Research Based
Math Interventions
Provision of Basic Research-Based Math Interventions
The North Carolina Department of Public Instruction has not designated specific researchbased interventions for use at the Intervention Team level. Instead, the Department of Public
Instruction indicates that the intensity of instruction or intervention should vary depending upon
student need. The curriculum, materials, and methods used should be research validated.
Intensity may be increased by:
1.)
2.)
3.)
4.)
5.)
Increasing instructional time
Decreasing the teacher to student ratio
Using alternative teaching procedures
Using alternative instructional materials
Using motivational strategies
When attempting to improve student performance, Intervention Teams may use more than one of
these strategies. The following pages contain alternative research-based teaching procedures,
instructional tools, and motivational strategies.
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.
Jim's Hints for Using...
Cover-Copy-Compare
When using CCC
worksheets, add an
occasional item (e.g.,
vocabulary word, math
problem) that the student
has already mastered.
These review items are
great for refreshing student
skills on learned material
and can also give the
teacher an indication of how
well the student is retaining
academic skills.
You can boost student
motivation by praising the
student for his or her efforts
in completing the
worksheets. You might also
want to have the student
build a portfolio of
completed CCC worksheets.
In reviewing this portfolio of
work periodically, the
student can see tangible
evidence of improvement in
his or her academic skills.
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.

(For spelling problems.) The student is instructed to cover the correct model on the left side of
the page with an index card and to spell the word in the space on the right of the sheet. The
student then uncovers the correct answer on the left to check his or her work.

(For vocabulary items.) The student is instructed to cover the correct model on the left side of
the page with an index card and to write both the word and its definition in the space on the
right side of the sheet. The student then uncovers the correct model on the left to check his or
her work.
Troubleshooting: How to Deal With Common Problems in Using 'Cover-Copy-Compare'
Q: How do I respond if the student simply copies the correct answers from the models into the answer
blanks and tries to pass this off as his or her own work?
An essential requirement of Cover-Copy-Compare is that the student cover the correct model and
attempt independently to solve the item using his or her own skills. if the student simply copies the
correct answer from the model math problem or spelling word, the review process is short-circuited
and the student will not benefit. If you suspect a student will copy rather than attempt to solve items
on a CCC worksheet, arrange to have a peer tutor, adult in the classroom, or parent sit with the
student to provide encouragement and monitoring.
Q: I have a student who is so disorganized that he will lose the index card before he can complete a
CCC worksheet. Any suggestions?
Here is an idea for getting rid of that index card: You can fold the worksheet in half length-wise so that
the answers appear on one side of the folded worksheet and the answer blanks appear on the other
side. For each item, the student will peer at the correct model, then flip the folded sheet over to the
right side to independently solve the item (with the correct model neatly folded out of sight).
Math Computation: Promote Mastery of Math Facts Through
Incremental Rehearsal
Incremental rehearsal builds student fluency in basic math facts ('arithmetic combinations') by
pairing unknown computation items with a steadily increasing collection of known items. This
intervention makes use of concentrated practice to promote fluency and guarantees that the student
will experience a high rate of success.
Materials

Index cards & pen
Steps to Implementing This Intervention
In preparation for this intervention:
1. The tutor first writes down on an index card in ink each math fact that a student is expected
to master-but without the answer. NOTE: Educators can use the A-Plus Math Flashcard
Creator, a free on-line application, to make and print flashcards in addition, subtraction,
multiplication, and division. The web address for the flashcard creator is:
http://www.aplusmath.com/Flashcards/Flashcard_Creator.html
2. The tutor reviews the collection of math-fact cards with the student. Any of the math facts that
the student can orally answer correctly within two seconds are considered to be known
problems and are separated into one pile. Math facts that the student cannot yet answer
correctly within two seconds are considered 'unknown' and collected in a second pile -- the
'unknown facts' deck.
3. The tutor next randomly selects 9 cards from the pile of known math facts and sets this
subset of cards aside as the 'known facts' deck. The rest of the pile of cards containing
known math facts is put away ('discard deck'), not to be used further in this intervention.
During each day of the intervention:
The tutor follows an incremental-rehearsal sequence each day when working with the student:
1. First, the tutor takes a single card from the 'unknown facts' deck. The tutor reads the math
fact on the card aloud, provides the answer, and prompts the student to read off and answer
the same unknown problem.
2. Next the tutor takes one math fact from the 'known facts' deck and pairs it with the unknown
problem. When shown the two problems in sequence, the student is asked during the
presentation of each math fact to read off the problem and answer it. The student is judged to
be successful on a problem if he or she orally provides the correct answer to that problem
within 2 seconds. If the student commits an error on any card or hesitates for longer than two
seconds, the tutor reads the math fact on the card aloud, gives the answer, then prompts the
student to read off the same unknown problem and provide the answer. This review
sequence continues until the student answers all cards within two seconds without errors.
3. The tutor then repeats the sequence--taking yet another problem from the 'known facts' deck
to add to the expanding collection of math facts being reviewed ('review deck'). Each time,
the tutor prompts the student to read off and answer the whole series of math facts in the
review deck, beginning with the unknown fact and then moving through the growing series of
known facts that follow it.
4. When the review deck has expanded to include one 'unknown' math fact followed by nine
'known' math facts (a ratio of 90 percent 'known' material to 10 percent 'unknown' material),
the last 'known' math fact that was added to the student's review deck is discarded (put away
with the 'discard deck'). The previously 'unknown' math fact that the student has just
successfully practiced in multiple trials is now treated as a 'known' math fact and is included
as the first item in the nine-card 'known facts' deck for future drills.
5. The student is then presented with a new math fact to answer, taken from the 'unknown facts'
deck. With each new 'unknown' math fact, the review sequence is again repeated as
described above until the 'unknown' math fact is grouped incrementally with nine math facts
from the 'known facts' deck-and on and on.
Daily review sessions are discontinued either when time runs out or when the student answers an
'unknown' math fact incorrectly three times.
References
Burns, M. K. (2005). Using incremental rehearsal to increase fluency of single-digit multiplication
facts with children identified as learning disabled in mathematics computation. Education and
Treatment of Children, 28, 237-249.
Math Computation: Increase Accuracy By Intermixing
Easy and Challenging Computation Problems
Teachers can improve accuracy and positively influence the attitude of students when completing
math-fact worksheets by intermixing 'easy' problems among the 'challenging' problems. Research
shows that students are more motivated to complete computation worksheets when they contain
some very easy problems interspersed among the more challenging items.
Materials

