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 - From any math operations page, select the

computation target. - 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.

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 - From any math operations page, select a

computation skill for the CCC worksheet. - Next, set the Number of Columns setting to 1.
- Then set the Number of Rows setting to the

number of CCC problems that you would like the

student to complete. - Click the Single-Skill Computation Probe

button. - Print off only the answer keyand use it as your

students 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.

Peer Tutoring in Math Computation with Constant

Time Delay pp. 20-26

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.

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.

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.

Peer Tutoring in Math Computation with Constant

Time Delay

- Select Student Participants (Cont.). Students

being considered 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 participatejust 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).

Peer Tutoring in Math Computation Teacher

Nomination Form

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.

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 tutees 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

tutees progress-monitoring chart.

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 progress-monitoring

components of the math peer tutoring.

Peer Tutoring in Math Computation Intervention

Integrity Sheet (Part 1 Tutoring Activity)

Peer Tutoring in Math Computation Intervention

Integrity Sheet (Part 2 Progress-Monitoring)

Peer Tutoring in Math Computation Score Sheet

Math Challenge The student has acquired math

computation skills but is not yet fluent.

What Does the Research Say?...

Benefits of Automaticity of Arithmetic

Combinations (Gersten, Jordan, Flojo, 2005)

- There is a strong correlation between poor

retrieval of arithmetic combinations (math

facts) and global math delays - Automatic recall of arithmetic combinations frees

up student cognitive capacity to allow for

understanding of higher-level problem-solving - By internalizing numbers as mental constructs,

students can manipulate those numbers in their

head, allowing for the intuitive understanding of

arithmetic properties, such as associative

property and commutative property

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Associative Property

- within an expression containing two or more of

the same associative operators in a row, the

order of operations does not matter as long as

the sequence of the operands is not changed - Example
- (23)510
- 2(35)10

Source Associativity. Wikipedia. Retrieved

September 5, 2007, from http//en.wikipedia.org/wi

ki/Associative

Commutative Property

- the ability to change the order of something

without changing the end result. - Example
- 23510
- 25310

Source Associativity. Wikipedia. Retrieved

September 5, 2007, from http//en.wikipedia.org/wi

ki/Commutative

How much is 3 8? Strategies to Solve

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Math Challenge The student has acquired math

computation skills but is not yet fluent.

- Solution Use these strategies
- Explicit Time Drills
- Self-Administered Arithmetic Combination Drills

With Performance Self-Monitoring Incentives

Explicit Time Drills p. 25 Math Computational

Fluency-Building Intervention

- Explicit time-drills are a method to boost

students rate of responding on math-fact

worksheets. - The teacher hands out the worksheet. Students

are told that they will have 3 minutes to work on

problems on the sheet. The teacher starts the

stop watch and tells the students to start work.

At the end of the first minute in the 3-minute

span, the teacher calls time, stops the

stopwatch, and tells the students to underline

the last number written and to put their pencils

in the air. Then students are told to resume work

and the teacher restarts the stopwatch. This

process is repeated at the end of minutes 2 and

3. At the conclusion of the 3 minutes, the

teacher collects the student worksheets.

Source Rhymer, K. N., Skinner, C. H., Jackson,

S., McNeill, S., Smith, T., Jackson, B. (2002).

The 1-minute explicit timing intervention The

influence of mathematics problem difficulty.

Journal of Instructional Psychology, 29(4),

305-311.

Fluency Stage Math Computation p. 30 Math

Computation Increase Accuracy and

Productivity Rates Via Self-Monitoring and

Performance Feedback

- The student is given a math computation worksheet

of a specific problem type, along with an answer

key Academic Opportunity to Respond. - The student consults his or her performance chart

and notes previous performance. The student is

encouraged to try to beat his or her most

recent score. - The student is given a pre-selected amount of

time (e.g., 5 minutes) to complete as many

problems as possible. The student sets a timer

and works on the computation sheet until the

timer rings. Active Student Responding - The student checks his or her work, giving credit

for each correct digit (digit of correct value

appearing in the correct place-position in the

answer). Performance Feedback - The student records the days score of TOTAL

number of correct digits on his or her personal

performance chart. - The student receives praise or a reward if he or

she exceeds the most recently posted number of

correct digits.

