How axioms can help pupils to understand algebra

Miren Jayapal
29th July 2019 at 09:00

Algebra - teacher in class

The first mathematics that children learn is algebraic. As babies, they stack cups, arrange objects and build with blocks. They develop by sorting and classifying these objects in a manner that makes most sense to them. They experiment, moderate their methods and repeat these experiments. These are the true foundations of mathematics and science.

The child’s symbolic algebra may not be the symbols we are accustomed to teaching, but rather colours, shapes, sizes and textures. As their toys become more complex, their play develops differently and this abstract approach is replaced with real-world scenarios, such as cars and figurines. This progression is important for development but it is a shame that after embracing a form of algebra so early on, it is some years until they meet it again. I wonder if, as they cultivate a sounder comprehension of the physical world, they lose that ability to grasp the abstract.

Pupils’ formal schooling begins by matching their tangible experiences with those of writing and manipulating numbers. These early experiences certainly make mathematics hands-on, where they can readily apply it to the world they see. They can count and combine objects, see areas in shapes and divide quantities into proportions. It may, therefore, come as something of a shock when algebra is introduced. This is a field built on abstract principles and it is unfortunately only when these have been mastered that their application to the real world can be appreciated.

There is, therefore, a leap that every pupil needs to make in their mathematics education, that from the tangible to the abstract. I feel that all too often this comes about as a sink or swim scenario. Some pupils go with it, find themselves carried along as they progress and the pay off comes later when they start to access more sophisticated ideas. Others make repeated attempts at it, gain an idea of how to answer certain questions mechanically but garner a certain fear of algebra. There are also those who cannot begin to access the subject throughout their schooling.

This is echoed in pupils’ sentiments on mathematics. Many pupils will cite it as their favourite subject at primary but I suspect fewer at secondary. This could be how the curriculum changes between these two stages. There would be an introduction to algebra at primary, but this soon becomes a major component of secondary mathematics. Relatively speaking, it is not necessarily that secondary mathematic is more challenging but that the challenges change. Why is it that some make this leap easier than others, and can we, as teachers, make this transition easier?

Algebra and axioms in maths

More often than not we feel the pressure of curriculum and time. One of our main concerns is to get through the content with enough time left to revisit and consolidate problematic areas. The spiral nature of the mathematics curriculum means that when each algebraic concept is revisited pupils do seem to better understand it. However, it is alarming when pupils make trivial mistakes in the basics even after having studied it for numerous years. They might multiply x and x to make 2x or simplify 2x+3y to 5xy. Pupils will generally realise their errors, but I wonder if the fact that they initially made them belies a weakness in the foundations, which could then have a knock-on effect as to how they progress thereafter.

A common way to approach algebra is to make it as applicable to the real world as possible, whether that is the infamous problem of Hannah’s sweets or a crocodile stalking its prey. It is not the lack of realism in this approach I take issue with, but rather I question how deep an appreciation of why they study algebra a pupil may gain from this. In turn, I worry that the attempt to relate this abstract field to the real world in a tokenistic way could take the focus away from the algebraic principles involved. If an activity is going to model phenomena using algebra, it should be meaningful and there needs to be a distinct focus on methodology. It needs to bridge the gap between formulating ideas and making generalisations.

Part of this leap is translating rules that apply with numbers to symbols. These symbols are used to represent quantities that vary or are unknown, but the very idea of working with something indefinite can be challenging – how is it possible to manipulate something you do not know? Perhaps more time should be spent on the fundamentals. Frequently pupils will unknowingly draw reference to these when they ask questions. "Why is x plus x equal to 2x and x times x the same as x squared?" In general, these classic "why" questions that pupils ask should be addressed – this is often where the deep learning arises and these ground rules are explored.

A good teacher gives a response that involves a discussion on the principles that underpin the topic or on how a result is derived. However, in some cases, a better answer would be "Because it does!". This is particularly applicable if a pupil is probing a convention of notation, but it may also be that they have unwittingly revealed a mathematical axiom.

Mathematics is based on axioms, self-evident truths that we take on face value and allow us to build a sophisticated system around. Pupils generally have an intuitive understanding of these axioms and use them readily when they start the formal study of the subject in primary school. They would learn that 2 + 3 is the same as 3 + 2, but likely do not know this is an example of an axiom at work. They may question what 0 times a number is and discover another axiom by doing so. Explicit reference to these axioms is not made in primary school, and this innate understanding arises out of using them in a practical way, probably for the best in the early stages.

However, as learners develop, there is more of an onus on developing mathematical reasoning. This is where a greater appreciation of mathematics as an axiomatic system could prove more useful. Tackling a problem by taking some given information and building a framework around it using known fundamentals mirrors how the entire subject is built upon axioms.

Like any axiomatic system, algebra needs to be built up incrementally, with plenty of opportunity for consolidation.  The past decade has seen a gradual increase in the amount of algebra learned at primary level, with most pupils exposed to it in Year 5 or 6. Perhaps there is scope for this to start even earlier, if only to develop a familiarity with the language of algebra or assuage the fear that could develop later. It does not need to be the most taxing of algebraic concepts, but having pupils do some straightforward simplifications, sums or statements at a younger age could make the transition easier. 

Or is there any value in expressly teaching pupils in an axiomatic way? To a degree, teachers already do this, but more of a focus could be on these foundations. Greater time could be spent on the rules and distinctions between them. Where reasons can be given, they should be, but if something exists by definition then we should say so. The discussion could be on what it is to be an axiom and how this enables us to build other results.

Primary teachers are already stretched with the mathematical content they need to cover, so were this to be viable something would need to be taken out of the curriculum, possibly a strand that could be covered later and more quickly at secondary level. Otherwise the secondary content could focus on these axioms. There could be a significant amount of time spent on them in the early stages. If pupils establish this firm basis in expressing concepts algebraically, they may then be able to proceed through the more sophisticated content at a faster, but more substantive, rate.

Much of our physical and societal structures are built on mathematics, whether that be the mechanics that creates our buildings, the code that allows us to buy things online or the data that informs our health service. Pupils at school can get a taste of how the subject works in these contexts, but there is still some way to go in their study before they can plainly see and understand its applications.

This is acutely the case with algebra. Whilst there is value in relating it as much as possible to sensory experience, at some point every pupil is going to have to be able to solve an equation or factorise an expression with the real world setting stripped down. They will need to face these purely abstract problems with a well-founded algebra skill set. The true application comes later, once they have built solid foundations.

Miren Jayapal is deputy head of mathematics at Fortismere School in London