Category: mathematics

Have you ever struggled with teaching statistics? Do you and your students share a sense of apprehension when data lessons appear on the scheme of work? You’re not alone. Anecdotally, many teachers tell me that statistics is one of the topics they like teaching the least, and I am no exception to this myself. In my mathematics degree I took the minimum number of statistics-related courses allowed following a very poor diet of data at school, and carried this negative association into my teaching. Looking back on my career in the classroom, I did not do a good job of teaching statistics, but having had the luxury of spending many years at Cambridge Mathematics immersed in research from excellent statistics teachers and education academics I now understand why!

So now of course, the question has been posed. Why is statistics hard to teach well? In part, I believe that it stems from viewing statistics through a mathematical lens – understandably, given that we are delivering it alongside quadratic equations, Pythagoras’ theorem, fractions, decimals and percentages. But while statistical analysis would not exist without the mathematical concepts and techniques underpinning it, we have a tendency within curricula to make the mathematical techniques the whole point, and reduce the statistical analysis part to an afterthought or an added extra. Students find the more subjective analysis hard, so it is tempting to make sure everyone can manage the techniques and then focus on the interpretation as something only the most able have time to spend on (although, there is always the additional temptation to move on to other, more properly ‘maths-y’ topics as soon as possible).

This approach is at odds with how education researchers suggest students should encounter statistical ideas. In the early 1990s, George Cobbⁱ and other researchers recommended that statistics should

• emphasise statistical thinking,
• include more real data,
• encourage the exploration of genuine statistical problems, and
• reduce emphasis on calculations and techniques.

Since then, much subsequent research has refined these recommendations to account for new technology tools and new ideas, but the core principles have remained the same. In much of my reading of education research, three ways of seeing or interacting with data keep appearing:

• Data modelling – the idea that data can be used to create models of the world in order to pose and answer questions
• Informal inference – the idea that data can be used to make predictions about something outside of the data itself with some attempt made to describe how likely the prediction is to be true
• Exploratory data analysis – the idea that data can be explored, manipulated and represented to identify and make visible patterns and associations that can be interpreted

In the abstract, these ways of seeing, while distinct, have a degree of overlap and all students may benefit from multiple experiences of all three approaches to data work from their very earliest encounters with data through to advanced level study.

Imagine the following classroom activity that could be given to very young students (e.g., in primary school). A class of students is given a list of snacks and treats and the students are asked to rank them on a scale of one to five based on how much they like each item. How could this data be worked with through each of the three approaches?

Firstly, we will consider data modelling. Students could be asked to plan a class party with a limited budget. They can buy some but not all of the items listed, and must decide what they should buy so that the maximum number of students get to have things they like. In this activity, students must create a model from the data that identifies those things they should buy more of, and those things they should buy least of, along with how many of each thing they should get – perhaps considering these quantities proportionally. This activity uses the data as a model but inevitably requires some assumptions and the creation of some principles. Is the goal to ensure everyone gets the thing they like most? Or is it to minimise the inclusion of the things students like least? What if everyone gets their favourite thing except one student who gets nothing they like?

Secondly, we will think about this as an activity in informal inference. Imagine a new student is joining the class and the class wants to make a welcome pack of a few treats for this student, but they don’t know which treats the student likes. Can they use the data to decide which five items an unknown student is most likely to choose? What if they know some small details about the student; would that additional information allow them to decide based on ‘similar’ students in the class? While the second part of this activity must be handled with a degree of sensitivity, it is an excellent primer for how purchasing algorithms, which are common in online shops, work.

Finally, we turn to exploratory data analysis. In this approach students are encouraged to look for patterns in the data, perhaps by creating representations. This approach may come from asking questions – e.g., do students who like one type of chocolate snacks rate the other chocolate snacks highly too? Is a certain brand of snack popular with everyone in the class? What is the least popular snack? Alternatively, the analysis may generate questions from patterns that are spotted – e.g. why do students seem to rate a certain snack highly? What are the common characteristics of the three most popular snacks?

