Alternative Teaching Techniques for Struggling Learners

1. A TEACHER AND TUTOR
GUIDE TO HELP THE OLDER
STUDENT WITH LIMITED
MATH SKILLS
Psychoeducation for Teachers

2. Background
 Children with low math skills typically evidence deficits in one or more of
these three main areas: recalling of math facts, computation, and/or word
problems. It is important to notice that most math skills overlap and a skill
deficiency in only one of the three domains has the potential of bringing
down the child’s whole math performance. Sometimes, we see children
struggling in one math area without realizing that the skill deficit is really in
a different area. When teachers and tutors work in developing students’
overall math skills, first, we need to identify (i.e. using diagnostic
assessment) in which of these areas the child is truly lacking math skills, so
that we target the real skill deficits, and do not waste precious time re-
teaching skills that the child already masters. In other words, first, we
determine the source of error and only then, we prepare a plan to
remediate. Remediation is the process of re-teaching the skill because the
student did not master the skill when it was taught, or the child forgot the
skill. Our remediation plan must include alternative teaching techniques and
compensatory strategies that we teach the student to help him or her profit
from traditional grade placement curriculum in the areas that are
developing adequately while the child is still strengthening skill deficits in
the areas of difficulty. Alternative and compensatory strategies are different
ways of doing the task, or using an assistive device, that allow the student
to complete the task, which the child otherwise would not be able to
perform.

3.  Children need to understand that, in handling math problems, it is
not the recalling of math facts and memorization of algorithms what
is more important, but the ability to use strategies to solve the
problem. For this reason, any remediation plan that we implement
should put less emphasis in memorization and more emphasis in
strategy using. Teachers and tutors get better results in developing
math skills in all kinds of learners when we see and teach math as a
planned and strategic way of thinking, rather than as a
disconnected collection of basic facts and computation skills. To
develop strategic thinking, we need to provide plenty of
discrimination practice in when to use a specific strategy as
opposed to using a different strategy. In other words, we help the
student identify when a strategy applies and when it does not apply.
With our struggling learner, a compensatory technique to develop
strategy using is to give the student the choice of two strategies,
asking the child, “Which strategy is better here, _____ or _____?” In
this guide, we discuss remediation activities and alternative math
techniques that we can teach children to compensate for skill
deficits in any of the three main areas.

4. Alternative Techniques for Recalling of
Math Facts
 Weak recalling of addition and multiplication facts is one of the most
common problems for children who struggle with higher-level math skills.
Because these students never memorized basic facts, they depend on
inadequate compensatory strategies such as counting on fingers, drawing
sticks or circles, and/or adding repeatedly to solve longer multiplication and
division problems. These inadequate strategies are simply too long to be
efficient, and for this reason, most of the time the child ends feeling
frustrated and giving up. We need to explain to the struggling learner that
the compensatory strategies that he is trying are simply inadequate, and we
teach the child alternative techniques that support and facilitate (instead of
frustrating) his ability in solving more sophisticated problems.
 Some remediation activities and alternative techniques that we can use
with struggling learners are:
 Do not press for speed until the child demonstrates accuracy at a slower
pace.
 Teach the student to draw a number line at the bottom of the paper, so that
he uses the number line to add, to subtract, or to tell which number is
bigger than or smaller than another number.

5. Rehearse for Mastery
 Use timed drills in addition (e.g. 9+3), subtraction (e.g. 9-3),
multiplication (e.g. 9*3), or division (e.g. 9/3) facts. Have the
child compete against her best time. Initially, timed drills
should include only a few facts at a time.
 Use the tracking technique to help the student memorize
math facts. Present a few facts at a time, gradually increasing
the number of facts the child must remember at a time.
 Rehearse the student for mastery, keeping in mind that it is
better that the child performs five problems with 100%
accuracy than performing 10 problems with 50% accuracy.
 Continue reviewing previously learned facts, even when it
appears that the child mastered the facts. Include at least two
known facts in the daily practice. This helps ensure success
with the new facts.
 Similar to the procedure for teaching spelling, verbalizing the
facts and then writing them from memory increases retention.

6. Build on What the Child Knows
 Build on what the student already knows. Teachers and tutors can
often turn a student’s failure into success if we build on what the
student already knows how to do.
 Use distributed practice, that is, teaching fewer facts that the child
practices more frequently. Or, teaching shorter tasks, but more of
them throughout the day. For example, split one longer task of
twenty problems into four shorter practices with five problems each.
Several shorter sessions are usually more effective than an
isolated, longer one.
 Have the student perform timed drills exercises to reinforce basic
math facts. The child competes against his own best time.
 Teach the child to use number tricks. This mnemonic technique
gives the child a visual or an auditory cue (e.g. music, rhyme, or a
visualization) to remember a particular fact. For example, 6+6= a
dozen eggs, and a dozen eggs equals 12. In another example,
8*7=56 and 56 is also the shirt’s number of the child’s favorite
football player.

7. Overcoming Addition and
Subtraction Deficits
 There is a strong correlation between knowing
addition facts and memorizing multiplication
facts. If the student is having difficulty with
addition, chances are that he will also have
problems with multiplication. The following
activities can help a struggling learner
overcome addition and subtraction deficits.

8. Pattern Recognition
To reduce
the demand
on memory,
teach the
student to
recognize
patterns.
Example:
 Doubles
o 7+7=14
 Doubles Plus One
o 7+8=
o 7+ (7+1) =
o 14+1=15
 Doubles Minus One
o 6+7=
o 7+7=
o 14-1=13

9. Pattern Recognition Needs to Be
Automatic
 At the beginning of this training, you may need to
point out the patterns to the student.
 In other words, first, you practice the student until
she learns to recognize immediately all double
patterns, i.e. 2+2, 3+3, 4+4, 5+5, 6+6, 7+7, 8+8,
and 9+9. Then, you teach the child to generalize
her knowledge of double patterns to solve quickly
other addition facts. In a problem like, 8+6, the
child can use either the 8+8 double pattern
(8+8=16, minus two, equals 14), or the 6+6
double pattern (6+6=12, plus two, equals 14).

