Table of Contents - Instructional...

Table of Contents Acknowledgments ....................................................................................................................................... 4

Chapter 1 ..................................................................................................................................................... 5

Introduction to Mathematics Learning Disabilities ................................................................................ 5

What is a Math Learning Disability? .................................................................................................... 5

Chapter 2 ..................................................................................................................................................... 8

Mathematics Learning Disability Subtype #1 – Number Sense ............................................................. 8

Number Sense and the Development of Number Sense Skills ............................................................... 8

Manifestation of a Number Sense Deficit in a Child with MLD ........................................................... 8

Chapter 3 ................................................................................................................................................... 11

Useful Interventions to Teach Number Sense ...................................................................................... 11

Interventions to Assist with Number Sense Deficits ............................................................................ 11

Number Sense: Promoting Basic Numeracy Skills through a Counting Game .............................. 11

Coach Card ......................................................................................................................................... 12

Number Sense: Promoting Basic Numeracy Skills through a Counting Game .................................. 15

Intervention Kit .................................................................................................................................. 15

Treatment Integrity Checklist: ........................................................................................................ 18

Baseline Data Form: ......................................................................................................................... 19

Progress Monitoring Form: ............................................................................................................. 20

.................................................................................................. 22

Chapter 4 ................................................................................................................................................... 23

Mathematics Learning Disability Subtype #2 – Counting Knowledge ................................................. 23

What is Counting Knowledge and how does it manifest in typical development? .............................. 23

Manifestation of a Counting Knowledge Deficit in a Child with MLD .............................................. 24

Assessing Children’s Counting Knowledge ........................................................................................ 24

Chapter 5 ................................................................................................................................................... 26

Useful Interventions to Teach Counting Knowledge ............................................................................ 26

Whole Number Foundations Level K .................................................................................................... 27

Coach Card ......................................................................................................................................... 27

Chapter 6 ................................................................................................................................................... 30

Mathematics Learning Disability Subtype #3 – Arithmetic .................................................................. 30

Manifestation of an Arithmetic Deficit in a Child with MLD ............................................................. 30

Arithmetic and the Development of Arithmetic Skills ......................................................................... 31

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Chapter 7 ................................................................................................................................................... 33

Useful Interventions to Teach Arithmetic ............................................................................................. 33

Underlying Neurocognitive Abilities .................................................................................................. 33

Interventions to Assist with Arithmetic Deficits .................................................................................. 34

Solve It! Say, Ask, Check. ...................................................................................................................... 35

Coach Card ......................................................................................................................................... 35

Solve It! Problem-Solving Skills ............................................................................................................ 42



Solve It! - Math Problem Solving Processes and Strategies Master Sheet ................................... 49

Sample Baseline Measure ................................................................................................................. 50

“Say, Ask, Check” Prompts Sheet ................................................................................................... 56

Folding-In Technique ............................................................................................................................ 58

Coach Card ......................................................................................................................................... 58

Folding-In Technique ............................................................................................................................ 61


Math-Facts SAFI: Student Checklist .............................................................................................. 66

Student Log: Mastered Math-Facts ................................................................................................ 68

Sample Baseline Measure ................................................................................................................. 70


Number Operations: Strategic Number Counting Instruction............................................................. 74

Coach Card ......................................................................................................................................... 74

Number Line Sheet ........................................................................................................................... 78

Strategic Number Counting Instruction Score Sheet .................................................................... 79

Concrete-Representational-Abstract (CRA) .......................................................................................... 80

Coach Card ......................................................................................................................................... 80

Concrete-Representational-Abstract CBM Probe Examples ....................................................... 83

Concrete-Representational-Abstract Examples ............................................................................. 84

Chapter 8 ................................................................................................................................................... 85

Useful Websites for Support in Mathematics ......................................................................................... 85

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Acknowledgments Thank you for taking the time to read this manual on interventions and related resources

for students, teachers, and parents encountering difficulties with learning mathematics. We hope

to provide you with valuable tools when researching other options to help both children and

adolescents in their educational development in the subject of mathematics. This booklet has

been divided into the following sections: mathematics learning disabilities, number sense

subtype, counting knowledge subtype, arithmetic subtype, and websites for educator and student

use. Within these sections there are also quick and effortless interventions, and intervention kits

that provide more in-depth information and resources. References are listed directly after each

section. If you have any additional questions, please feel free to contact the authors:

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Chapter 1

Introduction to Mathematics Learning Disabilities

When you typically think of learning disabilities the subjects that comes to mind are

reading and writing. Since most people struggle with mathematics, many people do not realize

there is a difference between having difficulties acquiring and using mathematical skills versus

students suffering from mathematics learning disabilities. Six perfect of the population has a

learning disability in mathematics (Mazzocco, 2007). Unfortunately, the identification and

treatment of mathematical disabilities have received less attention than problems associated with

reading and writing (Lerner & Johns, 2015).

What is a Math Learning Disability? A Mathematics Learning Disability (MLD) is a learning disability that manifests as

deficits in the ability to represent or process information in one or all of the mathematics

domains (Geary, 2004). When children score below the 10th percentile on standardized

mathematics achievement tests for at least two consecutive academic years they are categorized

as MLD (Flanagan & Alfonso, 2011). Heredity plays a huge role with the diagnosis of

MLD. Children of either parents or siblings with MLD are 10 times more likely to be diagnosed

with MLD than the general population (Flanagan & Alfonso, 2011). Individuals with MLD are

affected in all different areas such as information processing, dyscalculia and math anxiety, and

three separate subtypes which have an impact on how students learn mathematics.

Having difficulty in the area of information processing has a huge influence on how

successfully a student with an MLD will perform. These areas include problems in motor,

attention, memory, retrieval, visual-spatial processing, and auditory processing (Lerner & Johns,

2015). A student with motor issues may have problems with writing numbers as well as

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illegible, slow, and inaccurate responses (Lerner & Johns, 2015). This student can also have an

issue with writing numbers in small spaces. A student with attention problems may have

difficulty following the steps of mathematical calculations and following instructions in

mathematics (Lerner & Johns, 2015). Issues in memory and retrieval can lead to difficulty in

remembering math facts, recalling the sequence of steps, and remembering multiple steps in

word problems (Lerner & Johns, 2015). A student with difficulty in visual spatial processing

may have issues with a visual mathematical lesson and problems with aligning numbers (Lerner

& Johns, 2015). A student with a deficit in the area of mathematics auditory processing may

have issues remembering auditory arithmetic facts and “counting on” (Lerner & Johns, 2015).

Other specific MLD's are dyscalculia and math anxiety. Dyscalculia is one of the more

severe MLD that correlates with medical orientation. Dyscalculia can be defined as the lack of

ability to perform mathematical functions associated with neurological dysfunction (Lerner &

Johns, 2015). Students who have dyscalculia have extreme difficulties understanding basic

number concepts. The students may learn to memorize basic number facts, but they do not

understand the logic behind it (Lerner & Johns, 2015). By not understanding the logic behind

mathematics, it is difficult for these students to apply their knowledge to solving problems.

Math anxiety is an emotion-based reaction to mathematics that causes individuals to become

distressed when they confront math problems or when they take a math test (Lerner & Johns,

2015). The anxiety can develop from the fear of school failure and the loss of self-esteem.

Stress or anxiety can hit specific areas of the brain, which can block school performance of

students with mathematics disabilities (Lerner & Johns, 2015). This can interfere with their

abilities to transfer the mathematics knowledge they do have, and it becomes an obstacle when

they try to demonstrate their knowledge on tests (Lerner & Johns, 2015).

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There are three different subtypes of MLD discussed throughout this resource handbook.

These include number sense, counting knowledge, and arithmetic. Number sense refers to the

ability to think about quantity (Flanagan & Alfonso, 2011). Counting knowledge entails the

ability to count objects by numbers (Flanagan & Alfonso, 2011). Arithmetic uses both the

combination of number sense and counting skills to understand more abstract problem solving

questions (Flanagan & Alfonso, 2011). Further details will be discussed throughout this

intervention manual.

References

Flanagan, D.P., & Alfonso, V.C. (2011). Essentials of Specific Learning Disability Identification.

Hoboken, NJ: John Wiley & Sons, Inc.

Geary, D. C. (2004). Mathematics and learning disabilities. Journal of learning disabilities,

37(1), 4-15.

Lerner, J., & Johns, B. (2015). Learning Disabilities & Related Disabilities (13th ed.). Stamford,

CT: Cengage Learning.

Mazzocco, M. M. (2007). Defining and differentiating mathematical learning disabilities and

difficulties. In D. Berch & M. Mazzocco (Eds.), Why is math so hard for some children?

The nature and origins of mathematical learning difficulties and disabilities (pp. 29-48).

Baltimore, MD: Paul H. Brooks.

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Chapter 2

Mathematics Learning Disability Subtype #1 – Number Sense

Number Sense and the Development of Number Sense Skills Number Sense refers to the ability to understand the exact amount of small groups of

objects or symbols on paper, and being able to estimate the number of items in large groups

without counting. Geary (2009) reports that understanding the exact amount of items in a group

without counting them is part of the fundamental core knowledge of number sense. Number

sense is needed in order to carry out basic arithmetic. It is essential for students to be able to

readily and rapidly identify representations of numbers (Flanagan & Alfonso, 2011). Number

sense is the ability to understand what numbers mean and compare them to each other.

In the typical development of number sense, children are sensitive to differences between

groups of times. Infants as young as 6 months of age are able to discriminate between large

groups of objects. Subitizing is defined as judging the number of objects in a group quickly

without actually counting them. This can be done with 1-4 objects (Flanagan & Alfonso, 2011).

These skills provide the foundation for learning mathematics. Jordan. Glutting, and Ramineni

(2010) identified that core number sense competencies are predictive of later math achievement.

The way children make representations in their mind impacts their ability to accurately add and

subtract (Gersten & Chard, 2010).

Manifestation of a Number Sense Deficit in a Child with MLD Feifer (2016) identifies a Math Learning Disability (MLD) as a term that refers to

children who have below age and grade level math performance in the classroom. Children with

a MLD or persistent low achievement in mathematics (LA) have trouble subitizing and being

able to represent number quantities. More specifically, Dyscalculia is a term for students with

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specific math-related deficits. These deficits may include issues with language retrieval,

working memory and executive functioning. Basic number sense and concept development is

impacted with the impairment of executive functioning. Procedural Dyscalculia Subtype is when

there is an issue comprehending the syntax rules in sequencing numbers and counting numeric

information (Feifer, 2016). This impairs the development of number sense and retrieval of basic

math facts such as single digit addition and subtraction. Poor number sense is attributed to

Semantic Dyscalculia Subtype which impairs the brain’s ability to create symbolic relationships

of numbers causing trouble with number sense and spatial attention (Feifer, 2016).

References

Fiefer, S. (2016). The neuropsychology of mathematics: An introduction to the fam. Presentation

at the annual convention of School Psychologists, New Orleans, Louisiana.

Flanagan, D. R. & Alfonso, V. C. (2011). Essentials of specific learning disability identification.

Hoboken, NJ: John Wiley & Sons.

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.

Gersten, R., & Chord, D. (1999). Number sense: Rethinking arithmetic instruction for students

with mathematical disabilities. The Journal of Special Education, 33(1), 18-28.

Geary, D. C. (2010). Mathematical disabilities: Reflections on cognition, neuropsychological,

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and genetic components. Learning and Individual Differences, 20, 130-133.

Jordan, N. C., Glutting, J. & Ramineni, C. (2010). The importance of number sense to

mathematics achievement in first and third grades. Learning and Individual Differences,

20, 82-88.

