Frequently Asked Questions

Decimals are easy to teach and to learn, aren't they?

Answer


Why do you put the decimal point in the "ones"
column. This doesn't look neat.

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I have found teaching decimals using money is very successful.
Is this the best way?

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I teach students to compare decimals by adding zeros until
there are the same number of decimal places. Then they can
compare them as whole numbers. Is this the best way?

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Why should children learn to compare decimals of different
length? In real life this isn't needed.

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I wonder if primary school children need to be exposed to
decimals longer than one or two decimal places.

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To multiply decimal numbers, you get the number of decimal
places in the answer by adding the number of decimal places
in the two numbers being multiplied. Can this be explained
to children?

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How do you divide by a decimal number? Can it be explained
to children?

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The first priority of teaching mathematics is to make sure
children can work properly with whole numbers. Is it OK therefore
to leave complicated work with decimals out of the primary
curriculum?

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Why do builders measure in millimetres rather than metres?
For example they will give the dimensions of a room as 3150
mm by 4280 mm instead of 3.15m by 4.28m.

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What is a good teaching aid to show equivalences such as
0.387 = 3 tenths + 8 hundredths + 7 thousandths = 387 thousandths
= 3 tenths + 87 thousandths = 38 hundredths + 7 thousandths,
etc.?

Answer


I use the analogy of alphabetical ordering of words to assist
my students to order decimals. Is this a good strategy?

Answer


Why do you call the units column the ones column?

Answer


Why do you never just write .5 (point five) etc, but always
put a digit in the ones place?

Answer


I have a colleague who makes a big deal about the words "digit"
and "number". Is this just being pedantic?

Answer

Decimals are easy to teach and to learn,
aren't they?
Answer:
To some extent this is true. It is very easy, for example, to learn
to add and subtract with decimals. The rules for whole number addition
and subtraction can be applied with very little change. In contrast,
learning to add and subtract fractions is a new and challenging
task. For this reason, decimals may seem an easy topic to teach.
However, evidence from mathematics education studies across the
world indicates that both children and adults experience fundamental
difficulties with decimals. This is serious because decimals are
so widely used in practical situations. The main difficulties are

fully understanding the meaning of the decimal numbers,


being able to choose what operation with decimals to apply
in a given situation.

The best indication is that only about half of the population is
fully mastering decimals by the end of Year 10 (See research
results).Difficulties with decimals cause trouble solving problems,
making sense of answers to problems, rounding numbers off sensibly
and reading scales. Even though the rules for calculating with decimals
seem very simple, few children seem to be able to carry them out
correctly just by rote  they need to understand what they are doing.
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Why do you put the decimal point in the "ones"
column. This doesn't look neat.
Answer:
Many teachers explain the decimal point as the separator between
the whole number part and the fractional part of the number. We
think it is better to stress that a decimal is ONE number, not two
numbers put together. The role of the decimal point is to mark
where the ones column is. This is why we put it
in the ones column. It also helps students to understand where the
symmetry is. See here,
for example.
OUR MOTTO:
The decimal point marks the ones column.