Math computation worksheets & answer keys with a mixture of difficult and easy problems
Steps to Implementing This Intervention
1. The teacher first identifies one or more 'challenging' problem-types that are matched to the
student's current math-computation abilities (e.g., multiplying a 2-digit number by a 2-digit
number with regrouping).
2. The teacher next identifies an 'easy' problem-type that the students can complete very quickly
(e.g., adding or subtracting two 1-digit numbers).
3. The teacher then creates a a series of student math computation worksheets with 'easy'
computation problems interspersed at a fixed rate among the 'challenging' problems. (NOTE:
Instructions are included below for creating interspersal worksheets using a free online
application from www.interventioncentral.org.)
o
If the student is expected to complete the worksheet independently as seat work or
homework, 'challenging' and 'easy' problems should be interspersed at a 1:1 ratio (that
is, every 'challenging' problem in the worksheet is followed by an 'easy' problem).
o
If the student is to have the problems read aloud and then asked to solve the problems
mentally and write down only the answer, the items should appear on the worksheet at
a ratio of 3:1 (that is, every third 'challenging' problem is followed by an 'easy' one).
Directions for On-Line Creation of Worksheets With a Mix of Easy and Challenging
Computation Problems ('Interspersal Worksheets')
By following the directions below, teachers can use a free on-line Math Worksheet Generator to
create computation worksheets with easy problems interspersed among more challenging ones:

The teacher goes to the following URL for the Math Worksheet Generator:
http://www.interventioncentral.org/htmdocs/tools/mathprobe/allmult.php

Displayed on that Math Worksheet Generator web page is a series of math computation goals
for addition, subtraction, multiplication, and division. Teachers can select up to five different
problem types to appear on a student worksheet. Each problem type is selected by clicking on
the checkbox next to it.

It is simple to create a worksheet with a 1:1 ratio of challenging and easy problems (that is,
with an easy problem following every challenging problem). First, the teacher clicks the
checkbox next to an 'easy' problem type that the student can compute very quickly (e.g.,
adding or subtracting two 1-digit numbers). Next the teacher selects a 'challenging' problem
type that is instructionally appropriate for the student (e.g., multiplying a 2-digit number by a 2digit number with regrouping). Then the teacher clicks the 'Multiple Skill Computation Probe'
button. The computer program will then automatically create a student computation worksheet
and teacher answer key with alternating easy and challenging problems.

It is also convenient to create a worksheet with a higher (e.g., 2:1, 3:1, or 4:1) ratio of
challenging problems to easy problems. The teacher first clicks the checkbox next to an 'easy'
problem type that the student can compute very quickly (e.g., adding or subtracting two 1-digit
numbers). The teacher then selects up to four different challenging problem types that are
instructionally appropriate to the student. Depending on the number of challenging problem
types selected, when the teacher clicks the 'Multiple Skill Computation Probe' button, the
computer program will create a student computation worksheet and teacher answer key that
contain 2 (or 3 or 4) challenging problems for every easy problem.
Because the computer program generates new worksheets each time it is used, the teacher can
enter the desired settings and -in one sitting-- create and print off enough worksheets and answer
keys to support a six- or eight-week intervention.
References
Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal
ratios on fifth-grade students' mathematics accuracy. School Psychology Review, 34, 543-555.
Math Computation: Increase Accuracy and Productivity Rates Via
Self-Monitoring and Performance Feedback
Students can improve both their accuracy and fluency on math computation worksheets by
independently self-monitoring their computation speed, charting their daily progress, and earning
rewards for improved performance.
Materials

