Cover-Copy-CompareReviewed by Jennifer Wallace

Materials

One index card Math Fact Worksheet containing addition/subtraction/multiplication/division facts

written vertically on the left hand side of the paper. The right side of the paper is blank and contains enough space for the student to write.

Math Fact Worksheet containing addition/subtraction/multiplication/division facts without the answers spaced evenly over both halves of the paper.

Stopwatch Pencil

Directions

Place the first math fact worksheet with answers in front of the student. Instruct the student to do the following steps:

1. Look at the first problem on the worksheet.2. Cover the problem and the answer with the index card.3. Copy the problem in the white space next to it, while keeping original problem covered.4. Remove the index card from the problem and compare to original problem. 5. If the problem is written correctly, move onto the next problem6. If the problem is written incorrectly, provide the student with a separate sheet of paper

and instruct them to copy the problem three times then continue to the next problem.

Once all problems on the worksheet have been copied, give the student the second math fact worksheet with no answers. Use the stopwatch to time how many problems he completes from the time you say begin until one minute has passed. Compute the number of correct digits per minute and graph the data.

Tips

You can boost a student’s motivation to complete the worksheets by creating a portfolio of all of the student’s CCC worksheets. Every so often, review the portfolio with the student commenting on how much improvement the student is making.

This procedure can be combined with folding in (see later entry).

Trouble Shooting

What do if the student just copies the answer without covering it up?

Have a peer tutor, adult in the classroom, parent, or classroom teacher to sit with the child to ensure the procedure is being followed correctly

What if the student keeps losing their index card?

Try folding the worksheet in half lengthwise so the student is forced to flip the paper over and write his/her answer on the blank side.

References

http://www.interventioncentral.org/htmdocs/interventions/ccc.shtml

CCC worksheets can be made at:

Addition Worksheet Generator

http://www.lefthandlogic.com/htmdocs/tools/mathprobe/addsing.shtml

Subtraction Worksheet Generator

http://www.lefthandlogic.com/htmdocs/tools/mathprobe/subtsing.shtml

Multiplication Worksheet Generator

http://www.lefthandlogic.com/htmdocs/tools/mathprobe/multsing.shtml

Division Worksheet Generator

http://www.lefthandlogic.com/htmdocs/tools/mathprobe/divsing.shtml

Multi-Skill Worksheet Generator

http://www.lefthandlogic.com/htmdocs/tools/mathprobe/allmult.shtml

Touch Math

Touch math is a way for students to learn addition by teaching them to touch specific points on a written digit. The instructions as well as the touch points themselves can be found in the following sources. I have attached the two articles to the email I sent to you.

References

Scott, K.S. (1993). Multisensory mathematics for children with mild disabilities. Exceptionality,

4(2), 97-111.

Simon, R. & Hanrahan, J. (2004). An evaluation of the touch math method for teaching addition

to students with learning disabilities in mathematics. European Journal of Special Needs

Education, 19(2), 191-209.

http://www.touchmath.com

Following are

Reviewed by Amy Walter:

The Folding-In Technique

A more detailed explanation of The Folding-In Technique can be found in Shapiro’s Academic Skills Workbook (Step 3: Instructional Modification). The following example is an excerpt from that chapter.

Students: Two boys in third-grade special education class. Both boys referred for problems in learning multiplication facts.

Preassessment Phase: To determine the number of known and unkown facts, the students are administered a quiz in which they are asked to answer all computational problems with fact families 1-9. The number of problems not completed or incorrect provides an indication of the facts that have and have not been learned.

Instructional Structure: The procedure is set up as a peer-tutoring activity. The students are taught the procedure and are required to conduct 10-minute tutoring sessions in which they drill each other using the folding-in technique.

Step 1: Each student selects seven cards from their pile of preassessed known facts. Step 2: Each student selects one card from their unknown pile of presassessed facts. Step 3: The two students are informed by the teacher that they have 20 minutes to begin

tutoring. Step 4: After it is decided which student will begin the tutoring, the folding-in procedure

beings. The teacher of the pair presents the first unknown fact to the learner. The learner is required to write fact on a piece of paper, say it to him-/herself three times, and then turn the paper over.

Step 5: The teacher then presents a known fact, followed by the unknown fact, the first known fact, and another known fact. The unknown fact is presented sequentially in this fashion until all seven known facts have been presented and folded-in among the unknown facts. (See Figure 15 on page 121 for presentation sequence).

Step 6: The group of eight facts (one unknown and seven known) are shuffled. The second known fact is then presented and folded-in among the other eight facts. This is repeated again for the third unknown fact.

Step 7: If the student hesitates or is incorrect on any fact, the teacher instructs him/her to complete a brief correction procedure. The teacher tells him/her the correct answer and has him/her write the incorrect fact three times. The incorrect fact is then presented again to the learner.

Step 8: When all facts have been folded-in, the entire group of 10 facts is presented three times. Each time the packet of index cards is shuffled to prevent the learner from simply remembering the sequence of responses.

Step 9: The final step is a test of the 10 facts that the students have practiced. On this test, a mark is placed on the unknown fact cards if a student is correct on this trial. When an unknown fact attains three consecutive marks, it is considered as a learned fact.

Step 10: The number of new facts learned each week is graphed by the students. In addition, the teacher administers weekly curriculum-based measurement math probes taken from across all fact families. These data are also graphed.

Reciprocal Peer Tutoring to Improve Math Achievement

Appropriate Grade Level:Elementary and Middle school

Brief Description:The purpose of this intervention is to improve math performance and behavior during math instruction by means of peer tutoring, group rewards, and self-management procedures.Students monitor their academic progress in a group context, acting as instructional partners for each other, setting team goals, and managing their own group reward contingencies. Reciprocal peer tutoring has been demonstrated to improve not only math performance but also students’ perceptions of their own academic competence and self-control, and earns high satisfaction ratings from both teachers and students. The intervention takes approximately 30 minutes – 20 minutes for peer tutoring and 10 minutes for individual class drills and checking.