10. Numbers Within Numbers
 To master math facts, one of the first things that the child needs is to
recognize automatically the numbers within numbers. For example,
the child should be able to understand that five is made up of five
ones, two twos and a one, or a three and a two.
 Encourage the student to study the number combinations. For most
children, exploring number combinations helps in mastering number
facts. You can rehearse the child by giving a number, single or
multi-digits, and asking the child to write down as many number
combinations that she can find for that number. For example:
o 5=
o 4+1
o 3+2
o 2+2+1
o 3+1+1
o 1+1+1+1+1

11. Hidden Numbers
 Once the child recognizes numbers within numbers, you can teach her to
find hidden numbers, for example, there is a six hidden in eight (8=6+2),
and she will find a six, a seven, or an eight hidden in nine:
o 6+3=9
o 7+2=9
o 8+1=9
 Make sure the student performs automatically plus ones and minus ones,
or one more and one less, for example, 17+1 and 66-1. Then, practice the
child until he retrieves automatically plus twos (e.g. 17+2) and minus twos
(e.g. 66-2).
 Teach and emphasize number relationships. When the student has a good
grasping of number relationships, he can combine or use these patterns to
retrieve number facts faster. For example,
o 8+6=8+ (8-2) =16-2=14
o 7+9=7+ (7+2) =14+2=16

12. Hidden Tens
 Emphasize number relationships such as hidden tens.
Examples of hidden tens are:
 First Example
o 9+3= (9+1) +2=
o 10+2=12
 Second Example
o 7+5=2+ (5+5) =
o 2+10=12
 After learning the hidden tens, teach the child to recognize
that the nines are one less; the elevens are one more.

13. Strike Tens Sheets
Prepare
Strike Ten
Exercise
Sheets like
the one that
follows, and
give the child
three
minutes, then
two minutes,
and finally
one minute to
solve
different
worksheets
using the
hidden tens
strategy.
Example:
 Top
o 9
o 7
o 2
o 8
o 5
o 4
 Bottom
o 1
o 3
o 6
o 8
o 4
o 2

14. How to Solve
 In the previous example, the child circles 9
(top) and 1(bottom) and writes the first 10.
Next, she circles 7 (top) and 3 (bottom) and
writes the second 10. She has three additional
hidden tens to find, 2 top with 8 bottom, 8 top
with 2 bottom, and 4 top with 6 bottom. That
gives the child 5 tens or 50 with the five at the
top that was not matched, which is 55. Finally,
the child adds four more (bottom, not matched)
to 55, for the final answer of 59.

15. Make 10s
 Do Make 10s dictation. For example, you say
eight, and the child writes a two, or you say
four and the child writes a six.
 Once the child masters key addition patterns
such as doubles and hidden tens, teach her to
use these patterns as a reference for all the
other addition facts.

16. Prepare Cue Cards
 The number facts that add to ten are important
for the student to know by automatic recall.
The child can prepare a cue card that looks
like the following one. The five is repeated top
and bottom.
o 1 (space) 2 (space) 3 (space) 4 (space) 5
o 9 (space) 8 (space) 7 (space) 6 (space) 5

17. Use the Ten as a Base
 As you can see, all these combinations equal ten.
For recalling number facts involving nines, the
child simply uses the ten as a base. Practically, all
number facts can be retrieved faster using the ten
as a base, and then, adding or taking away ones,
twos, or threes.
 In the following examples, we are using the ten as
a base to find the number facts:
o 9+16=10+15=25
o 14+9=13+10=23
o 26+9=25+10=35

18. Number Keys
 Teach the child to use number keys. The numbers that
add to ten (e.g. 7+3 and 6+4) and the numbers
doubled (i.e. 7+7) are the number keys. The student
learns the “keys” and uses the keys to add or to
subtract.
 Teach the child to change the order of numbers to
make easy numbers.
o 23+14+5+6=
o 14+6+23+5=
o 20+23+5=
o 43+5=48

19. Look for Patterns
 Provide daily looking for patterns exercises, for
example,
 What should go next? Explain why
o 2, 4, 6, 8, ___ Explain why __________
o 20, 19, 18, 17, ___ Explain why __________
o 8, 12, 6, 10, 4, ___ Explain why _____

20. Answer
 The previous example is a plus four, then
minus six pattern:
o 8+4=12
o 12-6=6
o 6+4=10
o 10-6=4
o 4+4=8
 Therefore, the number that goes in the blank
space is the same as 4+4 or 8.

21. Turn Around Facts
 Drawing tallies or circles is a slow, tedious, and
inaccurate compensatory strategy. We can help
students speed this process by teaching them how to
draw tallies or circles for only one digit (the smallest),
and then, to count on from the biggest digit. For
example, to add 6+8, the student says and does, “8 is
in my head plus 1+1+1+1+1+1 (draws six tallies and
counts each) =14.”
 Encourage the student to draw a number line rather
than tally marks. With a number line, the child draws
only once, and it is not visually confusing like the tally
marks are.
 Teach turn around facts so that the child switches to
the math fact that requires drawing fewer tallies. For
example, the child turns around 6+8 and solves it as
8+6.

