# The power of mistakes in mathematics

The style of GCSE mathematics questions has changed dramatically in recent years. Five years ago topics were tested in separate questions in a very prescriptive manner. A pupil could learn a technique and employ it without fully understanding why it works. Pupils were awarded marks without answering questions in a logical and coherent manner. Topics would be split up. A pupil could look at a question and know it was “that type of question”, answering it by following a series of prescribed steps. They would carry out processes in their heads or on calculators and not commit it to their working out, with exams that allowed them to do so. Mark schemes would say marks can be awarded upon sight of a specific number, as opposed to showing the calculation that got there and why they are doing it.

However, recent changes have sought to tackle this. Pupils have to understand the processes in order to use them effectively. The focus in the new curriculum is on communicating mathematically.

Think back to the infamous "Hannah and her sweets" problem from 2015 and the furore that surrounded it. Despite the tenuous practical application of the mathematics (What 16-year-old would form a quadratic equation to complete the simple task of eating two sweets from a bag?), it signposted that changes were afoot. It was two years before the new GCSE was to be tested but exam boards were gradually trying to embed these new styles of question, making the mathematics as relevant as possible and merging disparate areas.

## Mathematics appreciation

Now a single question can test a variety of topics, all in unfamiliar contexts. As a mathematician, I welcome this. The subject is and should be appreciated as a coherent whole, with the common theme of mathematical rigour upholding all. But as a teacher, I know that pupils struggle with these new demands. Perhaps they should find this approach difficult, this could be part of the strategy to make a GCSE pass in mathematics more meaningful. It remains for us to teach pupils these concepts and skills to the extent that they can link them together when presented in an unusual manner. Pupils will need to have a far deeper understanding of the ideas that underlie a topic so that they can readily apply them. There are a number of programmes and ideas out there that assist teachers in doing so, maths mastery for instance. All of these have their benefits, as they allow pupils to think about mathematics from different perspectives. However, something that I have become particularly interested in over the last few years is how pupils can learn from their mistakes.

A typical pupil is taught a topic, they practise it, have their work assessed and practice again before taking it further. Perhaps the most useful step underpinning this cycle is the feedback they receive throughout. Different pupils could have got the same question wrong for a variety of reasons and it is only when their individual mistakes are addressed that they collectively move on. These mistakes are immensely powerful and where the true mathematics lies. Why can you not divide by zero? Why can you not solve this equation by square rooting it all? Why do you not sum the denominators when adding fractions? These are the deep-thinking questions that a good teacher will address.

Up to now this vital part of the learning process has been mostly undertaken by the maths teacher. Textbooks and websites provide model answers and questions. Pupils complete these and can check their final answer. But the most interesting mathematics is not the question or the answer, it is what occurs in-between. Simply checking a final answer does not make the pupil engage with the mathematics. If they are wrong most pupils are poor at evaluating their own work. This fundamental skill of mathematical reasoning is seen as onerous and irrelevant – but should be embraced.

## Engender discussions

We ultimately want to develop articulate and reflective mathematicians who are able to evaluate their own work effectively, especially when they do not have the solution in front of them. Simply telling pupils to check their answers is not enough. Instead, activities could be built around these mistakes that engender discussions. It is not anything new. The ‘spot the mistake’ exercise was used when I was in school, but perhaps it is a teaching tool that can be used more widely now. Both Edexcel and AQA have implemented this approach in their exam papers with ‘Explain why this is wrong’ or ‘Identify the error in this pupil’s answer’ appearing at GCSE.

Spending time with a class examining an error will emphasise how important it is to evaluate answers effectively. As teachers we should demonstrate that pupils often learn more from a wrong answer than from a correct one, provided it has been dissected. Many teachers do this intuitively, taking the time to unravel an error with the class. They discuss why it is incorrect. Does it lead to a contradiction or a fallacy? Is it breaking a fundamental rule? Is it reasonable? If some pupils do not resolve this misconception the first time, then perhaps a further examination is necessary. It is often the case that these misunderstandings need to be further picked apart. As a teacher you may have to rephrase your explanation so that different pupils can understand it. So they can explore the misconception further in an alternative and more detailed study. All pupils would ultimately benefit, whether or not they understood a concept initially or not, as they would be able to see it presented in a variety of ways. Throughout this process pupils have been compelled to reflect deeply on the processes at hand. They will appreciate how illuminating their mistakes can be and how meeting these head-on has progressed their mathematics.

We should aim to foster this skill and make it part of a pupil’s methodology. It is vital in producing reflective and independent learners, which would also do wonders for their enthusiasm and confidence in the subject.

*Miren Jayapal is deputy head of maths at a school in London*

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