Materials Needed:

Reinforcement Menus with activity rewards, one per student pair “Team Score Cards,” consisting of 3” by 5” index cards or sheets of paper, one per

student pair per week Stickers for team score cards Flash cards with math problems printed on the front and the problem plus computational

steps and answers printed on the back, one problem per card, one set of cards per student pair

Sheets of paper divided into four sections: “try 1,” “try 2,” “help,” “try 3” Instructional prompt cards or sheets with specific instructions related to common

mistakes in solving math problems, one per student pair Problem drill sheets with 10 or more problems, one per student per session Answer sheets for problem drill sheets, one per student per session (optional)

Procedure:

1. Assess students’ current level of math performance by calculating percent-correct scores on daily math drill sheets or weekly quizzes, administering Curriculum-Based Math Probes, and/or observing students’ behavior during math work periods.

2. Tell the students that they will be learning to work in teams to help each other do well in math.

3. Divide the class into pairs. Provide each pair with a Reinforcement Menu listing activity rewards. Help each pair select a reward for the day.

4. Meet weekly with each team to help the students select their team goal. 5. After each pair has chosen a team goal, have the pairs record their expected individual

contribution to the team (individual goals), the sum of the individual goals (team goal), and their choice of a reward on the team score card.

6. Give a set of flash cards to each pair, and tell the students to choose who will act as “teacher” first.

7. Have the “teachers” hold up the flash cards for the students, and tell the students to work the problem on their worksheets in the section marked “try 1” while their teachers observe their work.

8. If the problem is solved correctly, the teachers praise the students and present the next problem. If the solution is incorrect, the teachers give students instructional prompts read from a prompt card and tell them to try again in the worksheet section marked “try 2.”

9. If the students do not solve the problem correctly on the second try, teachers help them by computing the problem in the “help” section of the worksheet. As teachers work the problem, they explain what they are doing at each step and answer students’ questions. Then the teachers tell the students to work the problem again in the “try 3” section. If teachers have trouble answering students’ questions, they can ask the classroom teacher for help.

10. After 10 minutes, signal the pairs to switch roles for a second 10-minute tutoring session.11. During tutoring sessions, walk around the room supervising and identifying strategies

“teachers” can use to help their students.12. After the second tutoring session, give each student a problem drill sheet and have students

work on their own for a fixed period of time, such as 7 to 10 minutes.13. Have students switch papers with their team partner. Have them use an answer sheet to

correct their partner’s work or provide the correct answers yourself as students check papers.14. Have the pairs first determine their team’s total score by counting the number correct, and

then have them compare their team score with their team goal to see if they have “won” (met their goal).

15. If a team wins, give the students a sticker to put on their score card for that day. After five wins, schedule a time when the team can engage in the previously selected reward activity.

16. Evaluate the intervention by repeating the first step and comparing results.

Tips:

Rewards can also be provided on a weekly classwide basis rather than on a daily team basis when a pre-determined percentage of teams meet their goals 4 out of 5 days during the week. Deliver the rewards to the entire class on Friday.

References

Rathovan, Natalie (1999). Effective School Interventions. Guilford Press: New York, NY.

Resources:Fantuzzo, J.W., King, J.A., & Heller, L.R. (1992). Effects of reciprocal peer tutoring on

mathematics and school adjustment: A component analysis. Journal of Educational Psychology, 84, 331-339.

Fantuzzo, J.W., & Rohrbeck, C.A. (1992). Self-managed groups: Fitting self-management approaches into classroom systems. School Psychology Review, 21, 255-263.

Improving Math Performance with Reciprocal Peer Tutoring and Parental Involvement

Appropriate Grade Level:Elementary and Middle School

Brief Description:The purpose of this intervention is the improve math performance and classroom behavior by combining reciprocal peer tutoring with home-based rewards. Parents are invited to develop a system of home rewards to support their child’s academic performance. Compared with control groups, students participating in this intervention showed significantly greater gains on curriculum-based math measures and standardized math achievement tests and on teacher and student self-report measures of adjustment.

Materials Needed:

Reinforcement Menus with activity rewards, one per student pair Introductory parent letter, one per student “Team Score Cards,” one per student pair per week Stickers for team score cards Flash cards with math problems printed on the front and the problems plus computational

steps and the answers printed on the back, one problem per card, one set of cards per student pair

Sheets of paper divided into four sections: “try 1,” “try 2,” “help,” “try 3” Instructional prompt cards or sheets with specific instructions related to common

mistakes in solving math problems, one per student pair Problem drill sheets with 10 or more problems, one per student per session Answer sheets for problem drill sheets, one per student per session (optional) Reward Certificates to send home

Procedure:

1. Assess students’ current level of math performance by calculating percent-correct scores on daily math drill sheets or weekly quizzes, administering Curriculum-Based Math Probes, and/or observing students’ behavior during math work periods.

2. Tell the students that they will be learning to work in teams to help each other do well in mathematics and that their parents will be invited to provide support and rewards in that effort.

3. Send a letter to parents that provides information about the RPT intervention and invites them to consider several options for involvement.

4. Divide the class into pairs. Provide each pair with a Reinforcement Menu listing activity rewards. Help each pair select a reward for the day.

5. Meet weekly with each team to help the students select their team goal. 6. After each pair has chosen a team goal, have the pairs record their expected individual

contribution to the team (individual goals), the sum of the individual goals (team goal), and their choice of a reward on the team score card.