22. Find the Next Teen
 To add nine to any number, use find the next
teen technique, that is, the answer is one less
than the second addend plus the teen that
follows the first addend. Examples:
o 9+5 (Take away one from five=4). Find the
next teen (10) and add to four=14
o 19+7 (Take away one from seven=6). Find the
next teen (20) and add to six=26
o 59+8 (Take away one from eight=7). Find the
next teen (60) and add to seven=67

23. Write the Smaller Number First
We can teach
subtraction
facts similarly
to addition
facts. Tell the
child to write
the smaller
number, and
then count
from there
until he
reaches the
bigger
number. The
child can
draw tallies
or counts on
his fingers to
get the
answer.
Example:
o 14-6=
o 6+ one tally=7
o 7+ one tally=8
o 8+ one tally=9
o 9+ one tally=10
o 10+ one tally=11
o 11+one tally=12
o 12+one tally=13
o 13+one tally=14
 6+8 tallies=14.
 Therefore, 14-6=8

24. Rehearse
 Have the child rehearse timed sheets such as,
o 10-6=
o 10-4=
o 6+4=
o 4+6=

25. Partitioning
 After rehearsing the timed sheets, teach the
child to use partitioning. With this alternative
technique, the child solves subtraction facts by
recalling known addition combinations. For
example:
o 13-7=? The child breaks down the bigger
number or total (13) into two parts. He knows
that 7+6=13, so, 13-7 must be six.
o Teach the child to say, “Part is 17, total is 29.
The missing part is _____.”

26. Double and then Subtract
 Show the student that he can solve all subtraction facts as
addition by reading up, that is, from bottom to top, instead of
reading down.
 Teach the child to use doubles to subtract. For example, to
solve 14-6, the child doubles the six to make it twelve.
o 14-6=
o 14-(6+6) =
o 14-12=2
 To get 14, the child must adjust the computation, increasing
the six by two.
o 6+2=8
o 14-6=8

27. Second Example
o 15-7=
o 15-(7+7) =
o 15-14=1
 Adjusting the computation, we add one to
seven:
o 7+1=8
o 15-7=8

28. Number Grids
 Avoid timetables sheets, use number grids
instead. The difference is that, with a
timetables sheet, the student is not learning
multiples or sequences. On a number grid, the
student highlights or circles the multiples she
will use to multiply, e.g. all the multiples of
seven or 7, 14,21,28,35,42,49,56, and 64. This
helps in memorizing the multiples of a number
through repeated exposure. In addition,
number grids help the student see how the
addition of multiples relates to the
multiplication facts.

29. Transfer Knowledge
 Help the child understand that learning the
multiplication facts does not need to be an
overwhelming task when we take into consideration
the easy tables (2,3, and 5) and those facts that the
child can turn around (e.g. 3*9 and 9*3 are reversible).
A child that memorizes 7*3=21 already knows that
3*7=21. However, children with learning problems do
not transfer knowledge automatically, so we need to
make sure that the student applies knowledge of
mastered facts to solve new ones.
 Make sure the student masters easier tables first. This
will give the child a base or foundation to use when
she is computing harder tables.

30. Use Known Facts
 Teach the student to use the multiplication
facts that she knows to figure out new ones.
Some examples are:
o The child knows that 7*7 is 49, so, to solve
7*8, she counts seven more (49+7), getting
56. This is the lower one factor technique.
o The child knows that 2*9=18, so, to solve 4*9,
she uses 2*9=18 plus 18 more or 36. This is
the splitting and doubling technique.

31. Progression of Multiples
 Teach multiplication facts following the
progression of multiples. For example:
o Begin with the 2s, then the 4s, and then the
8s.
o Begin with the 3s, then the 6s, and then the
9s.
 The progression of multiples helps the child
retrieve multiplication facts that are yet to be
mastered using compensatory strategies such
as breaking down, splitting, or doubling.

32. The Doubling Strategy
 Make sure the child understands that, in
handling timetables, the four always doubles
the two, the six doubles the three, and the
eight doubles the four. For example, to solve
8*4, 9*6 and 7*8, the child can use the
following doubling strategy:
o 8*4=8*2=16+16=32
o 9*6=9*3=27+27=54
o 7*8=7*4=28+28=56

33. Nine Timetables
 To handle the nine timetables, teach the child to use
the ten timetables, for example,
o 9*4=10*4=40-4=36
o 9*7=10*7=70-7=63
 Another alternative technique to handle the nine
timetables is the one less technique. For example,
o 9*6 (one less is five) =5+4 (the number that added to
five equals nine) =54
o 9*9 (one less is eight) =8+1 (the number that added to
eight equals nine) =81
o 9*4 (one less is three) =3+6 (the number that added to
three equals nine) =36

34. Cue Cards
 Teach the student to turn around the multiplication facts. For
example, 7*5 and 5*7 produce the same answer.
 Have the student prepare a cue card (index card) with strategies to
use to recall timetables faster. For example,
o 2* skip count
o 3* skip count
o 4* double the 2*
o 5* skip count
o 6* double the 3*
o 7* turn around and/or use easier tables and then adjust or double
the answer
o 8* double the 4*
o 9* use the 10* or the one less technique

35. Skip Counting
 Teach extended facts. For example, as the student
learns 4*7, she also learns 40*7 and 70*4, 400*7 and
700*4.
 Put less emphasis in memorizing the tables, and more
emphasis in skip counting faster. Use the next
technique to reinforce timetables through skip
counting. Leaving the first circle empty to represent
zero, draw 13 circles numbered 1-to-12. Say, “Let’s
skip count by _____ (e.g. fours).” While you point to a
circle (e.g. the seventh circle), the child recites “28.”
Make sure that you give the child ample practice in
skip counting following the right sequence (e.g. 0, 4,
8,12,16,20,24,28,32, 36, 40, 44, and 48), before
asking for random sequences.

36. Multiplication is Repeated
Addition
 Make sure the child understands the concept of
multiplication as repeated addition. For example,
4+4+4+4+4 or five groups of four members each
is the same as 5*4.
 Provide repeated exposure to exercises such as,
o 3+3+3+3+3=
o 5 groups of three=
o 5*3=
o 15

37. Division is the Opposite of
Multiplication
 Make sure the student understands that
division is the opposite of multiplication. Teach
the division facts at the same time that you are
teaching the multiplication facts, so that the
student can see the reverse relationship. Use
exercises like, “If 6*4=24, then, 24 divided by
4= ___, and 24 divided by 6= ___.

