school help: a teacher and tutor guide to help the older student with limited math skills

8/13/2019 School Help: A Teacher and Tutor Guide to Help the Older Student with Limited Math Skills

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School Help

A Teacher and Tutor Guide

To Help the OlderStudent with

Limited Math Skills

Carmen Y. Reyes

Copyright 2011 by Carmen Y. Reyes

SolidRock Press


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ContentsBackground

1. Alternative Techniques for Recalling of Math Facts

2. Alternative Techniques to Develop Procedural Knowledge

3. Alternative Techniques for Problem Solving

Reference

Bibliography

About the Author

Connect with the Author Online


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Background

Children with low math skills typically evidence skill 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

childs 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 overallmath 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 techniquesand compensatory strategiesthatwe 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.

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 strategiesto solve the problem. For this reason, any remediation plan

that we implement should put less emphasis in memorization and more emphasis


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instrategy using. Teachers and tutors get better results in developing math skills in

all kindsof learners when we see and teach math as aplanned andstrategicway 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 book, 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.


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1

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, subtract, or tell which number is bigger than or smallerthan another number.

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.


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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.

Build on what the student already knows. Teachers and tutors can often turn a

students failure into success if we build on what the student already knowshow to

do it.

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 exercisesto 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


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another example, 8*7=56 and 56 is also the shirts number of the childs favorite

football player.

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.

To reduce the demand on memory, teach the student to recognizepatterns. For

example:

Doubles

7+7=14

Doubles Plus One

7+8=

7+ (7+1) =

14+1=15

Doubles Minus One

6+7=

7+7=

14-1=13

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 togeneralizeher knowledge of double patterns to solve

quickly other addition facts. In a problem like, 8+6, the child can use either the


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8+8 double pattern (8+8=16, minus two, equals 14), or the 6+6 double pattern

(6+6=12, plus two, equals 14).

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,

5=

4+1

3+2

2+2+1

3+1+1

1+1+1+1+1

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:

6+3=9

7+2=9

8+1=9


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Make sure the student performs automaticallyplus ones and minus ones, or one

moreand one less, for example, 17+1 and 66-1. Then, practice the child until he

retrieves automaticallyplus 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,

8+6=8+(8-2)=16-2=14

7+9=7+(7+2)=14+2=16

Emphasize number relationships such as hidden tens. Examples of hidden tens are:

First Example9+3= (9+1) +2=

10+2=12

Second Example7+5=2+ (5+5) =

2+10=12

After learning the hidden tens, teach the child to recognize that the nines are one

less; the elevens are one more.

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.

Top9

7

2


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8

5

4

Bottom1

3

6

8

4

2

In this 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.

DoMake 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.

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.

1 (space) 2 (space) 3 (space) 4 (space) 5


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9 (space) 8 (space) 7 (space) 6 (space) 5

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:

9+16=10+15=25

14+9=13+10=23

26+9=25+10=35

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 orto subtract.

Teach the child to change the order of numbers to make easy numbers.

23+14+5+6=

14+6+23+5+

20+23+5=

43+5=48

Provide daily looking for patterns exercises, for example,

What should go next? Explain why

2, 4, 6, 8, ___ Explain why __________

20, 19, 18, 17, ___ Explain why __________

8, 12, 6, 10, 4, ___ Explain why _____


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The last example is aplus four, then minus six pattern:

8+4=12

12-6=6

6+4=10

10-6=4

4+4=8

Therefore, the number that goes in the blank space is the same as 4+4 or 8.

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.

To add nine to any number, usefindthe next teen technique, that is, the answer is

one less than the second addend plus the teen that follows the first addend.

Examples:

9+5 (Take away one from five=4). Find the next teen (10) and addto four=14

19+7 (Take away one from seven=6). Find the next teen (20) and add to six=26

59+8 (Take away one from eight=7). Find the next teen (60) and add to seven=67


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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:

14-6=

6+ one tally=7

7+ one tally=8

8+ one tally=9

9+ one tally=10

10+ one tally=11

11+one tally=12

12+one tally=13

13+one tally=14

Alternatively, 6+8 tallies=14.

Therefore, 14-6=8

Have the child rehearse timed sheets such as,

10-6=

10-4=

6+4=

4+6=

After rehearsing the timed sheets, teach the child to usepartitioning. With this

alternative technique, the child solves subtraction facts by recalling known addition

combinations. For example:


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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.

Teach the child to say, Part is 17, total is 29. The missing part is _____.

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.