7. Give a set of flash cards to each pair, and tell the students to choose who will act as “teacher” first.

8. Have the “teachers” hold up the flash cards for the students, and tell the students to work the problem on their worksheets in the section marked “try 1” while their teachers observe their work.

9. If the problem is solved correctly, the teachers praise the students and present the next problem. If the solution is incorrect, the teachers give students instructional prompts read from a prompt card and tell them to try again in the worksheet section marked “try 2.”

10. If the students do not solve the problem correctly on the second try, teachers help them by computing the problem in the “help” section of the worksheet. As teachers work the problem, they explain what they are doing at each step and answer students’ questions. Then the teachers tell the students to work the problem again in the “try 3” section. If teachers have trouble answering students’ questions, they can ask the classroom teacher for help.

11. After 10 minutes, signal the pairs to switch roles for a second 10-minute tutoring session.12. During tutoring sessions, walk around the room supervising and identifying strategies

“teachers” can use to help their students.13. After the second tutoring session, give each student a problem drill sheet and have students

work on their own for a fixed period of time, such as 7 to 10 minutes.14. Have students switch papers with their team partner. Have them use an answer sheet to

correct their partner’s work or provide the correct answers yourself as students check papers.15. Have the pairs first determine their team’s total score by counting the number correct, and

then have them compare their team score with their team goal to see if they have “won” (met their goal).

16. If a team wins, give the students a sticker to put on their score card for that day. After three wins, deliver the reward and give them reward certificates to take home to their parents. Parents are to sign the certificates and indicate the type of reward provided (if any) along with any additional comments.

17. Remind students to return the reward certificates to you so that you can monitor the home-based rewards.

18. Evaluate the intervention by repeating the first step and comparing results.

Tips:

This intervention can be modified for implementation in any academic subject that includes fact drills, such as spelling, reading vocabulary, history, and geography.

Source:Rathovan, Natalie (1999). Effective School Interventions. Guilford Press: New York, NY.

Resources:Heller, L.R., & Fantuzzo, J.W. (1993). Reciprocal peer tutoring and parent partnership: Does parent involvement make a difference? School Psychology Rev

Discrete Trial Training for Number IdentificationProcedure:

1. Numbers are randomly presented on flashcards. 2. Student is asked, “What number is this” and given as much time as he needed to respond

to each number presentation.3. If the student identified the number correctly, they are provided with a “Good job”

response and the card was placed in the correct pile.4. If they answered incorrectly, they are provided with a “No the number is ___, now you

say it” response and the card was placed in the incorrect pile. 5. After the last number is presented the correct pile is pushed aside and the incorrect pile is

re-administered using steps 2 through 4. This is repeated until all numbers were named correctly.

*No deviations of these responses are ever made which is crucial to learning in a discrete trail training format.

Reference: Gary Cates, Ph.D. Reference: Miltenberger, R.G. (2004). Behavior Modification Principles and Procedures. Belmont, CA:

Wadsworth/Thomson Learning.

Folding-In Technique with Math FactsPreassessment Phase: To determine which number facts are know and unknown, the students are administered a quiz in which they are asked to answer computational problems. The number of problems not completed or incorrect provides an indication of the facts that have and have not been learned.

Instructional Structure: The students are the taught and participate in a 10-minute session in which they use peer-tutoring to drilled each other using the folding-in technique.

Step 1: Each student selects seven cards from their pile of preassessed known facts. Step 2: Each student selects one card from their unknown pile of preassessed facts. Step 3: The two students are informed by the teacher that they have 20 minutes to begin

tutoring. Step 4: After it is decided which student will begin the tutoring, the folding-in procedure

begins. The teacher of the pair presents the first unknown fact to the learner. The learner is required to write the fact on a piece of paper, say it to him-/herself three times, and then turn the paper over.

Step 5: The teacher than presents a known fact, followed by the unknown fact, the first known fact, and another known fact. The unknown fact is presented sequentially in this

fashion until all seven know facts have been presented and fold-in among the unknown facts.

Step 6: The group of eight facts (one unknown and seven known) are shuffled. The second unknown fact is then presented and folded-in among the other eight facts. This is repeated again for the third unknown fact.

Step 7: If the student hesitates or is incorrect on any fact, the teacher instructs him/her to complete a brief correction procedure. The teacher tells him/her the correct answer and has him/her write the incorrect fact three times. The incorrect fact is then presented again to the learner.

Step 8: When all facts have been folded in, the entire group of 10 facts is presented three times. Each time, the packet of index card is shuffled to prevent the learner from simply remembering the sequence of responses.

Step9: The final step is a test of the 10 facts that the students have practiced. On this test, a mark is placed on the unknown fact cards if a student is correct on this trial. When an unknown fact attains three consecutive marks, it is considered a learned fact.

Step 10: The number of new facts learned each week is graphed by the student. In addition, the teacher administers weekly curriculum-based measurement math probes taken from across all fact families. These data are also graphed.

Reference: Shapiro, E.S.  (2004).  Academic skills problems workbook (Rev. ed.).  New York: The Guilford Press.

The Folding-In TechniquePresentation # Unknown Item # Known Item # 1 12 13 14 15 26 17 18 29 310 111 112 213 314 415 116 117 218 319 420 5

Presentation # Unknown Item # Known Item #21 122 123 224 3

25 426 527 628 129 130 231 332 433 534 635 736 237 138 139 240 1

Macrocloze/Slotting for Sentence ExpansionMacrocloze (Whaley, 1981) and slotting (Poteet, 1987) may be used to assist with sentence expansion.  In macrocloze, sentences are deleted from stories and the student is then asked to fill in the missing content.  With the slotting technique, blanks are inserted where context can be elaborated in a sentence, and then the student brainstorm descriptive words, phrases, and clauses that could be added to make the writing more mature.  The student then can select the choices that make the writing most interesting. 