Each of these approaches could be engaged in as separate and isolated activities, but there is also the scope to combine them, and use the results of one approach to inform another. For example, exploratory data analysis may usefully contribute both to model building and inference making, and support students’ justifications for their decisions in those activities. Similarly, data modelling activities can be extended into inferential tasks very easily, simply by shifting the use of the model from the population of the data (e.g., the students in the class it was collected from) to some secondary population (e.g., another class in the school, or as in the example, a new student joining the class).

Looking back on my time in the classroom, I wish that my understanding of these approaches and their importance for developing statistical reasoning skills in my students had been better. While not made explicit as important in many curricula, there are ample opportunities to embed these approaches and make them a fundamental part of the statistics teacher’s pedagogy.

Do you currently use any of these approaches in your lessons? Can you see where you might use them in the future? And how might you adapt activities to allow your students opportunities to engage in data modelling, informal inference and exploratory data analysis?

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

*Credit for article given to Darren Macey*

Ground Picture/Shutterstock

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*

Photo by Markus Krisetya via Unsplash

Statistics professor Johan Ferreira was feeling overwhelmed by the amount of “screen time” involved in online learning in 2021. He imagined students must be feeling the same way, and wondered what he could do to inspire them and make his subject matter more appealing.

One of the topics in statistics is time series analysis: statistical methods to understand trend behaviour in data which is measured over time. There are lots of examples in daily life, from rainfall records to changes in commodity prices, import or exports, or temperature.

Ferreira asked his students to write a short, fictional “bedtime” story using “characters” from time series analysis. The results were collected into a book that is freely available. He tells us more about it.

Why use storytelling to learn about statistics?

I’m fortunate to be something of a creative myself, being a professional oboe player with the Johannesburg Philharmonic Orchestra. It’s a valuable outlet for self-expression. I reflected on what other activity could inspire creativity without compromising the essence of statistical thinking that was required in this particular course I was teaching.

Example of a time series, the kind of data analysed using statistical methods. Author provided (no reuse)

I invited my third-year science and commerce students at the University of Pretoria to take part in a voluntary storytelling exercise, using key concepts in time series analysis as characters. Students got some guidelines but were free to be creative. My colleague and co-editor, Dr Seite Makgai, and I then read, commented on and edited the stories and put them together into an anthology.

Students gave their consent that their stories could be used for research purposes and might be published. Out of a class of over 200 students, over 30 contributions were received; 23 students permitted their work to be included in this volume.

We curated submissions into two sections (Part I: Fables and Fairy Tales and Part II: Fantasy and Sci-Fi) based on the general style and gist of the work.

The project aimed to develop a new teaching resource, inspire students to take ownership of their learning in a creative way, and support them through informal, project-based peer learning.

This collection is written by students, for students. They used personal and cultural contexts relevant to their background and environment to create content that has a solid background in their direct academic interests. And the stories are available without a paywall!

What are some of the characters and stories?

Student Lebogang Malebati wrote Stationaryville and the Two Brothers, a tale about AR(1) and AR(2). In statistics, AR refers to processes in which numerical values are based on past values. The brothers “were both born with special powers, powers that could make them stationary…” and could trick an evil wizard.

David Dodkins wrote Zt and the Shadow-spawn. In this story, Zt (common notation in time series analysis) has a magic amulet that reveals his character growth through a sequence of models and shows the hero’s victory in the face of adversity. He is a function of those that came before him (through an AR process).

Then there’s Nelis Daniels’ story about a shepherd plagued by a wolf called Arma (autoregressive moving average) which kept making sheep disappear.

And Dikelede Rose Motseleng’s modern fable about the love-hate relationship between AR(1) (“more of a linear guy” with a bad habit of predicting the future based on the past) and MA(1), “the type of girl who would always provide you with stationarity (stability).”

What was the impact of the project?

It was a deeply enriching experience for us to see how students see statistics in a context beyond that of the classroom, especially in cases where students reformulated their stories within their own cultural identities or niche interests.

Three particular main impacts stand out for us:

• students have a new additional reference and learning resource for the course content
• new students can refer to the experiences and contextualisation of this content of former students, leading to informal peer learning
• students engage in a cognitive skill (higher-order and creative thinking) that is not frequently considered and included in this field and at this level.

