Month: September 2025

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Mathematics at A-level is going from strength to strength. Maths is the most popular subject choice, and further maths, which is a separate A-level course, has seen the most growth in uptake. Despite this, concerns still remain about the mathematical skills of young people who do not choose to study maths after they are 16.

Students in England who have passed GCSE maths at grade four or above, but who are not taking A-level or AS-level maths, are eligible to take a core maths qualification.

Core maths was introduced in 2014-15 to attempt to remedy a lack in mathematics education after 16. But the number of entries remains well short of what they could be. Many students who would benefit from maths after 16 are not taking this subject.

A 2010 report from the Nuffield Foundation found students in the UK lag their peers in other countries in participation in mathematics after the age of 16. Further research from the Royal Society and higher education charity AdvanceHE showed that as a consequence, many were not well prepared for the demands of their university courses or careers. Survey data has also found that over half of UK adults’ maths skills are low.

Many courses at university include mathematical or quantitative elements, but do not require AS or A-level maths for entry. These include psychology, geography, business and management, sociology, health sciences, biology, education and IT. When many students have not studied mathematics since GCSE, this results in a lack of fluency and confidence in using and applying it.

Core maths consolidates and builds on students’ mathematical understanding. The focus is on using and applying mathematics to authentic problems drawn from study, work and life. This includes understanding and using graphs, statistics and tools such as spreadsheets, as well as understanding risk and probability.

Core maths includes topics such as probability. EF Stock/Shutterstock

Take-up remains low despite incentives – schools receive an additional £900 in funding for each student who studies core maths. In 2025, 15,327 students took core maths – a 20% increase on 12,810 entries in 2024, which is very encouraging. However, research from the Royal Society in 2022 found that fewer than 10% of the number of A-level students who were not taking A-level mathematics had taken core maths, which will not have changed significantly even with the current numbers.

Increasing enrolment

There remains strong commitment from the government for increasing participation in mathematics after 16 in England through core maths. Many schools and colleges have embraced the subject, and universities have expressed support too.

However, a real incentive for teenagers to study this subject would be if it was rewarded in entry to university. Universities can allow students entry to a course with lower A-level grade profiles than normally required if they also passed core maths, for instance. But the number of universities making this kind of offer is low.

Schools and colleges need stronger signals from universities to induce them to offer students the opportunity to study for a core maths qualification, and to encourage their students to do so. Shifting today’s landscape to one where the vast majority of learners aged 16 to 19 in England are studying some form of mathematics which is relevant to their current and future interests and needs will require reform.

The Royal Society’s 2024 report on mathematical and data education sets out several reforms necessary to develop the mass mathematical, quantitative and data skills needed for the careers of the future. These include compulsory maths and data education in some form until 18. Extending the take up of core maths would be an excellent way to begin achieving this.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Paul Glaister CBE*

Gorodenkoff/Shutterstock

In 2025, more young people than ever have opened their A-level results to find out how they did in their maths exam. Once again, maths has been the most popular A-level subject, with 112,138 entries in 2025.

This is up by more than 4% compared with 2024. Entries in further maths, an A-level that expands on the maths curriculum, have also risen – an increase of 7% since 2024, with over 19,000 entries this year.

As a professional mathematician this is pleasing news. Some of these students will be happily receiving confirmation of their place to study maths at university.

The joy I experienced when I discovered in my maths degree that many of the subjects I studied at school – chemistry, biology, physics and even music – are woven together by a mathematical fabric, is something I’ve never forgotten.

I’m excited by the idea that many young people are about to experience this for themselves. But I am concerned that fewer students will have the same opportunities in the future, as more maths departments are forced to downsize or close, and as we become more reliant on artificial intelligence.

There are a number of differences between studying maths at university compared with school. While this can be daunting at first, all of these differences underscore just how richly layered, deeply interconnected and vastly applicable maths is.

At university, not only do you learn beautiful formulas and powerful algorithms, but also grapple with why these formulas are true and dissect exactly what these algorithms are doing. This is the idea of the “proof”, which is not explored much at school and is something that can initially take students by surprise.

But proving why formulas are true and why algorithms work is an important and necessary step in being able discover new and exciting applications of the maths you’re studying.