Reference: Mather, N. & Roberts, R.  (1995).  "Informal Assessment and Instruction in Written Language:  A practitioner's Guide for Students with Learning disabilities."  Clinical Psychology Publishing Co.

Integrated Writing InstructionStudents with writing disabilities typically find the act of writing to be both

difficult and unrewarding. These students' resulting lack of motivation to write can lock them into a downward spiral, in which they avoid most writing tasks and fail to develop those writing skills in which they are deficient. Indeed, for some students, a diagnosed writing disability may not be neurologically based but instead can be explained by the student's simple lack of opportunities to practice and build competent writing skills.

MacArthur and colleagues (MacArthur, Graham, & Schwartz, 1993) have developed an integrated approach to classroom writing instruction designed to accommodate the special needs of disabled writers, as well as those of their non-disabled peers. In this instructional approach, the student writes about authentic topics that have a 'real-world' purpose and relevance. Student writing is regularly shared with classmates and the instructor, with these audiences creating a sustaining social context to motivate and support the writer. Students receive instruction and feedback in an interactive manner, presented both in lecture format and through writing conferences with classmates. Technology (particularly computer word processing) is harnessed to help the writing disabled student to be more productive and to make use of software writing tools to extend his or her own capabilities in written expression.

The instructor follows a uniform daily instructional framework for writing instruction. First, the instructor checks in with students about the status of their current writing projects, then teaches a mini-lesson, next allows the group time to write and to conference with peers and the teacher, and finally arranges for the group to share or publish their work for a larger audience. Status-checking. At the start of the writing session, the instructor quickly goes

around the room, asking each student what writing goal(s) he or she plans to accomplish that day. The instructor records these responses for all to see.

Mini-Lesson. The instructor teaches a mini-lesson relevant to the writing process. Mini-lessons are a useful means to present explicit writing strategies (e.g., an outline for drafting an opinion essay), as well as a forum for reviewing the conventions of writing. Mini-lessons should be kept shore (e.g., 5-10 minutes) to hold the attention of the class.

Student Writing. During the session, substantial time is set aside for students to write. Their writing assignment might be one handed out by the instructor that day or part of a longer composition (e.g., story, extended essay) that the student is writing and editing across multiple days. When possible, student writers are encouraged use computers as aids in composing and editing their work. (Before students can compose

efficiently on computers, of course, they must have been trained in keyboarding and use of word-processing software).

Peer & Teacher Conferences. Writers need timely, gentle, focused feedback from readers of their work in order to improve their compositions. At the end of the daily writing block, the student may sit with a classmate to review each other's work, using a structured peer editing strategy. During this discussion time, the teacher also holds brief individual conferences with students to review their work, have students evaluate how successfully they completed their writing goals for the day, and hear writers' thoughts about how they might plan to further develop a writing assignment.

Group Sharing or Publishing. At the end of each session, writing produced that day is shared with the whole class. Students might volunteer to read passages aloud from their compositions. Another method of sharing might be for the students to post their work on the classroom wall or bulletin board for everyone to read and respond to. Periodically, polished student work might be displayed in a public area of the school for all to read, published in an anthology of school writings, read aloud at school assemblies, or published on the Internet.

Reference: MacArthur, C., Graham, S., & Schwarz, S. (1993). Integrating strategy instruction and word processing into a process approach to writing instruction. School Psychology Review, 22, 671-681).

Following are reviewed by Chris Ratajski

Intervention BankMath

Planning, Attention, Simultaneous, Successive (PASS) Theory

This is especially helpful for students who have poor planning skills

Students are presented with math problems and asked to solve as many as they can in 10

minutes.

Self-reflection phase:

Goal: to have child realize the need to plan how to solve problem and use and

efficient method

The teacher will select some of the students’ work and display it anonymously to

spark class discussion

Teachers encourage children to

o Realize how they did the problems

o Explain thoughts

o Verbalize which methods worked and which did not

o Be self-reflective

o The following questions may be used by the teacher to encourage this

reflection:

“Can anyone tell me anything about these problems?”

“Let’s talk about how you did the work sheet.”

“Why did you do it that way?”

“How did you do the problems?”

“What could you have done to get more correct?”

“What did it teach you?”

“What else did you notice about how this page was done?”

“What will you do next time?”

“I noticed many of you did not do what you said was important.

What do you think of that?”

The teacher does not:

o Give the correct answers

o Tell which method to use

o Give math instructions

Naglieri, J. A. & Gottling, S. H. (1997). Mathematics instruction and PASS Cognitive processes: An intervention study. Journal of Learning Disabilities, 30, 513-520.

The MASTER Program: Division and multiplication for children with mathematics disabilities

Children with mathematics disabilities often do not make the connection between what

they know and a new task. (Ex. 2 X 3 = 6 and 6 / 3 = 2)

The goals of the program are:

To realize multiplication is repeated additions

To understand reversibitiliy (ex: 3 X 8 = 8 X 3), associations, and doubling

To memorize the multiplication facts below 100

To realize addition is repeated subtraction

To understand how division and multiplication are related

To memorize all division facts below 100

To use multiplication in division in real and true-to-life problems.

The program consists of 23 multiplication sessions and 19 division sessions. The

children may manipulatives at first and then are asked to solve problems mentally.

Multiplication Lessons (23):

Basic procedures – 8

Multiplication tables – 11

Specific problems – 4

Division Lessons (19):

Division – 7

Division with remainders – 6

Division without remainders – 5

Denominators between 10 and 20 – 1

Curriculum based test: Student needs to obtain an 80% or more to go on to next series.

VanLuit, J. E. H., & Naglieri, J. A. (1999). Effectiveness of the MASTER program for teaching special children multiplication and division. Journal of Learning Disabilities, 32, 98-107.