Collection of student math computation worksheets & matching answer keys (NOTE:
Educators can use a free online application to create math computation worksheets and
answer keys at http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php)
Student self-monitoring chart (Click to view a sample progress-monitoring chart)
Steps to Implementing This Intervention
In preparation for this intervention:

the teacher selects one or more computation problem types that the student needs to
practice. Using that set of problem types as a guide, the teacher creates a number of
standardized worksheets with similar items to be used across multiple instructional days. (A
Math Worksheet Generator that will create these worksheets automatically can be accessed
at http://www.interventioncentral.org).

the teacher prepares a progress-monitoring chart. The vertical axis of the chart extends from
0 to 100 and is labeled 'Correct Digits' The horizontal axis of the chart is labeled 'Date'.

the teacher creates a menu of rewards that the student can choose from on a given day if the
student was able to exceed his or her previously posted computation fluency score.
At the start of the intervention, the teacher meets with the student. The teacher shows the student a
sample math computation worksheet and answer key. The teacher tells the student that the student
will have the opportunity to complete similar math worksheets as time drills and chart the results.
The student is told that he or she will win a reward on any day when the student's number of
correctly computed digits on the worksheet exceeds that of the previous day.
During each day of the intervention:
1. The student is given one of the math computation worksheets previously created by the
teacher, along with an answer key. The student first consults his or her progress-monitoring
chart and notes the most recent charted computation fluency score previously posted. The
student is encouraged to try to exceed that score.
2. When the intervention session starts, the student is given a pre-selected amount of time
(e.g., 5 minutes) to complete as many problems on the computation worksheet as possible.
The student sets a timer for the allocated time and works on the computation sheet until the
timer rings.
3. The student then uses the answer key to check his or her work, giving credit for each correct
digit in an answer. (A 'correct digits' is defined as a digit of the correct value that appears in
the correct place-value location in an answer. In this scoring method, students can get partial
credit even if some of the digits in an answer are correct and some are incorrect.).
4. The student plots his or her computational fluency score on the progress-monitoring chart
and writes the current date at the bottom of the chart below the plotted data point. The
student is allowed to select a choice from the reward menu if he or she exceeds the most
recent, previously posted fluency score.
References
Bennett, K., & Cavanaugh, R. A. (1998). Effects of immediate self-correction, delayed selfcorrection, and no correction on the acquisition and maintenance of multiplication facts by a fourthgrade student with learning disabilities. Journal of Applied Behavior Analysis, 31, 303-306.
Shimabukuro, S. M., Prater, M. A., Jenkins, A., & Edelen-Smith, P. (1999). The effects of selfmonitoring of academic performance on students with learning disabilities and ADD/ADHD.
Education and Treatment of Children, 22, 397-414.
Math Review: Balance Massed & Distributed Practice
(Carnine, 1997)
Teachers can best promote students acquisition and fluency in a newly taught math skill by
transitioning from massed to distributed practice.
When students have just acquired a math skill but are not yet fluent in its use, they need lots of
opportunities to try out the skill under teacher supervision—a technique sometimes referred to as
‘massed practice’. Once students have developed facility and independence with that new math
skill, it is essential that they then be required periodically to use the skill in order to embed and
retain it—a strategy also known as ‘distributed practice’. Teachers can program distributed
practice of a math skill such as reducing fractions to least common denominators into instruction
either by (a) regularly requiring the student to complete short assignments in which they practice
that skill in isolation (e.g., completing drill sheets with fractions to be reduced), or (b) teaching a
more advanced algorithm or problem-solving approach that incorporates--and therefore requires
repeated use of--the previously learned math skill (e.g., requiring students to reduce fractions to
least-common denominators as a necessary first step to adding the fractions together and
converting the resulting improper fraction to a mixed number).
References
Carnine, D. (1997). Instructional design in mathematics for students with learning disabilities.
Journal of Learning Disabilities, 30, 130-141.
Copyright ©2009 Jim Wright
Applied Problems: Encourage Students to Draw to Clarify
Understanding
(Van Essen & Hamaker,1990; Van Garderen, 2006)
Making a drawing of an applied, or ‘word’, problem is one easy heuristic tool that students can use to
help them to find the solution.
An additional benefit of the drawing strategy is that it can reveal to the teacher any student
misunderstandings about how to set up or solve the word problem. To introduce students to the
drawing strategy, the teacher hands out a worksheet containing at least six word problems. The
teacher explains to students that making a picture of a word problem sometimes makes that problem
clearer and easier to solve. The teacher and students then independently create drawings of each of
the problems on the worksheet. Next, the students show their drawings for each problem, explaining
each drawing and how it relates to the word problem. The teacher also participates, explaining his or
her drawings to the class or group. Then students are directed independently to make drawings as
an intermediate problem-solving step when they are faced with challenging word problems. NOTE:
This strategy appears to be more effective when used in later, rather than earlier, elementary grades.
References
Van Essen, G., & Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word
problems. Journal of Educational Research, 83, 301-312.
Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of
students with varying abilities. Journal of Learning Disabilities, 39, 496-506.
Copyright ©2009 Jim Wright
Math Computation: Boost Fluency Through Explicit Time-Drills
(Rhymer, Skinner, Jackson, McNeill, Smith & Jackson, 2002; Skinner, Pappas & Davis, 2005;
Woodward, 2006)
Explicit time-drills are a method to boost students’ rate of responding on math-fact worksheets.
The teacher hands out the worksheet. Students are told that they will have 3 minutes to work on
problems on the sheet. The teacher starts the stop watch and tells the students to start work. At the
end of the first minute in the 3-minute span, the teacher ‘calls time’, stops the stopwatch, and tells the
students to underline the last number written and to put their pencils in the air. Then students are told
to resume work and the teacher restarts the stopwatch. This process is repeated at the end of
minutes 2 and 3. At the conclusion of the 3 minutes, the teacher collects the student worksheets.
TIPS: Explicit time-drills work best on ‘simple’ math facts requiring few computation steps. They are
less effective on more complex math facts. Also, a less intrusive and more flexible version of this
