Summary
By Dr Fiona Campbell
Introduction
In 2013, Chinn asked: is the population really woefully bad at maths? His answer was unequivocal. At the time of writing, the curriculum and pedagogy did not equip all learners equitably with the numerical knowledge and skills they needed to cope with the everyday demands of modern living. More than ten years have passed since Chinn’s article (2013) and an enormous amount of time and money has been spent trying to drive up numeracy standards in the UK. Pass rates in GCSE maths have gone up. Employers are no longer so dissatisfied with the academic qualifications of job-seeking school leavers. Yet recent GCSE data show that 40% of students still do not achieve the pass (grade 4) in mathematics necessary to qualify for further study. This article asks: what can be done to help these learners?
Was the Population Woefully Bad at Maths?
In his article, Chinn (2013) drew on three sources to answer the question of whether the population was really so woefully bad at maths. They were: a report on the state of mathematics education commissioned by the Conservative Party; an analysis of data on the numeracy levels of 13- to 19-year-olds from national and international surveys; and an annual employment trends survey conducted by the Confederation of British Industry (CBI). Key findings were:
- in June 2010, almost half of pupils taking GCSE mathematics failed to achieve grade C or above;
- 22% of 16- to 19-year-olds in England had only basic numerical competence;
- and almost 50% of employers were dissatisfied with the numeracy levels of school, college and university leavers.
The picture painted by these results was clear. Not only were young people leaving school ill-equipped to deal confidently with the mathematical challenges posed by everyday life, but this was also exerting long-term effects in the workplace.
Chinn (2013) supplemented this analysis with his own data: results obtained from the standardisation of a fifteen-minute mathematics test, focussing on four age groups: 10- (primary school leavers), 13- (end of key stage 3), 15- (the year before GCSEs) and 16–19-year-olds (post-school). His findings were:
- the percentage of correct responses on the fifteen-minute mathematics test increased steadily across age groups for all items;
- the highest percentage of correct responses was awarded to addition and subtraction calculations; multiplication and division did not lead to similar levels of calculation success;
- anything less than a straightforward presentation of calculations resulted in a substantial drop in the percentage of correct responses.
Ill-equipped school leavers, it seemed, were ill-equipped pupils whilst at school.
Has the Answer Changed?
More than ten years have passed since Chinn’s article (2013) and an enormous amount of time and money has been spent trying to drive up numeracy standards in the UK, including: the introduction of a new curriculum in 2014; a reformed GCSE based on the new curriculum with a grading scale of 1 to 9; and a drive to introduce a maths mastery approach in schools. This begs another question: has the answer changed? It would appear it has.
In 2024, the percentage of entries receiving a pass or higher (grade 4/C) in maths was 59.5%, which was down slightly from 61.1% in 2023 (Joint Council for Qualifications, JCQ, 2024). This may not seem like cause for celebration. However, pass rates are dragged down by older students who are resitting exams. If considering data from 16-year-olds taking the exam for the first time, results are broadly similar between the two years (71.2% in 2024 and 71.7% in 2023) and much more optimistic.
Furthermore, employer satisfaction with the academic results and qualifications of school-leaving job seekers, including literacy and numeracy, is higher than ever (74% and 53% respectively) and “may reflect changes to GCSEs in England, with the reformed qualifications designed to ensure students are better prepared for work or further study” (Confederation of British Industry, CBI, 2019). Yet nearly three in every ten 16-year-olds still fail to achieve a pass at GCSE maths. What can be done to help these learners?
How to Support Pupils Who Struggle With Maths
In his paper, Chinn (2013) stated that there are certain characteristics of individuals with specific learning difficulties (SpLD) such as dyscalculia and dyslexia that make traditional methods of teaching maths ineffective. These include: poor working memory and long-term memory for mathematical facts. SpLDs of this kind are on a spectrum. Hence, these same characteristics will be present to a greater or a lesser degree in all learners, including those who have not been formally diagnosed with a SpLD. If this is true, then what works for learners who have a SpLD that makes the acquisition of maths challenging may work for other pupils too who have encountered failure in maths. But what does work?