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Chapter 3

Useful Interventions to Teach Number Sense

Interventions to Assist with Number Sense Deficits Feifer lists interventions to improve lower working memory (2016). Of these strategies,

attaching a number-line to a student’s desk and increasing number sense through games

involving dice, domino’s and cards were listed. The numbers allow for graphic representations

to help students learn distances between numbers (Flanagan & Alfonso, 2011). The primary step

in reducing math difficulties is developing an understanding of number sense and how it is a

basic skill needed to succeed in math (Gersten & Chard, 2010). Interventions such as “Number

Operations: Strategic Number Counting Instruction” and “Number Sense: Promoting Basic

Numeracy Skills through a Counting Game” work on improving number sense (Fuchs et al.

2009; Sieglar, 2009).

References

Fiefer, S. (2016). The neuropsychology of mathematics: An introduction to the fam. Presentation

at the annual convention of School Psychologists, New Orleans, Louisiana.

Gersten, R., & Chord, D. (1999). Number sense: Rethinking arithmetic instruction for students

with mathematical disabilities. The Journal of Special Education, 33(1), 18-28.

Sieglar, R. S. (2009). Improving numerical understanding of children from low-income families.

Child Development Perspectives, 3(2), 118-124.

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Number Sense: Promoting Basic Numeracy Skills through a Counting Game

Coach Card Description: This intervention involves engaging the student (in Kindergarten through second

grade) in playing a board game with numbers in the boxes to build number skills. These skills

include number identification, counting, estimating skills, and the ability to visualize and access

specific number values by using the number line board game.

Target Skills: The goal of the Number Sense Intervention is to increase early math fluency

skills, such as the ability to identify numbers 1-10, increase counting ability, and estimating the

distance from lower numbers to 10. Target skills include the ability to identify numbers 1-10,

increasing adding “1” and “2” to numbers and estimating the distance from lower numbers to 10.

Location: This intervention can be done in the classroom as a game, outside the classroom with

a teaching assistant or paraprofessional, as well as at home with parents or caregivers.

Materials:

• The Great Number Line Race!

• A spinner divided into two equal parts, labeled with a “1” and a “2”. (NOTE: if a spinner

is not available, a wooden block with three sides labeled “1” and three sides labeled “2”

can be used.

• Game pieces for all players.

Frequency: This intervention can be implemented as often as necessary to increase number

sense skills. The game lasts between 4-5 minutes and is played three to four times in one

session.

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Baseline Measures: Prior to beginning the intervention, the interventionist will document the

number of digits (1-10) that the student can accurately identify. The interventionist will circle

the digits that the student can identify and underline the digits that the student is unable to

identify. There must be 3-5 data points for baseline data, prior to implementing the intervention.

Progress Monitoring Measure: In order to progress monitor, the interventionist will collect at

least two data points per week, for 60 minutes of total play. The game can be repeated as often

as necessary. The interventionist will document (on the progress monitoring form) the number

of digits the student can identify, circle the digits that the student can identify, and underline the

digits that the student is unable to identify.

Directions:

1. Tell the student that he/she will be playing a counting game and the goal is to beat their

opponent.

2. Have the players pick their game pieces.

3. Explain that each player takes a turn by spinning the spinner (or tossing the block) to

identify how many spaces they move their game piece on The Great Number Line Race!

board.

4. The student moves the game piece forward by either “1” or “2” (depending on their spin

or toss) through the numbered boxes on the board.

5. While moving the game pieces forward, the player must call out the number of each box

as they pass over it. For example, if a player is on box 4 and spins a “2”; they move two

spaces, calling out “5” and “6” as their piece lands on that box.

6. The player who reaches the “10” box wins.

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7. At the end of the game, the interventionist records the winner using The Great Number

Line Race! form. The session continues by repeating game play until time is up (after

12- 15 minutes).

8. The board game continues until the student has a played the game for a total of at least 60

minutes across multiple days. The amount of play can be added using the daily time spent

record on The Great Number Line Race! form.

References

Siegler, R. S. (2009). Improving the numerical understanding of children from low-income

families. Child Development Perspectives, 3(2), 118-124.

Wright, J. (n.d.). Number Sense: Promoting Basic Numeracy Skills through a Counting Board

Game. Retrieved March 10, 2016, from

http://www.interventioncentral.org/academic-interventions/math/number-sense-

promoting-basic-numeracy-skills-through-counting-board-ga-0

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Number Sense: Promoting Basic Numeracy Skills through a Counting Game

Intervention Kit Appropriate Grade Level: Early Elementary School grades, Kindergarten through 2nd grade.

Brief Description: This intervention involves engaging the student in playing a board game with

numbers in the boxes to build number skills. These skills include number identification,

counting; estimating skills, and the ability to visualize and access specific number values by

using the number line board game.

Home Implementation: This activity can also be a fun game that parents can play with their

children at home. This can be assigned as homework to play a few rounds and record the scores

to show to the teacher the next day. Utilizing this intervention in this manner can create

consistency between school and home and help parents take a role in their child’s learning in a

fun and educational way.

Intervention Goal: The goal of the Number Sense Intervention is to increase early math fluency

skills, such as the ability to identify numbers 1-10 and increase counting ability and estimating

the distance from lower numbers to 10.

Materials:

• The Great Number Line Race! (Attached)

• A spinner divided into two equal parts, labeled with a “1” and a “2”. (NOTE: if a

spinner is not available, a wooden block with three sides labeled “1” and three sides

labeled “2” can be used (provided in kit)

• Game pieces for all players (options provided in this kit)

Time: 12-15 minutes per session (can play multiple rounds)

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Group Size: 1-2 players

Script:

1. Tell the student that he/she will be playing a counting game and the goal is to beat their

opponent.

2. Have the players pick their game pieces.

3. Explain that each player takes a turn by spinning the spinner (or tossing the block) to

identify how many spaces they move their game piece on The Great Number Line Race!

board.

4. The student moves the game piece forward by either “1” or “2” (depending on their spin

or toss) through the numbered boxes on the board.

5. While moving the game pieces forward, the player must call out the number of each box

as they pass over it. For example, if a player is on box 4 and spins a “2”; they move two

spaces, calling out “5” and “6” as their piece lands on that box.

6. The player who reaches the “10” box wins.

7. At the end of the game, the interventionist records the winner using The Great Number

Line Race! form. The session continues by repeating game play until time is up (after 12-

15 minutes).

8. The board game continues until the student has a played the game for a total of at least 60

minutes across multiple days. The amount of play can be added using the daily time spent

record on The Great Number Line Race! form.

Baseline Measures: Prior to beginning the intervention, the interventionist will document the

number of digits (1-10) that the student can accurately identify. The interventionist will circle

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the digits that the student can identify and underline the digits that the student is unable to

identify. There must be 3-5 data points for baseline data, prior to implementing the intervention.

Progress Monitoring Measure: In order to progress monitor, the interventionist will collect at

least two data points per week, for 60 minutes of play. The game can be repeated as often as

necessary. The interventionist will document (on the progress monitoring form) the number of

digits the student can identify, circle the digits that the student can identify, and underline the

digits that the student is unable to identify.

References

Siegler, R. S. (2009). Improving the numerical understanding of children from low-income

families. Child Development Perspectives, 3(2), 118-124.

Wright, J. (n.d.). Number Sense: Promoting Basic Numeracy Skills through a Counting Board

Game. Retrieved March 10, 2016, from

http://www.interventioncentral.org/academic-interventions/math/number-sense-

promoting-basic-numeracy-skills-through-counting-board-ga-0

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Treatment Integrity Checklist: Treatment Integrity Checklist

Student: ___________________________

Teacher: ___________________________

Was the interventionist sufficiently prepared to administer the

intervention? YES NO

Did the interventionist use the correct forms and attached documents for

the intervention? YES NO

Was baseline data collected using the baseline data form? YES NO

Did the interventionist clearly introduce and explain the rules of the game

to the student? YES NO

Did the interventionist use the progress monitoring form to track

progress? YES NO

Was the intervention continued for 60 minutes of play in total? YES NO

Did the interventionist record the winner using the Great Number Line

Race Form? YES NO

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Baseline Data Form: Circle the digits the student can identify. Underline the digits the student cannot identify.

Date: ___________

3 8 5 10 7 2 9 1 4 6

Date: ___________

9 8 1 6 3 5 2 7 4 10

Date: ___________

10 7 8 4 2 3 6 7 9 1

Date: ___________

9 1 4 6 5 8 2 7 3 10

Date: ___________

6 4 8 3 5 10 9 2 7 1

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Progress Monitoring Form: Circle the digits the student can identify. Underline the digits the student cannot identify.

Date: ___________

2 4 10 8 3 5 6 7 9 1

Date: ___________

7 9 2 8 5 1 4 3 10 6

Date: ___________

4 8 6 9 1 3 5 7 10 2

Date: ___________

5 8 2 3 7 1 6 4 9 10

Date: ___________

5 8 2 3 7 1 6 4 9 10

Date: ___________

7 3 4 10 8 5 9 6 2 1

Date: ___________

3 9 6 4 5 1 2 7 10 8

Date: ___________

3 1 7 5 9 4 6 10 8 2

Date: ___________

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1 3 6 2 5 10 9 7 8 4

Date: ___________

2 7 3 4 6 1 8 10 5 9

Date: ___________

3 2 5 4 6 7 1 9 8 10

Date: ___________

8 4 1 9 3 2 5 6 7 10

Date: ___________

5 3 1 10 8 4 6 7 9 2

Date: ___________

4 1 2 5 10 3 6 9 7 8

Date: ___________

4 5 3 8 7 6 1 9 2 10

Date: ___________

8 7 3 5 1 4 10 2 6 9

Date: ___________

2 9 6 4 8 5 10 3 7 1

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Date: ____________________ Start Time: ____:____ End Time: ____:____ Directions: Mark the winner for each game with an ‘X’ in the table below.

Players Game 1 Game 2 Game 3 Game 4 Game 5 Game 6 Game 7

1:________

2:________

1 2 3 4 5 6 7 8 9 10 Start

Chapter 4

Mathematics Learning Disability Subtype #2 – Counting Knowledge

What is Counting Knowledge and how does it manifest in typical development? Counting knowledge involves five core principles such as one-to-one correspondence,

stable order, cardinality, abstraction, and order irrelevance. According to Geary (2004), one-to-

one correspondence is when only one word tag is assigned to each counted object (i.e., “one”,

“two” etc.). Stable order is when the order of the word tags is the same across counted sets,

while cardinality is when the values of the final word tag represents the total amount of items in

the counted set. Abstraction involves any set of objects that can be collected together and

counted, and order irrelevance is when items in a set are tagged in any sequence (Geary, 2004).

These five principles provide the foundation for children to understand and learn the rules of

counting. Additionally, there are several unessential features of counting which elaborate on

these principles (Geary, 2004). Some of these unessential features are standard direction and

adjacency. Standard direction is when the counting of a set of objects starts at one of the

endpoints and adjacency is the incorrect belief that items must be counted consecutively from

one neighboring item to the next (Geary, 2004). According to Flanagan and Alfonso (2011),

these unessential features are common errors that children make when counting.

These five principles of counting develop during the preschool years and mature during

the early elementary-school years. During this time, children also observe others’ counting

which allows them to make inferences about the basic characteristics of counting (Flanagan &

Alfonso, 2011). Typically by the age of 5, many children understand the essential features of

counting. However, this understanding can be rigid and immature and is influenced by the

observation of standard counting procedures (Geary, 2004).