38. Mental Organization
 When we teach children to organize mentally
math facts, we are both reducing the demands
on memory and maximizing retrieval. Activities
that involve mental organization are linking
strategies (turn around facts and extended
facts), number relationships (e.g. hidden tens
and near tens), and patterns like doubles, skip
counting, and multiples.

39. Alternative Techniques to Develop Procedural
Knowledge
 Let the child use prompt cards with the sequenced
steps.
 Work on fewer problems, (e.g. five rather than 20) and
have the learner spend more time talking through the
steps at the conceptual level.
 When you are teaching algorithms, that is, steps or
procedures, use verbal organizational cues such as
first, second, third, and last step. When you are
rehearsing the student in talking through the steps,
make sure the child also uses organizational cues.
 To prevent the student in learning faulty algorithms, do
not allow her to practice errors. Monitor the child
closely so that you can catch and correct mistakes
immediately.

40. Self-monitoring
 Teach the student a self-monitoring strategy, for
example, when solving a long division problem,
the child asks, “Does my answer make sense?”
Train the child in automatically looking for answers
that are too high (e.g. 26+7=83), or too low (e.g.
85*46=410) for the problem that she is solving.
 Give the child breaks, that is, sandwich easier
computation in between harder problems.
 Every time you introduce a new algorithm or a
new concept, talk more slowly than you would do
when you are teaching familiar information.

41. Paraphrase and Explain
 Reinforce the information that you present verbally with visuals such
as pictures and graphic organizers. Pictures help the child visualize
(see in her mind) the information; graphic organizers (e.g. flow
charts, comparing and contrasting frames, and sequence frames),
help in forming associations and connections among ideas and
concepts, also between the material the child already knows and
the new information.
 Provide practice in paraphrasing by having the child restate the
steps in her own words. This way of processing information
strengthens the child’s memory.
 Use the turn to your partner and explain technique. Being able to
explain the new procedure or concept to a peer not only enhances
memory, but also is a good measure of the child’s understanding.
Alternatively, you can ask the child to explain the steps or concept
to you.
 Have the child recite the steps in the long multiplication or long
division algorithm, without actually performing the computation.

42. One Problem at a Time
 Have the student fold the paper to create four squares and write
only one problem inside each square.
 For students with attention deficits and/or impulsive behaviors, use
the one problem at a time approach. For example, you copy the first
problem inside the first square, or an index card. Only after the child
works on that problem, and you check the answer, present the next
problem, and so on.
 For children that confuse directionality, use visual cues such as
arrows or color dots to indicate progressions such as from right to
left and from top to bottom.
 For children with alignment problems (i.e. the digits are “all over the
place” rather than aligned properly in columns), you can use graph
paper to force the child to write one digit only inside each square.
Alternatively, you can use sheets of lined notebook paper turned
vertically so the lines run up and down. Tell the child to use the lines
on the paper as a guide for keeping the numerals in the correct
columns.

43. Rounding
 If the student is having difficulty rounding numbers
the traditional way, try using this progression: start
rounding only numbers that end in nine (e.g. 59)
or in one (e.g. 41). Once the child masters this,
teach rounding numbers that end in eight, like 38,
and numbers that end in two, like 22. With this
foundation, extend rounding to bigger numbers,
e.g. “If you can round 59, you can round 359” and
“If you can round 359, you can round 2,359…”
You can teach rounding to a different place value
(i.e. tens, hundreds, thousands) applying the
same technique.

44. Difficult-step Segregation
 A number line is also useful in rounding numbers,
because the number line helps the child see
whether a given number is closer to _____ or to
_____. Teach the child that, if the target number is
halfway or bigger, she rounds up.
 Use the difficult-step segregation technique, that
is, have the child work only on the one-step that
he struggles (for example, borrowing) without
having to deal with any of the other steps of the
problem at the same time. You supply the other
steps. Then you can switch to a different difficult
step in the same algorithm.

45. Spread Out the Digits
 Use partially solved problems, so that the child focuses only
on the targeted sub-skill.
 Break down the algorithm into each step and teach each step
separately, e.g. at a different time or in a different day. Later,
show the student how the steps combine into one algorithm.
 Computation that requires multiple renaming can be
confusing visually to the child. To help simplify the visual
information, make worksheets with numbers that are bigger
and spread out the numbers, for example, 8975 + 2376 will
be,
o 8 (space) 9 (space) 7 (space) 5+
o 2 (space) 3 (space) 7 (space) 6

46. Circle or Highlight
 Another way to simplify the visual information is telling the child to
leave at least one empty line or a blank space between the problem
and the carried numbers. Alternatively, teach the child to place the
carried numbers at the bottom of the problem, that is, between the
problem and the answer.
 Children that perform fluently plus ones, plus twos, doubles, and
hidden tens have an easier time adding and/or subtracting multiple
digits. Before computing, have the child circle or highlight the
doubles, hidden tens, etc. that she sees in the problem. For
example, in the problem 8971 + 2376, six and one are a plus one,
seven and seven are a double, there is a hidden ten in nine plus
three (10 +2), and finally, eight and two are also a hidden ten.
 Teach the child to use easier numbers and then to transfer the
answer to the bigger numbers. For example, to solve 4287 + 2619,
first, the child solves 42 + 26=68; so, the final answer must be at
least 6,800.

47. Organize and Recognize
 Most children with learning problems handle computation as
simply adding or taking away ones (e.g.
63+8=63+1+1+1+1+1+1+1+1), failing to perceive the number
patterns and/or number combinations that help them perform
longer computation faster and easily. Help the child perceive
numbers as a range of number combinations that interrelate
differently depending on the situation. For example, a
hundred will equal two units of fifty in one situation, four units
of twenty five in another situation, and ten units of ten in a
third situation. A child that is able to organize and reorganize
numbers depending on which organization best fits the
particular situation, will be able to handle computation that is
more sophisticated with less struggle. In the example above,
the student can reorganize sixty-three as six units of ten or
sixty, moving all the ones together. Now the child has 60+3+8
or 60+10+1, which is the same as 71. Alternatively, the child
can reorganize the numbers as 50+10+10+1, etc.

