NOT SO SIMPLE MATH

Gordon Scott, March, 1998.

1.

Part 1, Word Problems:

A short story containing a question that can be answered by working with numbers that
are directly and indirectly given in the story can simply be called a problem.
The child’s task is to get the correct number answer, and relate it back to the story.

EXAMPLE: A 7 year old child picks up 5 shiny pebbles on the way to school. On the
way home he picks up some more. At home he shows his mother that he has 11
pebbles.
How many did he pick up on the way home?

Easy?? Not for most of the youngest children.

What follows is an attempt to make adults aware that solving word problems is not so
simple for many young children, particularly when reading is still a difficult task and
logical thought isn’t well developed. This attempt isn’t intended to meet university
standards for thoroughness and accuracy; just to provoke thought on the process
involved in finding an answer to what appears to adults to be a simple question.

Remember, these are my opinions. You will often see ‘I’ and ‘my’ used, and you can
take that as a reminder that these are just the thoughts of one teacher.
If you have your own thoughts I’ll be happy to have provoked some thinking.

Problem solving requires a number of steps, and the failure to handle any one of these
may result in an inability to complete the problem, or an inaccurate answer.

1. First you have to read the words, and for most of the very youngest children this
means read aloud. Many very young children must be able to read, in the sense of
‘say’, all of the words contained in the problem. They are unable or unwilling to try to
really understand a short story when they cannot decipher some of the words.

2. Next you have to interpret the words. Some young children have difficulty
understanding the main ideas of the story if there are words whose meaning is not
clear to them. Those words stop them from putting the ideas together to form a story.
Every time they run into them the story falls apart as they focus on word meaning
instead of story meaning.

Some children could answer the problem without knowing what “pebbles” are, but
others would lose sight of the overall picture when they read that word. They may not
have the confidence to say that “pebbles” are obviously something you find, and get
on with the rest of it.

3.Then there’s the question. It has to be related back to the story. Most adults wouldn’t
notice that “pebbles” isn’t used in the question, but this could puzzle some children.

4. Then they have to sort out the numbers that relate to the question asked. Some
children want to skip directly to the next step, working with the numbers, without

2.

relating them to real ideas. They will grab all the numbers they see, regardless if they
have anything to do with the question or not. They would want to use the 7 without
thinking of it as counting years of age.

5. The numbers have to be fitted together in an incomplete number sentence that mirrors the
original word story, and the question.
5 + ___ = 11
This may be a mental, or mental and written, process.

6. The number sentence must be solved, and sometimes this is not so easy when the
answer does not pop into the mind. Many children have difficulty turning an additive
situation into a subtractive one.

Some may even need permission to do this before attempting it. Remember, they are
just starting all this, and they may lack confidence.

It may also depend to some extent on how they have been taught the basic number
facts.

Some may have to say to themselves that they will pretend to start with the 11 and take
away the part they know was found on the way to school. In other words they will go
back to the word story ideas to do 11 - 5 = ____ .

7. When this work is done correctly, it has to be related to the word story. The number
answer may be 6, but what is the 6 in terms of the word story?


In a sense, the child has to go into the word story world, find what’s missing, take what
is needed from the story into the world of numbers, and come back out again into the
word story world with something that will complete the story.

I hope you can see that this is not so simple for many young children.

EXAMPLE: A 7 year old child picks up 5 shiny pebbles on the way to school. On the
way home he picks up 4 more pebbles. At home he adds them to his collection and
finds he now has 15. How many did he have before he went to school?

This one has two steps and can be done in more than one way.

Then there’s the word problem that has numbers counting different important things.
Here the child has to keep in mind what job each number is doing.
The numbers are small, but for many children this word problem would be confusing.

EXAMPLE: A 7 year old child picks up 5 shiny pebbles on the way to school. On the
way home he picks up some more. At home he shows his older brother that he has 11
pebbles. His older brother says he will trade him 10 pennies for 10 of the pebbles.
If they do trade, what will the younger brother have then?

3.

This next one has two steps requiring number work, and requires a child to separate
the number of pennies from the number of pebbles.

EXAMPLE: A 7 year old child picks up 5 shiny pebbles on the way to school. On the
way home he picks up some more. At home he shows his brother that he has 11
pebbles.
His brother says he will give him 2 pennies for each of the pebbles he found on the
way home. If the younger brother agrees to this, what will he have then?


Can children be taught problem solving strategies?

Most children will need help to become confident in their ability to break word
problems into manageable, ordered steps. I hope you can see that the examples I
have given here are much too difficult to start with. They were only intended as
examples for adults.