Manifestation of a Counting Knowledge Deficit in a Child with MLD According to Fuchs (2010), children with a mathematics learning disability (MLD)

experience greater difficulty with counting. Children who have MLD in elementary school

understand most of the basic principles for counting, but have difficulty when counting deviates

from the standard left to right counting of neighboring items (Flanagan & Alfonso, 2011). These

children also have difficulty detecting errors when the first object, in an array of objects, is

counted twice (i.e. first item tagged as “one” and “two”). However, they are able to pick up on

this error when it occurs with the last item in the set. When the double counting occurs on the

first object, the child has trouble holding onto this counting error in working memory during the

count. In other words, the child forgets that the error occurred. This forgetting can pose

problems for children who are learning to use counting in order to solve an arithmetic problem

(Geary, 2004). Even though children with MLD have a difficult time understanding some of the

basic principles of counting, such as order irrelevance and adjacency, they do understand the

concepts of stable order and cardinality.

Assessing Children’s Counting Knowledge A common way to assess children’s counting knowledge is to ask them to watch a puppet

count a set of objects. The puppet will sometimes count the objects correctly and sometimes the

puppet will violate the counting principles or unessential features of counting (Flanagan &

Alfonso, 2011). The child must determine if the puppet’s count was correct or incorrect. With

this method, the puppet performed the procedural aspect of counting and the child’s response is

based on his/her understanding of the counting principles (Flanagan & Alfonso, 2011).

References

Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities,

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37(1), 4-15.






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

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Chapter 5

Useful Interventions to Teach Counting Knowledge

Several important skills can be taught in order to help a child with an MLD improve their

counting knowledge. As previously mentioned, counting knowledge involves skills such as one-

to-one correspondence, stable order, and cardinality (Flanagan & Alfonso, 2011). These skills

can be worked on and taught through the use of interventions. Below can be found a Whole

Number Foundations coach card. This intervention has 50 lessons that can be used to help

students work on counting from left to right, how to use number lines, count items in numerical

order, and become more fluent in counting in general. The more repetition and direct instruction

a child experiences the better their counting knowledge.

References



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Whole Number Foundations Level K

Coach Card Description: Counting knowledge consists of skills related to assigning just one number to a

counted object, counting from left to right, and the ability to state the total number of items

counted in a group of several (Flanagan & Alfonso, 2011). Through a 50-lesson program, Whole

Number Foundations Level K (WNF-K) aims to help struggling kindergarteners develop

procedural fluency with an understanding of whole number concepts (Clarke, Doabler,

Smolkowski, Baker, Fien, & Strand, 2016).

Target Skills:

• Conceptual Understanding: A basic knowledge of number sense is built through the use

of mathematical models such as number lines, ten-frames, finger models, and base-ten

models

• Procedural Fluency and Automaticity: A skill developed through the systematic practice

and review throughout the lessons. Checks for mastery occur at the end of each activity.

• Vocabulary and Discourse: Key mathematical vocabulary is taught and reviewed

throughout the scripted lessons.

Location: Small group instruction (4-5 students) in the classroom at the kindergarten level.

Materials:

The materials are offered in two different formats once the program is purchased. The

teacher books, worksheets, and other materials can be downloaded in PDF format and printed out

as needed, or pre-printed copies can be ordered.

• 50 lesson plans found in two “Teacher Books”

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• Math practice worksheets

• Program Support materials (number cards, place value mats, ten-frames, and number

chart)

• Manipulatives (linking cubes, teddy bear counters, number line, dice, place value models)

Frequency: Three to five times per week for 20 minutes each lesson.

Progress Monitoring: Math checkpoints are incorporated into the program. These are

individually administered progress monitoring assessments that are given to students every 5th

lesson through to lesson 25. These assessments are then given at lessons 34, 42, and 50. Each

lesson also incorporates a check for mastery at the end of each activity to ensure student

understanding.

Directions: Sample Lesson Order Numerals 1-5

1. Give each child a set of numeral cards 1-5.

2. Tell children that they will arrange their cards in order from 1-5

a) “I’m giving each of you a set of numeral cards 1-5 to put in order just like the

number line.”

3. Be sure the children start at the left with numeral 1. If children start at the right, show

them where to begin. You may wish to place a small circle at the left to prompt the

children

4. If an error occurs, for example a child places numeral 5 immediately after numeral 2,

correct with a counting strategy: “Listen to me count. One. Two. What comes next?

Three. Your turn. One. Two. What comes next? (“three”) Yes, three comes next. Fix

your number line so numeral 3 comes after numeral 2.”

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5. When the children have the correct sequence in front of them, confirm by having the

children touch and name numerals 1-5

a) “Let’s touch and count the numerals starting with 1.”

6. If any child made an error, have the children mix up their cards and repeat the task.

7. Congratulate the children for being able to make their own number line. The children

should leave their cards in the correct order for the next task.

References

Clarke, B., Doabler, C. T., Smolkowski, K., Baker, S. K., Fien, H., & Cary, M. S. (2016).

Examining the efficacy of a tier 2 kindergarten mathematics intervention. Journal

of Learning Disabilities, 49(2), 152-165. Doi: 10.1177/0022219414538514

Clarke, B., Doabler, C. T., Smolkowski, K., Kurtz-Nelson, E., Fien, H., & Baker, S.

(2015). Testing the immediate and long-term efficacy of a tier 2 kindergarten

mathematics intervention (Technical Report 1502). Eugene, OR: University of

Oregon.

Center on Teaching and Learning (2016). Whole Number Foundations Level K. Moving

Up! Mathematics. UO Dibels Data System: University of Oregon.

https://dibels.uoregon.edu/market/movingup/kfoundation#



29 | P a g e

https://dibels.uoregon.edu/market/movingup/kfoundation

Chapter 6

Mathematics Learning Disability Subtype #3 – Arithmetic

Manifestation of an Arithmetic Deficit in a Child with MLD Fletcher, Lyon and Fuchs (2006) indicated that there are two hypothesized subtypes in

arithmetic in which students display difficulty. Semantic Memory and Procedural Competence

are two arithmetic areas in which common errors are made by students with MLD (Flanagan &

Alfonso, 2011). The semantic memory area involves difficulties in learning, representing, and

retrieving mathematics facts (Fletcher, Lyon, & Fuchs, 2006). These difficulties manifest

themselves in a few different ways. A child with a difficulty in the area of semantic memory

with show slow, inaccurate, or inconsistent computational problem solving skills (Fletcher,

Lyon, & Fuchs, 2006). These children will have difficulty with learning basic arithmetic facts or

retrieving facts from long-term semantic memory (Flanagan & Alfonso, 2011). Children with

MLD will have a hard time learning basic arithmetic combinations and have difficulty

memorizing the answers to these combinations (Flanagan & Alfonso, 2011). These children will

revert back to simple counting strategies to help them solve problems, which can result in

inaccurate answers and incorrect calculations (Fletcher, Lyon, & Fuchs, 2006).

Students’ difficulty in the area of arithmetic can manifest in a breakdown in their

procedural competence. Procedural competence is essentially the ability to remember and utilize

the necessary procedures in order to solve an arithmetic problem (Fletcher, Lyon, & Fuchs,

2006). This involves difficulties with the concepts that underlie different math procedures, such

as the problem-solving process, and the base 10-system (Fletcher, Lyon, & Fuchs, 2006).

Children with a difficulty in this area typically revert to utilizing more simplistic procedures that

are characteristic of younger children (Fletcher, Lyon, & Fuchs, 2006). This would include

30 | P a g e

finger counting and sum-counting strategies (Flanagan & Alfonso, 2011). For example, when a

child is given a multi-step arithmetic problem, students with this deficit will typically make

errors in the area of misalignment of numbers, carrying and borrowing from the wrong column,

and subtracting when they should have added (Flanagan & Alfonso, 2011). Students with this

deficit use developmentally immature strategies that do not suit the given problem and make

calculation mistakes due to an inability to follow the prescribe procedure for problem solving

(Fletcher, Lyon, & Fuchs, 2006).

Arithmetic and the Development of Arithmetic Skills According to Flanagan and Alfonso (2011) one of the consistently found deficits for

students with a Mathematics Learning Disability is in the area of procedural competence, or

arithmetic. Arithmetic is a mathematical skill that utilizes various cognitive processes, as well as

incorporates number sense and counting knowledge abilities for the purpose of solving a

mathematical problem (Bartelet, Ansari, Vaessen, & Blomert, 2014). Counting strategies are

skills learned as early as kindergarten which can help a student work towards solving simple

addition or subtraction problem (Flanagan & Alfonso, 2011). These strategies can include finger

counting or verbal counting which can help a child combine two addends in a problem in order

to find the sum (Flanagan & Alfonso, 2011). The use of these counting strategies early on can

lead to the development of memory representations of basic facts (Flanagan & Alfonso, 2011).

Essentially, these are skills that are formed in the long-term memory and can be retrieved when

the child is completing a math problem (Flanagan & Alfonso, 2011). Flanagan and Alfonso

(2011) indicate that this process becomes known as direct retrieval of arithmetic facts. The

complexity in the development of arithmetic skills is in the fact that it is not a linear progression,

meaning children do not develop skills and strategies from the less sophisticated to the more

31 | P a g e

sophisticated one right after the other (Flanagan & Alfonso, 2011). These skills and strategies

are developed through direct instruction and practice. A child who does not properly learn a

strategy or recall a learned strategy will not begin to use it frequently. This child will then

heavily rely on the less sophisticated method and begin to struggle in the area of arithmetic, and,

ultimately, their mathematic skills will suffer (Flanagan & Alfonso, 2011).

References

Bartelet, D., Ansari, D., Vaessen, A., & Blomert, L. (2014). Cognitive subtypes of mathematics

learning difficulties in primary education. Research in Developmental Disabilities, 35,

pp. 657-670.

Fletcher, J. M., Lyon, G. R., & Fuchs, L. S. (2006). Learning Disabilities. New York, US: The \

Guilford Press.



32 | P a g e

Chapter 7

Useful Interventions to Teach Arithmetic

Underlying Neurocognitive Abilities

In order to appropriately intervene it is important to be aware of the potential underlying

causes to the child’s deficits. There are a few separate neurocognitive skills that contribute to a

child’s ability to comprehend mathematical concepts and accurately solve problems in math.

Working memory is one area that plays a key role in a student’s ability to succeed in

mathematics (Flanagan & Alfonso, 2011). Working memory’s main function is to hold

information in the mind while engaging in other mental processes at the same time (Flanagan &

Alfonso, 2011). It has been found that children with MLD display poor working memory skills

compared to typically achieving same-aged peers (Mabbott & Bisanz, 2008). It has also been

found that performance in the area of arithmetic has been linked to working memory ability

(Mabbott & Bisanz, 2008). The ability to hold onto recently presented information and

manipulate that information to solve a problem is required in the area of arithmetic with such

tasks as simple calculation, solving word problems, and using complex equations. It has been

found that working memory predicts a student’s word problem solution accuracy in children with

MLD (Mabbott & Bisanz, 2008).

Speed of processing is another area that is affected in students with MLD. It has been

found that speed of processing in children with MLD is slower than that of children who are

typically achieving (Flanagan & Alfonso, 2011). This slower processing speed can result in

performance deficits in several areas such as identifying and naming numbers and automaticity

in calculating simple addition and subtraction problems (Flanagan & Alfonso, 2011).

33 | P a g e

Interventions to Assist with Arithmetic Deficits Interventions can be found in this section pertaining to improving a student’s word

problem solving abilities as well as their understanding of basic math computations. The “Solve

It!” intervention helps students routinize their math problem solving skills in order to support

their working memory skills and help them more easily solve math equations and word

problems. The “Folding-In” intervention helps students build automaticity in the area of simple

math facts which can help if there is a weakness in their processing speed. Having an idea of the

underlying neurocognitive causes to a child’s deficit in the area of arithmetic can help shape the

intervention that is used to assist the child.

