The art of manipulation
I am a mature PGCE student, having just completed a maths degree. At the end of last summer I helped in a school and got really frustrated by students who seemed unable to remember the difference between the expression (ab)2 and ab2. Is this a common phenomenon?
Yes, it is a common phenomenon, and often because students don't understand the use of brackets and the notation, or haven't had much experience of writing the algebra out in full or in manipulating algebraic expressions.
On reading your letter, I consulted an algebra textbook from 1957 by GH Leslie, which stated: "The notation of algebra presents considerable difficulty to most pupils, and many of them still lack the necessary confidence and facility in symbolic expression quite late in the school course".
So this difficulty has been with us for quite some time. Dr Tony Barnard, lecturer in maths at King's College London, wrote three extensive articles for the Mathematical Association's magazine Mathematics in School in 2002, which tackled misconceptions in algebra. The first of these is a "must read" and can be found at www.mth.kcl.ac.ukstaffad_barnard.html
The other two articles have been developed into a booklet, which is referenced below. I wrote to Dr Barnard about your letter and he replied:
"Although a number of strategies are suggested, it is an error that is likely to recur, unless an expression like (3x)2 has meaning for the pupil, and such "meaning" includes being able to think of 3x as a unified single entity - something with a life of its own. Some suggestions for addressing this underlying aspect (objectification of algebraic expressions, or parts of such) were given in the second article (September, 2002)."
In his article a number of suggestions are made. The way in which algebra is spoken should offer a clue to its interpretation. For example, (xy)2 should perhaps be read as "xy all squared". Dr Barnard suggests that the xy should be read at speed, so that it is thought of as a single entity. When reading out xy2, this should be "x (slight pause), y-squared".
A useful lesson starter could be algebraic dictation. Give each pupil an A4 wipeable whiteboard or some scrap paper, so they can show their answers.
Read out an algebraic expression and ask them to write it down. Ask them to show their answers and discuss their responses. You might find they have some interesting suggestions on how each expression should be enunciated.
An alternative is to invite pupils in turn to read out the expression given to them in the original notational form and see how they respond. You might like to try the following sentences. Some of these are not the correct way to dictate them, which provides an opportunity to open up a discussion about the meaning of each expression.
1. Four a b squared.
2. Two x times a plus b.
3. Minus xy all squared.
4. p plus q divided by 5.
5. x plus x over two.
6. Half of three mn squared.
7. The square root of 16 ab all squared. (The answers are given below.) Brackets are a useful tool to help us in understanding notation, so in this exercise xy2 would be re-written by the pupils as x(y)2 showing that just the y is squared.
Creating questions where pupils have to work backwards can be fun and useful. An activity that encourages pupils to evaluate expressions is one in which expressions have to be matched to their values. On an interactive whiteboard the expressions can easily be moved. Otherwise, create them on pieces of card and stick them to the board.
Another activity that helps students understand the differences between the expressions is one in which they substitute values for the expressions, in this case x = 2 and y = 3, and for each expression written on a card. When they have determined the value of each expression they have to arrange the cards in order, from lowest to highest value. These cards are available as A4 sheets if you want to email me for copies.
* Answers to Algebraic Dictation: 1. (4ab)2 2. 2x (a + b) 3. - (xy)2 4. p + q
5. x + x 6. 3mn2 7.C16(ab)2 5
The booklet Hurdles and Strategies in the Teaching of Algebra is on the Mathematical Association's website www.m-a.org.ukresourcespublications