Since we are constantly inflicting tests on our pupils, let us begin with a simple test for maths teachers. Look at these four mental arithmetic questions involving decimal fractions, but instead of giving the correct answers, note the most common error that pupils make with each.
* 5 x 0.3
* Which is greater: 0.35 or 0.4?
Common responses to the first pair of questions are 0.12 and 0.15, which pupils often call "point 12" and "point 15". The calculation has been done with the whole numbers, ignoring the significance of the decimal point. If placed in the context of money - add together pound;0.30 and pound;0.90 - there would be fewer wrong answers. It is likely that the use of a decimal point in expressing an amount of money does little to reinforce understanding of decimals. Many people see pound;12.36 as two numbers - 12 pounds and 36 pence - with the decimal point to separate them, rather than as the single number 12.36.
Money is familiar to people and simple calculations are not usually problematic when they arise in everyday contexts. Learning about decimals in the classroom is another matter. Money offers a useful reference point but that is all. The underlying problem is linked to understanding place value, but I suspect that much talk of tenths and hundredths falls on deaf ears because it generates little understanding about what decimal fractions mean and how they behave.
Here are some ways in which a graphical calculator can be used to develop a "feel" for decimals.
A graphical calculator has a large screen so that several numbers or calculations can be displayed simultaneously. A single calculator linked to a display unit placed on an overhead projector offers scope for interactive whole-class teaching and costs much less than a computer. Pupils do not always need access to individual calculators - the advantage of a single calculator display as a focus for discussion is that pupils can be asked to think what will happen before pressing a key to reveal the answer. Left to themselves the temptation to press the key before thinking is too great.
All graphical calculators have a simple facility for generating sequences. Start by entering 0 and successively adding 0.3. After each number stop, ask what comes next and ask pupils to justify their responses. If there are misconceptions about adding 0.9 and 0.3, then asking what follows 0.9 provides an opportunity to discuss the issue. Finally, confirm the right response by pressing the enter key. Continuing the sequence, a similar discussion can take place at 2.7. Some pupils may suggest 2.10 follows 2.7, and there is also the question of whether 3 and 3.0 ae the same. (Figure 1.) A number line is another valuable aid in making sense of decimals - it is clearly very useful to represent the sequence of 0.3s on a number line as a picture of what the numbers mean and how they behave. Different representations of the same ideas are a powerful reinforcement of understanding. (Figure 2.) There is much more that we can get from this simple sequence. It is the "0.3 times table" and, as such, products such as 5x0.3 can be considered, together with the related division questions: 1.5V0.3 and 1.5V5. Pupils' understanding of these as the number of 0.3s in 1.5 and 1.5 divided into five equal parts is reinforced by seeing the sequence of 0.3s and its representation on a number line. This can help pupils to do such calculations mentally with confidence.
The second pair of calculations in my initial test for teachers involves the second decimal place.
If we present the simple sequence that goes up in intervals of 0.1, we can ask what numbers go half way between 0 and 0.1, between 0.1 and 0.2, and so on. This leads us to a sequence where the interval is 0.05. (Figure 3, above.) Moving through this sequence and along a number line at intervals of 0.05 reinforces the interplay between the first and second decimal place. That helps to make it clear why 0.17 is not a sensible response to 0.15+0.2 and that 0.35 is clearly less than 0.4. The common error with the latter is to say that 0.35 is greater than 0.4 because pupils ignore the point and rely on the fact that 35 is greater than 4. Pupils who make this mistake will quite happily respond with 0.35 when they are asked for a number halfway between 0.3 and 0.4. It is as though the context of "between" acts as a trigger for the right thinking - emphasising sequences and number lines provides valuable contexts for making sense of decimals. Using trial and improvement to solve equations does serve to reinforce understanding of decimals, whatever its role may be in relation to learning algebra.
Varying the number used for the interval will create a wealth of further examples, but the opportunities for discussion with a class are not limited to the simple arithmetic sequences considered here. Multiplying and dividing provide a range of possibilities related to understanding these operations in relation to decimals and to questions of growth and decay. Beyond this there are applications of sequences to equation solving and a variety of other areas of math-ematics. (Figures 4 and 5.) Using a large screen offers many opportunities for provoking interest in maths ideas and enhancing mental skills.
Doug French is a lecturer in education at the University of Hull and is chair of the Mathematical Association's teaching committeeMA website: www.m-a.org.uk