Algebra – Equations

This may be used when initially teaching students about balancing algebraic equations or

to help students who do not understand that both sides of the equation need to be equal.

Place a scale between the teacher and the student

Explain that both sides of the scale need to be equal to balance

Put a numbered cube on the scale and an object of equal weight on the side of the scale.

Ask the student what the object must equal to be balance. (The answer is the same

number as is on the cube)

Repeat this at least one time with different numbered cubes

Tell the student that the object will be called X and that X must always have the same

value in the equation. This will help the student learn about variables.

Put 2 objects on one side of the scale and a cube with an 8 on the other side of the scale

Ask what X must be for the scale to balance.

Repeat this with other cubes until the student gets it correct 5 times.

Write an algebraic equation on a piece of paper and have the student determine the

answer using his or her scale, objects and cubes. (ex.: 2x = x + 3).

Help the student set up the objects and cubes on the scale.

Ask the child to guess what X is.

Have the child check the guess by plugging it into the equation.

www.usd.edu/cpe/math.htm

Algebra – Mnemonic

This mnemonic may be useful for students who have difficulty remember the rules or the

order of operations.

Please excuse my dear Aunt Sally

1st: Parenthesis: solve what is in Parentheses

2nd: Exponents

3rd: Multiply

4th: Divide

5th: Add

6th: Subtract

www.abcteach.com

Setting goals and improving students’ attitudes towards mathDetermine student’s instructional level

Have the student help set challenging but attainable goals.

Guarantee accomplishment of the math goals by

Decreasing the difficulty of the instructional sequence

Building on previous skills

Using word problems that relate to student’s life

Use charts to let students know how doing

Reinforce student for effort

Explain that mistakes are part of the learning process

www.usd.edu/cpe/math.htm

Following reviewed by Sarah Reck

Sarah ReckDiagnostic Procedures

Intervention Bank: Mathematics

Introduction

Mathematics interventions for elementary-aged children tend to focus on direct

instruction techniques (Allsopp, 1999). Because mathematics is very rule-oriented in

nature, and because virtually every skill builds on previous skills, it makes sense that

interventions generally focus on ensuring that the student knows and understands the

rules and the order of the rules. Allsopp (1999) makes the following comments about

direct instruction:

“It does not have to be boring. You should implement direct instruction

systematically, but with enthusiasm, with creativity, and in a fluid manner.

Providing ample amounts of specific feedback, which includes both positive

feedback and corrective feedback, is also a critical feature of direct

instruction. You should provide feedback throughout the direct instruction

process.”

However, if the step of direct instruction is absent from a student’s repertoire, or if they

missed parts of direct instruction for any reason, interventions need to be put in place.

There are few empirically validated direct interventions for mathematics that don’t involve

teaching math facts. Many interventions for teaching math facts have been found;

however, I have choose to focus on some more complex problems that involve more than

learning simple math (addition, subtraction, multiplication, or division) facts. Problem-

solving interventions generally involve teaching students new ways of memorizing the

rules (i.e., mnemonics or a mathematics dictionary).

There has been some support found for using self-monitoring strategies when

children are completing independent math seatwork (McDougall & Brady, 1998).

However, these interventions tend to be intrusive. It is suggested that as students catch

on to the procedure, the equipment, such as the tape player and headphones, can be

gradually faded.

Some tried-and-true methods of increasing mathematics achievement to try are

interspersing of problems that a student already knows how to do with unknown problems

(Wildmon, Skinner, & McDade, 1998), the cover-copy-compare method, and peer

tutoring. Touchmath (www.touchmath.com) is recommended as a secondary

instructional program for student struggling in math. Touchmath is a systematic,

multisensory program used to improve basic computational skills.

References for IntroductionAllsopp, D. H. (1999). Using Modeling, Manipulatives, and Mnemonics with Eighth-Grade Math Students.

Teaching Exceptional Children, 32(2), 74-81.McDougall, D., & Brady, M. P. (1998). Initiating and fading self-management interventions to increase math

fluency in general education classes. Exceptional Children, 64, 151-166.www.touchmath.comWildmon, M. E., Skinner, C. H., McDade, A. (1998). Interspersing additional brief, easy problems to increase

assignment preference on mathematics reading problems. Journal of Behavioral Education, 8(3), 337-346.

From the University of South Dakota School Psychology Program Intervention Site:

TIC TAC MATH1. Do Math Lesson  2. Divide class into two groups. 

3. Draw a tic tac toe grid on the board. 

4. Write an X on the right side of the board, and an O on the left side of the board as each group’s logo. 

5. Have one student from each group approach the board. 

6. Teacher calls out a question having to do with the lesson. 

7. Students write the correct answer as quickly as possible. 

8. The teacher asks the class who is correct and why. 

9. The first student with the first correct answer writes their group logo on the tic tac toe grid. 

10. The first team to get a straight line of 3 is the winning team. 

11. During the game students are told the real winning team is the one who gave the most positive reinforcement.  Competition should not be emphasized. 

12. During the game, the teacher will put a tally mark by students name for each positive comment.  Each student should come up with 5.  Reminders throughout the game may be necessary. 

13. End with a discussion on the importance of positive speech between peers. 

Reference: Stensland, T. (1996). Lesson Plan #: AELP-ATH0021

Tic Tac Math works because it involves a reinforcement contingency. By adding an element of competition, students become motivated to work quickly and correctly.

Mathematics DictionaryFor student struggling in math understanding the technical language of math, Dr. Mel Levine (2005) recommends a “Math Dictionary.” Many children have trouble remembering terms like exponent, hypothesis, and denominator, which can cause lower mathematics performance. They are more likely to become confused with word problems and verbal explanations of the processes.