intervention is to use time-prompts while students are working independently on math facts to speed
their rate of responding. For example, at the end of every minute of seatwork, the teacher can call the
time and have students draw a line under the item that they are working on when that minute expires.
References
Rhymer, K. N., Skinner, C. H., Jackson, S., McNeill, S., Smith, T., & Jackson, B. (2002). The 1-minute
explicit timing intervention: The influence of mathematics problem difficulty. Journal of Instructional
Psychology, 29(4), 305-311.
Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing
opportunities for responding and influencing students to choose to respond. Psychology in the
Schools, 42, 389-403.
Woodward, J. (2006). Developing automaticity in multiplication facts integrating strategy instruction
with timed practice drills. Learning Disability Quarterly, 29, 269-289.
Copyright ©2009 Jim Wright
Math Computation: Motivate With ‘Errorless Learning’ Worksheets
Reluctant students can be motivated to practice math number problems to build computational
fluency when given worksheets that include an answer key (number problems with correct answers)
displayed at the top of the page.
In this version of an ‘errorless learning’ approach, the student is directed to complete math facts as
quickly as possible. If the student comes to a number problem that he or she cannot solve, the
student is encouraged to locate the problem and its correct answer in the key at the top of the page
and write it in. Such speed drills build computational fluency while promoting students’ ability to
visualize and to use a mental number line. TIP: Consider turning this activity into a ‘speed drill’. The
student is given a kitchen timer and instructed to set the timer for a predetermined span of time (e.g.,
2 minutes) for each drill. The student completes as many problems as possible before the timer rings.
The student then graphs the number of problems correctly computed each day on a time-series
graph, attempting to better his or her previous score.
References
Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282.
Copyright ©2009 Jim Wright
Math Problem-Solving: Help Students Avoid Errors With the
‘Individualized Self-Correction Checklist’
(Zrebiec Uberti, Mastropieri & Scruggs, 2004)
Students can improve their accuracy on particular types of word and number problems by using
an ‘individualized self-instruction checklist’ that reminds them to pay attention to their own specific
error patterns.
To create such a checklist, the teacher meets with the student. Together they analyze common
error patterns that the student tends to commit on a particular problem type (e.g., ‘On addition
problems that require carrying, I don’t always remember to carry the number from the previously
added column.’). For each type of error identified, the student and teacher together describe the
appropriate step to take to prevent the error from occurring (e.g., ‘When adding each column,
make sure to carry numbers when needed.’). These self-check items are compiled into a single
checklist. Students are then encouraged to use their individualized self-instruction checklist
whenever they work independently on their number or word problems. As older students become
proficient in creating and using these individualized error checklists, they can begin to analyze
their own math errors and to make their checklists independently whenever they encounter new
problem types.
References
Zrebiec Uberti, H., Mastropieri, M. A., & Scruggs, T. E. (2004). Check it off: Individualizing a math
algorithm for students with disabilities via self-monitoring checklists. Intervention in School and
Clinic, 39, 269-275.
Copyright ©2009 Jim Wright
DRAW & FAST DRAW
The Math strategy, draw, is used for teaching students with special needs to solve multiplication facts
that are not yet committed to memory.
Discover
• Discover the Sign
• The student looks at the sign to figure out what operation to perform
Read
• Read the problem
• The student says the problem aloud or to himself or herself
Answer
• Answer, or draw, and check
• The student thinks of the answer or draws lines to figure out the answer
• The student checks his or her drawing and counting
Write
• Write the answer
• The student writes the answer in the answer space
•
The FASTDRAW strategy can help students make the transition from pictures
to abstract numbers. First use FAST then use DRAW.
Find
• Find what you are solving for
• Students look for the question in the problem
Ask
• Ask yourself, “What are the parts of the problem?”
• Students identify the number of groups and the number of objects in each
group
Set
• Set up the numbers
• Students write the two numbers in the problem in a vertical format
Tie
• Tie down the sign
• Students add the multiplication sign to the problem
Harris, C. A., Miller, S.P., & Mercer, C.D. (1995). Teaching initial multiplication skills to students with
disabilities in general education classrooms. Learning Disabilities Research & Practice, 10(3), 180195.
Multiplication Attack Strategy
Cullinan, D., Lloyd, J., & Epstein, M.H. (1981). Strategy training: A structured approach to arithmetic
instruction. Exceptional Education Quarterly, 2, 41-49.
This Math strategy is designed help to teach multiplication facts. Before using the strategy it is
essential that the teacher determine what the learner has to do to implement the strategy and
determine whether or not the student possesses the necessary skills; this can be done by using a
task analysis (Figure 1).
Steps in Attack Strategy
Example:
1. Read the problem 2 x 5 =
2. Point to a number that you can count by: student points to 2
3. Make the number of marks indicated
by the other number. ///// (5 in this example)
4. Begin counting by the number you know
how to count by and count up once for “2, 4, 6, 8, 10”
each mark, touching each mark.
5. Stop counting when you’ve touched the
last mark.
6. Write the last number you said in the 2 x 5 = 10
answer space.
Preskills for Multiplication Attack Strategy
1. Say the numbers 0 to 100.
2. Write the numbers 0 to 100.
3. Name x and = signs.
4. Make the number of marks indicated by numerals 0 to 10.
5. Count by numbers 1 to 10.
6. End counting-by sequences in various positions.
7. Coordinate counting-by and touching-marks actions.
Self-Monitoring Strategies in Arithmetic
Frank, A.R., & Brown, D., (1992, Winter). Self-Monitoring strategies in Arithmetic.
Teaching Exceptional Children, 52-53.
This math strategy is designed to help students who have been able to learn basic single-digit
arithmetic facts but have difficulty in remembering how to solve problems that involve several steps. It
helps them remember the steps involved in computing answers to arithmetic problems. It is designed
to teach students to use a self-monitoring procedure that guides responding as arithmetic problems
are completed. The strategy employs a mnemonic device to help students remember how to solve
arithmetic problems once the self-monitoring procedure has been faded.
Procedure
• Provide students with a guide that prompts them to remember how to complete a task.
• Checklist:
1. Carefully analyze the skill to be taught and identify the critical steps to be performed.
2. Write down steps at a level that can be understood by the student(s).
3. Organize the written statements in sequential order.
4. Place the complete checklist next to each problem as it is done.
• Explain the checklist and give a concrete example of how to use the self-monitoring component.