In 2024, shortly after the book was published, we asked students in the time series analysis course of that year to read any one of four stories (related to concepts that were already covered in the course material at that point in time). We asked them to complete a short and informal survey to gauge their experience and insights regarding the potential of this book as a learning resource for them.

The 53 responses we got indicated that most students saw the book as a useful contribution to their learning experience in time series analysis.

Student perceptions of value of stories. Author supplied, Author provided (no reuse)

One positive comment from a student was:

I will always remember that the Random Walk is indeed not stationary but White Noise is. I already knew it, but now I won’t forget it.

Will you build on this in future?

It is definitely valuable to consider similar projects in other branches of statistics, but also, in other disciplines entirely, to develop content by students, for students.

At this stage, we’re having the stories and book translated into languages beyond English. In large classes that are essential to data science (such as statistics and mathematics), many different home languages may be spoken. Students often have to learn in their second, third, or even fourth language. So, this project is proving valuable in making advanced statistical concepts tactile and “at home” via translations.

Our publisher recently let us know that the Setswana translation is complete, with the Sepedi and Afrikaans translations following soon. To our knowledge, it’ll be the first such project not only in the discipline of statistics, but in four of the official languages in South Africa.

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

*Credit for article given to Johan Ferreira*

Rachel Moore

From hand-sized lantern sharks that glow in the deep sea to bus-sized whale sharks gliding through tropical waters, sharks come in all shapes and sizes.

Despite these differences, they all face the same fundamental challenge: how to get oxygen, heat and nutrients to every part of their bodies efficiently.

Our new study, published today in Royal Society Open Science, shows that sharks follow a centuries-old mathematical rule – the two-thirds scaling law – that predicts how body shape changes with size. This tells us something profound about how evolution works – and why size really does matter.

What is the two-thirds scaling law?

The basic idea is mathematical: surface area increases with the square of body length, while volume increases with the cube. That means surface area increases more slowly than volume, and the ratio between the two – crucial for many biological functions – decreases with size.

This matters because many essential life processes happen at the surface: gas exchange in the lungs or gills, such as to take in oxygen or release carbon dioxide, but also heat loss through skin and nutrient uptake in the gut.

These processes depend on surface area, while the demands they must meet – such as the crucial task of keeping the body supplied with oxygen – depend on volume. So, the surface area-to-volume ratio shapes how animals function.

Whale sharks are as big as buses, while dwarf lanternsharks (pictured here) are as small as a human hand. Chip Clark/Smithsonian Institution

Despite its central role in biology, this rule has only ever been rigorously tested in cells, tissues and small organisms such as insects.

Until now.

Why sharks?

Sharks might seem like an unlikely group for testing an old mathematical theory, but they’re actually ideal.

For starters, they span a huge range of sizes, from the tiny dwarf lantern shark (about 20 centimetres long) to the whale shark (which can exceed 20 metres). They also have diverse shapes and lifestyles – hammerheads, reef-dwellers, deep-sea hunters – each posing different challenges for physiology and movement.

Plus, sharks are charismatic, ecologically important and increasingly under threat. Understanding their biology is both scientifically valuable and important for conservation.

Sharks are ecologically important but are increasingly under threat. Rachel Moore

How did we test the rule?

We used high-resolution 3D models to digitally measure surface area and volume in 54 species of sharks. These models were created using open-source CT scans and photogrammetry, which involves using photographs to approximate a 3D structure. Until recently, these techniques were the domain of video game designers and special effects artists, not biologists.

We refined the models in Blender, a powerful 3D software tool, and extracted surface and volume data for each species.

Then we applied phylogenetic regression – a statistical method that accounts for shared evolutionary history – to see how closely shark shapes follow the predictions of the two-thirds rule.

Sharks follow the two-thirds scaling rule almost perfectly, as seen in this 3D representation. Joel Gayford et al

What did we find?

The results were striking: sharks follow the two-thirds scaling rule almost perfectly, with surface area scaling to body volume raised to the power of 0.64 – just a 3% difference from the theoretical 0.67.

This suggests something deeper is going on. Despite their wide range of forms and habitats, sharks seem to converge on the same basic body plan when it comes to surface area and volume. Why?