14-6=

14-(6+6) =

14-12=2

To get 14, the child must adjust the computation, increasing the six by two. For

example:

6+2=8

14-6=8

Second Example:15-7=

15-(7+7) =

15-14=1

Adjusting the computation, we add oneto seven:

7+1=8

15-7=8

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


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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.

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 makesure 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.

Teach the student to use the multiplication facts that she knows to figure out new

ones. Some examples are:

1. 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.

2. The child knows that 2*9=18, so, to solve 4*9, she uses 2*9=18 plus 18more or 36. This is thesplitting and doubling technique.

Teach multiplication facts following the progression of multiples. For example:

1. Begin with the 2s, then the 4s, and then the 8s.2. 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.


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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:

8*4=8*2=16+16=32

9*6=9*3=27+27=54

7*8=7*4=28+28=56

To handle the nine timetables, teach the child to use the ten timetables, for

example,

9*4=10*4=40-4=36

9*7=10*7=70-7=63

Another alternative technique to handle the nine timetables is the one less

technique. For example,

9*6 (one less is five) =5+4(the number that added to five equals nine) =54

9*9 (one less is eight) =8+1(the number that added to eight equals nine) =81

9*4 (one less is three) =3+6(the number that added to three equals nine) =36

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,

2* skip count

3* skip count

4* double the 2*

5* skip count


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6* double the 3*

7* turn around and/or use easier tables and then adjust or double the answer

8* double the 4*

9* use the 10* or the one less technique

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 inskip 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, Lets 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.

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,

3+3+3+3+3=

5 groups of three=

5*3=

15

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


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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= ___.

When we teach children to organizementally math facts, we are 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), andpatterns like doubles, skip

counting, and multiples.


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2

Alternative Techniques to Develop

Procedural Knowledge

Let the child useprompt 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 stepsat 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.

Teach the student aself-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.


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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 inparaphrasingby having the child restate the steps in her own

words. This way ofprocessing informationstrengthens the childsmemory.

Use the turn to your partner and explaintechnique. Being able to explain the new

procedure or concept to a peer not only enhances memory, but also is a good

measure of the childs 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.

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 asfrom right to leftandfrom top to bottom.

For children with alignmentproblems (i.e. the digits are all over the place rather

than aligned properly in columns), you can usegraph paperto force the child to

write one digit only inside each square. Alternatively, you can usesheets of lined


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notebook paper turned verticallyso 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.

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 tobigger 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.

A number line is also useful in rounding numbers, because the number line helps

the childsee 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.

Usepartially 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,


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8 (space) 9 (space) 7 (space) 5+

2 (space) 3 (space) 7 (space) 6

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 bottomof 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 aplus 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 numbersand 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.

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 organizeand reorganizenumbers 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


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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.

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,

36+25=

36+ (20+5) =

36+20=56

56+5=61

Similarly, the child can break up or expand two numbers. Example:

155+34=

(150+5)+ (30+4) =

150+30=180

5+4=9

180+9=189

The child can use the same approach (breaking down and expanding numbers) to

subtract. For example,

91-67=

91-(60+7) =


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(91-60)-7=

31-7=24

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:87+9=

87+10=

97-1=96

Second Example:77+8=

77+10=

87-2=85

Children who understandplace value are more efficient in computing multi-digits.

Expose the child to exercises such as:

246 + 457=

200+40+6 plus 400+50+7=

(Ones column) 6+7=13 or 10+3; the ten moves to the next column and the three

stays

(Tens column) 10+40+50=100; one hundred moves to the next column

(Hundreds column) 100+200+400=700

The final answer is 700+0+3 or 703

Some children confuse the digits namewith the digits value. Make sure the

student understands that, in a digit like 3,469, the nine is the same as nine ones, the


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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 below), and keep the chart

visibly posted, so that the child can use it as a reference.

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:777=

7 (biggest) 7 (bigger) 7 (big) =

700 (biggest) + 70 (bigger) + 7 (big) =

7 hundreds (biggest) + 7 tens (bigger) + 7 ones (big)

Second Example:4469=

4 (jumbo size) 4 (biggest) 6 (bigger) 9 (big) =

4000 (jumbo) + 400 (biggest) + 60 (bigger) + 9 (big) =

4 thousands (jumbo) + 4 hundreds (biggest) + 6 tens (bigger) + 9 ones (big)

This approach helps the studentseeand understand that the closer a digit is to the

right, the smaller its value.

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 onethat she carries, but one ten or one thousand.

Edit the childs worksheets so that he performs multi-digits subtraction with only

one borrowing.