The dictionary should be designed by the child. The additional opportunity for rehearsing learned information will help the information stick in the child’s head. The dictionary should include examples of every term that appears in the dictionary. Students need practice looking at word problems and just identifying what process (such as subtraction) will be needed – without having to solve the problem. To help with problems understanding verbal explanations, teachers should give these kids correctly solved problems (demonstration models) to analyze and talk about.

Reference: Levine, M. (2005) The pathways of math’s ways. Retrieved 20 April 2005 from http://www.allkindsofminds.org/ArticleDisplay.aspx?articleID=4

Self-Monitoring and Self-Reinforcing Using Systematic Self-Management Training Procedures

McDougall, D., & Brady, M. P. (1998). Initiating and fading self-management interventions to increase math fluency in general education classes. Exceptional Children, 64, 151-166.

1. Record student’s voice on a cassette tape, first saying “Am I paying attention?” Rearrange the tape to have it repeated on average, once a minute. 2. Intermittently (approximate one cue per minute, with the range between 50 – 70 s), they will hear their own tape-recorded voices asking, “Am I paying attention?” Then participants self-monitor by (a) deciding (i.e., self-assessing) whether their behavior at the time of the cue met objective criteria for on-task behavior, and (b) writing (i.e., self-recording) a check mark by the word “Yes” or “No” printed on a form taped to their desks. When participants used the second package (i.e., self-monitoring of productivity) they heard their own tape-recorded voices3. When this increases a student’s work completion for mathematics, re-record the tape to have the student’s voice say, “Am I working quickly?” Then students self-assess and self-record their responses on a form taped to their desks.4. Students can earn “bonus points” when they beat their own personal best.5. Students should be working toward a tangible reinforcer. Once a certain number of point is earned (the amount determined appropriate by both the teacher and the student—it should be an amount that is not impossible to obtain, but one that will force the child to work hard), the student will earn a self-selected reinforcer. McDougall and Brady

(1998) suggest sugarless gum, a blank audio cassette, or an announcement on the school’s intercom system that acknowledged the student’s name and math fluency performance.

McDougall and Brady (1998) found support for such an intervention in increasing math fact fluency and academic engaged time for the period that the intervention was in place in both typical and learning disabled students in a regular education classroom.

Procedure for Self-Instruction in Math

Shapiro, E. S., & Cole, C. L. (1994). Behavior change in the classroom: Self-management interventions. New

York: Guildford Press.

This is an intervention for early elementary school years, when children are learning addition problems.

Materials needed: Marked number line, pencil, & sheet of addition problems1. Ask, “What is my problem?”2. Circle the “+” sign.3. Match “+” sign to sign on number line4. Copy arrow over problem5. Find top/largest number on number line6. Hold place with finger or pencil7. Find bottom/smallest number and move in direction of arrow that many

spacers on the line.8. Read answer from the number line.9. Copy answer below the equals sign.10. Ask, “How did I do?”

Using MnemonicsTaken from Allsopp, D. H. (1999). Using modeling, manipulatives, and mnemonics with eighth-grade math students. Teaching Exceptional Children, 32(2), 74-81.

In doing so, use the following steps:    * Model when to use the mnemonic (its purpose).    * Model what each letter in the mnemonic stands for and how it relates to solving the math problem.    * Model how to apply it to what students have already learned about solving problems.    * Provide students with cue cards to place on their desks or to place in their math folders.    * Put up posters of a mnemonic on the walls of the classroom.    * Use rapid-fire-verbal-rehearsal to help students remember mnemonics.    * Rapid-fire-verbal-rehearsal is a method used in the Kansas Learning Strategies Series (Deshler & Lenz, 1989; Deshler & Schumaker, 1986) and entails the teacher asking students randomly what a particular mnemonic is used for and what specific letters stand for. As students demonstrate the ability to correctly answer the prompts, begin to speed up the rate of questioning. When done with enthusiasm and dramatic flare, rapid-fire drills can be very motivating for students, as well as for the teacher. You may also select individual students to lead the rapid-fire-verbal-rehearsal.

Following is an example of a mnemonic to use when solving story problems…

The Sir Right Problem-Solving StrategyGive a problem solving strategy the student can follow for word problems, checking off each step as they goes. For example, the letters SIR RIGHT stand for:

1. Start by reading the problem 2. Identify all numbers (digits and words) 3. Reread problem and draw a picture of diagram 4. Reread problem again to find the question 5. Inquire "What do I have to do to answer the question?"6. ‘Guesstimate,’ or estimate, an answer (use smaller numbers if puzzled by

large numbers)

7. Have a go at computing an answer 8. Take a check back to see if the answer makes sense

From All Kinds Of Minds: Understanding Differences in Learning (http://www.allkindsofminds.org/CaseStudy.aspx?casestudyid=6)

Teacher-Student Think AloudAllsopp, D. H. (1999). Using Modeling, Manipulatives, and Mnemonics with Eighth-Grade Math Students. Teaching Exceptional Children, 32(2), 74-81.

An academic teaching strategy to use in a large group setting that gets students involved is the Teacher-Student Think Aloud. The students are able to hear the problem-solving strategy out loud.

Teacher Think Aloud: Teacher asks a question and teacher answers question. Teacher-Student Think Aloud: Teacher asks a question and students help

provide answer.

(An Example of Teacher Think Aloud)Now I see I have the fraction 8/12 on the board. I know that when I have a

fraction, I should reduce it to its lowest possible terms. I know that I have to find the biggest number that goes into both the top number and the bottom number evenly.(Example of Teacher-Student Think Aloud)

Now I wonder what I do with the 4, the biggest number that goes into both the top number and the bottom number evenly? (Teacher elicits student response, “You divide both the numerator and the denominator by that number!”). That’s right! Hmm, now I wonder what the answer is? (Teacher elicits student response, "It is 2/3.") Great thinking!