References



Mabbott, D. J., & Bisanz, J. (2008). Computational skills, working memory, and conceptual

knowledge in older children with mathematics learning disabilities. Journal of Learning

Disabilities, 41 (1).

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Solve It! Say, Ask, Check.

Coach Card Description: Solve It! is a research-based instructional program that was developed for students

who display difficulty in solving mathematical word problems (Montague, Warger, & Morgan,

2000). This intervention focuses on helping struggling students learn and use cognitive

processes and self-regulation strategies to improve their word problem solving abilities in

mathematics (Montague, Warger, & Morgan, 2000).

Target Skills: Solve It! is a program designed to provide students with the cognitive tools and

self-regulation strategies that are necessary to achieve success in mathematical problem solving.

Location: This intervention can be implemented in whole class or small group instruction in the

classroom for students in the upper-elementary grades on through high school.

Materials:

• “Say, Ask, Check” Prompt Sheet (see kit)

• Master Chart (see kit)

• Baseline Measure Assessments (see kit)

• Treatment Integrity Checklist (see kit)

• Progress Monitoring Measure (see kit)

Frequency: The Solve It! program consists of 8 scripted lessons that are taught by teachers over

the course of a few weeks.

Progress Monitoring: At the beginning of the program the student is given a 5 question

criterion-referenced test which consists of one-, two-, or three-step word problems. The pretest

is given to determine a baseline score, and then students will be tested throughout the

35 | P a g e

instructional program using the same format in order to ensure continual progress monitoring

(Montague, Warger, & Morgan, 2000).

Directions / Sequence of Lessons:

1. Lesson 1: Students are engaged in a discussion about the importance of math problem

solving. The master chart is presented to students which incorporates the 7 steps to the

problem-solving model followed in the Solve It! program. As the teacher instructs on

each step of the model there is an emphasis placed on the techniques of SAY, ASK, and

CHECK. Students practice reciting the strategies by reading through the chart

individually and then in a choral reading manner.

2. Lesson 2: Students learn to recite the processes and strategies from memory. The teacher

demonstrates mathematical problem solving using the methods outlined on the chart and

emphasizing the use of SAY, ASK, and CHECK.

3. Lesson 3: Students are tested using a progress monitoring measure to determine if they

have mastered the cognitive processes and skills taught in the problem-solving model and

SAY, ASK, CHECK. During this lesson the teacher also leads the group in a recitation

of the processes and the SAY, ASK, CHECK strategies. The students are then asked to

solve a practice problem independently and are instructed to think aloud and verbalize the

processes and strategies.

4. Lesson 4: The lesson begins with another test of students’ mastery of the processes to

track progress. Students are then given problems to solve individually, while the teacher

assists them in thinking aloud and following the learned cognitive processes. At the end

of the lesson a student volunteer is asked to model the proper problem-solving method on

the board for the whole group or class.

36 | P a g e

5. Lesson 5: The lesson begins with another test of mastery of the processes in order to track

progress. Students are then paired off to solve problems. Students and teachers will take

turns modeling correct solutions for the class after all pairs have completed their

problems.

6. Lesson 6: The group moves to lesson 6 only if all students have shown mastery of the

cognitive strategies (i.e. ability to recite the cognitive processes with 100% accuracy, and

ability to work through math problems with relative comfort and confidence). If this

criteria has not been met then lessons 3-4 are repeated. If the criteria has been met then

lesson 6 involves providing students with 10 practice problems to solve. Modeling of the

correct procedures is done by both the teacher and the students during this lesson.

7. Lesson 7: During this lesson students are required to solve an entire set of 10 problems

before modeling is used. Students are encouraged to ask questions and engage in

discussion about the solutions to the set of problems.

8. Lesson 8: Students are given a “progress check”. Students are allowed to either grade

their own papers or exchange their work with a peer for grading. Emphasis during this

lesson is placed on the fact that learning effective problem solving is an ongoing process

and students will continue to improve over time.

Script:

1. Teacher introduces the activity by writing a word problem on the board in front of the

class. Teacher says, “Now watch me as I say everything I am thinking and doing as I

solve this problem”, then reads the word problem out loud "Mrs. Hilt read 21 books.

Each book had exactly 2,010 words in it. She sold five of her books for $4.95 each. How

many words did Mrs. Hilt read?”

37 | P a g e

2. The teacher then explains, “First, I am going to read the problem for understanding

a. SAY: “Read the problem, okay. (Reads problem out loud one more time). If I

don’t understand it, I will read it again. Huh, I think I need to try reading it one

more time, (reads problem aloud once more)”

b. ASK: “Have I read and understood the problem? I think so.”

c. CHECK: “I have checked for understanding as I read the problem, and will

continue to do this as I solve the problem.”

3. The teacher then moves on to the next problem-solving step. Teacher states, “I am going

to paraphrase the question by putting the problem into my own words.”

a. SAY: “Put the problem into my own words, okay. Mrs. Hilt read 21 books. Each

book she read had 2,010 words in it. I should underline the important

information; I will underline the number of books she read, and the number of

words in each book. Since the question only asks about the number of words, the

$4.95 is not important to underline.”

b. ASK: “Have I underlined the important information? Let’s see, yes I did. What is

the question? The question is how many words Mrs. Hilt read. What am I looking

for? I am looking for the number of words that Mrs. Hilt read in all 21 of her

books.”

c. CHECK: “I have underlined all the important information. I have the number of

books she read and the number of words in each book underlined.”

4. The teacher instructs “I will now visualize the problem by making a drawing or a

diagram.”

38 | P a g e

a. SAY: “Make a drawing or diagram. Maybe I’ll draw 21 rectangles for books and

label them each with 2,010 words. Then I will draw a circle around all of the

books to tell myself that I will need to find the total.”

b. ASK: “Does the picture fit the problem? Yes, I believe it does tell the story.”

c. CHECK: “I need to double check that my picture does match. And yes, I have 21

rectangles and the number 2,010 in each rectangle.”

5. The teacher explains “I will now hypothesize by making a plan to solve the problem.”

a. SAY: “I need to decide how many steps and operations are needed. Let me see.

There is really only one step to solving this problem. I need to find the total

number of words read. I can add 2,010 21 times, or I can multiply 2,010 by 21.”

b. ASK: “If I multiply 21 books and 2,010 words, I will get the total number of

words read. Doing this will solve my problem in one step.”

c. CHECK: “I believe my plan makes sense. If I don’t think it will work, or I am

unsure, I should ask for help.”

6. Next, the teacher states, “I need to estimate by predicting the answer to this problem.”

a. SAY: “I should round the numbers and do the math in my head, and then write

down my estimate. So 21 can be rounded down to 20, and 2,010 can be rounded

down to 2,000. So now I can multiply 20 by 2000 and get an estimated answer of

40,000 words read.”

b. ASK: “Did I round the numbers up and down correctly? Yes, I did. Also, did I

write the estimate? Yes.”

c. CHECK: “I need to be sure I used all the needed information and followed the

step that I planned to use. I did.”

39 | P a g e

7. The teacher explains, “Now I need to compute the actual answer by doing the

arithmetic.”

a. SAY: “I need to use the correct numbers and do the operations in order. Okay,

so, first I will write down 2,010 on top, and then 21 below the 2,010. I must make

sure it’s all lined up properly. Now I am ready to solve the problem. 2,010 times

21 is 42,210 words.”

b. ASK: “How does my answer compare to my estimate? Well, I estimated 40,000

words, and my actual answer is 42,210, this is close. Does my answer make

sense? Yes, all 21 books have a combined total of 42,210 words. Are any

decimals, units, or signs needed for this answer? No, I do not need any of those

for this problem.”

c. CHECK: “I should make sure my operations were done in the right order, and all

my calculations were correct. Yes, they are.”

8. The teacher explains, “Okay, now I really get to check to see if the answer is correct.”

a. SAY: “I should check the computation. Let’s see. I will go backward. 42,210

divided by 21 equals 2,010. Which is what I started with.”

b. ASK: “Have I checked every step? Yes, the operations chosen were correct, and

the computation was correct. Is my answer right? Yes.”

c. CHECK: “I must check that everything is right. I have checked everything. If I

had not checked everything, I would go back. Or I could ask for help if I need it.”

9. Teacher allows time for students to ask any questions. Finally, the activity ends with the

teacher giving the students a problem to solve independently. They are instructed to say

everything they are thinking and doing and to use the processes and strategies that have

40 | P a g e

been taught. A student is then selected to model the solution on the board just as the

teacher had done.

References

Montague, M., Warger, C., & Morgan, T. H. (2000). Solve It! Strategy instruction to improve

mathematical problem solving. Learning Disabilities Research and Practice, 15, 110-

116.

41 | P a g e

Solve It! Problem-Solving Skills

Intervention Kit Appropriate Grade Level: This intervention can be utilized with grades 4 through 12 on

mathematical material involving: Word problems, expressions and equations, algebraic

equations, and ratios and proportional relationships.

Brief Description: Solve It! is an academic intervention that can be used to help students learn

effectively solve mathematical word problems. Solve It! uses a mathematical program that

specifically teaches students to understand the task, interpret and solve the problem, and evaluate

the answer from mathematical problems through a series of strategic steps.

Home Implementation: Parents can provide support for their children at home using this

method with some help from teachers. Teachers can provide parents with a copy of the “Say,

Ask, Check” prompts sheet to use at home with their children when completing assigned

homework. This will create repetition and consistency between school and home and will help

to reinforce the problem-solving strategy that is being implemented in the school.

Intervention Goal: The purpose of Solve It! is to teach students how to use a 7 step problem

solving model to accurately solve mathematical word problems.

Materials Needed:

• Master Chart

• Baseline Measure Assessments

• Treatment Integrity Checklist

• Progress Monitoring Measure

• “Say, Ask, Check” Prompts Sheet

42 | P a g e

Script:

1. Teacher introduces the activity by writing a word problem on the board in front of the

class. Teacher says, “Now watch me as I say everything I am thinking and doing as I

solve this problem”, then reads the word problem out loud "Mrs. Hilt read 21 books.

Each book had exactly 2,010 words in it. She sold five of her books for $4.95 each. How

many words did Mrs. Hilt read?”

2. The teacher then explains, “First, I am going to read the problem for understanding

a. SAY: “Read the problem, okay. (Reads problem out loud one more time). If I

don’t understand it, I will read it again. Huh, I think I need to try reading it one

more time, (reads problem aloud once more)”

b. ASK: “Have I read and understood the problem? I think so.”

c. CHECK: “I have checked for understanding as I read the problem, and will

continue to do this as I solve the problem.”

3. The teacher then moves on to the next problem-solving step. Teacher states, “I am going

to paraphrase the question by putting the problem into my own words.”

a. SAY: “Put the problem into my own words, okay. Mrs. Hilt read 21 books. Each

book she read had 2,010 words in it. I should underline the important

information; I will underline the number of books she read, and the number of

words in each book. Since the question only asks about the number of words, the

$4.95 is not important to underline.”

b. ASK: “Have I underlined the important information? Let’s see, yes I did. What is

the question? The question is how many words Mrs. Hilt read. What am I looking

43 | P a g e

for? I am looking for the number of words that Mrs. Hilt read in all 21 of her

books.”

c. CHECK: “I have underlined all the important information. I have the number of

books she read and the number of words in each book underlined.”

4. The teacher instructs “I will now visualize the problem by making a drawing or a

diagram.”

a. SAY: “Make a drawing or diagram. Maybe I’ll draw 21 rectangles for books and

label them each with 2,010 words. Then I will draw a circle around all of the

books to tell myself that I will need to find the total.”

b. ASK: “Does the picture fit the problem? Yes, I believe it does tell the story.”

c. CHECK: “I need to double check that my picture does match. And yes, I have 21

rectangles and the number 2,010 in each rectangle.”