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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 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.

Similar to addition, the child can benefit from exposure toplace value subtraction

such as

6423-2585=

6000 (space) 400 (space) 20 (space) 3-

2000 (space) 500 (space) 80 (space) 5

To eliminate borrowing, teach the student to add the same amount to both

numbers. For example,

82-67 can be solved as 82+3(85) minus 67+3(70).

85-70=15

82-67=15

Another example,

71-33=

71+7or 78 minus 33+7or 40

78-40=38

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.


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Another technique to help the student with difficulty carrying numbers is to teach

the child to rewrite the problem using zeros. For example:

42

68

73

96

84

The child rewrites this problem the following way:

40

60

70

90

80

The partial addition is 340. To get the second partial answer, the child adds what is

left:

2

8

3

6

4

The second partial answer is 23. Finally, the child adds the two partial answers

(340+23), and gets the final answer, or 363.


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Second Example:369+827=

300+800=1100

60+20=80

9+7=16

Finally, the child adds the partial answers, that is, 1100+80+16=1,196.

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.

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,

235*7

459*7

803*7

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:


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235*47

459*74

803*44

672*747

You can apply the same technique to rehearse the division algorithm.

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 afinal 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.

The following visual multiplication algorithmis ideally suited for children whostruggle 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


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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

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).

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 asequential 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 borderaround 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.

To teach the long multiplication and long division algorithms, follow these steps:

Step One: You model and the child seesStep Two: An example worked together. You can say, Tell me what I am

going to write or I do ithere and you do it on this second copy.


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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 create a set of separation between 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.


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3

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 students

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 childs difficulties are

rooted.

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, Doesmy answer make

sense? and that they self-monitor if they understand the story problem, so that

they can ask for help. Bothself-help strategies (e.g. cue cards), and classroom-

based strategies (e.g. peer assistance and charts visibly posted) need to be in place.


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If language skills are an area of concern, we can simplify the language, and we can

clarify the vocabulary using synonyms and rephrasing. In addition, we can use

shorter sentences. To measure comprehension, we can ask the child to restate the

problem in his own words.

Be aware of any linguistic complexityin the problem, so that you can clarify

difficult language terms to the child. Levine and Reed (2001) provide the following

list of linguistic complexitiesthat students find frequently when solving math word

problems.

1.Direct statement. Sam had four apples. Inez had three apples. How manyapples did Sam and Inez have in all?

2.Indirect statement.Sam had four apples. Inez had the same number as Sam.How many apples did Sam and Inez have?

3.Inverted sequence. After June went to the store, she had three dollars. Shespent five dollars on groceries. How much money did June take to the store?

4.Inverted syntax. Seven puppies were given to Jack. Rachel had six puppies.Togetherhow many puppies did they have?

5. Too much information. John and Brittany bought eight cookies. The cookiescost twenty cents each. They ate five cookies on the way home from the

store. How many cookies were left when they got home?

6. Semantic ambiguities (misleading cue words).Davon has twelve pens. Hehas three morepens than Sheila has. How many pens does Sheila have?

7.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?

8.Multiple steps. Patrick sold 410 tickets to the play. He sold twice as many asEllis. How many tickets did they sell in all?


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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.

Ask the child to restate or 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

altogether how many in all what is the sum or total what is the difference how many more than how many less than

Discuss with the child what these key words and key phrases mean.


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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 thepartsin the word problem. That is, the child identifies the

background of the problem orsetting,the information he needs to solve the

problem orfacts, what he needs to find out or the question, and finally, the

distracters, that is, information that is irrelevant in solving the word problem.

Teach the child aprocedure for solving story problems. The child can follow these

steps:

1. Read the problem2. 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 problem5. Use objects to solve the problem and to identify the operations you need to

use

6. Write the problem7. Work the problem

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 onproblems 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.


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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.

Arrange the word problem so that it is clear that it requires more than one-step. To

guide the childs thinking, provide answer blanks for the child to write specific

information or steps. Two examples:

1. Ms. Andersons class is planning a Thanksgiving party for 93 residents at asenior citizens 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?

Cookies for the senior citizens_____

2. There are 27 students in Ms. Andersons class, and each child wants twocookies. How many cookies the class needs for the students?

Cookies for the students_____

How many cookies altogether the class needs for the party?

Cookies needed altogether_____

For some children, performance improves when you allow them to solve the word

problem orally.

Have the student replace operational words with the correspondent computationsymbols (i.e. + - * / =).