Application of Learn Unit framework from

Heward, W.L. (1996). Three low-tech strategies

for increasing the frequency of active student

response during group instruction. In R. Gardner,

D. M.S ainato, J. O. Cooper, T. E. Heron, W. L.

Heward, J. W. Eshleman, T. A. Grossi (Eds.),

Behavior analysis in education Focus on

measurably superior instruction (pp.283-320).

Pacific Grove, CABrooks/Cole.

Self-Monitoring Performance Feedback Examples

of Student Worksheet and Answer Key

Worksheets created using Math Worksheet

Generator. Available online at http//www.interve

ntioncentral.org/htmdocs/tools/mathprobe/addsing.p

hp

Self-Monitoring Performance Feedback

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

(Supplemental Packet)

- DESCRIPTION The student monitors and records

her or his work production on math computation

worksheets on a daily basiswith a goal of

improving overall fluency (Maag, Reid, R.,

DiGangi, 1993). This intervention can be used

with a single student, a small group, or an

entire class.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- MATERIALS
- Student self-monitoring audio prompt Tape /

audio file with random tones or dial-style

kitchen timer - Math computation worksheets containing problems

targeted for increased fluency - Student Speed Check! recording form

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Speed Check! Form

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - Decides on the Length and Frequency of Each

Self-Monitoring Period. The instructor decides on

the length of session and frequency of the

student's self-monitoring intervention. NOTE One

good rule of thumb is to set aside at least 10

minutes per day for this or other interventions

to promote fluent student retrieval of math facts

(Gersten et al., 2009).

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - Selects a Math Computation Skill Target. The

instructor chooses one or more problem types that

are to appear in intervention worksheets. For

example, a teacher may select two math

computation problem-types for a student

Additiondouble-digit plus double-digit with

regrouping and Subtractiondouble-digit plus

double-digit with no regrouping.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - Creates Math Computation Worksheets. When the

teacher has chosen the problem types, he or she

makes up sufficient equivalent worksheets (with

the same number of problems and the same mix of

problem-types) to be used across the intervention

days. Each worksheet should have enough problems

to keep the student busy for the length of time

set aside for a self-monitoring intervention

session.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - Determines How Many Audio Prompts the Student

Will Receive. This intervention relies on student

self-monitoring triggered by audio prompts.

Therefore, the teacher must decide on a fixed

number of audio prompts the student is to receive

per session. NOTE On the attached Student Speed

Check! form, space is provided for the student to

record productivity for up to five audio prompts

per session.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - Selects a Method to Generate Random Audio

Prompts. Next, the teacher must decide on how to

generate the audio prompts (tones) that drive

this intervention. There are two possible

choices (A) The teacher can develop a tape or

audio file that has several random tones spread

across the time-span of the intervention session,

with the number of tones equaling the fixed

number of audio prompts selected for the

intervention. For example, the instructor may

develop a 10-minute tape with five tones randomly

sounding at 2 minutes, 3 minutes, 5 minutes, 7

minutes, and 10 minutes.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- Preparation To prepare for the intervention the

teacher - (B) The instructor may purchase a dial-type

kitchen timer. During the intervention period,

the instructor turns the dial to a randomly

selected number of minutes. When the timer

expires and chimes as a student audio prompt, the

teacher resets the timer to another random number

of minutes and repeats this process until the

intervention period is over.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- INTERVENTION STEPS Sessions of the productivity

self-monitoring intervention for math computation

include these steps - Student Set a Session Computation Goal. The

student looks up the total number of problems

completed on his or her most recent timed

worksheet and writes that figure into the 'Score

to Beat' section of the current day's Student

Speed Check! form.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- INTERVENTION STEPS Sessions of the productivity

self-monitoring intervention for math computation

include these steps - Teacher Set the Timer or Start the Tape. The

teacher directs the student to begin working on

the worksheet and either starts the tape with

tones spaced at random intervals or sets a

kitchen timer. If using a timer, the teacher

randomly sets the timer randomly to a specific

number of minutes. When the timer expires and

chimes as a student audio prompt, the teacher

resets the timer to another random number of

minutes and repeats this process until the

intervention period is over.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- INTERVENTION STEPS Sessions of the productivity

self-monitoring intervention for math computation

include these steps - Student At Each Tone, Record Problems

Completed. Whenever the student hears an audio

prompt or at the conclusion of the timed

intervention period, the student pauses to - circle the problem that he or she is currently

working on - count up the number of problems completed since

the previous tone (or in the case of the first

tone, the number of problems completed since

starting the worksheet) - record the number of completed problems next to

the appropriate tone interval on the attached

Student Speed Check! form.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Speed Check! Form