One explanation is that what are known as “developmental constraints” – limits imposed by how animals grow and form in early life – make it difficult, or too costly, for sharks to deviate from this fundamental pattern.

Changing surface area-to-volume ratios might require rewiring how tissues are allocated during embryonic development, something that evolution appears to avoid unless absolutely necessary.

The scale of sharks

A study of 54 species of sharks shows the ratio of the surface areas and volumes of their bodies follows a mathematical rule called the two-thirds scaling law.

But why does it matter?

This isn’t just academic. Many equations in biology, physiology and climate science rely on assumptions about surface area-to-volume ratios.

These equations are used to model how animals regulate temperature, use oxygen, and respond to environmental stress. Until now, we haven’t had accurate data from large animals to test those assumptions. Our findings give researchers more confidence in using these models – not just for sharks, but potentially for other groups too.

As we face accelerating climate change and biodiversity loss, understanding how animals of all sizes interact with their environments has never been more urgent.

This study, powered by modern imaging tech and some old-school curiosity, brings us one step closer to that goal.

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

*Credit for article given to Jodie L. Rummer & Joel Gayford*

A painless, non-invasive brain stimulation technique can significantly improve how young adults learn maths, my colleagues and I found in a recent study. In a paper in PLOS Biology, we describe how this might be most helpful for those who are likely to struggle with mathematical learning because of how their brain areas involved in this skill communicate with each other.

Maths is essential for many jobs, especially in science, technology, engineering and finance. However, a 2016 OECD report suggested that a large proportion of adults in developed countries (24% to 29%) have maths skills no better than a typical seven-year-old. This lack of numeracy can contribute to lower income, poor health, reduced political participation and even diminished trust in others.

Education often widens rather than closes the gap between high and low achievers, a phenomenon known as the Matthew effect. Those who start with an advantage, such as being able to read more words when starting school, tend to pull further ahead. Stronger educational achievement has been also associated with socioeconomic status, higher motivation and greater engagement with material learned during a class.

Biological factors, such as genes, brain connectivity, and chemical signalling, have been shown in some studies to play a stronger role in learning outcomes than environmental ones. This has been well-documented in different areas, including maths, where differences in biology may explain educational achievements.

To explore this question, we recruited 72 young adults (18–30 years old) and taught them new maths calculation techniques over five days. Some received a placebo treatment. Others received transcranial random noise stimulation (tRNS), which delivers gentle electrical currents to the brain. It is painless and often imperceptible, unless you focus hard to try and sense it.

It is possible tRNS may cause long term side effects, but in previous studies my team assessed participants for cognitive side effects and found no evidence for it.

Participants who received tRNS were randomly assigned to receive it in one of two different brain areas. Some received it over the dorsolateral prefrontal cortex, a region critical for memory, attention, or when we acquire a new cognitive skill. Others had tRNS over the posterior parietal cortex, which processes maths information, mainly when the learning has been accomplished.

Before and after the training, we also scanned their brains and measured levels of key neurochemicals such as gamma-aminobutyric acid (gaba), which we showed previously, in a 2021 study, to play a role in brain plasticity and learning, including maths.

Some participants started with weaker connections between the prefrontal and parietal brain regions, a biological profile that is associated with poorer learning. The study results showed these participants made significant gains in learning when they received tRNS over the prefrontal cortex.

Stimulation helped them catch up with peers who had stronger natural connectivity. This finding shows the critical role of the prefrontal cortex in learning and could help reduce educational inequalities that are grounded in neurobiology.

How does this work? One explanation lies in a principle called stochastic resonance. This is when a weak signal becomes clearer when a small amount of random noise is added.

In the brain, tRNS may enhance learning by gently boosting the activity of underperforming neurons, helping them get closer to the point at which they become active and send signals. This is a point known as the “firing threshold”, especially in people whose brain activity is suboptimal for a task like maths learning.

It is important to note what this technique does not do. It does not make the best learners even better. That is what makes this approach promising for bridging gaps, not widening them. This form of brain stimulation helps level the playing field.

Our study focused on healthy, high-performing university students. But in similar studies on children with maths learning disabilities (2017) and with attention-deficit/hyperactivity disorder (2023) my colleagues and I found tRNS seemed to improve their learning and performance in cognitive training.