Once the student is able to use the procedure without assistance, the teacher can discuss a
mnemonic strategy for remembering the steps in the checklist.
• Once the student is showing success with the checklist they should be encouraged try and use the
mnemonic device and wean them off of the checklist. First remove the steps from the top of the
worksheet and later omit the checklist adjacent to individual problems.
• When transitions are difficult the checklist can either be placed on note cards or a chart on the wall.
A Subtraction Illustration
• 4 critical steps to solving multi-digit subtraction problems
1. Starting in the 1st column
2. Determine which numeral in each column is bigger
3. Regrouping if the bottom numeral is bigger
4. Check basic facts
The mnemonic strategy “4Bs” helps students remember the 4 steps: Begin, Bigger, Borrow, and
Basic Facts.
• The student begins the first problem by looking at the first step at the list of directions. Starting with
Begin – the student then places a check mark on the line by Begin.
• Next, the student refers to the second step at the top of the page, which prompts them to determine
which numeral in the 1st column is bigger. The student places a check mark on the line next to the
word Bigger above the first column, and continues to do so for the 10’s and 100’s columns,
completing each step and checking it off in sequential order.
Subtraction Worksheet Including a
Self-Monitoring Strategy
Name
SUBTRACT. Remember the 4 B’s:
Begin? In the 1s column.
Bigger? Which number is bigger?
Borrow? If bottom number is bigger I must borrow.
Basic Facts? Remember them. Use Touch Math if needed.
Date
Modifications for Use with Addition Problems
• 4 critical steps to solving multi-digit addition problems:
1. Start in the 1s column
2. Add together the numerals in each column
3. Determine whether or not regrouping is necessary
4. Check to see whether or not the correct numeral has been carried to the next column when
regrouping is necessary.
• The word SASH can be used as a mnemonic strategy to help students remember the steps: (Start in
the 1s column; Add the numerals together; Should I carry a numeral? and Have I carried the correct
numeral?)
Addition Worksheet Including a
Self-Monitoring Strategy
Name Date
ADD. Remember SASH
Start In the 1s column.
Add together the numerals in each column.
Should I carry a numeral?
Have I carried the correct numeral?
Modifications for Use with Multiplication Problems
• There are also 4 critical steps in solving multi-digit multiplication problems.
1. Multiply the 1s column.
2. Carry any 10s over to the 10s column.
3. Multiply the bottom 1s digit with the top 10s digit.
4. Add any number that was carried in step 2 to the product of step 3.
• The word MAMA can be used as a mnemonic strategy to help students remember the steps:
(Multiply the 1s column; Across to the 10s; Multiply the bottom 1s digit with the top 10s digit; Add any
number that was carried in step 2)
Multiplication Worksheet Including a
Self-Monitoring Strategy
Name Date
MULTIPLY. Remember MAMA
Multiply the 1s column.
Across to the 10s.
Multiply the bottom 1s digit with the top 10s digit
Add any number that was carried in step 2
SLOBS & LAMPS
Reetz, L., & Rasmussen, T. (1988). Arithmetic mind joggers. Academic Therapy,
24(1), 79-82.
These math strategies are designed to help students remember the regrouping process of borrowing
and carrying. Slobs is used in subtraction and lamps is used in addition.
Smaller
• Smaller. Follow steps.
Larger
• Larger. Leap to subtract.
Off
• Cross off the number in the next column.
Borrow
• Borrow, by taking one ten and adding to he next column.
Subtract
• Subtract.
This is how one would follow these steps with the following problem:
72
-46
Look at the top number on the right (2) and wee if it is smaller or larger than the lower number (6). If it
is larger, the student will leap to subtract. If it is smaller, as in this example, the student must follow
the steps. The next step is to cross off the number in the next column, which in this problem is the
seven. The “B” of
SLOBS stands for borrow, which is the next step. Now borrow one ten from that column by reducing
the number by one and adding ten to the other number (12). The last step is to subtract the six from
the twelve. Repeat the steps if there are more digits to be subtracted.
Line
• Line up the numbers
• This is particularly important with extensive columns of numbers and with
numbers with decimal points.
Add
• Add the right column of numbers and ask. . .
More
• More than nine? If so, do more steps.
Put
• Put the ones below the column.
Send
• Send the tens to the top of the next column.
The application of this process is demonstrated in the following problem:
62.1
42.
7.1
+ 4.4
The numbers have been lined in columns according to their decimal points. In adding the right column
the sum is 13 and more than nine. Therefore, follow all the steps. Put the ones (3) below the column
and send the tens (1) to the top of the next column. Repeat from Add for each of the next columns.
Number Writing Strategy
Boom, S.E., & Fine, E. (1995, Winter). Star – A number writing strategy. Teaching Exceptional
Children, 42-45.
A strategy used to assist students in recalling, reciting, and writing numerals.
Step 1: Pretest and Obtain a Commitment to Learn the Strategy
• Observation or formal assessment – the Brigance Diagnostic Inventory of Early Development
(Brigance, 1978).
• After the child has written 0-9, the teacher, showing a chart of correctly formed numerals, brings the
differences to the child’s attention.
• Show numbers as prices to explain how unacceptable it is to make them incorrectly.
• The student can make a commitment to learning a strategy for writing numbers.
Step 2: Describe the Strategy
• Formation of numerals is taught first using multisensory experiences.
• The teacher demonstrates the formation of a selected numeral while reciting its associated saying.
• The child repeats the saying while forming the numeral:
1. In the air with large motor movements
2. Using sand, clay, sandpaper, crayon on paper held over a screen, and a marker
3. Finally, pencil and paper
• Teach the STAR strategy for when the student is unable to recall the correct formation.
S = Stop. Stop and ask myself what I am expected to do (for example, write the number that the
teacher is saying).
T = Think. Think of using a saying to help in forming the number.
A = Ask. Ask myself which saying should be used for this number.
R = Recite. Recite the saying while I write the number.
• The teacher should point out various times the STAR strategy can be used, and demonstrate each
step.
• A STAR Strategy Card can be made with the steps of the strategy on one side and textured
numerals with the saying on the other side. If the child is unable to read, a simple drawing of each
step can be used instead of writing the words.
Step 3: Model the Strategy
• Demonstrate how the strategy is used, i.e. I have to write a seven. I get mad when I write it
backwards. I want to do it right so my mom will be proud of me. I’m going to use my new STAR
strategy. The first thing, I have to do is Stop and ask myself what I have to do. O.K., I have to write a
good seven on this line. Now I have to Think. Let’s see, to remember which way the seven goes, I
can use one of the Sayings. Now I have to Ask myself which Saying to use. Which one is it? I know,
seven is the one with the man who made a line at the