5. The teacher explains “I will now hypothesize by making a plan to solve the problem.”

a. SAY: “I need to decide how many steps and operations are needed. Let me see.

There is really only one step to solving this problem. I need to find the total

number of words read. I can add 2,010 21 times, or I can multiply 2,010 by 21.”

b. ASK: “If I multiply 21 books and 2,010 words, I will get the total number of

words read. Doing this will solve my problem in one step.”

c. CHECK: “I believe my plan makes sense. If I don’t think it will work, or I am

unsure, I should ask for help.”

6. Next, the teacher states, “I need to estimate by predicting the answer to this problem.”

a. SAY: “I should round the numbers and do the math in my head, and then write

down my estimate. So 21 can be rounded down to 20, and 2,010 can be rounded

44 | P a g e

down to 2,000. So now I can multiply 20 by 2000 and get an estimated answer of

40,000 words read.”

b. ASK: “Did I round the numbers up and down correctly? Yes, I did. Also, did I

write the estimate? Yes.”

c. CHECK: “I need to be sure I used all the needed information and followed the

step that I planned to use. I did.”

7. The teacher explains, “Now I need to compute the actual answer by doing the

arithmetic.”

a. SAY: “I need to use the correct numbers and do the operations in order. Okay,

so, first I will write down 2,010 on top, and then 21 below the 2,010. I must make

sure it’s all lined up properly. Now I am ready to solve the problem. 2,010 times

21 is 42,210 words.”

b. ASK: “How does my answer compare to my estimate? Well, I estimated 40,000

words, and my actual answer is 42,210, this is close. Does my answer make

sense? Yes, all 21 books have a combined total of 42,210 words. Are any

decimals, units, or signs needed for this answer? No, I do not need any of those

for this problem.”

c. CHECK: “I should make sure my operations were done in the right order, and all

my calculations were correct. Yes, they are.”

8. The teacher explains, “Okay, now I really get to check to see if the answer is correct.”

a. SAY: “I should check the computation. Let’s see. I will go backward. 42,210

divided by 21 equals 2,010. Which is what I started with.”

45 | P a g e

b. ASK: “Have I checked every step? Yes, the operations chosen were correct, and

the computation was correct. Is my answer right? Yes.”

c. CHECK: “I must check that everything is right. I have checked everything. If I

had not checked everything, I would go back. Or I could ask for help if I need it.”

9. Teacher allows time for students to ask any questions. Finally, the activity ends with the

teacher giving the students a problem to solve independently. They are instructed to say

everything they are thinking and doing and to use the processes and strategies that have

been taught. A student is then selected to model the solution on the board just as the

teacher had done.

Baseline Measures: Prior to beginning the intervention, the interventionist should assess each

student’s ability to accurately solve a mathematical word problem. Baseline data can be

gathered by utilizing a 5 question word-problem worksheet at the student’s instructional level.

This worksheet, and at least two others should be given on at least three separate occasions in

order to gather an appropriate amount of baseline data. There must be 3-5 data points for

baseline data, prior to implementing the intervention. An example of a worksheet is included

below, almost any classroom worksheet from the student’s grade level curriculum would also

work for this purpose. The student’s scores should be recorded on the sheets and saved for

comparison purposes during the progress monitoring process.

Progress Monitoring: Educators should assign weekly, untimed, probes to their students which

follow the mathematical reasoning and concepts they are learning in the classroom. Versions of

these probes can be used to collect baseline data. The fundamental goal would be checking for

accuracy in both process and answers. This progressing monitoring probe should be at an

instructional level for the student, similar to the baseline measure, to ensure the student is

46 | P a g e

working towards improving their skills in problem-solving. These probes should be

administered weekly throughout the implementation of the 8 lesson intervention until the student

reaches mastery. Success is determined when a student is able to complete an entire progress

monitoring sheet by accurately using each of the “Say, Ask, Check” steps and earning a score

above 80%.

References:

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the

mathematical problem solving of middle school students with learning disabilities.

Journal of Learning Disabilities, 25, 230-248.

Montague, M. & Dietz, S. (2009). Evaluating the evidence base for cognitive strategy instruction

and mathematical problem solving. Exceptional Children, 75, 285-302.

Montague, M., Warger, C., & Morgan, T. H. (2000). Solve It! Strategy instruction to improve

mathematical problem solving. Learning Disabilities Research and Practice, 15, 110-

116.

Wright, J. (n.d.). Math Problem-Solving: Combining Cognitive & Metacognitive Strategies

Retrieved March 11, 2016, from

http://www.interventioncentral.org/academicinterventions/math/math-problem-solving-

combining-cognitive-metacognitive-strategies

47 | P a g e

Treatment Integrity Checklist: Student: ___________________________ Week Of: _________________

Teacher: ___________________________ Start: __________ End: ________

0 = No / not completed 1 = done / completed

Circle the corresponding number and calculate the treatment integrity by following the given equation.

Criteria Mon. Tue. Wed. Thu. Fri. Total %age (Total/5)

X100

Student completed all steps in “Say- Ask-Check” for Step 1: Reading the

Problem

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 2: Paraphrasing

the Problem

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 3: “Drawing”

the Problem

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 4: Creating a

Plan to Solve the Problem

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 5: Predicting/

Estimating the Answer

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 6: Computing

the Answer

0 1

0 1

0 1

0 1

0 1

Student completed all steps in “Say- Ask-Check” for Step 7: Checking the

Answer

0 1

0 1

0 1

0 1

0 1

Total

Percentage (Total/5) X 100

Adapted from: Comprehensive, Integrated, Three-Tiered Model of Prevention (2015). Systematic

Screening. Retrieved December 14, 2015 from: http://www.ci3t.org/pl.html#tier2

48 | P a g e

Solve It! - Math Problem Solving Processes and Strategies Master Sheet

Montague, M. (2003). Solve it! A mathematical problem-solving instructional program.

Reston, VA: Exceptional Innovations

READ (for understanding) Say: Read the problem. If I don’t understand, read it again. Ask: Have I read and understood the problem? Check: For understanding as I solve the problem. PARAPHRASE (your own words) Say: Underline the important information. Put the problem in my own words. Ask: Have I underlined the important information? What is the question? What am I looking for? Check: That the information goes with the question. VISUALIZE (a picture or a diagram) Say: Make a drawing or a diagram. Show the relationships among the problem parts. Ask: Does the picture fit the problem? Did I show the relationships? Check: The picture against the problem information. HYPOTHESIZE (a plan to solve the problem) Say: Decide how many steps and operations are needed. Write the operation symbols (+, -, x, and /). Ask: If I …, what will I get? If I …, then what do I need to do next? How many steps are needed? Check: That the plan makes sense. ESTIMATE (predict the answer) Say: Round the numbers, do the problem in my head, and write the estimate. Ask: Did I round up and down? Did I write the estimate? Check: That I used the important information. COMPUTE (do the arithmetic) Say: Do the operations in the right order. Ask: How does my answer compare with my estimate? Does my answer make sense? Are the decimals or money signs in the right places? Check: That all the operations were done in the right order. CHECK (make sure everything is right) Say: Check the plan to make sure it is right. Check the computation. Ask: Have I checked every step? Have I checked the computation? Is my answer right? Check: That everything is right. If not, go back. Ask for help if I need it.

49 | P a g e

Sample Baseline Measure Name: ___________________________ Date: ___________________ Score: __________________

Word Problems

1. Nancy wants several different color plates for her birthday. Nancy wants to get 132 cyan plates, 144 green plates, and some amount of yellow plates. It total, Nancy wants 372 plates, so how many yellow plates should she get?

i. Answer: __________________________

2. Jess ants 156 cupcakes for her party. Jess has already made 36 mocha cupcakes, and 72 fudge cupcakes. How many more cupcakes does Jess need to make?

i. Answer: __________________________

3. For softball season, Mary decided to buy a ball for $7.30, new shorts for $15.40, as well as a pair of softball shoes for $48.10. Mary currently has $21.00, and a coupon for $10.00 off her purchase. How much more money does Mary need to complete her purchase?

i. Answer: _________________________

4. Jason likes to collect coins. Jason got 25 coins from his brother, 30 coins from his mother, as well as 33 coins from benny. However, Jason lost 19 coins before putting those coins into his piggy bank. How many coins does Jason have in his piggy bank?

i. Answer: ________________________

5. Nancy made 3 Different sacks of wooden blocks. The first stack was 7 blocks high, the second stack was 5 blocks higher than the first, and the final stack was 7 blocks higher than the second. In total, how many blocks did Nancy use for all 3 stacks?

i. Answer: ________________________

Total Incorrect:

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Directions: Use this to complete the word problems on the previous page using the 7 Step Problem-Solving Model and Say, Ask, Check.

Problem #1 Problem #2


Problem #5

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Sample Baseline Measure

Name: ______Answer Key_______ Date: ___________________

Score: __________________

Word Problems

6. Nancy wants several different color plates for her birthday. Nancy wants to get 132 cyan plates, 144 green plates, and some amount of yellow plates. It total, Nancy wants 372 plates, so how many yellow plates should she get?

i. Answer: ___96 Plates______

7. Jess ants 156 cupcakes for her party. Jess has already made 36 mocha cupcakes, and 72 fudge cupcakes. How many more cupcakes does Jess need to make?

i. Answer: ____48 Cupcakes____

8. For softball season, Mary decided to buy a ball for $7.30, new shorts for $15.40, as well as a pair of softball shoes for $48.10. Mary currently has $21.00, and a coupon for $10.00 off her purchase. How much more money does Mary need to complete her purchase?

i. Answer: ____$39.00_______

9. Jason likes to collect coins. Jason got 25 coins from his brother, 30 coins from his mother, as well as 33 coins from benny. However, Jason lost 19 coins before putting those coins into his piggy bank. How many coins does Jason have in his piggy bank?

i. Answer: _____69 Coins______

10. Nancy made 3 Different sacks of wooden blocks. The first stack was 7 blocks high, the second stack was 5 blocks higher than the first, and the final stack was 7 blocks higher than the second. In total, how many blocks did Nancy use for all 3 stacks?

i. Answer: ____38 Blocks_________

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Sample Progress Monitoring Measure

Name: ___________________________ Date: ___________________

Score: __________________

Word Problems

1. Uncle Ben has 440 chickens on his farm. 39 are roosters and the rest are hens. 15 of his hens do not lay eggs. The rest lay eggs. How many egg-laying hens does Uncle Ben have on his farm?

i. Answer: _________________________________

2. Aunt May mils her Holstein cows twice a day. This morning she got 365 gallons of milks. This evening she got 380 gallons. She sold 612 gallons to the local ice cream factory. How many gallons of milk does she have left?

i. Answer: __________________________________

3. Mr. Parker has 982 pounds of grain. He feeds 240 pounds to his pigs and 460 to his cows. How much grain does he have left?

i. Answer: ___________________________________

4. Peter has four horses. Each one eats 4 pounds of oats, twice a day. How many pounds of oats does he need to feed his horses for 3 days?

i. Answer: ___________________________________

5. The bakers are the Beverly Hills Bakery baked 200 loaves of bread on Monday morning. They sold 93 loaves in the morning and 39 loaves in the afternoon. How many loaves of bread did they have left?

i. Answer: ____________________________________

Total Incorrect:

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Directions: Use this to complete the word problems on the previous page using the 7 Step Problem-Solving Model and Say, Ask, Check.