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- INTERVENTION STEPS Sessions of the productivity

self-monitoring intervention for math computation

include these steps - Teacher Announce the End of the Intervention

Period. The teacher announces that the

intervention period is over and that the student

should stop working on the worksheet.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Self-Monitoring of Productivity to

Increase Fluency on Math Computation Worksheets

- INTERVENTION STEPS Sessions of the productivity

self-monitoring intervention for math computation

include these steps - Student Tally Day's Performance. The student

adds up the problems completed at the

tone-intervals to give a productivity total for

the day. The student then compares the current

day's figure to that of the previous day to see

if he or she was able to beat the previous score.

If YES, the student receives praise from the

teacher if NO, the student receives

encouragement from the teacher.

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Student Speed Check! Form

Source Maag, J. W., Reid, R., DiGangi, S. A.

(1993). Differential effects of self-monitoring

attention, accuracy, and productivity. Journal of

Applied Behavior Analysis, 26, 329-344.

Math Challenge The student is often

inconsistent in performance on computation or

word problems and may make a variety of

hard-to-predict errors.

What Does the Research Say?...

Profile of Students With Significant Math

Difficulties p. 4

- Spatial organization. The student commits errors

such as misaligning numbers in columns in a

multiplication problem or confusing

directionality in a subtraction problem (and

subtracting the original numberminuendfrom the

figure to be subtracted (subtrahend). - Visual detail. The student misreads a

mathematical sign or leaves out a decimal or

dollar sign in the answer. - Procedural errors. The student skips or adds a

step in a computation sequence. Or the student

misapplies a learned rule from one arithmetic

procedure when completing another, different

arithmetic procedure. - Inability to shift psychological set. The

student does not shift from one operation type

(e.g., addition) to another (e.g.,

multiplication) when warranted. - Graphomotor. The students poor handwriting can

cause him or her to misread handwritten numbers,

leading to errors in computation. - Memory. The student fails to remember a specific

math fact needed to solve a problem. (The student

may KNOW the math fact but not be able to recall

it at point of performance.) - Judgment and reasoning. The student comes up with

solutions to problems that are clearly

unreasonable. However, the student is not able

adequately to evaluate those responses to gauge

whether they actually make sense in context.

Source Rourke, B. P. (1993). Arithmetic

disabilities, specific otherwise A

neuropsychological perspective. Journal of

Learning Disabilities, 26, 214-226.

Activity Profile of Math Difficulties p. 4

- Review the profile of students with significant

math difficulties that appears on p. 4 of your

handout. - For each item in the profile, discuss what

methods you might use to discover whether a

particular student experiences this difficulty.

Jot your ideas in the NOTES column.

Math Challenge The student is often

inconsistent in performance on computation or

word problems and may make a variety of

hard-to-predict errors.

- Solution Use this strategy
- Increase Student Math Success with Customized

Math Self- Correction Checklists (Supplemental

Packet)

Increase Student Math Success with Customized

Math Self-Correction Checklists

- DESCRIPTION The teacher analyzes a particular

student's pattern of errors commonly made when

solving a math algorithm (on either computation

or word problems) and develops a brief error

self-correction checklist unique to that student.

The student then uses this checklist to

self-monitorand when necessary correcthis or

her performance on math worksheets before turning

them in.

Sources Dunlap, L. K., Dunlap, G. (1989). A

self-monitoring package for teaching subtraction

with regrouping to students with learning

disabilities. Journal of Applied Behavior

Analysis, 229, 309-314. Uberti, H. Z.,

Mastropieri, M. A., Scruggs, T. E. (2004).

Check it off Individualizing a math algorithm

for students with disabilities via

self-monitoring checklists. Intervention in

School and Clinic, 39(5), 269-275.

Increase Student Math Success with Customized

Math Self-Correction Checklists

- MATERIALS
- Customized student math error self-correction

checklist - Worksheets or assignments containing math

problems matched to the error self-correction

checklist

Sources Dunlap, L. K., Dunlap, G. (1989). A

self-monitoring package for teaching subtraction

with regrouping to students with learning

disabilities. Journal of Applied Behavior

Analysis, 229, 309-314. Uberti, H. Z.,

Mastropieri, M. A., Scruggs, T. E. (2004).

Check it off Individualizing a math algorithm

for students with disabilities via

self-monitoring checklists. In