I am a specialist dyscalculia assessor and tutor of young people with mathematical learning difficulties. Most of my work involves pupils in secondary schools or colleges. The pupils I work with have failed in maths (often repeatedly) or they are predicted to fail. Below are some of the strategies I employ in my practice. While I cannot promise they will guarantee a pass at GCSE maths, nor will it be possible to employ everyone in a busy school environment where there is pressure to constantly move on, they do facilitate the acquisition of greater reasoning skills, number sense and more enjoyment of the subject.
For further ideas of what works, practitioners are advised to connect with the Dyscalculia Network, founded in 2019 by Catherine Eadle and Rob Jennings. This is a growing body of professional, qualified teachers and tutors who work with individuals who have SpLDs to improve their outcomes in maths. They have a vast knowledge and experience of what works and share this information through their website and social media accounts.
1) Encourage Pupils to See Themselves as Problem Solvers
Faced with an unfamiliar problem on a GCSE maths paper, it is easy for a pupil to feel overwhelmed. Believing that they do not know what to do, or how to begin, pupils confess “I skip it” and move on. I want my pupils to see themselves as active problem solvers. That is why I start every lesson with a logical reasoning or a visual puzzle. They may involve numbers, but they often do not. The puzzles start easy, but as the pupil grows in confidence, they become increasingly more challenging. I encourage the pupil to take the problem line by line, highlighting any important information? Can the problem be viewed from a different perspective? What assumptions have they made? Are they correct? If a pupil is having difficulties understanding the words, I encourage them to draw the problem out.
And while the links to maths may not appear immediately obvious, I make it so by telling the pupils how these same skills – logical reasoning, critical thinking, visualising a problem, chunking and highlighting important information, checking your assumptions – can all be used in any maths paper. More importantly, it means that every session starts with a smile and a feeling of having succeeded, which they do not traditionally associate with maths lessons.
2) Develop Estimation Skills
Pupils with mathematical learning difficulties may not have a good sense of the magnitude of the numbers they are working with. Consequently, they will not know if an answer to a calculation is reasonable or not. For example, when assessing a pupil before I started tutoring them, I asked them what 400 add 600 was. They answered 10,000. Evidently, they had misremembered a rule – “add the zeros” – and misapplied it in this context. This not only illustrates the problem of teaching short cuts without understanding to pupils who struggle with maths, but had they had a better grasp of quantities, they would have understood their answer was unrealistic. Hence, I build estimation activities into every lesson, starting with small amounts and growing gradually larger and larger.
How many books are on the bookcase? How many sheep are in the field? How much does a pint of milk cost? What does a litre look like? How many of these one millilitre drops would it take to fill this jug? What is the population of the UK, of the world? Pupils are then routinely encouraged to estimate the answers to calculations first, and then check them against the final answer at the end.
3) Explicitly Teach Mathematical Vocabulary
At the start of each new topic, I explicitly teach mathematical vocabulary. There are some students with which I do this every lesson, and they keep a vocabulary bank of flashcards at home, to which they refer daily. The importance of teaching mathematical vocabulary was highlighted to me recently when I asked a pupil, “What is volume?” To which he answered, “The button on my remote control.” There are words in maths that have one meaning inside the classroom and an entirely different meaning outside the classroom. While as teachers or tutors, this may not cause us any difficulties, we cannot underestimate how confusing it can be for pupils. Add to this the fact that each operation in maths has multiple synonyms. Addition alone could be referred to as add, more, plus, sum, more than, and, together.
Knowing these synonyms can give important cues to what procedure or operation a word problem is asking you to carry out. There are also some words that will only ever be encountered in a maths classroom (e.g. scalene, parabola) and if not enough time is devoted to understanding these words, they will not be memorable and they will be forgotten. Explicitly teaching mathematical vocabulary also allows pupils to understand what the teacher is saying, to construct their own understanding and to communicate what they do (and do not) know to those who are trying to support them.