Problem #5

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Sample Progress Monitoring Measure

Name: _____Answer Key__________ Date: ___________________

Score: __________________

Word Problems

6. Uncle Ben has 440 chickens on his farm. 39 are roosters and the rest are hens. 15 of his hens do not lay eggs. The rest lay eggs. How many egg-laying hens does Uncle Ben have on his farm?

i. Answer; ________386 Egg-Laying Hens__________

7. Aunt May mils her Holstein cows twice a day. This morning she got 365 gallons of milks. This evening she got 380 gallons. She sold 612 gallons to the local ice cream factory. How many gallons of milk does she have left?

i. Answer: _________133 Gallons of milk__________

8. Mr. Parker has 982 pounds of grain. He feeds 240 pounds to his pigs and 460 to his cows. How much grain does he have left?

i. Answer: ___________282 Pounds of grain__________

9. Peter has four horses. Each one eats 4 pounds of oats, twice a day. How many pounds of oats does he need to feed his horses for 3 days?

i. Answer: _______96 Pounds of oats____________

10. The bakers are the Beverly Hills Bakery baked 200 loaves of bread on Monday morning. They sold 93 loaves in the morning and 39 loaves in the afternoon. How many loaves of bread did they have left?

i. Answer: _________68 Loaves_______________

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“Say, Ask, Check” Prompts Sheet

‘Say-Ask-Check’ Metacognitive Prompts Tied to a Word-Problem Cognitive Strategy (Montague, 1992) Cognitive Strategy Step

Metacognitive ‘Say-Ask-Check’ Prompt Targets

Sample Metacognitive ‘Say- Ask-Check’ Prompts

1. Read the Problem.

‘Say’ (Self-Instruction) Target: The student reads and studies the problem carefully before proceeding. ‘Ask’ (Self-Question) Target: Does the student fully understand the problem? ‘Check’ (Self-Monitor) Target: Proceed only if the problem is understood.

Say: “I will read the problem. I will reread the problem if I don’t understand it.” Ask: “Now that I have read the problem, do I fully understand it?” Check: “I understand the problem and will move forward.”

2. Paraphrase the problem.

‘Say’ (Self-Instruction) Target: The student restates the problem in order to demonstrate understanding. ‘Ask’ (Self-Question) Target: Is the student able to paraphrase the problem? ‘Check’ (Self-Monitor) Target: Ensure that any highlighted key words are relevant to the question.

Say: “I will highlight key words and phrases that relate to the problem question.” “I will restate the problem in my own words.” Ask: “Did I highlight the most important words or phrases in the problem?” Check: “I found the key words or phrases that will help to solve the problem.”

3. ‘Draw’ the problem.

‘Say’ (Self-Instruction) Target: The student creates a drawing of the problem to consolidate understanding. ‘Ask’ (Self-Question) Target: Is there a match between the drawing and the problem? ‘Check’ (Self-Monitor) Target: The drawing includes in visual form the key elements of the math problem.

Say: “I will draw a diagram of the problem.” Ask: “Does my drawing represent the problem?” Check: “The drawing contains the essential parts of the problem.”

4. Create a Plan to solve the problem.

‘Say’ (Self-Instruction) Target: The student generates a plan to solve the problem. ‘Ask’ (Self-Question) Target: What plan will help the student to solve this problem? ‘Check’ (Self-Monitor) Target: The plan is appropriate to solve the problem.

Say: “I will make a plan to solve the problem.” Ask: “What is the first step of this plan? What is the next step of the plan?” Check: “My plan has the right steps to solve the problem.”

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5. Predict/ Estimate the Answer.

‘Say’ (Self-Instruction) Target: The student uses estimation or other strategies to predict or estimate the answer. ‘Ask’ (Self-Question) Target: What estimating technique will the student use to predict the answer? ‘Check’ (Self-Monitor) Target: The predicted/estimated answer used all of the essential problem information.

Say: “I will estimate what the answer will be.” Ask: “What numbers in the problem should be used in my estimation?” Check: “I did not skip any important information in my estimation.”

6. Compute the answer.

‘Say’ (Self-Instruction) Target: The student follows the plan to compute the solution to the problem. ‘Ask’ (Self-Question) Target: Does the answer agree with the estimate? ‘Check’ (Self-Monitor) Target: The steps in the plan were followed and the operations completed in the correct order.

Say: “I will compute the answer to the problem.” Ask: “Does my answer sound right?” “Is my answer close to my estimate?” Check: “I carried out all of the operations in the correct order to solve this problem.”

7. Check the answer.

‘Say’ (Self-Instruction) Target: The student reviews the computation steps to verify the answer. ‘Ask’ (Self-Question) Target: Did the student check all the steps in solving the problem and are all computations correct? ‘Check’ (Self-Monitor) Target: The problem solution appears to have been done correctly.

Say: “I will check the steps of my answer.” Ask: “Did I go through each step in my answer and check my work?” Check: “”

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the

mathematical problem solving of middle school students with learning disabilities.

Journal of Learning Disabilities, 25, 230-248.

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Folding-In Technique

Coach Card Description: The “Folding-In” intervention can be used either by self-administration by a

student or administered with an adult. This intervention can be used in the classroom and within

the home setting. This intervention is used to help teach and practice basic math fact skills.

“Folding-In” uses flashcards to slowly incorporate unknown math fact cards in with known math

facts.

Target Skill: Improve skills in math facts, which can include addition, subtraction,

multiplication, and division.

Location: “Folding-In” can be implemented by a teacher, parent, and also self-administered by

a student. This intervention can be used at home and in the classroom.

Materials:

• Flash Cards (i.e. Index cards with math facts)

• Rubber Bands (to sort known and unknown cards)

• Math-Facts SAFI: Student Checklist (Self-Administered)

• Dry-Erase Board, Markers, and Eraser (Self-Administered)

• Student Log: Mastered Math-Facts (Self-Administered)

• Baseline and Progress Monitoring Measures

Frequency: Two to three times a week for 10-20 minutes each session.

Progress Monitoring: Curriculum Based Measurement (CBM) probes should be administered

two times per week to assess the student’s overall progress. Different versions of these probes

can be used to collect baseline data as well.

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Directions:

1. The teacher determines the instructional level of the student based on Curriculum Based

Measurement (CBM) probes.

2. The teacher should create known and unknown flash cards based on the curriculum they

are teaching and based on the results from the CBM.

3. 10-16 flashcards (70% known and 30% unknown) will be selected and the folding-in

procedure will begin.

4. As the teacher, parent, or student presents the flashcards, a particular sequence should be

followed for the folding-in unknown basic math facts.

a. Each unknown card should be presented between each known math fact before a

new unknown math fact is presented.

5. When unknown math facts are introduced, the teacher, parent, or student should say the

math fact, and its answer. The student should then repeat the math fact and its answer.

a. When the math fact is presented again, the student will repeat this method. If the

student answers the problem incorrectly, the teacher, parent or student should

briefly correct him or her, and have the student write the correct answer three

times before moving on.

6. Once an entire folding-in session is complete, the cards should be shuffled and presented

one more time.

a. The unknown cards are marked correct or incorrect. Once an unknown fact is

correctly answered consecutively for two to three days, it becomes a new known

fact and a new unknown fact is introduced.

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7. After the flashcard session (10-20 minutes), the student will redo the same math probe,

record any math facts that they have transferred to the student’s “known” weekly stack,

and complete their student checklist.

References:

Wright, J. (n.d). How to: Improve proficiency in math-facts through a self-administered

folding-in technique. Retrieved March 10, 2016, from

http://www.interventioncentral.org/node/965168

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Folding-In Technique

Intervention Kit Appropriate Grade Level: This method can be utilized with grades 1 through 8 on basic math

facts such as addition, subtraction, multiplication, and division.

Brief Description: The folding-in technique can be used to teach and practice factual

information, such as basic math facts that involves slowly incorporating unknown math facts to

known ones. This intervention can be self-administered by the student or administered by an

adult and can be utilized in the classroom as well as within the home setting.

Home Implementation: Practice with folding-in can be assigned as homework. A direction

sheet outlining the steps and script can be sent home with the child, along with the child's

flashcards. By using this technique, parents can help increase their child's knowledge of basic

math facts.

Intervention Goal: Improve proficiency in math-facts such as addition, subtraction,

multiplication, and division.

Materials Needed:

• Flash Cards (i.e. Index cards with math facts)

• Rubber Bands (to sort known and unknown cards)

• Math-Facts SAFI: Student Checklist (Self-Administered)

• Dry-Erase Board, Markers, and Eraser (Self-Administered)

• Student Log: Mastered Math-Facts (Self-Administered)

• Baseline and Progress Monitoring Measures

Script:

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For grades 1st- 4th administered by teacher:

1. The teacher should determine the student’s instructional level on solving basic

math facts using Curriculum Based Measurement (CBM) probes.

2. After the student completes a CBM probe the teacher should identify the student’s

known and unknown basic math facts from the results.

3. The teacher will then create flashcards with both unknown and known facts from

the math probe.

4. The teacher should reassess material for known and unknown facts by introducing

all the flashcards and marking them.

5. The teacher will then select around 10-16 flashcards (70% known and 30%

unknown) and begin the folding-in procedure.

6. As the teacher presents the flashcards, a specific sequence should be followed for

the folding-in of unknown basic math facts.

a. Each unknown card should be presented between each known math fact

before a new unknown math fact is presented. Unknown math facts should be

repeatedly presented among known math facts so the child uses repetition

experiences frequent success by using known material.

7. When unknown material is first introduced, the teacher should first say the math

fact, and its answer. The child should then repeat what was said. When the math fact is

presented again, the student will repeat this procedure. If the student answers the

problem incorrectly, the teacher should briefly correct it and the student should write the

correct answer three times before moving on.

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8. Once the entire folding-in sequence is complete, the cards are shuffled and all of

the cards are presented once more. The unknown cards are then marked correct or

incorrect. Once an unknown fact is correctly answered for two to three consecutive days,

it becomes a new known fact and a new unknown fact is introduced.

9. After the flashcard session (10-20 minutes), the child should redo the same math

probe and record correct and incorrect responses.

10. Administer CBM math probes two times per week for progress monitoring of

overall student progress and graph the results using the attached data collection sheets.

For grades 3rd-8th Self-Administered Folding-In Technique

1. Determine the student’s instructional level using Curriculum Based Measurement

(CBM) probes.

2. Start with “working pile” of 10 cards from the last session, or create a new

"working pile" by taking 7 cards from your weekly "known" stack and 3 cards from your

weekly "unknown" stack and shuffling them.

3. Take the first card from the top of the working pile and place it flat on the table.

4. Read the math-fact on the card and write the answer on the dry-erase board.

5. Turn the card over and compare your answer to the answer on the card.

6. If your answer is correct, sort that card into a "working known" pile. If your

answer is incorrect, sort that card into a "working unknown" pile, then practice by writing

the math-fact and correct answer on your dry-erase board three times in a row.

7. Continue until you have answered all 10 flashcards. Then look at the working

"known" and "unknown" card stacks. If all working cards are in the "known" stack, draw

a star in the bottom left corner of your dry-erase board.

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8. Shuffle the 10 cards in the working card deck.

9. Continue reviewing all 10 cards in the working deck as explained in steps 2-7

until you have drawn three stars in the bottom left corner of the dry-erase board.

a. Continue until you have answered all 10 cards correctly in a single run-

through and have accomplished this feat a total of three times in the session.

10. When you have earned 3 stars, consider the entire working stack to be "known"

cards. So it's now time to update the working deck.

11. Take any 3 cards from your current working 10-card deck and transfer them to the

weekly "known" deck. Then, on the Student Log: Mastered Math-facts form, record the

math-facts and date for the 3 cards that you transfer. Congratulations! These now count

as mastered math-facts!

12. Next, take 3 cards from the weekly "unknown" stack and add them to your current

working deck to bring it back up to 10 cards.

13. Begin reviewing the working stack again (as outlined in steps 2-7) until your time

runs out.