Maths degrees involve finding out why mathematics works the way it does. Gorodenkoff/Shutterstock

A maths degree can lead to careers in finance, data science, AI, cybersecurity, quantum computing, ecology and climate modelling. But more importantly, maths is a beautifully creative subject, one that allows people to be immensely expressive in their scientific and artistic ideas.

A recent and stunning example of this is Hannah Cairo, who at just 17 disproved a 40-year old conjecture.

If there is a message I wish I knew when I started studying university mathematics it is this: maths is not just something to learn, but something to create. I’m continually amazed at how my students find new ways to solve problems that I first encountered over 20 years ago.

Accessiblity of maths degrees

But the question of going on to study maths at university is no longer just a matter of A-level grades. The recent and growing phenomenon of maths deserts – areas of the country where maths degrees are not offered – is making maths degrees less accessible, particularly for students outside of big cities.

Forthcoming research from The Campaign for Mathematical Sciences (CAMS), of which I am a supporter, shows that research-intensive, higher tariff universities – the ones that require higher grades to get in – took 66% of UK maths undergraduates in 2024, up from 56% in 2006.

This puts smaller departments in lower-tariff universities in danger of closure as enrolments drop. The CAMS research forecasts that an additional nine maths departments will have fewer than 50 enrolments in their degrees by 2035.

This cycle will further concentrate maths degrees in high tariff institutions, reinforcing stereotypes such as that only exceptionally gifted people should go on to study maths at university. This could also have severe consequences for teacher recruitment. The CAMS research also found that 25% of maths graduates from lower-tariff universities go into jobs in education, compared to 8% from higher tariff universities.

Maths in the age of AI

The growing capability and sophistication of AI is also putting pressure on maths departments

With Open AI’s claim that their recently released GPT-5 is like having “a team of PhD-level experts in your pocket”, the temptation to overly rely on AI poses further risks to the existence and quality of future maths degrees.

But the process of turning knowledge into wisdom and theory into application comes from the act of doing: doing calculations and forming logical and rigorous arguments. That is the key constituent of thinking clearly and creatively. It ensures students have ownership of their skills, capacities, and the work that they produce.

A data scientist will still require an in-depth working knowledge of the mathematical, algorithmic and statistical theory underpinning data science if they are going to be effective. The same for financial analysts, engineers and computer scientists.

The distinguished mathematician and computer scientist Leslie Lamport said that “coding is to programming what typing is to writing”. Just as you need to have some idea of what you are writing before you type it, you need to have some idea of the (mathematical) algorithm you are creating before you code it.

It is worth remembering that the early pioneers in AI – John McCarthy, Marvin Minsky, Claude Shannon, Alan Turing – all had degrees in mathematics. So we have every reason to expect that future breakthroughs in AI will come from people with mathematics degrees working creatively in interdisciplinary teams.

This is another great feature of maths: its versatility. It’s a subject that doesn’t just train you for a job but enables you to enjoy a rich and fulfilling career – one that can comprise many different jobs, in many different fields, over the course of a lifetime.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Neil Saunders*

Mississippi’s reforms have led to significant gains in reading and math, despite the state being one of the lowest spenders per pupil in the U.S. Klaus Vedfelt/Getty Images

In a surprising turnaround, Mississippi, once ranked near the bottom of U.S. education standings, has dramatically improved its student literacy rates.

As of 2023, the state ranks among the top 20 for fourth grade reading, a significant leap from its 49th-place ranking in 2013. This transformation was driven by evidence-based policy reforms focused on early literacy and teacher development.

The rest of the country might want to take note.

That’s because Mississippi’s success offers a proven solution to the reading literacy crisis facing many states – a clear road map for closing early literacy gaps and improving reading outcomes nationwide.

As an expert on the economics of education, I believe the learning crisis is not just an educational issue. It’s also economic.

When students struggle, their academic performance declines. And that leads to lower test scores. Research shows that these declining scores are closely linked to reduced economic growth, as a less educated workforce hampers productivity and innovation.

The Mississippi approach

In 2013, Mississippi implemented a multifaceted strategy for enhancing kindergarten to third grade literacy. The Literacy-Based Promotion Act focuses on early literacy and teacher development. It includes teacher training in proven reading instruction methods and teacher coaching.

Relying on federally supported research from the Institute of Education Science, the state invested in phonics, fluency, vocabulary and reading comprehension. The law provided K-3 teachers with training and support to help students master reading by the end of third grade.