48. Numbers are Flexible
 Your emphasis should be in helping the student understand both
that numbers are flexible, and that numbers relate to one another.
 The child that is limited to counting tallies is not adding; she is
sequencing rather than grouping numbers. By definition, addition is
the grouping of numbers, and these numbers can be single digits,
multiple digits, and/or columns of numbers.
 A compensatory addition technique is to teach the student to break
up one number or expanding. For example,
o 36+25=
o 36+ (20+5) =
o 36+20=56
o 56+5=61

49. Expand
 Similarly, the child can break up or expand two numbers. Example:
o 155+34=
o (150+5)+ (30+4) =
o 150+30=180
o 5+4=9
o 180+9=189
 The child can use the same approach (breaking down and
expanding numbers) to subtract. For example,
o 91-67=
o 91-(60+7) =
o (91-60)-7=
o 31-7=24

50. Transfer
 Once the child handles math facts using the ten as a base or
key, she can transfer this strategy to perform longer
computation. For example:
 First Example
o 87+9=
o 87+10=
o 97-1=96
 Second Example
o 77+8=
o 77+10=
o 87-2=85

51. Place Value
 Children who understand place value are more
efficient in computing multi-digits. Expose the
child to exercises such as:
o 246 + 457=
o 200+40+6 plus 400+50+7=
o (Ones column) 6+7=13 or 10+3; the ten moves to
the next column and the three stays
o (Tens column) 10+40+50=100; one hundred
moves to the next column
o (Hundreds column) 100+200+400=700
 The final answer is 700+0+3 or 703

52. The Digit’s Value
 Some children confuse the digit’s name with
the digit’s value. Make sure the student
understands that, in a digit like 3,469, the nine
is the same as nine ones, the six is the same
as 60 or six tens, the four is the same as 400
or four hundreds, and the three is the same as
3,000 or three thousands. Ask questions such
as, “In the digit 3,469, point to the digit with the
highest value” and “Point to the digit with the
smallest value.” Prepare a chart with
examples (see next), and keep the chart
visibly posted, so that the child can use it as a
reference.

53. Post a Chart
 Prepare and post a chart with numbers such as
777 and 4469 scaffolded visually, that is, each
place value is of a different size.
 First Example:
o 777=
o 7 (biggest) 7 (bigger) 7 (big) =
o 700 (biggest) + 70 (bigger) + 7 (big) =
o 7 hundreds (biggest) + 7 tens (bigger) + 7 ones
(big)

54. Second Example
o 4469=
o 4 (jumbo size) 4 (biggest) 6 (bigger) 9 (big) =
o 4000 (jumbo) + 400 (biggest) + 60 (bigger) + 9
(big) =
o 4 thousands (jumbo) + 4 hundreds (biggest) +
6 tens (bigger) + 9 ones (big)
 This approach helps the student see and
understand that the closer a digit is to the right,
the smaller its value.

55. Extra Visual Structure
 Make sure that the student refers to numbers correctly. For
example, to add 7+5, children typically say, “Two goes down,
and I carry one.” It is important that the child understands that
is not a one that she carries, but one ten or one thousand.
 Edit the child’s worksheets so that he performs multi-digits
subtraction with only one borrowing.
 Provide extra visual structure drawing mini frames or circles
where the child must place the carried numbers.
 Circle or color the number the child needs to change or to
rename.
 Color-code each column to show what numbers belong in
each column. When a number is regrouped to the next
column, color-code it to match the column is coming from so
that the child sees it has been moved.

56. Place Value Subtraction
 Similar to addition, the child can benefit from
exposure to place value subtraction such as
o 6423-2585=
o 6000 (space) 400 (space) 20 (space) 3-
o 2000 (space) 500 (space) 80 (space) 5

57. Eliminate Borrowing
 To eliminate borrowing, teach the student to add the same amount
to both numbers.
 Example
o 82-67 can be solved as 82+3 (85) minus 67+3 (70).
o 85-70=15
o 82-67=15
 Another example
o 71-33=
o 71+7 or 78 minus 33+7 or 40
o 78-40=38
o 71-33=38
 The key in using this compensatory strategy is that the child must
add the same amount to both numbers. The child can recite the
phrase, “What I do to one number, I do to the other number.”

58. Use Zeros
 Another technique to help the student with
difficulty carrying numbers is to teach the child
to rewrite the problem using zeros.

59. Example 1
o 42
o 68
o 73
o 96
o 84
 The child rewrites this problem the following way
o 40
o 60
o 70
o 90
o 80
 The partial addition is 340. To get the second partial answer, the child adds what is left
o 2
o 8
o 3
o 6
o 4
 The second partial answer is 23. Finally, the child adds the two partial answers (340+23), and gets the final
answer, or 363.

60. Example 2
o 369+827=
o 300+800=1100
o 60+20=80
o 9+7=16
 Finally, the child adds the partial answers, that
is, 1100+80+16=1,196.

61. Visualize
 Teach the child to visualize (see) a number line in her
mind. For example, to subtract 105-98, the child uses
the number line to see that 105 are five away from
one hundred, and 98 are two away from one hundred.
The child adds five and two, and gets seven. So, 105-
98=7. In this second example, 221-89, 221 are 21
away from 200, and 89 are 111 away from 200.
Adding 21 and 111, the child gets 132. The answer to
221-89 is 132.
 To teach the multiplication algorithm, use timetables
that the child already knows, or use easier tables like
the twos, threes, and fives. This way, the child can
focus on procedure.