Beginning problems must be very simple, with no numbers unrelated to the question.

If you look at the seven points above you can see that the ability to read well is
most important in doing this work independently.

Some children will have to have the problems read to them, and will have to
have someone available to reread sections, and to explain words.

When reading ability is not the chief difficulty, then there are some strategies that
children can learn to help them through the problem to an accurate answer.

Most beginning problems are situations where people or things are being
put together, or taken apart.

Children can practise recognizing each of these situations, without
having to worry about working with numbers.

When children have been taught, as in my Simple Math booklet, to think in terms of
two situations, PART + PART = WHOLE and WHOLE - PART = PART , they
should find it easier to think, “Are parts going together, or being taken apart?”

Children can practise this with numbers that are impossible for most to work with. The
example problem given first could have had 517 pebbles found on the way to school
and 978 counted out at home.

The answer might not leap out at them, but the process remains the same, and the
process is most important.

Children like to discover that some things do not change, that no matter what the
numbers are, the problems remains essentially the same.

4.

It gives them a very important feeling of confidence, and being confident in one’s
ability is a very large factor in one’s success in math.

Children could practise a few simple problems over and over, using different numbers
to reinforce the idea that problems can be very similar.

When they are competent at recognizing addition and subtraction situations, then they
can practice pulling out the relevant numbers from the story and putting them into a
number sentence that mirrors the story.

Again, this need not mean that the children focus on finding the answer. Finding the
answer may take away from developing this separate skill.
Putting in large numbers that they cannot be expected to add or subtract, just so they
focus on the process, may be a benefit to them.

I think this is most easily seen when one speaks in terms of PARTS and WHOLES.
What is missing? In most of the earliest word problems it's one or the other.

If they think in terms of
    PART + PART = WHOLE and WHOLE - PART = PART ,
the number sentences then become a choice between six variations on the above:

The most simple, PART + PART = __ or WHOLE - PART = __ ,

or the more difficult, PART + __ = WHOLE or WHOLE - __ = PART ,

and __ + PART = WHOLE or __ - PART = PART .

The first problem given as an example is more difficult because it is an additive
situation that, given large enough numbers, requires subtraction to find the number
answer.
517 + __ = 978 is not a question most young children are going to tackle.
978 - 517 = ___ is a question they will someday learn how to place in a vertical form
so that a number answer can be found.

By practising word problems that result in each of the six types of incomplete number
sentences, children can develop confidence.

They can learn that additive situations sometimes require subtraction to solve, and
subtractive situations sometimes require addition to solve. They can see that it is
permissible to subtract sometimes when things are being put together, and add
sometimes when things are really being taken apart.
It all depends on what is missing.

N.B. This does not mean that the first problems should be of these types.
The first should be about additive situations that require adding to find the number
answer, and subtractive situations that require subtracting.

5.

This effort put into learning a process for handling most beginning problems has a
carry over to multiplication and division problems later if it is done well. The
previous comments apply to these as well.

In my Simple Math booklet I speak of

    Number of Parts x Size of Parts = WHOLE , and

    WHOLE -:- Size of Parts = Number of Parts .

Children can learn to recognize these situations in word problems when they see that
things are going together, or apart, in equal groups.

Knowing that the Number of Parts and the Size of Parts are both factors of the
Whole, and that factors can change places with their partner in a number
sentence allows an even simpler representation:

FACTOR x FACTOR = WHOLE , or WHOLE -:- FACTOR = FACTOR .

Knowing this would allow one to see the possible word problems falling into types
which are very similar to those for addition and subtraction:

The simple FACTOR x FACTOR = __ , and WHOLE -:- FACTOR = __ ,

or the more difficult FACTOR x __ = WHOLE , and WHOLE -:- __ = FACTOR ,

or __ x FACTOR = WHOLE , and __ -:- FACTOR = FACTOR .

Here children can be led to see that given two FACTORS, one multiplies to find the
WHOLE, and given a WHOLE and one FACTOR one divides to find the other FACTOR.

It can all be handled much the same as with addition and subtraction word problems,
with one important difference.
One FACTOR counts groups or parts in the world of the word problem.
The other counts the same objects as the WHOLE.
This can make it much more difficult to relate the number answer to the word problem
and tell just what it is a number of.


Word problems are not always included in the math or arithmetic curriculum for young
children. If they are they may not be given much time. But ,when they are there, I hope
you will view them as something new to be taught, and you will consider the reading
ability required to do them independently.

I haven't yet seen any math texts that really have problems written in language that
most children would find easy to read.

6.