It includes provisions for reading coaches, parent communication, individual reading plans and other supportive measures. It also includes targeted support for struggling readers. Students repeat the third grade if they fail to meet reading standards.

The state also aligned its test to the NAEP, or National Assessment of Educational Progress, something which not all states do. Often referred to as “The Nation’s Report Card,” the NAEP is a nationwide assessment that measures student performance in various subjects.

Mississippi 4th graders’ reading improved the most from 2013 to 2022

According to federal data, fourth graders’ reading scores improved by nine points in Mississippi from 2013 to 2022. At the other end of the spectrum, Maryland fourth graders’ reading levels fell by 20 points over the same period.

Mississippi’s reforms have led to significant gains in reading and math, with fourth graders improving on national assessments.

I believe this is extremely important. That’s because early reading is a foundational skill that helps develop the ability to read at grade level by the end of third grade. It also leads to general academic success, graduating from high school prepared for college, and becoming productive adults less likely to fall into poverty.

Research by Noah Spencer, an economics doctoral student at the University of Toronto, shows that the Mississippi law boosted scores.

Students exposed to it from kindergarten to the third grade gained a 0.25 standard deviation improvement in reading scores. That is roughly equivalent to one year of academic progress in reading, according to educational benchmarks. This gain reflects significant strides in students’ literacy development over the course of a school year.

Another study has found an even greater impact attributed to grade retention in the third grade – it led to a huge increase in learning in English Language Arts by the sixth grade.

But the Mississippi law is not just about retention. Spencer found that grade retention explains only about 22% of the treatment effect. The rest is presumably due to the other components of the measure – namely, teacher training and coaching.

Other previous research supports these results across the country.

Adopting an early literacy policy improves elementary students’ reading achievement on important student assessments, with third grade retention and instructional support substantially enhancing English learners’ skills. The policy also increases test scores for students’ younger siblings, although it is not clear why.

Moreover, third grade retention programs immediately boost English Language Arts and math achievements into middle school without disciplinary incidents or negatively impacting student attendance.

These changes were achieved despite Mississippi being one of the lowest spenders per pupil in the U.S., proving that strategic investments in teacher development and early literacy can yield impressive results even with limited resources.

The global learning crisis

Mississippi’s success is timely. Millions of children globally struggle to read by age 10. It’s a crisis that has worsened after the COVID-19 pandemic.

Mississippi’s early literacy interventions show lasting impact and offer a potential solution for other regions facing similar challenges.

In 2024, only 31% of U.S. fourth grade students were proficient or above in reading, according to the NAEP, while 40% were below basic. Reading scores for fourth and eighth graders also dropped by five points compared with 2019, with averages lower than any year since 2005.

In 2013, Mississippi ranked 49th in fourth grade reading scores. Klaus Vedfelt/Getty Images

Mississippi’s literacy program provides a learning gain equal to a year of schooling. The program costs US$15 million annually – 0.2% of the state budget in 2023 – and $32 per student.

The learning gain associated with the Mississippi program is equal to about an extra quarter of a year. Since each year of schooling raises earnings by about 9%, then a quarter-year gain means that Mississippi students benefiting from the program will increase future earnings by 2.25% a year.

Based on typical high school graduate earnings, the average student can expect to earn an extra $1,000 per year for the rest of their life.

That is, for every dollar Mississippi spends, the state gains about $32 in additional lifetime earnings, offering substantial long-term economic benefits compared with the initial cost.

The Mississippi literacy project focuses on teaching at the right level, which focuses on assessing children’s actual learning levels and then tailoring instruction to meet them, rather than strictly following age- or grade-level curriculum.

Teaching at the right level and a scripted lessons plan are among the most effective strategies to address the global learning crisis. After the World Bank reviewed over 150 education programs in 2020, nearly half showed no learning benefit.

I believe Mississippi’s progress, despite being the second-poorest state, can serve as a wake-up call.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Harry Anthony Patrinos*

Ten! Alamy/Insidefoto

Humans are finite creatures. Our brains have a finite number of neurons and we interact with a finite number of people during our finite lifetime. Yet humans have the remarkable ability to conceive of the infinite.

This ability underlies Euclid’s proof that there are infinite prime numbers as well as the belief of billions that their gods are infinite beings, free of mortal constraints.

These ideas will be well known to Pope Leo XIV since before his life in the church, he trained as a mathematician. Leo’s trajectory is probably no coincidence since there is a connection between mathematics and theology.