62. Use the Same Multiplier
 Rewrite the multiplication problems, so that the student deals with only one
timetable at a time. You can prepare a worksheet where all the problems
have the same multiplier
 For example
o 235*7
o 459*7
o 803*7
o 672*7
 The idea is to have the child rehearsing the same timetables. On the next
worksheet, four and seven are the only multipliers that we used:
o 235*47
o 459*74
o 803*44
o 672*747
 You can apply the same technique to rehearse the division algorithm.

63. Minimize Visual Confusion
 Match the multiplication problem to parallel division
problems. For example, the child solves 803*44
(answer is 35,332), and then solves 35,332/803 and
35,332/44.
 Show the student that 203*54 is the same as 203*50
plus 203*4. In other words, the child first gets partial
answers, and then he adds the partial answers to get
a final answer.
 To minimize visual confusion and/or the child skipping
steps, teach the student to mark the steps as she
does the steps. The child can draw boxes or circles
around the multiplier and place value that she is
using, and then, crosses out the digits already used.

64. Visual Multiplication Algorithm
The following
visual
multiplication
algorithm is
ideally suited for
children who
struggle
memorizing the
traditional long
multiplication
algorithm. It
helps children
because they
can perform the
computation in
any order, or
from any
direction, and
minimizes
regrouping. In
addition, this
algorithm
reinforces
knowledge of
place value.
 426*53=
 400+20+6*
 50+3
 First Sub-Step
 400*50=20,000
 400*3=1,200
 Second Sub-Step
 20*50=1,000
 20*3=60
 Third Sub-Step
 6*50=300
 6*3=18

65. Answer
 Then, the child adds the partial answers
(20,000+1,200+1,000+60+300+18) in any
order he likes to get the final answer (22,578).

66. Use Color and Draw Frames
 Clarify the different steps in a long multiplication
or division problem using color. For example, the
first step in long division is always red, the second
step is always blue, and the third step is always
green. When you use a sequential color
procedure, you can tell at a glance where the child
is stuck in the computation algorithm.
 An alternative approach is to draw a frame or a
border around each major section in the problem
and/or the different steps.
 Show the student parallel multiplication and
division problems and say, “This is how your
completed problem will look.”

67. Algorithms
 To teach the long multiplication and long division algorithms, follow these
steps:
 Step One: You model and the child sees
 Step Two: An example worked together. You can say, “Tell me what I am
going to write” or “I do it here and you do it on this second copy.”
 Step Three: The child solves a third problem while you give feedback.
 In addition, make sure that the child rehearses algorithms, first, saying
aloud the steps, next, whispering, and finally silently.
 To visually separate the steps, set up different locations in the room or your
classroom for each different step (i.e. long division). On each location, label
the step. The student walks to each location to perform the corresponding
step.
 To help the student memorize the sequence in long division, have him
describe the procedure without computing answers. You can do the
computation, and you can give the partial and final answers. This approach
helps the child focus in the sequence of steps until he masters the
sequence.

68. Alternative Techniques for Problem
Solving
 Children have difficulty solving math word problems
for several reasons. Math story problems are
language laden, and children with either low
vocabularies or weak reading skills are negatively
affected. Some children struggle deciding which
operation or operations they need to use, or the child
may know the steps but confuses the correct
sequence, that is, what to do first, second, third, etc.
Problem solving is an area where students with weak
computational skills are affected the most. If the word
problem requires computation that is beyond the
student’s current skills, the child is not going to be
able to solve the problem. To plan remediation, it is
important that we know first where the child’s
difficulties are rooted.

69. Self-help and Classroom-based
Strategies
 Most of the strategies that help remediate procedural
knowledge deficits are useful also in remediating word
problems difficulties. Among them, giving the child
fewer story problems to solve paired with more time
spent talking through the steps at the conceptual
level. In addition, when both the teacher and the
student consistently use verbal organizational cues,
the sequencing of steps becomes easier to the child.
We need to teach children to ask regularly, “Does my
answer make sense?” and that they self-monitor if
they understand the story problem, so that they can
ask for help. Both self-help strategies (e.g. cue cards),
and classroom-based strategies (e.g. peer assistance
and charts visibly posted) need to be in place.

70. Simplify Language
 Be aware of any linguistic complexity in the problem, so that
you can clarify difficult language terms to the child. Levine
and Reed (2001) provide the following list of linguistic
complexities that students find frequently when solving math
word problems.
 Direct statement. Sam had four apples. Inez had three
apples. How many apples did Sam and Inez have in all?
 Indirect statement. Sam had four apples. Inez had the same
number as Sam. How many apples did Sam and Inez have?
 Inverted sequence. After June went to the store, she had
three dollars. She spent five dollars on groceries. How much
money did June take to the store?
 Inverted syntax. Seven puppies were given to Jack. Rachel
had six puppies. Together how many puppies did they have?

71.  Too much information. John and Brittany bought eight cookies. The cookies
cost twenty cents each. They ate five cookies on the way home from the
store. How many cookies were left when they got home?
 Semantic ambiguities (misleading cue words). Davon has twelve pens. He
has three more pens than Sheila has. How many pens does Sheila have?
 Important “little” words. Connie, Ray, and Ralph bought tacos for supper.
They each ate three, and there were four left. How many tacos did they
buy?
 Multiple steps. Patrick sold 410 tickets to the play. He sold twice as many
as Ellis. How many tickets did they sell in all?
 Implicit information. A plane flies east between two cities at 150 miles per
hour. The cities are 300 miles apart. On its return flight the plane flies at
300 miles per hour. What was the plane’s average flying speed? The first
trip was twice as long as the second was. (The student needs to know the
formula distance=rate*time so that she can find the time of the first and
second trip, and then, she needs to combine the results to find the average
speed.)