top. Now I have to Recite the saying while I do it. Here’s my pencil. The man made a line across the
top, and then he slid down the hill to the left. That’s a good seven. I know it’s facing the right way. I’ll
check it with the card to make sure. The teacher should model other numerals, this time stopping
after the varying STAR steps to ask the children, “What should I tell myself to do next?” –thus giving
the children practice in self-verbalization.
Step 4: Memorization of the Strategy
• Through verbal rehearsal, students can memorize the sayings and the STAR strategy. The STAR
Strategy Card can be used as a cue during rehearsal.
Step 5: Practice with Controlled Materials
• Practice one number at a time: dictate, simple math, or answer a question
• The teacher should provide feedback on the elements of the strategy that are being done correctly
and corrective feedback to improve performance.
Step 6: Practice with Grade-Appropriate Tasks
• The child should apply the strategy to classroom materials.
• Frequent opportunities for practice are needed:
-digit numbers, numbering pages or lines, completing math problems,
writing the date, or pretending to operate a store
Step 7: Administration of a Posttest
• Require the child to write the numerals and compare the results with the pretest.
• Show the child the results and explain that they need to remember and use the strategy whenever
they need to write a number.
Step 8: Generalization
• Monitor the use of the strategy in other situations
• Children can report when they have use the strategy outside the classroom
• Review the steps periodically to encourage generalization
Star: A Number Writing Strategy
Self-instructions for forming numerals
To make 0: The woman went around in a circle until she got home.
To make 1: The man went straight down, like a stick.
To make 2: The woman went right and around, slid down he hill to the left, then make a line across
the ground.
To make 3: The man went right and around, then around again.
To make 4: The woman went down the street, turned to the right, then back to the top for a straight
ride down.
To make 5: The man went down the street, around the corner, and his hat blew off.
To make 6: The woman made a curve and then a circle at the bottom
To make 7: The man made a line across the top, then slid down the hill to the left.
To make 8: The woman made a half circle to the left, another to the right, and then she found her way
back up to the top again.
To make 9: The man made a small circle and then a straight line down.
Mathematics Strategy Instruction (SI) for
Middle School Students
STAR is an example of an empirically validated (Maccini & Hughes, 2000; Maccini & Ruhl, 2000) firstletter mnemonic that can help students recall the sequential steps from familiar words used to help
solve word problems involving integer numbers.
The steps for STAR include:
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Search the word problem;
Translate the problem;
Answer the problem; and
Review the solution (see Figure 1).
Figure 1: Instructional steps for a classroom lesson
1. Provide an Advance Organizer
The teacher provides an advance organizer of the strategy to help:
a. relate previously mastered information to the new lesson;
b. state the new skill/information that is to be presented; and
c. provide a rationale for learning the new information.
Yesterday, we used the problem solving strategy, STAR, with word problems involving integer
numbers. We used our Algebra tiles to demonstrate the problem and our STAR worksheets to keep
track of the steps.
Today, we are going to use the strategy and draw pictures to demonstrate the problems on our
worksheets. This will be useful because we will not always have the math tiles available to help us
solve subtraction problems involving integer numbers.
It is important to learn how to solve these problems in order to solve many real-world problems,
including money and exchange problems, temperature differences, and keeping track of yardage lost
or gained in a game.
2 . Provide Teacher Modeling of the Strategy Steps
The teacher first thinks aloud while modeling the use of the strategy with the target problems. Then
the teacher checks off the steps and writes down the responses on an overhead version of the
structured worksheet, while the students write their responses on individual structured worksheets.
Next, the teacher models one or two more problems while gradually fading his or her assistance and
prompts and involving the students via questions (e.g., "What do I do first?") and written responses
(i.e., having students write down the problems and answers on their structured worksheet).
Watch and listen as I solve the problem using the STAR strategy and the structured worksheet.
The problem states, "On a certain morning in College Park, Maryland, the low temperature was -8° F,
and the temperature increased by 17° F by the afternoon. What was the temperature in the afternoon
that day?"
(See Figure 2 for a copy of the structured worksheet).
S: Okay, so the first step in the STAR strategy is for me to search the word problem. That means I
need to read the problem carefully, and write down what I know and what I need to find. In this
problem, I know that I have two temperatures and I need to find the temperature by the afternoon.
T: My next step is to translate the problem into picture form. First, I'll draw 8 tiles in the negative area
and then I'll draw 17 tiles in the positive area.
A: Then I need to answer the problem. I know one positive and one negative cancel each other. I can
cancel -8 and +8, which results in +9 remaining. Therefore, the answer is +9.
R: Finally, I need to check my answer. OK, I'll reread the word problem and check the
reasonableness of my answer. Yes, my answer is +9°F and it is a reasonable answer.
3. Provide Guided Practice
The teacher provides many opportunities for the students to practice solving a variety of problems
using their structured worksheets. Guidance is gradually faded until the students perform the task
with few prompts from the teacher.
4. Provide Independent Student Practice
Students perform additional problems without teacher prompts or assistance, and the teacher
monitors student performance.
5. Feedback and Correction
The teacher monitors student performance and provides both positive and corrective feedback using
the following guidelines:
a. checks for error patterns;
b. reteaches if necessary and provides additional problems for students to practice corrections;
and
c. closes the session with positive feedback.
d. documents student performance (percent correct)
6. Program for Generalization
The teacher provides a cumulative review of problems for maintenance over time (weekly, monthly)
and provides opportunities for students to generalize the strategy to other problems (see Figure 3).
Teachers can use self-monitoring forms or structured worksheets to help students remember and
organize important steps and substeps. For example, students can keep track of their problem
solving performance by checking off (√) the steps they completed (e.g., "Did I check the
reasonableness of my answer?" √ ).
Figure 2: Structured worksheet of the STAR strategy
Problem: On a certain morning in College Park, Maryland, the low temperature was -8°F, and the
temperature increased by 17°F by the afternoon. What was the temperature in the afternoon that
day?
Strategy Questions
S earch the word problem
a. Read the problem carefully
b. Ask yourself questions: "What do I know? What do I need to find?"
c. Write down the facts