14. Before ending the session, place rubber-bands around the weekly "known" and

"unknown" decks and the working stack that you are currently on. Also, be sure that

your Student Log: Mastered Math-facts form is up-to-date.

Sample Baseline Measures: 3-5 Curriculum Based Measurement (CBM) probes should be

implemented to determine the student's instructional level. An example of a CBM probe is found

below. Further probes for different operations (i.e. addition, subtraction, multiplication, and

division) can be found on the aimsweb website (aimsweb.pearson.com) at a cost. Additional

probes can also be found at interventioncentral.com at no cost. Students are given 8 minutes to

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http://www.interventioncentral.org/sites/default/files/pdfs/pdfs_blog/self_management_math_SAFI_2.pdf

complete the CBM probe and are told to put an X through any problems that they do not

understand.

Progress Monitoring: Randomly selected CBM probes should be administered two times per

week to assess the student’s overall progress. Versions of these probes can be used to collect

baseline data. Same as the baseline measure these probes can be found on either the aimsweb

website, or intervention central. The directions are the same for the progress monitoring probes,

students have 8 minutes to complete the probe and should put an X through any problem they do

not understand.

References:

Wright, J. (n.d). How to: Improve proficiency in math-facts through a self-administered folding-

in technique. Retrieved March 10, 2016, from


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Math-Facts SAFI: Student Checklist Treatment Integrity Checklist for student Self-Administration (Grades 3-8)

Carried Out? Intervention Step

__Y __N 1. Start with the daily stack of cards from the last session. Or create a new "daily stack" by

taking 7 cards from your weekly "known" stack and 3 cards from your weekly "unknown"

stack and shuffling them.

__Y __N 2. Take the first card from the top of the daily stack and place it flat on the table.

__Y __N 3. Read the math-fact on the card and write the answer on the dry-erase board within 3

seconds.

__Y __N 4. Turn the card over and compare the answer that you wrote to the answer on the card.

__Y __N 5. If your answer is correct, sort that card into a "daily known" pile. If your answer is

incorrect, sort that card into a "daily unknown" pile--then practice by writing the math-fact

and correct answer on your dry-erase board three times in a row.

__Y __N 6. Continue until you have answered all 10 daily cards. Then look at the daily "known" and

"unknown" card stacks. If all daily cards are in the "known" stack, draw a star in the bottom

left corner of your dry-erase board.

__Y __N 7. Shuffle the 10 cards in the daily card deck.

__Y __N 8. Continue reviewing all 10 cards in the daily deck as explained in steps 2-7 until you have

drawn three stars in the bottom left corner of the dry-erase board. (In other words, continue

until you have answered all 10 cards without error in a single run-through and have

accomplished this feat a total of three times in the session.)

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__Y __N 9. When you have earned 3 stars, consider the entire daily stack to be "known" cards. So it's

now time to update the daily deck.

__Y __N 10. Take any 3 cards from your current daily 10-card deck and transfer them to the weekly

"known" deck. Then, on the Student Log: Mastered Math-facts form, record the math-facts

and current date for the 3 cards that you transfer. Congratulations! These now count as

mastered math-facts!

__Y __N 11. Next, take 3 cards from the weekly "unknown" stack and add them to your current daily

deck to bring it back up to 10 cards.

__Y __N 12. Begin reviewing the daily stack again (as outlined in steps 2-7) until your time runs out.

__Y __N 13. Before ending the session, place rubber-bands around the weekly "known" and

"unknown" decks and the daily stack that you are currently working on. Also, be sure that

your Student Log: Mastered Math-facts form is up-to-date

Math-Facts SAFI: Student Checklist (Hulac, Dejong, & Benson, 2012).

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Student Log: Mastered Math-Facts

Student Log: Mastered Math-facts Student: School Yr: Classroom/Course:

Item 1: Date: / /

Item 25: Date: / /

Item 2: Date: / / Item 26: Date: / /























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Student Log: Mastered Math-facts Directions to the Student: Record any math-facts that you are transferring to the 'known' weekly stack.

Item 49: Date: / /

Item 75: Date: / /


























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Sample Baseline Measure

Curriculum-Based Assessment Mathematics Single-Skill Computation Probe: Student Copy

Student:

Date: ____________________

3 + 3

| | | | |

1 + 1

| | | | |

6 + 5

| | | | |

7 + 7

| | | | |

8 + 2

| | | | |

2 + 9

| | | | |

2 + 4

| | | | |

2 + 8

| | | | |

4 + 7

| | | | |

www.interventioncentral.com

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Student:

Date: ____________________

6 + 1

| | | | |

6 + 1

| | | | |

5 + 1

| | | | |

1 + 2

| | | | |

2 + 5

| | | | |

4 + 5

| | | | |

1 + 6

| | | | |

4 + 7

| | | | |

5 + 3

| | | | |


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Student:

Date: ____________________

3 + 4

| | | | |

5 + 2

| | | | |

1 + 1

| | | | |

8 + 1

| | | | |

4 + 2

| | | | |

9 + 8

| | | | |

8 + 6

| | | | |

3 + 6

| | | | |

2 + 1

| | | | |


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Treatment Integrity Checklist: Treatment Integrity Checklist for adult administration (Grades 1-4)

Student: ___________________________ Week Of: _________________

Teacher: ___________________________ Start: __________ End: ________

Carried Out? Intervention Step

__Y __N Determine the student’s instructional level using Curriculum Based Measurement (CBM) probes.

__Y __N Teacher identified basic math facts that are known and unknown based on results from CBM, to create flashcards.

__Y __N The teacher reassessed material for known and unknown by introducing all the flashcards to the student and marking them.

__Y __N The teacher selected around 10-16 flashcards (70% known and 30% unknown) and began the folding-in procedure.

__Y __N Each unknown card was presented between each known math fact before a new unknown math fact was presented.

__Y __N The teacher said the unknown math fact and its answer when it was first introduced to the student. The student repeated the math fact and its answer.

__Y __N When the unknown math fact was presented again, the student repeat this procedure, if the student answered the problem incorrectly, the teacher corrected it and the student wrote correct answer three times before continuing.

__Y __N Once the entire folding-in sequence was completed, the cards were shuffled and all of the cards were presented one more time. The unknown cards were then marked correct or incorrect.

__Y __N Unknown fact correctly answered in three consecutive days, become a new known fact and a new unknown fact was introduced.

__Y __N After the flashcard session (10-20 minutes), the student retook the same math probe and the teacher recorded digits correct and unknown facts.

__Y __N CBM math probes were administered two times per week to assess the student’s overall progress.

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Number Operations: Strategic Number Counting Instruction

Coach Card Description: The student is taught explicit counting strategies for basic addition and subtraction.

With a tutor, the specific number counting strategies are practiced. This instruction was adapted

from Fuchs et al. (2009).

Target Skills: Addition and Subtraction facts, through strategic counting instruction.

Location: This intervention can be taught individually or in small groups, but the tutoring portion

needed to practice must be done individually. This can be done in the classroom at the elementary

school level

Materials:

• Number-line (attached)

• Number combination (math fact) flashcards for basic addition and subtraction

• Strategic Number Counting Instruction Score Sheet (attached)

Frequency: This intervention should be conducted for as long as necessary to ensure the student

learns the basic addition and subtraction facts. The original study conducted by Fuchs et al.

(2009) lasted 16 weeks with 3 sessions per week for 20-30 minutes.

Progress Monitoring: In order to gauge the progress being made, the tutor keeps track of the

students previous scores on the number of flashcards the student correctly answered in the three

minute time period, as well as the errors made. The tutor keeps track of the necessary progress

monitoring information on the Strategic Number Counting Instruction Score Sheet (attached).

Directions:

Prior to beginning the flash card stage, the tutor trains the student to use two counting strategies

for addition and subtraction:

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• Addition: The student is given a copy of the appropriate number-line (1-10 or 1-20—see

attached handout). 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.

• Subtraction: The student is given a copy of the appropriate number-line (1-10 or 1-20—

see attached handout). 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 is

directed to start at the minus number on the number-line and to count 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.

1. Create Flashcards. The tutor creates addition and/or subtraction flashcards of problems

that the student is to practice. Each flashcard displays the numerals and operation sign

that make up the problem but leaves the answer blank.

2. Review Count-Up Strategies. At the start 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.

3. Complete Flashcard Warm-Up. The tutor reviews addition/subtraction flashcards with

the student for three minutes. Before beginning, the tutor reminds the student that, when

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shown a flashcard, the student should try to recall the answer from memory—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, then 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 number of cards that the student

(a) identified from memory, (b) solved using the count-up strategy, and (c) was not able to

correctly answer. Totals are recorded on the Strategic Number Counting Instruction Score Sheet.

4. Repeat Flashcard Review. The tutor shuffles the math-fact flashcards, encourages the

student to try to beat his or her previous score, and again reviews the flashcards with the

student for three minutes. As before, whenever the student makes an error, the tutor

directs the student to use the appropriate count-up strategy. Also, if the student

completes all cards in the stack with time remaining, the tutor shuffles the stack and

continues presenting cards until the time is elapsed. At the end of the three minutes, the

tutor again counts up the number of cards reviewed and records the number of cards that

the student (a) identified from memory, (b) solved using the count-up strategy, and (c)

was not able to correctly answer. These totals are again recorded on the Strategic

Number Scoring Sheet (attached).

5. Provide Performance Feedback. The tutor orally provides about whether the student’s

performance on the second trial exceeded the first, and by how much. The tutor also

provides praise if the student beat the previous score. The tutor provides encouragement

if the student failed to beat the previous score.

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References




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

Attached References

Number Line

http://www.interventioncentral.org/sites/default/files/pdfs/pdfs_interventions/number_line_1_10

_and_1_20.pdf

Scoring Sheet

http://www.interventioncentral.org/sites/default/files/pdfs/pdfs_interventions/strategic_number_c

ounting_instruction_score_sheet.pdf

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Number Line Sheet

Strategic Number Counting Instruction: Number-Lines

0 1 2 3 4 5 6 7 8 9 10

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‘How RTI Works’ Series © 2012 Jim Wright www.interventioncentral.org

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http://www.interventioncentral.org/

Strategic Number Counting Instruction Score Sheet Student: __________________________ Interventionist(s):___________________________

Directions: During the strategic number counting instruction intervention, use this sheet to tally student responses: Number of Flash-Cards Known from Memory; Number of Flash-Cards Answered Correctly with Count-Up Strategy (with or without assistance); Number of Flash-Cards Unknown or Answered Incorrectly (even with assistance).

Date:

[Optional] Type/Range of Addition/Subtraction Math-Fact Flash-Cards Reviewed This Session:

Trial 1: Math Flash-Card Warm-Up: 3 Minutes

Number of Flash-Cards Known From Memory

Number of Flash-Cards Answered Correctly With Count-Up Strategy

Number of Flash-Cards Unknown or Answered Incorrectly

Trial 2: Math Flash-Card Review: 3 Minutes




Date:

[Optional] Type/Range of Addition/Subtraction Math-Fact Flash-Cards Reviewed This Session:

Trial 1: Math Flash-Card Warm-Up: 3 Minutes


Number of Flash-Cards Answered Correctly With Count-Up Strategy

Number of Flash-Cards Unknown or Answered Incorrectly

Trial 2: Math Flash-Card Review: 3 Minutes




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Concrete-Representational-Abstract (CRA)

Coach Card Description: CRA is a process in which the teacher guides the students through a math concept

and its corresponding computational process through the use of manipulatives and visual

representations (Agrawal & Morin, 2016). First, the student works hands-on with materials that

represent mathematical problems (concrete), then the student uses pictures which represent

mathematical problems (representational), and, finally, the student uses symbols or numbers to

model the mathematical concepts being learned (abstract) (Steedly, Dragoo, Arafeh, & Luke,

2008). The teacher must also develop a connection between the concrete, representational and

abstract representations of the mathematics problems.

