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Models for teaching decimals
To understand a mathematical concept, students need to build a
mental model that faithfully represents its structure. Concrete
representations are an important intermediary, which students can
use to learn and to help solve problems.
The models described here represent decimal numbers in several
different ways, none representing all aspects. Some of these are
concrete materials which can be physically manipulated, others are
symbolic. Throughout their schooling students will benefit from
careful and planned exposure to several models.
Green font is
used to indicate activities and models that can be accessed from
the summary chart.
Representing size of numbers by length
Length is the simplest way to represent the size of a decimal number.
The best structured material for teaching
about decimals is Linear Arithmetic
Blocks (LAB). These are
homemade structured materials which can be used like MAB,
but they are easier for students to understand. The size of the
number is represented by the length of plastic pipes. LAB can be
used to illustrate how the size of the number 1.234 depends on the
number of ones (1), tenths (2), hundredths (3) and thousandths (4)
and also to illustrate the steps of algorithms for addition and
subtraction, multiplication and division. Note that we are talking
about making the general number 1.234 (not 1.234 metres!).
Just like MAB uses volume (but is not described in cubic cm), LAB
uses length but should not be limited to a discussion of metres,
centimetres or millimetres! (Our research shows LAB may be the best
model for decimals. Click here
to read why we think they are best.)
Number lines initially represent a number by the length
of a line segment from zero to a point. The point then comes to
represent the number instead. Number lines are useful for ordering
decimals but do not show the component parts of a decimal number
well.
Reading scales is an important real-life
application of number lines.
Many activities and aids are based on the number line model, including
Decimal Delivery, Number
Between, Stickers game, Flying Photographer
and Decimal Line.
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Representing size of numbers by volume
Representing decimal numbers by volume (or equivalently by mass)
is most commonly done with MAB, multi-base arithmetic blocks. The MAB can
be used to illustrate the number of ones, tenths and hundredths
in the number 3.14 and also to illustrate the steps of algorithms
such as addition and subtraction. Activities using MAB blocks include
Long Line of Blocks and Moving
Closer.
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Representing size of numbers by area
Area Cards provide another representation
of decimals in which the size of the number is related to the physical
size of the model.
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Representing decimals symbolically
Some features of the number system can be modelled other than with
physical attributes. By means of a clever folding pattern, a Number
expander displays expanded notation and the unitising and
reunitising that is required to see 20 hundredths as 2 tenths etc.The
Number slide demonstrates the process
of multiplication and division by ten and powers of ten.
Although these models are not concretely manipulable, they can
build strong visual images for children which support their thinking
about number. A strength of expanders and slides is that there is
no limit to the number of columns that they display. This means
that they can show the base 10 relationships between the columns
of all sizes. For example, the number 1 234 567.123 can be seen
as a little over 1 million or as 1 234 567 123 thousandths. Such
a number could never be modelled by length, area or volume.
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Money as a model for decimals
In countries with a decimal system of money, children learn a lot
about decimals by dealing with money. However, this is not an adequate
model. Firstly, there are no visual clues from the coins that 45
cents (say) is a fraction of a dollar. Secondly, the dollars and
cents are often seen as two separate systems of whole numbers and
the relationship that 100 cents = 1 dollar does not, of itself,
create a strong feeling of cents as a fraction of a dollar. This
can lead to whole number thinking. Thirdly, the money system is
discrete and so the smaller decimal places are not modelled.
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Advice for using manipulative materials effectively
Materials that can be physically manipulated are extremely important
for teaching mathematics. They include structured materials such
as MAB which is "structured" to mirror the base ten features of
our numeration system and unstructured materials such as counters,
which can be bundled up into groups of ten etc. English
and Halford (1995) describe how all these materials serve basically
as an analogy. The child sees the structure in the material and
is to build an abstract conceptual structure which mirrors it. The
book also reviews the research literature and lists principles for
deciding when analogues are likely to be successful for teaching.
Teaching with manipulative materials can help children understand
and reduce anxiety. However, they are not a simple panacea, but
require skillful use by teachers.
Models provide mental images and analogies to which students can
return to when thinking about the structure of numbers. Each model
has its own benefits and shortcomings, although some are certainly
better than others for children of a particular stage or for demonstrating
particular ideas.
It is important to use a model to the full and not to swap models
too often. However, in the long term, using more than one model
is important to illustrate the full nature of place value ideas.
The mapping between what is to be done with numbers and what is
done with materials needs to be simple and clear. Children need
careful help to draw parallels between physical, verbal, symbolic
and pictorial representations. Some children do not link what is
done with materials with what is done with symbols. Every step should
be able to be mapped across.
Teachers' and children's explanations need to be clear and apply
to both the physical materials and the target symbols. Teachers
must be careful not to use one type of language with the
physical models (e.g. trading) and another language (e.g. borrow
and pay back) with the numbers.
It is best if a model is simple to understand. This is why we recommend
using LAB, based on length, rather than MAB, based on volume.
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