72. Show Rather than Tell
 Use a show rather than tell approach. Both
you and the student can use objects to
demonstrate the problem.
 As you read and describe the operations in the
word problem, the child manipulates objects
like toy cars, oranges, or sticks.

73. Translate the Problem to
Pictures
 Most story problems translate easily to pictures. To do
this, have the child read each sentence, stopping at
the period. Tell the child to draw a picture of what the
sentence says.
 Teach the student to draw -or to sketch- diagrams,
pictures, or flowcharts so that he can see the answer
to the problem. This strategy is especially helpful
when the child is working in word problems involving
fractions.
 Help the student develop mental images of the word
problem.
 As an intermediate step, teach the child to create a
situational model of the word problem (using pictures
or diagrams) before trying to set up the quantitative
representation (equation) of the word problem.

74. Draw Attention to Important
Details
 If the child’s reading skills are low, read aloud the problem
with the student following along. Draw the student’s attention
to the important details by placing vocal stress on key
information, and/or by saying, “This is important.”
 Before computing, set up the problem by having the child
eliminate (cross out) any extraneous or irrelevant information.
The student can highlight, underline, or color-code the
important details in the story.
 With the student, go through every sentence in the word
problem, asking if the information in the sentence is
necessary in solving the word problem. If it is not, the child
crosses out that sentence.
 Teach the child to read the whole problem first, and then, go
back and reread the problem, looking for the question. The
child underlines the question so that he can look back at it as
he works on the problem. Require from the child to always
identify the question part of the problem and circle or
underline it.

75. Key Words
 Ask the child to restate or to rewrite the question using her
own words.
 Let the student circle key information or use color
highlighters.
 Teach the student to highlight, circle or underline the key
words in the word problem, i.e. add, subtract, multiply, divide,
estimate, round, etc. Tell the child to do exactly what the key
words tell her to do. Some common key words and key
phrases in word problems are
o altogether
o how many in all
o what is the sum or total
o what is the difference
o how many more than
o how many less than

76. Identify Parts
 Discuss with the child what these key words and key phrases
mean.
 Give examples with the important information already
highlighted or underlined, so that the child has models to use
as a visual reference. In addition, show the child the specific
words in the examples that are telling which operations to
use.
 For each word problem, before attempting any computation,
begin by having the child identify the parts in the word
problem. That is, the child identifies the background of the
problem or setting, the information he needs to solve the
problem or facts, what he needs to find out or the question,
and finally, the distracters, that is, information that is irrelevant
in solving the word problem.

77. Teach a Procedure
 Teach the child a procedure for solving story
problems. The child can follow these steps:
1. Read the problem
2. Reread the problem to identify what is given (What
do I know?)
3. Decide what the problem is asking you to do (What
do I need to find out?)
4. Draw one or more pictures to represent the problem
5. Use objects to solve the problem and to identify the
operations you need to use
6. Write the problem
7. Work the problem

78. Write Notes in the Margin
 Teach the student to write helpful rules in the
margins of her paper. For example, the child
can write the order of operations for long
division, the sequence of steps in the word
problem, or any other useful information. If the
child is working on problems that require
knowledge of place value, in the margins of
the paper, she can write a place value chart
with ones, tens, hundreds, etc. This is a
timesaving technique, particularly when the
child is answering a test, because she does
not need to think of the rule each time she
begins to work a problem.

79. Number the Information
 Before computing, have the child hypothesize
the number of steps the problem requires, and
in which order. The child can color-code or
number the sequence of steps. To make it
easier to the child, you can reorder the steps in
the problem.
 Number the information in the word problem
according to the order in which the child needs
to do the steps.

80. Break the Problem into Steps
 Arrange the word problem so that it is clear that it requires more
than one-step. To guide the child’s thinking, provide answer blanks
for the child to write specific information or steps. Examples:
 Ms. Anderson’s class is planning a Thanksgiving party for 93
residents at a senior citizen’s home. The children want to put a play
for the seniors, and they want to give cookies and refreshments. If
each resident gets two cookies, how many cookies the class needs
for the senior citizens?
o Cookies for the senior citizens _____
 There are 27 students in Ms. Anderson’s class, and each child
wants two cookies. How many cookies the class needs for the
students?
o Cookies for the students _____
o How many cookies altogether the class needs for the party?
o Cookies needed altogether _____

81. Solve the Word Problem Orally
 For some children, performance improves
when you allow them to solve the word
problem orally.
 Have the student replace operational words
with the correspondent computation symbols
(i.e. + - * / =).
 Teach the student to recognize the hidden
facts that will help solve the problem, for
example, days in a week, items in a dozen, or
how many feet in a yard.

82. The Word Problem Type
 Teach the student to recognize the word
problem type.
 Change: Jenny baked _____ cupcakes. She
ate _____. How many cupcakes are left?
 Group or from part to whole: There are _____
blue crayons and _____ red crayons. How
many crayons are there altogether?
 Compare: Shawn has 41 baseball cards. Eric
has 27 more baseball cards than Shawn has.
How many baseball cards Eric has?

83. Simplify the Computation
 For practice, group similar problems together. Prepare a variety of
the same problems type so that the child has plenty of practice with
each type.
 Use simple calculations to control the effect of low computational
skills on problem solving.
 You can simplify the computation in the word problem by replacing
harder computation with easier numbers. For example, if the word
problem requires computing 684*925, the child can try first, 68*92,
or round to 70*90 to get an approximation of the answer.
 Give the child story problems with the final answer included and
have him discuss the steps used to solve each problem.
 Give the child a word problem and the steps to solve the problem,
but in random order. The child must arrange the steps in the correct
order and get the final answer.
 Tell the student to think of a similar problem and use the same
steps.