In the morning it was -8° F.
During the day, the temperature rose by +17° F.
One positive and one negative cancel each other out.
I need to find what the temperature was in the afternoon.
T ranslate the words into an equation in picture form
A nswer the problem
I can cancel -8 and +8, which leaves me with +9 tiles remaining, therefore:
(-8°F) + (+17°F) = +9°F
R eview the Solution
a. Reread the problem
b. Ask yourself questions: "Does the answer make sense? Why?"
c. Check the answer
I checked my answer.
+9 remains when I cancel -8 and +8 and I keep my units of 9°F.

In addition to its application to problem solving involving integer numbers, the STAR strategy
can be generalized across math topics (see Figure 3 for an example involving area).
Figure 3: Area Example
Problem: Matt is buying wall-to-wall carpeting for his bedroom, which measures 12 feet by 16 feet. If
he has $40 to spend, will he have enough money to buy the carpet that costs $2 per square yard?
Strategy Questions
Search the word problem
a. Read the problem carefully
b. Ask yourself questions: "What do I know? What do I need to find?"
c. Write down the facts





The bedroom is 12 feet by 16 feet.
Matt has $40.
Carpet costs $2 per yard.
First, I need to find out the area of the room in yards.
o Area is calculated by multiplying the width by length.
o There are three feet in a yard.
Then I need to find out how much the carpet will cost, and compare that number with the
amount of money that Matt has.
Translate the words into an equation in picture form
Answer the problem
Area of the room: 12 ft x 16 ft = 192ft²
I know that 3 ft = 1 yd, and (3ft)² = (1yd)² so 9 ft² = 1 yd². I will divide 192 ft² by 9 to get yd²: 192 ÷ 9 =
21.3 yd²
The carpet costs $2/yd², so I will need to multipy the square yardage of the room by $2: $2 x 21.3 yd²
= $42.60.
$42.60 is more than $40. Matt does not have enough money.
Review the Solution
a. Reread the problem
b. Ask yourself questions: "Does the answer make sense? Why?"
c. Check the answer
I checked my answer and it makes sense – Matt needs $2.60 more in order to buy the carpet for his
room.
Elements of Successful Secondary Math Interventions
The purpose of this report was to synthesize and summarize the research on interventions conducted
between 1963 and 1997. Fifty-eight intervention studies were analyzed according to the age and
intelligence of the adolescents, the characteristics of the intervention (e.g. number of instructional
interventions, components of instruction) and the methods used by the original investigators. It was
hoped that this synthesis of the research literature would help identify instructional models for
adolescents that predict success in improving problem-solving skills.
It was clear from earlier research that not all interventions work equally well in this population of
students, and two instructional methods seemed superior to others: direct instruction and strategy
instruction. It was considered important to distinguish the points of commonality and distinction of
these two approaches and identify the components of instructional models that predict the best
outcome for adolescents with LD.
For purposes of this study, direct instruction was categorized as those that employed the following
techniques:
Breaking down a task into small steps
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Administering probes
Supplying repeated feedback
Providing students with diagrammatic or pictorial presentations
Allowing independent practice and individually paced instruction
Breaking instruction down into simpler phrases
Instructing in a small group
Teacher modeling of skills
Providing set materials at a rapid pace
Providing instruction for individual children
Having the teacher ask skill-related questions
Having the teacher provide new materials
Breaking down a task into small steps
Studies included in this synthesis that were categorized as strategy instruction had the following
components:

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Teacher modeling of processes
Elaborate explanations
Reminders to use certain strategies
Step-by-step prompts
Teacher-student dialogue
Teacher asks process-oriented questions
Teacher provides only necessary assistance
Findings
Three main questions were posed as the core of this review of the literature on problem solving in
adolescent learners:
Do studies that use direct and/or strategy instruction produce better effects on problem-solving skills
than those that do not?
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Effective instruction can be either a bottom-up or a top-down approach, as long as certain
components are included in the intervention.
In contrast to other methods, interventions were more effective when studies included
derivatives of cognitive and/or direct instruction.
No statistical advantages of the direct instruction or strategy instruction were apparent.
A clear orientation to task, drill-repetition-practice, sequencing, teacher modeling, and
systematic probing may supersede effects related to distinctive qualities of either the strategy or
direct instruction methods.
Is the magnitude of the beneficial effect related to certain components of the intervention?
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Regardless of the overall model of instruction, only a few specific instructional components
increased treatment effectiveness.
The following components predicted the magnitude of treatment outcomes: sequencing (eg,
breaking down the task), step-by-step prompts; drill-repetition-practice; directed questioning and
responses; individualization combined with small group instruction; segmentation (breaking down a
targeted skill into smaller units and then synthesizing the parts into a whole); technology (e.g.,
computer use); and small interactive group instruction.
The drill-repetition-practice-review component was an important variable in predicting
effectiveness.
This document was prepared for the Keys to Successful Learning Summit held in May 1999 in
Washington, D.C. Keys to Successful Learning is an ongoing collaboration sponsored by the National
Center for Learning Disabilities in partnership with the Office of Special Education Programs (US
Department of Education) and the National Institute of Child Health & Human Development (National
Institutes of Health).
The purpose of this initiative is to translate research and policy on learning disabilities into high
standards for learning and achievement in the classroom, and to take action at the local, state and
federal levels to ensure that all students, including those with learning disabilities, are afforded the
highest quality education.
Keys to Successful Learning is supported by a coalition of national and regional funders as well as a
broad range of participating education organizations.