Target Skills: All areas of math from kindergarten (K) to high school (HS) including: Counting

and Cardinality (K), Operations and Algebraic Thinking (K-5), Numbers and Operations in Base

Ten (K-5), Numbers and Operations-Fractions (3-5), Measurement and Data (K-5), Geometry

(K-HS), Ratios and Proportional Relationships (6-7), The Number System (6-8), Expressions and

Equations (6-8), Statistics and Probability (6-HS), Functions (8-HS), Numbers and Quantity

(HS), Algebra (HS), and Modeling (HS)

• Acquisition- the student is having difficulty solving the math problem because the task is

too hard for them.

• Proficiency- the student is having mathematical difficulties because they need to become

more fluent with the math skill.

• Generalization- the student is having difficulties solving mathematic problems because

they have never solved the task in that specific way.

Location: Whole Class, Small Group or Individual. In the classroom at any level (K-HS).

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Materials:

• Hands-on materials such as household materials (i.e. apples, beans, noodles) or classroom

materials (i.e. paperclips, stickers, cups, etc.). The National Library of Virtual

Manipulatives (www.nlvm.usu.edu) can be used to supplement hands-on materials

(Powell & Seethaler, 2011).

• Pictorial representations can be generated by word-processing programs or students can

draw pictures to represent the math problems (i.e. graphic organizers, number lines, tally

marks).

Frequency: As long as necessary for the students to understand the conceptual and procedural

skills related to specific math topics (Powell & Seethaler, 2011).

Progress Monitoring: Formal assessments (i.e. CBM probes and standardized assessments) and

informal assessments (i.e. classwork and homework) should be given during each stage of CRA

(example attached).

Directions: Explicit Instruction used during each stage. Stage 1= Use of hands-on activities,

Stage 2= Use of visual representation, Stage 3= Students demonstrate the ability to comprehend

numbers and symbols without concrete and representational phases (example attached).

• Advance organizer- the teacher links the new lesson with the previously taught lesson by

reviewing objectives of the previous lesson. Teacher gives rationale for learning the skill

of the day.

• Teacher demonstration- The teacher demonstrates the skill while describing the steps.

• Guided practice- The teacher gives prompts to solve a couple of the problems together.

• Independent practice- Students solve a few problems independently.

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http://www.nlvm.usu.edu/

• Problem-solving practice- The students and teacher solve the first word problem together

and the students solve the second word problem independently.

• Feedback- The students’ understanding is monitored and the decision to continue the

lesson or go back to step one of the lesson is made.

References:

Agrawal, J., & Morin, L. L. (2016). Evidence-based practices: Applications of concrete

representational abstract framework across math concepts for students with mathematics

disabilities. Learning Disabilities Research & Practice, 31(1), 34-44.

Powell, S. R., & Seethaler, P. M. (2011). Brief concrete-representational-abstract.

Evidence Based Intervention Network: University of Missouri. Retrieved from

http://ebi.missouri.edu/?page_id=805

Steedly, K., Dragoo, K., Arafeh, S., & Luke, S. D. (2008). Effective mathematics

instruction. Evidence for Education, 3(1), 1-12.

Attached Resources:

http://www.interventioncentral.org/teacher-resources/math-work-sheet-generator

http://ebi.missouri.edu/wp-content/uploads/2013/08/EBI-Brief-Template-Concrete-

Representational-Abstract.pdf

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http://www.interventioncentral.org/teacher-resources/math-work-sheet-generator

http://ebi.missouri.edu/wp-content/uploads/2013/08/EBI-Brief-Template-Concrete-Representational-Abstract.pdf

http://ebi.missouri.edu/wp-content/uploads/2013/08/EBI-Brief-Template-Concrete-Representational-Abstract.pdf

Concrete-Representational-Abstract CBM Probe Examples

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Concrete-Representational-Abstract Examples

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Chapter 8

Useful Websites for Support in Mathematics www.webmath.com

WebMath can be used by students who are having difficulty completing homework

problems by providing users with step-by-step instructions on how to solve the problem.

WebMath uses “fill-in-forms” where the student can type in the math problem on which they are

working. Within these forms are “math-solvers” which can instantly analyze the problem.

WebMath can be used at all ages. This site provides help for kindergarten all the way through

college Trigonometry and Calculus.

This site uses a variety of methods to help students solve their math problems. WebMath

uses a question-and-answers math database, tutorials, specific math sites, and math reference

sites. WebMath prides itself on giving the user the answer while providing step-by-step

instruction on how to get to the solution. This site does not just want to give immediate help but

also provides an in depth understanding of the mathematical problem.

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WebMath does not use link after link of static web pages containing information on

mathematics. This site uses a sophisticated computer “math engine” that can both recognize and

help show how to do the math on the particular problem.

WebMath understands that using computers to solve math problems has its

disadvantages. The overall development goal of the site is to keep increasing and improving its

math-solving capabilities to help students and give them a place that helps to answer their math

problem at any time.

illuminations.nctm.org

Illuminations is designed by the National Council of Teachers of Mathematics (NCTM)

to provide resources to math teachers and students. The website provides lesson plans, games

and brain teasers for teachers to use with their students. Standards-based resources are provided

for math teachers to use with students from pre-kindergarten through twelfth grade. There are

over 700 lesson plans with 100 corresponding activities. These activities encourage the use of

games to reinforce the material covered in the lessons.

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Once a teacher selects a lesson plan, they are provided with the grade level the lesson

corresponds with, the Common Core State Standard it relates to, and the time it takes to complete

the lesson. The lesson lists the prerequisite knowledge (if any is required) and then lists the

learning objectives that will be met. The materials required are listed with the instructional plan

and then a detailed lesson plan is provided. Following the lesson there are reflection questions

for both teachers and students. Illuminations provides ideas and stencils for manipulatives that

can be used along with the math lesson; especially useful for hands on learners.

This website is beneficial for teachers as it provides detailed lesson plans with time

requirements and possible activities with manipulatives for all grade levels that correspond to the

Common Core Standards. Using manipulatives adheres to the needs of visual and kinesthetic

learners while making real life connections to abstract concepts. There are interactive games and

brain teasers to encourage students to think critically. The brain teasers provide the correct

answer and explicitly state, with pictures if possible, how the answer was produced. The brain

teasers and the answers can be downloaded for teachers to use in class as a way to make math

interesting. The website also has links to five mobile apps for students to play games requiring

counting skills, maze skills, fraction knowledge and concentration skills.

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ebi.missouri.edu

Evidence Based Intervention (EBI) Network is a website that was created in 2009, as a

result of the East Carolina University Evidence Based Intervention Project and the Indiana

University School Psychology program. The website provides teachers and other educational

professionals with guidance in selecting and implementing evidence-based interventions,

pertaining to reading, math, and behavior, in a classroom setting. In 2013, Dr. Erica Lembke at

the University of Missouri, Dr. Sarah Powell at the University of Texas, Dr. Pamela Seethaler at

Vanderbilt University, and Elizabeth Hughes at Duquesne University joined the project with the

intention of developing a section of the EBI Network dedicated specifically to math

interventions.

The math interventions that are offered on this site are evidence-based and incorporate

both a focus on content areas such as Counting & Cardinality or Operations & Algebraic

Thinking, and the type of problem the child is having such as acquisition, proficiency, or

generalization. When navigating the site, the teacher/educational professional can search “math

interventions” by using the search engine located on the right hand side of the page. The

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teacher/educational professional is provided with a full list of about 15 interventions, which can

be broken down based on the reason the student is having difficulty (i.e. acquisition

interventions, proficiency interventions, generalization interventions and motivation

interventions). The name of the intervention is provided along with a description of how it is

related to the Common Core Standards, the setting it can be used in (i.e. whole class, small

group, individual), the focus area (i.e. acquisition, fluency, generalization, etc.), and a brief

overview. If the teacher is interested in the intervention, he/she can click on “Full Brief Link”

which will provide him/her with a printable document that contains details about the intervention

such as: the function of the intervention, a brief description, examples, procedures, materials, and

references.

This website also provides teachers and educational professionals with evidence-based

interventions in reading and behavior. Overall, the EBI website is useful for teachers and

educational professionals who are searching for evidence-based classroom interventions in

reading, math, and behavior.

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www.adaptedmind.com

Adapted Mind is created by graduates of Stanford, Berkeley, and Harvard who consulted

with teachers, parents, and children in order to make a unique website to help students learn math

in a fun and interactive manner. Adapted Mind was created for students in grades one through

eight. Each grade level has numerous lessons in different topics like number sense, basic

operations, money, geometry, algebra, and decimals; among others. There are multiple lessons

in each topic area with 500 problems in each lesson and an instructional video for extra

support. While the website appears to be geared towards parents, it could also be used in a

classroom by teachers. Visitors to this website can access one lesson for free with the use of an

email address before they must purchase a membership. Memberships cost $9.95 per month for

access for up to five children.

Some highlights of the website and its program include adaptive learning, tailor-made

worksheets, and progress monitoring. First, as a child goes through the lessons and plays the

games, the program adapts to the student’s abilities. The program notices if a child has mastered

a topic, or is displaying difficulty in an area, and will alter the lessons to

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accommodate. Repetition and reinforcement are also used to ensure retention and mastery. The

program tracks a child’s progress through the lessons and levels and will supply progress reports

to parents and teachers. Printable worksheets can also be created for the child based on his or

her progress and needs.

The games are based around an adorable and colorful monster theme. The child can

choose his or her monster at the beginning of the game. As the child plays, he or she must

answer math questions correctly to earn points and coins, and move through the levels. If the

child answers a problem incorrectly there is a video that will pop up and teach how to solve that

exact problem correctly. When the child earns enough coins, he or she is able to buy different

items for his or her monster, like balloons or a skateboard.

Overall, this website’s games and lessons provide children with interactive and engaging

activities all while teaching and reinforcing new math skills. It also provides tools to help

teachers and parents monitor student progress and track improvement as the child progresses

through the levels.

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www.insidemathematics.org

Inside Mathematics is a website created as a resource for teachers who have mathematics

as a part of their curriculum. This site contains useful resources for teachers to use to improve

their teaching skills, learn how to teach common core, and free lessons and mathematics

activities to print and use in the classroom. The website also provides teachers with a

professional learning community to contact and discuss challenges in the classroom. This

website will most benefit teachers as well as administrators who are looking for instructional

support.

At the top of the main screen there are several tab options. The first option is labeled

"Tools for Educators". This section contains a large number of free resources and lessons for

teachers. A lot of the resources are videos recorded by teachers instructing on how to teach and

implement different instructional strategies and lessons. Some activities that are included under

this section are free plans and instructions on how to implement "Problem of the Month". These

resources include lesson plans organized by grade and math topic. All lesson plans are aligned

with the Common Core State Standards (CCSS) and are free to print and reuse. The "Tools for

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Educators" section also contains links to videos, documents, and sample plans for principals and

math content area leaders to utilize when planning school curriculum or interventions.

Another useful tab at the top is the "Common Core Resources" tab. With the introduction

of Common Core, teachers needed to relearn how to teach the content to students. This tab

provides numerous resources for teachers to use in their classrooms. Videos have been created

to show teachers how to teach different topics within the CCSS math curriculum. Content within

this section is organized by grade level and then CCSS strand for easy use and navigation.

Overall, this website contains a large amount of valuable information for both teachers

and administrators. The information and resources found can be utilized when working towards

appropriately implementing the CCSS and teaching mathematics in a manner in which all

students can succeed.

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