I argue our findings could open a new direction in education. The biology of the learner matters, and with advances in knowledge and technology, we can develop tools that act on the brain directly, not just work around it. This could give more people the chance to get the best benefit from education.

In time, perhaps personalised, brain-based interventions like tRNS could support learners who are being left behind not because of poor teaching or personal circumstances, but because of natural differences in how their brains work.

Of course, very often education systems aren’t operating to their full potential because of inadequate resources, social disadvantage or systemic barriers. And so any brain-based tools must go hand-in-hand with efforts to tackle these obstacles.

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

*Credit for article given to Roi Cohen Kadosh*

Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho/Moment via Getty Images

You scrambled up a Rubik’s cube, and now you want to put it back in order. What sequence of moves should you make?

Surprise: You can answer this question with modern algebra.

Most folks who have been through high school mathematics courses will have taken a class called algebra – maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x. The word “algebra” may evoke memories of complicated-looking polynomial equations like ax² + bx + c = 0 or plots of polynomial functions like y = ax² + bx + c.

You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x-axis, too.

Graph of a quadratic equation and its roots via the quadratic formula. Jacob Rus, CC BY-SA

Equations and plots like these are part of algebra, but they’re not the whole story. What unifies algebra is the practice of studying things – like the moves you can make on a Rubik’s cube or the numbers on a clock face you use to tell time – and the way they behave when you put them together in different ways. What happens when you string together the Rubik’s cube moves or add up numbers on a clock?

In my work as a mathematician, I’ve learned that many algebra questions come down to classifying objects by their similarities.

Sets and groups

How did equations like ax² + bx + c = 0 and their solutions lead to abstract algebra?

The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x was three or four. But they couldn’t do it for five. It took mathematician Évariste Galois and techniques he developed – now called group theory – to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more.

So what is a group, anyway?

It starts with a set, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don’t have too many properties – that is, characteristics – but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.

In clock addition, 3 + 12 = 3. OpenStax, CC BY-SA

Let’s call this set of numbers 1 through 12 “clock numbers.” Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say “3 + 11 = 2” is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock.

Clock addition has some nice properties. It satisfies:

closure, where adding things in the set gives you something else in the set,

identity, where there’s an element that doesn’t change the value of other elements in the set when added – adding 12 to any number will equal that same number,

associativity, where you can add wherever you want in the set,

inverses, where you can undo whatever an element does, and

commutativity, where you can change the order of which clock numbers you add up without changing the outcome: a + b = b + a.

By satisfying all these properties, mathematicians can consider clock numbers with clock addition a group. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can’t be made into a group easily – what’s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.

Rings and fields

Along with groups, the two other fundamental types of algebraic objects you would study in an introduction to modern algebra are rings and fields.

We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call a ring. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where a(b + c) = ab + ac. Lastly, fields are rings that satisfy even more conditions.

At the turn of the 20th century, mathematicians David Hilbert and Emmy Noether – who were interested in understanding how the principles in Einstein’s relativity worked mathematically – unified algebra and showed the utility of studying groups, rings and fields.

It’s all fun and games until you do the math

Groups, rings and fields are abstract, but they have many useful applications.

For example, the symmetries of molecular structures are categorized by different point groups. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.

The water molecule H₂O can be flipped horizontally and the end result is indistinguishable from the original position. Courtney Gibbons, CC BY-SA

But let’s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to describe a Sudoku puzzle: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board.

To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using x₃₅ to represent the cell in the third row and fifth column.

The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x₁₁ – 1)(x₁₁ – 2)(x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that’s already 81 equations just to say, “Don’t plug in anything other than 1 through 9.”

The rule “1 through 9 each appear exactly once in the top row” can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say x₁₁ + x₁₂ + ⋅⋅⋅ + x₁₉ – 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say x₁₁ x₁₂ ⋅⋅⋅ x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero.

If you’re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you’re not wrong.

Turning Sudoku into algebra takes a fair bit of work. Courtney Gibbons

What do we get by doing this complicated translation into algebra? Well, we get to use late-20th century algorithms to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring – called an ideal – these game board clues make within the larger ring. The algorithms will tell you if there’s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all.