The Ask-Do-Do-Ask Procedure for Increasing Reducing Fractions Skills

(Taken from personal communication with Gary Cates, Ph.D. on 3/7/2005)The Ask-Do-Do-Ask Procedure

1. Ask: What number will both of these divide into evenly?2. Do: Divide the top number by this LCD.3. Do: Divide the bottom number by this LCD.4. Ask: Can this be reduced more? If so, go back to number 2. If not, write your

answer.

The Ask-Do-Do-Ask Procedure—Modified for students without vocabulary knowledge (i.e., LCD)

1. Ask: What number will go evenly into both the bottom number and the top number?

2. Do: Divide the top number by this new number. [Additional prompt: How many times does this new number go into the old number?]

3. Do: Divide the bottom number by this new number. [Additional prompt: How many times does this new number go into the old number?]

4. Ask: Is there any other number that can go evenly into these new numbers? If so, then go back to number 2. If not, write your answer.

If the student possesses the vocabulary, use the first version of the outline; if the student does not, use the second version of the outline. Include modeling and prompting in the procedure: begin by modeling 3 problems, while saying the steps out loud, and answering yourself. Then, ask the student to try a problem, providing the necessary prompts (i.e., say the steps out loud). Then model 3 more problems. As the student becomes successful, fade back how many problems you do versus how many they do, until they are doing the majority of the problems.

A case study completed for Diagnostic Procedures found this to be an effective intervention for increasing one child’s fraction reduction skills.

Cover, Copy, Compare ProcedureShapiro, E. S., & Cole, C. L. (1994). Behavior change in the classroom: Self-management interventions. New

York: Guildford Press.

1. Students are given a sheet of math problems with the problems and their answer listed in a column on the left hand side of the paper.

2. Students are instructed to look at the first problem and its answer.3. Students then cover the answer with a piece of construction paper.4. Students then write the problem and answer on the right hand side of the original

math problems.5. Students uncover the problem and evaluate if their answer was correct.6. If correct, students place a “+” mark next to the problem.7. If incorrect, students repeat the procedure until the response is correct.

Math Problem-Solving StrategyMontague, M., & Bos, C. S. (1986). The effect of cognitive strategy training on verbal math problem solving

performance of learning disabled adolescents. Journal of Learning Disabilities, 19, 26-33.

1. Read the problem aloud. Have a teacher help you identify any unknown words.2. Paraphrase the problem. Reread the problem, identify the question that is

asked, and summarize the information that will be important for solving the problem.

3. Visualize. Draw a picture of the problem or visualize the situation and tell what it is about.

4. State the problem. Underline the most important information in the problem and then complete the sentence: “What I know is…” and “What I want to find out is…”

5. Hypothesize. Complete the sentence: “If…then..”6. Estimate. Estimate an answer that would make sense. Write it down.7. Calculate. Calculate the answer and label it.8. Self-check. Review the problem, check computation, and ask if your answer

makes sense.

Touch MathMather, N, & Jaffe, L. (2002). Woodcock-Johnson III: Reports, recommendations, and strategies. New York: John Wiley & Sons.www.touchmath.com

Because Touch Math is a popular teaching strategy in elementary schools, I will provide a brief overview. Touch Math is a systematic, multisensory program used to introduce and improve basic computational skills. This program is effective for students who have difficulty memorizing math facts and the steps of the four basic operations. This supplemental program is used in conjunction with the existing math program in kindergarten through third grade or with students at any level who need help with basic math skills.

Each number is assigned the number of Touchpoints that are associated with it’s value. When the students no longer need them, the Touchpoints are gradually removed from the numbers (the student continues to “touch” them from memory),, at which point the student can beign to use general classroom materials while continuing instruction in the Touch Math materials.

At each stage, visual cues and simple rule statements reinforce the student’s learning of the sequence of steps. For example, in Step 1 addition, the student begins by touching and counting all of the Touchpoints. In step 2, the student names the larger number and touches points of the smaller number, point by point, while counting on. The student also verbalizes the procuedre: “I touch the larger number, say its name and continue counting.” When the student advances to two-digit addition, an arrow is drawn over the units column and the student learns, “I start on the side with the arrow. The arrow is on

the right side.” A square over the tens column serves as a reminder to “carry over” a number.

To help you visualize what TouchMath looks like, I’m attaching an example of a worksheet for second grade Touchmath.

Following reviewed by Tara Dickey

Cover-Copy-Compare

Students who can be trusted to work independently and need extra drill and practice with math computational problems, spelling, or vocabulary words will benefit from Cover-Copy-Compare.

Intervention:Preparing Cover-Copy-Compare Worksheets:The teacher prepares worksheets for the student to use independently:

For math worksheets, computation problems with answers appear on the left side of the sheet. The same computation problems appear on the right side of the page, unsolved. Here is a sample CCC item for math:

For spelling words, correctly spelled words are listed on the left of the page, with space on the right for the student to spell each word.

For vocabulary items, words and their definitions are listed on the left side of the page, with space on the right for the student to write out each word and a corresponding definition for that word.

Using Cover-Copy-Compare Worksheets for Student Review:When first introducing Cover-Copy-Compare worksheets to the student, the teacher gives the student an index card. The student is directed to look at each correct item (e.g., correctly spelled word, computation problem with solution) on the left side of the page.

(For math problems) The student is instructed to cover the correct model on the left side of the page with an index card and to copy the problem and compute the correct answer in the space on the right side of the sheet. The student then uncovers the correct answer on the left and checks his or her own work.