Infinity is undoubtedly of central importance to both. Virtually all mathematical objects, such as numbers or geometric shapes, form infinite collections. And theologians frequently describe God as a unique, absolutely infinite being.

Despite using the same word, though, there has traditionally been a vast gap between how mathematicians and theologians conceptualise infinity. From antiquity until the 19th century, mathematicians have believed that there are infinitely many numbers, but – in contrast to theologians – firmly rejected the idea of the absolute infinite.

The idea roughly is this: surely, there are infinitely many numbers, since we can always keep counting. But each number itself is finite – there are no infinite numbers. What is rejected is the legitimacy of the collection of all numbers as a closed object in its own right. For the existence of such a collection leads to logical paradoxes.

A paradox of the infinite

The most simple example is a version of Galileo’s paradox and leads to seemingly contradictory statements about the natural numbers 1,2,3….

First, observe that some numbers are even, while others are not. Hence, the numbers – even and odd – must be more numerous than just the even numbers 2,4,6…. And yet, for every number there is exactly one even number. To see this, simply multiply any given number by 2.

But then there cannot be more numbers than there are even numbers. We thus arrive at the contradictory conclusion that numbers are more numerous than the even numbers, while at the same time there are not more numbers than there are even numbers.

Because of such paradoxes, mathematicians rejected actual infinities for millennia. As a result, mathematics was concerned with a much tamer concept of infinity than the absolute one used by theologians. This situation dramatically changed with mathematician Georg Cantor’s introduction of transfinite set theory in the second half of the 19th century.

Georg Cantor, mathematical rebel. Wikipedia

Cantor’s radical idea was to introduce, in a mathematically rigorous way, absolute infinities to the realm of mathematics. This innovation revolutionised the field by delivering a powerful and unifying theory of the infinite. Today, set theory provides the foundations of mathematics, upon which all other subdisciplines are built.

According to Cantor’s theory, two sets – A and B – have the same size if their elements stand in a one-to-one correspondence. This means that each element of A can be related to a unique element of B, and vice versa.

Think of sets of husbands and wives respectively, in a heterosexual, monogamous society. These sets can be seen to have the same size, even though we might not be able to count each husband and wife.

The reason is that the relation of marriage is one-to-one. For each husband there is a unique wife, and conversely, for each wife there is a unique husband.

Using the same idea, we have seen above that in Cantor’s theory, the set of numbers – even and odd – has the same size as the set of even numbers. And so does the set of integers, which includes negative numbers, and the set of rational numbers, which can be written as fractions.

The most striking feature of Cantor’s theory is that not all infinite sets have the same size. In particular, Cantor showed that the set of real numbers, which can be written as infinite decimals, must be strictly larger than the set of integers.

The set of real numbers, in turn, is smaller than even larger infinities, and so on. To measure the size of infinite sets, Cantor introduced so-called transfinite numbers.

The ever-increasing series of transfinite numbers is denoted by Aleph, the first letter of the Hebrew alphabet, whose mystic nature has been explored by philosophers, theologians and poets alike.

Set theory and Pope Leo XIII

For Cantor, a devout Lutheran Christian, the motivation and justification of his theory of absolute infinities was directly inspired by religion. In fact, he was convinced that the transfinite numbers were communicated to him by God. Moreover, Cantor was deeply concerned about the consequences of his theory for Catholic theology.

Pope Leo XIII, Cantor’s contemporary, encouraged theologians to engage with modern science, to show that the conclusions of science were compatible with religious doctrine. In his extensive correspondence with Catholic theologians, Cantor went to great lengths to argue that his theory does not challenge the status of God as the unique actual infinite being.

On the contrary, he understood his transfinite numbers as increasing the extent of God’s nature, as a “pathway to the throne of God”. Cantor even addressed a letter and several notes on this topic to Leo XIII himself.

Pope Leo XIII. Wikipedia/Braun et Compagnie

For Cantor, absolute infinities lie at the intersection of mathematics and theology. It is striking to consider that one of the most fundamental revolutions in the history of mathematics, the introduction of absolute infinities, was so deeply entangled with religious concerns.

Pope Leo XIV has been explicit that Leo XIII was his inspiration for his choice of pontifical name. Perhaps among an infinite number of potential reasons for the choice, this mathematical link was one.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Balthasar Grabmayr*