4) Make Maths as Meaningful as Possible
As maths teachers, how many times have you heard the question, “When am I ever going to need this in real life?” In my own practice, I like to pre-empt this question by providing real, concrete examples of how pupils are already using the skill I am about to teach. For example, did they realise that when working out what is the latest they can get out of bed if it takes them 15 minutes to get ready, 20 minutes to walk to school and they must be there at 8:50am, that this is algebra?
I give examples of jobs where this skill may be necessary, or some exciting examples of how it has been applied in the real-life world. Social media algorithms, for example, are complex mathematical formulas calculated based on how a person interacts with a website that determines what content they see. Trigonometry was used to calculate the height of Mount Everest, while understanding quadratic equations (and practice) can help you improve your shot accuracy in basketball.
5) Build Strong Mathematical Foundations
Pupils with dyscalculia and mathematical learning difficulties must have strong mathematical foundations in place before they can move forward. As Steven Pinker (1997) wrote: “Mathematics is ruthlessly cumulative, all the way back to counting to ten.” A student who does not understand numbers will not be able to count. A student who cannot count will not be able to add or subtract. This student will struggle with algebra. They may also struggle in other subjects that rely on algebra such as science. Eadle and Jennings call this the ‘Jenga Effect’ (2022), referring to a game in which players take turns removing one of 54 blocks from a carefully crafted tower (see Figure 1). The more blocks that are removed, the more unstable the tower becomes.
And so it is with maths; if the foundations are insecure, so too will be every ‘layer’ of maths that gets added to the tower. The more maths that gets added, the more the pupil will struggle. They will feel overwhelmed. The tower may topple and they may refuse to engage in any maths at all. Jennings and Eadle state that: “It is vitally important for pupils with dyscalculia and maths difficulties to have the early foundations of maths put in place – no matter how long that takes. No matter how old the pupil is.” It could be added, no matter whether the pupil has been diagnosed with a SpLD or not.
6) Use Manipulatives
Pupils who struggle with maths will find it hard to engage with it in it at a purely abstract level. Using manipulatives enables pupils to touch, hold and see the maths. Examples include: tokens, glass beads, dominoes, Cuisenaire rods, base 10 blocks, linking cubes, fraction blocks, algebra tiles and double-sided counters. Manipulatives typically refer to concrete objects, but virtual manipulatives based on concrete ones are becoming increasingly popular due to the growth in online learning. They may also be less stigmatising for older learners. Manipulatives alone, however, are not enough to bring any mathematical insight, so the teacher or tutor plays a vital role in facilitating the pupil to make the connection between the manipulatives and the mathematical concept it represents. With this support, teachers will be able to help learners who have encountered repeated failure in maths model, understand and solve complex, abstract mathematical problems.
7) Fade Manipulatives Out to Visual Representations (and Abstract Symbols)
I have often read that manipulatives should be used with caution as pupils become over reliant on them. This is to fail to understand their purpose. Once the pupil can touch, feel and see the maths, they should be encouraged to draw visual representations of the problem. This too in time can be faded out to work with purely abstract symbols. However, if the pupil needs to stay with drawings, then that too should be encouraged and may result in greater success than pure reliance on an algorithm. Take the rule “keep, change, flip”, for example, when dividing fractions which is likely to be forgotten, partially remembered or confused with other algorithms. Whereas drawing bar models to represent the division can result in success first time, every time.
8 )Reduce the Need to Memorise by Teaching for Deep Conceptual Understanding
I recently worked with a pupil who had tried and failed to learn how to calculate simple and compound interest using a formula. Teaching for deep conceptual understanding of a topic, means that pupils do not need to memorise long, complicated algorithms. They can apply logic and reasoning to the problem, draw models to support their thinking and gradually work their way through the problem. In this instance, I explicitly taught the vocabulary associated with interest.
