As a mathematician, algebra is a key tool of my trade which enables me to solve complex problems (and it can be quite beautiful too!). Nevertheless, when new acquaintances discover that I’m a mathematician, they sometimes reflect that algebra was one of the reasons they dropped mathematics from their school timetable as soon as they were allowed to do so. Research among high school drop-outs in the US indicates that many blamed failing their compulsory algebra course for leaving college prematurely. I believe it’s time to look again at how we teach algebra, from primary school upwards, so that future generations fully embrace this essential mathematical tool.
What’s the point of algebra?
Anyone who has taught GCSE-level mathematics will have come across this question on a fairly frequently basis, and I think it says a lot about the previous experiences of the students asking the question. I suspect that their earlier mathematics curriculum may have focused on developing their algorithmic fluency rather than promoting their conceptual understanding. To help overcome these issues, I believe in starting algebra much earlier than Year Six. I’m a great supporter of early algebra with young learners, which is not necessarily the same as bringing algebraic notation early into primary schools. When I talk about early algebra, I’m thinking about posing challenges which allow primary-aged learners to reach beyond the constraints of simple arithmetic.
Let’s look at an example of an early algebra activity adapted from a research project (Kaput et al., 2008): Imagine that two friends, Max, and Bryony, both have identical boxes full of sweets. On top of Bryony’s box, there are three more sweets. Without counting out their sweets, what do you already know about how many sweets they have?
When we posed this 'sweetie box question' on NRICH, schools were encouraged to share their classwork with our team. The nature of this task enabled many pupils to use their past experiences to estimate the number of sweets in the box. For example, if Max’s box held 10 sweets, the pupils could state that Bryony had 13 sweets in total and justify their answer. Using specific examples enabled the children to improve their understanding of the problem. By repeating this approach several times, and encouraging to record their answers, many noticed a pattern in their pairs of answers; one student submitted this response to the team: We know that Bryony has more sweets than Max because if they had the same number of sweets in each box Bryony would still have the 3 extra sweets on top of her box so she will have more.
This student has gone beyond using specific numbers, he’s noticed a pattern about his pairs of answers which is enabling him to generalise his response. I would argue that they are working algebraically; given the number of sweets of either child, they could now work out how many the other child has got too. This move from using specific numbers to generalising is the move from arithmetic to algebra; after all, elementary algebra is often known as generalised arithmetic. When the student knows one quantity, they can calculate the other by manipulating or balancing their numbers so that their answers make sense. Exploring activities such as the 'sweetie box', and reflecting on what they’ve learnt from the challenge, helps younger learners to understand why generalising is so important, and perhaps offers an opportunity to explain why we use the word ‘algebra’ for this way of working (‘algebra’ is derived from the words ‘al-jabr’ which refer to fixing something which is broken or missing).
The benefits of algebra
As students develop and embed their confidence and skills exploring early algebra activities, the usefulness of generalising their skills becomes clear. From calculating the number of slabs needed to surface a new patio, to working out the exchange rate for their holiday currency, algebra is a tool which makes life easier even among young learners. Learning algebra also enables students to rehearse and develop skills, work logically and apply rules consistently – all transferable skills whichever career path they might follow.
The history of algebra
So far, we have focused on algebraic tasks which use words rather than using symbols. More complex problems do not necessarily need symbols, but they do require an understanding of the situation to be able to generalise it. Consider the following problem posed by Robert Coolman (2015): I have two fields that total 1,800 square yards. Yields for each field are ⅔ gallon of grain per square yard and ½ gallon per square yard. The first field gave 500 more gallons than the second. What are the areas of each field?
Clearly, this is much more challenging than our earlier problem about the sweets, but it is not a new problem; it was adapted from a question discovered by archaeologists on a clay tablet from ancient Mesopotamia (Sesiano, 1999). Although several thousand years old, it shows how ancient civilisations valued algebra for manging their everyday lives, just as we should do today.
The solution accompanying the problem was very different from those you see today in the mathematics classroom - just like the early algebra example above, it used words rather than symbols, carefully taking the reader through each step of the process as the writer manipulated the variables and balanced the numbers to reach a solution. That said, a modern solution using symbols seems much more efficient (and familiar) to me. Nevertheless, the clay tablet question and accompanying solution demonstrate that study of algebra is nothing new, it has been a valued approach for managing our lives for millennia.
Our algebraic approaches may have changed over the years, but its usefulness remains. If we can support our students to understand its usefulness, and the skills they develop practising it, then perhaps more students will continue with their education. Hopefully enjoying an algebra topic or two along the way!