Teach the student to recognize the hidden facts that will help solve the problem, for

example,


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days in a week, items in a dozen, or how many feet in a yard.

Teach the student to recognize the word problem type.

A.Change: Jenny baked _____ cupcakes. She ate _____. How many cupcakesare left?

B.Grouporfrom part to whole: There are _____ blue crayons and _____ redcrayons. How many crayons are there altogether?

C.Compare: Shawn has 41 baseball cards. Eric has 27 more baseball cardsthanShawn has. How many baseball cards Eric has?

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 childstory problems with the final answer includedand 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.

Prepare a chart with aparallel 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


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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 thestepwise approach, helping the child develop the mindset that solving a

math word problem always involves asequence of stepsrather than something the

child does all at once.

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,

1) What am I doing?And

2) What I did already?Give the child opportunities to verbalize the problem and to talk about possible

solutions. The child needs to practice the language.


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Use the discovery approach; asking questions such as:

1. How did you solve this problem?2. Why did that strategy work? Alternatively,3. Why the strategy did not work?

In addition, ask, Can you think of another way of solving this problem?

Have the child prepare a cue card, or index card, with a list ofplanning 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

Teach the student to complete aself-monitoring checklist:

1. I answered what I know2 I answered what I need to know

3. I answered the problems question4. I pictured the problem in my mind5. I made drawings


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6. I recognized the number of steps the problem needsYou can prepare a similar checklist or aproblem-solving frame for the student to

follow or to fill-in while she is solving the problem.

When you check the childs 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.

Do exercises that require from the student to check and fix work samples. These

exercises increase the childs 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.


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Reference

Levine, M. D., & Reed, M. (2001).Developmental variation and learning

disorders. Cambridge, MA: Educational Publishing Service.

Bibliography

To prepare this eguide, the following sources were consulted

Ashlock, R. B. (2006).Error patterns in computation(9thed.). Upper Saddle

River, NJ: Pearson Education.

Choate, J. S. (2000). Successful inclusive teaching: Proven ways to detect and

correct special needs (3rded.). Needhan Heights, MA: Allyn and Bacon.

Cooper, R. (2005).Alternative math techniques: When nothing else seems to work.

Longmont, CO: Sopris West.

Currie, P. S., & Wadlington, E. M. (2000). The source for learning disabilities.

East Moline, IL: Linguisystems.

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.

Goldish, M. (1991).Making multiplication easy: Strategies for mastering the

tables through 10. Broadway, NY: Scholastic.


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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.

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.

Jitedra, A. (2002). Teaching students math problem-solving through graphic

representations. Teaching Exceptional Children, 34(4), 34-38.

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.

Lyle, M. (2000). The LD teachers IDEA companion. East Moline, IL:

Linguisystems.

Mather, N., & Jaffe, L. E. (1992). Woodcock-Johnson psychoeducational battery-

revised: Recommendations and reports.New York, NY: John Wiley.

Mather, N., & Jaffe, L. E. (2002). Woodcock-Johnson III: Reports,

recommendations, and strategies. New York, NY: John Wiley.

Miller, S. P., & Hudson, P. J. (2006). Helping students with disabilities understand

what mathematics means. Teaching Exceptional Children, 39(1), 28-35.

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.

Witt, J., & Beck, R. (1999). One-minute academic functional assessment and

interventions. Longmont, CO: Sopris West.


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About the Author

Carmen Y. Reyes, The Psycho-Educational Teacher, has more than twenty years

of experience as a self-contained special education teacher, resource room teacher,

and educational diagnostician. Carmen has taught at all grade levels, from

kindergarten to post-secondary. Carmen is an expert in the application of behavior

management strategies, and in teaching students with learning or behavior

problems. Her classroom background, in New York City and her native Puerto

Rico, includes ten years teaching emotionally disturbed/behaviorally disordered

children and four years teaching students with a learning disability or low

cognitive functioning. Carmen has a bachelors degree in psychology (University

of Puerto Rico) and a masters degree in special education with a specialization in

emotional disorders (Long Island University, Brooklyn: NY). She also has

extensive graduate training in psychology (30+ credits). Carmen is the author of

60+ books and articles in child guidance and in alternative teaching techniques for

low-achieving students. To read the complete collection of articles, downloadfreelesson plans, and preview her books, visit Carmensblog The Psycho-Educational

Teacher.


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Connect with the Author OnlineBlog

http://thepsychoeducationalteacher.blogspot.com/

Twitter

http://twitter.com/psychoeducation

Email

[email protected]
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