I ensured that the foundations necessary to tackle simple and compound interest problems were in place, including using the division bar model to calculate percentages of an amount, and calculator skills. I then broke a simple interest problem down into chunks. How do you calculate simple interest using a division bar model and on a calculator? How much money was in the bank account after 1 year? What about after 2 years and 3 years? How much interest was earned overall?
We practiced the same approach over different questions. When it became apparent that the pupil was becoming confused between simple and compound interest, I ran an experiment with real money in which they invested £10 in a bank account, one with simple interest and one with compound interest. The pupil went through several cycles of calculating the interest earned, so that they saw the difference it made. I showed them what simple and compound interest looked like when plotted on a graph. With time and patience, the pupil said what every tutor of learners with mathematical difficulties longs to hear, “I get it now.”
9) Use External Memory Aids
In maths classrooms, there is sometimes a premium placed on the ability to do mental maths and to do so quickly. This may disadvantage pupils who are already at a disadvantage. When teaching a new topic, it is more important that pupils understand what is being done, why and how, than trip over the numbers. Full use of external memory aids should be encouraged. This includes calculators, number lines, multiplication squares and place value charts. Yes, pupils will eventually have to sit a non-calculator paper. But they can also be shown how to create their own external memory aids for use in an exam including completing a multiplication square from scratch, drawing a number lines or sketching a place value chart.
10) Promote Overlearning and Constant Cycling
Rote learning has fallen out of favour, and for good reason. However, the principle that repeatedly practicing a skill increases the chances that it will be transferred to long-term memory is sound. Hence repetition of a skill should be practiced, but only in the presence of deep conceptual understanding. This is called overlearning. Overlearning practices include: giving students time to consolidate learning before moving onto new or more challenging material; starting and ending each lesson with a recap of previous learning; creating flashcards for information a learner has trouble remembering; and incorporating regular revision sessions to revisit content learners have previously covered.
11) Have Fun
Many pupils who struggle with maths experience mathematics anxiety. Maths anxiety often has more to do with how the subject is presented than the maths itself and teachers and tutors of learners who struggle with numeracy employ a games-based approach to reduce anxiety and free up valuable working memory space. I end every lesson with a game. Sometimes it may be no longer than a few minutes, but the game always reinforces the skill we have been learning in the session or another skill I believe the pupil needs to reinforce. Far more importantly, however, is that every session finishes with a smile and a feeling of having succeeded.
Conclusion
In conclusion, while the population may no longer be so woefully bad at maths as in 2013 when Chinn posed the question, there appears to be a cohort of learners that remain immune to the time and money that has been spent by the government trying to drive up standards of numeracy in the UK. Worse still, they may be doomed to repeat this cycle of failure and underachievement.
Chinn (2013) argued that an understanding of how to teach maths to pupils with a SpLD would benefit all pupils who possess, to a greater or lesser extent, many of the characteristics that makes maths difficult for some. While this would constitute what Chinn called a cultural shift in the pedagogy of maths, the Dyscalculia Network is a body of highly trained, experienced professionals with knowledge of what works that can act as a source of information and advice for teachers and parents who wish to improve outcomes for learners who struggle with numeracy.
References
Confederation of British Industry (2019). Education and learning for the modern world: CBI/Pearson Education and Skills Survey report, https://www.cbi.org.uk/media/3841/12546_tess_2019.pdf
Chinn, S. (2013) Is the population really woefully bad at maths. Mathematics Teaching, 232, 25 – 28.
Eadle, C. & Jennings, R. (2022). What is the Jenga effect? https://dyscalculianetwork.com/what-is-the-jenga-effect/
Joint Council for Qualifications. (2024). GCSE summer – 2024. https://www.jcq.org.uk/gcse-level-1-and-level-2-results-summer-2024/
Pinker, S. (1997). How the mind works. Norton.
For more information on supporting learners in the classroom go to – https://dyscalculianetwork.com/dyscalculia-for-educators/
