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. Why we think they are the 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.
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.
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.
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.
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.
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.