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I have found teaching decimals using money
is very successful. Is this the best way?
Answer:
Money can indeed be very useful, but there are important traps.
With money, although 26 cents is 26 hundredths of a dollar, people
often do not think about $5.26 in this way. They think instead of
5 dollars and 26 cents with the dollars and cents representing two
separate systems of units, albeit with a conversion between them.
Although people know that there are 100 cents in a dollar, the fractional
relationship is not uppermost in their consciousness and so often,
5.26 is thought of as two numbers, not as one. As a teacher, if
you use money to teach about decimals you need to be careful not
to teach two parallel systems of whole numbers, without strong links
between the dollar and cent components.
There is also some evidence that teaching which relies too much
on money encourages children to be "money
thinkers". These children do not
appreciate that there are decimals beyond a fixed number of places
(usually 2). This is reinforced by too many questions in secondary
school requiring rounding to two decimal places.
Another point for caution with using money to teach decimals is
that is $5.26, the 2 as a decimal represents 2 tenths. As
money it represents 2 tens (of a different unit).
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Comparing decimals by adding zeros
I teach students to compare decimals by adding zeros until there
are the same numbers of decimal places. Then they can compare them
as whole numbers. For example, to compare 0.186 and 0.3 change it
to a comparison of 0.186 and 0.300. It is easy then to see that
300 is bigger than 186 so 0.3 is bigger than 0.186. Is this the
best way?
Answer:
This is a mathematically correct strategy for comparing decimals.
However, Resnick et al
(1989) suggest that using this approach can mask other misconceptions
that students may have about decimals. We have interviewed some
Grade 5 and 6 children who completed the Decimal Comparison Test
completely correctly with the adding zeros strategy. However, difficulty
with some interview questions reveals they had an incomplete understanding
of decimal notation.
The strategy also encourages some children to think in a whole
number fashion. It is important to reinforce the size of the
"fractional part" of the number.
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Why should children learn to compare decimals
of different length? In real life this isn't needed.
Answer:
Calculator multiplication and division can result in answers with
more than two decimal places (for example, 2/7=0.2857142857...).
The conversion of metric units can also lead to decimal measurements
with different numbers of decimal places. A person intending to
borrow money for a home loan could be presented with interest rates
given to different numbers of decimal places, eg 4.6% and 4.35%.
Have you ever noticed signs in large libraries which say "do
not reshelve books"? Well, now we know why... Many people are
unable to reshelve a book in the correct location, as it requires
comparing decimal numbers of different lengths. A university librarian
told us of the difficulties experienced by some new staff (employed
as "reshelvers") during their training.
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I wonder if primary school children need
to be exposed to decimals longer than one or two decimal places.?
Answer:
For many children it takes several years for full understanding
of the decimal number system to develop. Children need to work with
decimal numbers that contain more than two decimal places to foster
this understanding. They need to be able to see the general patterns
of the endless base
ten chain, not just a few isolated facts they can learn off
by heart. There are many contexts (in measurement topics, for example)
where such numbers are encountered and this should be encouraged
in the primary school classroom.
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Can the rule for multiplying decimals
be explained to children?
To multiply decimal numbers, you get the number of decimal places
in the answer by adding the number of decimal places in the two
numbers being multiplied. For example to multiply 3.1 (one decimal
place) by 4.02 (two decimal places), first multiply 31 by 402 and
put the decimal point in so that there are three (1 & 2) decimal
places in the answer. Can this be explained to children?
Answer:
The algorithm described here is often used in schools to carry out
pen and paper calculations of decimal long multiplication. Following
the rule does not "justify" or "prove" why it works. A full justification
of the rule is given in Foundations
for Teaching Arithmetic (Marston and Stacey, 2001).
It is important to develop the meaning of decimals by numbers less
than one. Area models are a useful way of showing how decimals can
be combined through multiplication, while estimation helps children
to decide the correct order of magnitude of a result. For example,
0.5 x 0.8 cannot equal 4 or 40 because both the multipliers
are less than one. Also 0.17 x 65.983871 must be less 65 but more
than 6.5 (one tenth of 65), so answers of 1.12172 or 112.1725 cannot
be correct).
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How do you divide by a decimal number? Can
it be explained to children?
Answer:
One difficulty that children have with dividing by decimal
numbers is that when dividing by a decimal between 0 and 1, the
result is larger than the original amount. This is counterintuitive
to many children who associate division with "getting smaller".
Being able to use various interpretations of division may assist
in making sense of division by a decimal. For example, 0.6
0.2 may be read as "0.6 divided by 0.2" or "how many point 2's in
point 6?" (the latter is more likely to help a student find the
correct answer 3). A pen and paper algorithm that can be used to
divide two decimals involves converting the decimals into whole
numbers and then doing a usual short or long division. So long as
the decimal point is moved an equal number of places in each number
than the ratio of the number stays the same and the division result
will therefore be correct. For more information, see Foundations
for Teaching Arithmetic by Marston and Stacey..
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Can I omit decimals if students aren't good
at whole numbers?
The first priority of teaching mathematics is to make sure children
can work properly with whole numbers. Is it OK therefore to leave
complicated work with decimals out of the primary curriculum?
Answer:
Many of the difficulties that we have encountered with children
in the field of decimals result from overgeneralisations about
whole numbers. This would suggest that students could benefit from
being introduced to decimals and the meaning of decimal notation
earlier rather than later in their primary school lives. It is clear
that this area is complicated for children, but the importance of
the fundamental numeracy represented by understanding decimal notation
means that they must have as many opportunities to practise as possible.
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Why don't builders and other tradesmen use
decimals more?
Why do builders measure in millimetres rather than metres? For
example they will give the dimensions of a room as 3150 mm by 4280
mm instead of 3.15m by 4.28m.
Answer:
One reason for using millimetres is probably to avoid decimal numbers!
However the most important reason is to make the precision in measurements
clear. If building measurements were quoted in centimetres then
it may not always be clear that a measurement to the nearest mm
is required. For example, reporting the shelf width as 35 cm does
not tell you whether the shelf is exactly 35 cm, or if it is actually
35.2 cm rounded down. Writing 352 mm makes it clear that the measurement
is to the nearest millimetre, a suitable precision for most building
tasks. (See more information about indicating the accuracy
of measurements)
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What is a good teaching aid to show
equivalences?
What is a good teaching aid to show equivalences such as 0.387
= 3 tenths + 8 hundredths + 7 thousandths = 387 thousandths = 3
tenths + 87 thousandths = 38 hundredths + 7 thousandths, etc.?
Answer:
Number expanders are great! (Full
details and instructions on how to make/buy and use are provided.)
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Is decimal order like alphabetical
order?
I use the analogy of alphabetical ordering of words to assist
my students to order decimals. Is this a good strategy?
Answer:
The process of ordering words alphabetically (eg bear, ear, eat,
eats, edible) is similar to that of ordering numbers, however, care
needs to be taken with this analogy. While letters in words are
identified with a position relative to the start of the word
(leftjustified), digits within whole numbers are identified with
columns determined by the position of the ones column at the end
of the number (rightjustified). The following table demonstrates
how the correct alphabetical ordering of words corresponds with
a correct ordering of decimals but an incorrect ordering of whole
numbers. (Here the nine letters used are replaced by the digits
1 to 9.)
Alphabetical order:
Correct

