Many muddled ideas about algebra arisewhen pupils associate letters with objects. Doug French points out the pitfalls
We are not very successful at teaching algebra. Many of our pupils see it as having little meaning or purpose. Pupils' motivation is clearly not helped if they find algebra difficult to understand and its only apparent purpose seems to be to do the questions in the next test, with a vague suggestion that it will be useful later when they have learned more of it - a promise which is not fulfilled for most of them.
Although much good work has been done in recent years to encourage interesting approaches to algebra, particularly through exploring number patterns, this is not always followed up and linked appropriately to the important matter of developing fluency in handling algebraic expressions. Many textbooks have endless exercises in mindless manipulation rather than tasks that reinforce the link between letters and numbers and give some sense of purpose.
One source of difficulty is that pupils commonly acquire wrong ideas about the meaning to be attributed to letters. They are often given the erroneous notion that letters stand for objects, with an expression such as 5b + 3c interpreted as 5 bananas and 3 carrots. What are they supposed to make of 5b - 3c + 2 or of bc if they understand b and c to mean bananas and carrots? Does b2 really mean square bananas? Letters should be used to stand for numbers, so c could certainly be a number of carrots, but linking letters to the names of objects in this way may not be a good initial teaching strategy.
I observed a lesson where a pupil had written 7k as the answer to 2k + k + 4 in a simplification exercise. The teacher had some difficulty explaining what was wrong. If pupils interpret k as kangaroos, then their instinctive thought is likely to be "4 what?", and the obvious answer is 4 kangaroos, giving 7k altogether. When this is coupled with the fact that 3k + 4 does not look like an "answer" in the pupils' terms, it is not surprising that such errors occur.
If k is understood to be a number then substituting some values in the expressions 2k + k + 4, 3k + 4 and 7k makes it abundantly clear, as Figure 1 shows, that the first two are not equivalent to the last. Reference to rules about gathering together like terms is not sufficient to address the misconception: pupils need to see what is wrong in a way that reinforces a correct interpretation of the expressions.Figure 1 Odd numbers provide a good starting point for some simple algebra. It is easy to find the 10th odd number because you can count up on your fingers. The 20th is a bit more difficult, but the 100th, or the 143rd may present a real challenge. On the other hand finding the 143rd even number is no problem - just double it. Finding the 143rd odd number is now easy - just one less than the even number. In algebraic terms the even numbers are given by 2n and the odd numbers by one less, 2n - 1. The n tells you the position in the sequence.
The key point is that letters are used in algebra to denote numbers in order to represent relationships in simple ways. Exploring these relationships is one valuable way of stimulating an interest in algebra and developing understanding and fluency in the subject.
Try adding any pair of consecutive odd numbers together and you will see that the answer is a multiple of 4. Figure 2 shows some exaples with column headings representing the sequences algebraically. With 2n - 1 as an odd number the next odd number is 2 more, and we can simplify 2n - 1 + 2 to give 2n + 1. Adding the pair of numbers corresponds to simplifying 2n - 1 + 2n + 1 which gives us 4n. The table of values helps to make sense of the simplification which proves that the sum is a multiple of 4.
Try writing down the first five numbers in the 3 times table and alongside them the first five odd numbers. Now add up the corresponding pairs of numbers and look at the results. You see 4, 9, 4, 9, 4 - the numbers go up by 5 each time and each is 1 less than a multiple of 5. Figure 3 shows the patterns with appropriate column headings.
The approach to algebra adopted by so many school textbooks results in pupils spending much time simplifying expressions such as 3n + 2n - 1 without developing any sense of where such things come from or what an answer such as 5n - 1 might mean. It would help their understanding a lot if the numerical links were emphasised much more. Tables of values can be generated easily with pencil and paper, and a spreadsheet or the table facility of a graphic calculator offers a powerful alternative medium.
Confusion caused by misinterpreting the meaning of letters arises in other ways. It is all too easy blithely to translate the statement that a week is seven days into the plausible looking w = 7d. If w is supposed to stand for the number of weeks and d the number of days, taking a value of 70 for d seems to suggest that 70 days are the same as 490 weeks! Since the number of days is 7 times the number of weeks the formula should be d = 7w. This statement is counter-intuitive if you do not interpret it as a relationship between two sets of numbers.
Last autumn, I asked a group of PGCE secondary mathematics students each to write down a formula linking k, the number of kilometres, to m, the number of miles, using the usual approximation that 5 miles is equivalent to 8 kilometres. One common answer was k = 58m. I asked what that would give for 8 miles: I was pleased that the answer of 5 kilometres, found by putting m = 8, worried them! They had been led astray by wrongly translating the link between the units as 5m = 8k. An algebraic formula is a succinct way of expressing a relationship between sets of numbers, not a shorthand for the words in a verbal or written statement. Realising that 1 kilometre is equivalent to 58 mile is the key to arriving at a correct relationship in the form m = 58k.
These examples show the scope for confusion when the meaning given to letters in algebra is not established at an early stage and constantly reinforced. The Teacher Training Agency is certainly right to focus considerable attention on the need for student teachers to have sound subject knowledge and an awareness of pupils' common misconceptions, but it is also important for new teachers to realise how such misunderstandings arise in the first place and to have a wide range of strategies to reinforce the right ideas.
Since textbooks have a profound influence on how algebra is taught, there is also an important message here for writers and publishers, who need to ensure that materials reduce misconceptions rather than reinforce them.
Doug French is alecturer in education at the University of Hull and is chair of the Mathematical Association's teaching committee