Source: www.interventioncentral.com

Help Signal

The time that students spend in the classroom actually working on academic subjects is sometimes referred to as 'engaged time.' During independent seatwork, difficult-to-teach students may not have effective strategies to ask for teacher help. Instead, when these students encounter a problem or work example that they cannot complete on their own, they may start to act out, distract peers seated around them, interrupt the teacher (who may be working with another group of students), or simply sit passively doing nothing. The help-signal is a flexible procedure that the student can use to get teacher assistance during independent seatwork. It allows the student to signal the teacher unobtrusively for help while continuing to work productively on alternative assignments.

Intervention: Step 1: Select a Student Signal. Decide on a way that the student can signal that they require teacher help. One approach is to prepare a 'help-flag' (a strip of colored, laminated posterboard) with the word 'Help' or similar word written on it. Attach a Velcro tab to the flag and affix a corresponding adhesive Velcro strip to the student's desk.

Step 2: Create an Alternative Work Folder. Create a student work folder and fill it with alternate assignments or worksheets that the student can work on independently. For example, you might insert into the folder math worksheets, a writing assignment, or lists of reading vocabulary words to be practiced.

Step 3: Introduce the Program to the Student. Set aside time to meet with the student to introduce the help-signal routine.

Show the student how to post the help-flag or other help signal. Instruct the student that he or she should post the help-signal whenever he

or she becomes stuck on seatwork and needs instructor assistance. Tell the student that--after posting the help-signal--the student should next

check over the current work assignment to see if their other problems or items that he or she can work on while waiting for the teacher.

Show the student the alternative-work folder. Tell the student that, if he or she cannot continue on any part of the seatwork, the student should pull out the folder and to begin to work on an alternative assignment. The student is to continue working on that assignment until the teacher or other staff member can get to the student's desk to provide assistance. Also, be sure that your student knows during what activities and times during the school day that he or she is to use the help-signal to indicate that adult attention is needed.

Give the student a chance to try out the help-signal under your guidance, and offer feedback about the performance. Let students know that if they stand and approach you for help directly rather than posting the help-signal, you will remind them to use the signal and then send them back to their seat

Step 4: Begin the Intervention. Start the help-signal as soon as you feel that the student understands and will comply with the system. Take care to scan the room periodically when you are free during student independent seatwork to see if any students might need your assistance.

Source: www.interventioncentral.com

Immediate Self-Correction

Intervention: The immediate self-correction intervention involves the learner self-corrects her work, using answer keys, immediately following the completion of items. The learner circles each error, then immediately writes the correct answer below each circled response.

Source: Bennett, K., & Cavanaugh, R. A. (1998). Effects of immediate self-correction, delayed self-correction, and no correction on the acquisition and maintenance of multiplication facts by a fourth-grade student with learning disabilities. Journal of Applied Behavior Analysis, 31, 303-306.

Making Estimations

Materials: 2 identical sized containers (e.g., beakers A & B, which are short and wide) and 1 different sized container (e.g., beaker C, which is taller & thinner); liquid (e.g., water); measuring cup

Intervention: Step 1: Two identical beakers (beakers A & B), each filled with the same amount of liquid, are presented to the students. Step 2: Then the teacher pours the liquid from beaker B into beaker C. Step 3: The students are asked if these beakers (A and C) have the same amount of liquid. Step 4: Discuss the students’ answers and thoughts.

Reference: Santrock, J. W. (1994). Child development. Madison, WI: Brown & Benchmark Publishers.

Source: South Dakota Mathematics Intervention Website ( http://www.usd.edu/cpe/math.htm#35 )

Promoting Positive Attitudes Toward Math

Intervention: Involve students in setting challenging but attainable instructional goals. Goal setting has a powerful influence on student involvement and effort.  Ensure success by building on prior skills and using task analysis to simplify the instructional sequence of a math skill or concept. Use charts to give students feedback on how well they are doing.  Use word problems that are part of a student's daily life. Reinforce students for effort on math work and stress that errors are learning opportunities. 

Reference: Mercer, C. D., Mercer, A. R. (1998). Teaching Students with Learning Problems, 5th edition. Prentice-Hall, Inc., p. 476. 

Source: South Dakota Mathematics Intervention Website (http://www.usd.edu/cpe/math.htm#35)

Integrate Math With Other Content Areas

Intervention: Teachers can integrate math with other content areas in a meaningful way. Students who can link math to other subject fields are able to apply previously acquired information to new situations.  Teachers have many opportunities to integrate math with other areas of the curriculum and enhance learning.  For example, students cold be asked to analyze measurements collected during a science experiment or to accumulate data on a presidential candidates success in parts of the country as indicated by the polls. Another example that shows how important mathematics is and how it is integrated in all aspects of a students life is to challenge the students to go through their school day WITHOUT using mathematics anywhere except in math class can open there eyes to the vital connections among math and other subjects. 

Reference: Beyth-Maron, R., & Dekel, S. (1983). A curriculum to improve thinking under uncertainty. Instructional Science, 12, 67-82.

Source: South Dakota Mathematics Intervention Website (http://www.usd.edu/cpe/math.htm#35)

Drilling with FlashcardsIntervention:Step 1: Obtain flashcards with the problem on the front (e.g., 8x6=__) and the answer on

the back (e.g., 48).Step 2: Hold up one flash card and wait for a response. Step 3: If the student responds correctly, place the flashcard in a “correct” pile and repeat

Step 2. If the student responds incorrectly, correct the student (e.g., “No, the answer is 48”) and instruct the student to repeat the entire problem (e.g., “Eight times six is forty-eight”) and place this card in an “incorrect” pile. Repeat steps 2 and 3 until the student completes the entire stack.

Step 4: At the end, repeat steps 2 and 3 with the cards in the “incorrect” pile until all the cards are in the “correct” pile.

Source: Gary Cates, Ph.D.

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