84. Find Part of the Answer
 Prepare a chart with a parallel word problem, that is, a similar word
problem with the same steps in the same sequence, and the final
answer. Tell the child, “Your word problem and the steps that you
must follow look like this example, but with a different answer. If
your problem does not look like the example, you did something
incorrectly. The example also shows you where you are going to
write the final answer.” The child follows the example to solve the
word problem. Repeat this exercise using word problems that
require different steps.
 Break one longer, complex problem into two or three simpler
problems that the child solves separately, and then, the child
combines the partial answers to find the final answer.
 Tell the child to try to find part of the answer and see if she can
continue from the partial answer.
 Use the stepwise approach, helping the child develop the mindset
that solving a math word problem always involves a sequence of
steps rather than something the child does all at once.

85. Teach the Child to Self-question
 Teach the child to verbalize what she is doing
while she is solving the word problem (talking
through the steps).
 Similar to performing longer multiplication or
division, you can set up separate spaces in the
room for each step. For example, tape five lines
on the floor for the five steps required to solve the
problem, and let the child walk to each line when
handling the step.
 Teach the child how to decide what to do. While
working on the problem the child answers the self-
questions: what am I doing? and what I did

86. Use the Discovery Approach
 Give the child opportunities to verbalize the
problem and to talk about possible solutions. The
child needs to practice the language.
 Use the discovery approach; asking questions
such as:
 How did you solve this problem?
 Why did that strategy work? Alternatively,
 Why the strategy did not work?
 In addition, ask,
 Can you think of another way of solving this
problem?

87. Planning Strategies
Have the
child prepare
a cue card,
or index
card, with a
list of
planning
strategies for
solving word
problems.
For
example:
 Draw a picture or a diagram
 Make a model
 Make a chart
 Visualize (see) the word problem in your mind
 Work backwards (starting from the end)
 Use your own words to restate the problem in a
different way
 Break one longer problem into two or three
smaller problems
 Act it out
 Think of a similar problem and borrow the steps
from that problem

88. The Self-monitoring Checklist
 Teach the student to complete a self-monitoring
checklist:
 I answered what I know
 I answered what I need to know
 I answered the problem’s question
 I pictured the problem in my mind
 I made drawings
 I recognized the number of steps the problem needs
 You can prepare a similar checklist or a problem-
solving frame for the student to follow or to fill-in while
she is solving the problem.

89. Focus on Models
 When you check the child’s work, mark the
problems and the steps that the child did
correctly, and do not mark the errors. This way,
the student focuses on good examples or
models, and you reinforce self-monitoring
skills. Then, tell the child that he will earn extra
credit for each error that he can identify and
correct (self-monitoring). You can tell the child
the number of errors he must find, for
example, “Find and fix the two errors in this
(computation or word) problem.”

90. Daily Experiences
 Do exercises that require from the student to
check and to fix work samples. These exercises
increase the child’s attention to detail, strengthen
knowledge of algorithms and the right sequence
of steps, improve self-monitoring, develop
automatic recall of math facts, and enhance
identification of salient information in word
problems. (Levine and Reed, 2001)
 Give the child credit for correct reasoning even if
the computation in the word problem is incorrect.
 Do not limit math problem solving to paper-and-
pencil activities. Incorporate math word problems
into daily experiences.

91. Reference
 Levine, M. D., & Reed, M. (2001).
Developmental variation and learning
disorders. Cambridge, MA: Educational
Publishing Service.

92. Bibliography (1)
 To prepare this guide, the following sources were consulted:
o Ashlock, R. B. (2006). Error patterns in computation (9th ed.). Upper
Saddle River, NJ: Pearson Education.
o Choate, J. S. (2000). Successful inclusive teaching: Proven ways to
detect and correct special needs (3rd ed.). Needhan Heights, MA:
Allyn and Bacon.
o Cooper, R. (2005). Alternative math techniques: When nothing else
seems to work. Longmont, CO: Sopris West.
o Currie, P. S., & Wadlington, E. M. (2000). The source for learning
disabilities. East Moline, IL: Linguisystems.
o ERIC/OSEP Special Project (Fall 2002). Knowing and doing math
improves mathematic achievement. Research Connections in
Special Education (number 11). Arlington, VA: The Eric
Clearinghouse on Disabilities and Gifted Education.
o Goldish, M. (1991). Making multiplication easy: Strategies for
mastering the tables through 10. Broadway, NY: Scholastic.

93. Bibliography (2)
o Harwell, J. M. (1995). Complete learning disabilities resource
library: Ready-to-use information and materials for assessing
specific learning disabilities. Volume I. West Nyack, NY: The Center
for Applied Research in Education.
o Harwell, J. M. (1995). Complete learning disabilities resource
library: Ready-to-use tools and materials for remediating specific
learning disabilities. Volume II. West Nyack, NY: The Center for
Applied Research in Education.
o Jitedra, A. (2002). Teaching students math problem-solving through
graphic representations. Teaching Exceptional Children, 34(4), 34-
38.
o LUCIMATH Project. Multidigit multiplication and division (PDF).
UCLA math content program for teaching multidigit multiplication
and division. Appendix B. Available on line at
www.math.ucla.edu/Luci/Lausd.
o Lyle, M. (2000). The LD teacher’s IDEA companion. East Moline, IL:
Linguisystems.
o Mather, N., & Jaffe, L. E. (1992). Woodcock-Johnson
psychoeducational battery-revised: Recommendations and reports.

94. Bibliography (3)
o Mather, N., & Jaffe, L. E. (2002). Woodcock-Johnson
III: Reports, recommendations, and strategies. New
York, NY: John Wiley.
o Miller, S. P., & Hudson, P. J. (2006). Helping students
with disabilities understand what mathematics means.
Teaching Exceptional Children, 39(1), 28-35.
o Sherman, H. J., Richardson, L. I., & Yard, G. J.
(2005). Teaching children who struggle with
mathematics: A systematic approach to analysis and
correction. Upper Saddle, NJ: Pearson.
o Witt, J., & Beck, R. (1999). One-minute academic
functional assessment and interventions. Longmont,
CO: Sopris West.

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