Whole number order:
Incorrect

Decimal order:
Correct

bear

2416

0.2416

ear

416

0.416

eat

418

0.418

eats

4187

0.4187

edible

435294

0.435294

Other analogies such as money, sports
scores, remainders in division and the Dewey
Decimal System (used to arrange books in libraries) are discussed
in the Background section.
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Why do you call the units column the ones
column?
A very hard choice. If we use numbers (hundred, ten, tenth) etc
for the other column name it seems odd not to use "one".
The disadvantage with "ones" is that the abbreviation
O as in Hundreds, Tens, Ones (H, T, O) must NOT be used. Too many
students confuse one and zero (bizarre, but common, see Stacey,
Helme and Steinle (2001))
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Why do you never just write .5 (point five)
etc, but always 0.5 etc?
This is good practice because sometimes the decimal point is hard
to see. As my wonderful Grade 4 teacher said, "You can never
tell if it is a decimal point or just a spot made when a fly crawled
out of the inkwell and onto your page". Not such a problem
since the invention of ball point pens, but still a good idea!
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I have a colleague who makes a big deal
about the words "digit" and "number". Is this
just being pedantic?
We expect children to know the difference between letters and
words! Any child who did not understand this difference would
have trouble when their teacher (or parent) was using these terms
in later teaching.
Yet we find children in interviews who do not know the difference
between digits and numbers! These children do not know whether
to focus on the individual digits or the entire number when their
teacher is explaining some mathematics.
The 26 letters (a to z) are the building
blocks used
to form a word (the complete quantity)

The 10 digits (0 to 9) are the building
blocks used
to form a number (the complete quantity)

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