This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle problems in artificial intelligence, robotics, cryptography, quantum computing and so much more – all with the same bag of tricks you’d use to solve the Sudoku puzzle or Rubik’s cube.

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

*Credit for article given to Courtney Gibbons*

Prime numbers are numbers that are not products of smaller whole numbers. Jeremiah Bartz

A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique – the etchings, lines like tally marks, may have represented prime numbers. Similarly, a clay tablet from 1800 B.C.E. inscribed with Babylonian numbers describes a number system built on prime numbers.

As the Ishango bone, the Plimpton 322 tablet and other artifacts throughout history display, prime numbers have fascinated and captivated people throughout history. Today, prime numbers and their properties are studied in number theory, a branch of mathematics and active area of research today.

A history of prime numbers

Some scientists guess that the markings on the Ishango bone represent prime numbers. Joeykentin/Wikimedia Commons, CC BY-SA

Informally, a positive counting number larger than one is prime if that number of dots can be arranged only into a rectangular array with one column or one row. For example, 11 is a prime number since 11 dots form only rectangular arrays of sizes 1 by 11 and 11 by 1. Conversely, 12 is not prime since you can use 12 dots to make an array of 3 by 4 dots, with multiple rows and multiple columns. Math textbooks define a prime number as a whole number greater than one whose only positive divisors are only 1 and itself.

Math historian Peter S. Rudman suggests that Greek mathematicians were likely the first to understand the concept of prime numbers, around 500 B.C.E.

Around 300 B.C.E., the Greek mathematician and logician Euclid proved that there are infinitely many prime numbers. Euclid began by assuming that there is a finite number of primes. Then he came up with a prime that was not on the original list to create a contradiction. Since a fundamental principle of mathematics is being logically consistent with no contradictions, Euclid then concluded that his original assumption must be false. So, there are infinitely many primes.

The argument established the existence of infinitely many primes, however it was not particularly constructive. Euclid had no efficient method to list all the primes in an ascending list.

In the middle ages, Arab mathematicians advanced the Greeks’ theory of prime numbers, referred to as hasam numbers during this time. The Persian mathematician Kamal al-Din al-Farisi formulated the fundamental theorem of arithmetic, which states that any positive integer larger than one can be expressed uniquely as a product of primes.

From this view, prime numbers are the basic building blocks for constructing any positive whole number using multiplication – akin to atoms combining to make molecules in chemistry.

Prime numbers can be sorted into different types. In 1202, Leonardo Fibonacci introduced in his book “Liber Abaci: Book of Calculation” prime numbers of the form (2p – 1) where p is also prime.

Prime numbers, when expressed as that number of dots, can be arranged only in a single row or column, rather than a square or rectangle. David Eppstein/Wikimedia Commons

Today, primes in this form are called Mersenne primes after the French monk Marin Mersenne. Many of the largest known primes follow this format.

Several early mathematicians believed that a number of the form (2p – 1) is prime whenever p is prime. But in 1536, mathematician Hudalricus Regius noticed that 11 is prime but not (211 – 1), which equals 2047. The number 2047 can be expressed as 23 times 89, disproving the conjecture.

While not always true, number theorists realized that the (2p – 1) shortcut often produces primes and gives a systematic way to search for large primes.

The search for large primes

The number (2p – 1) is much larger relative to the value of p and provides opportunities to identify large primes.

When the number (2p – 1) becomes sufficiently large, it is much harder to check whether (2p – 1) is prime – that is, if (2p – 1) dots can be arranged only into a rectangular array with one column or one row.

Fortunately, Édouard Lucas developed a prime number test in 1878, later proved by Derrick Henry Lehmer in 1930. Their work resulted in an efficient algorithm for evaluating potential Mersenne primes. Using this algorithm with hand computations on paper, Lucas showed in 1876 that the 39-digit number (2127 – 1) equals 170,141,183,460,469,231,731,687,303,715,884,105,727, and that value is prime

Also known as M127, this number remains the largest prime verified by hand computations. It held the record for largest known prime for 75 years.

Researchers began using computers in the 1950s, and the pace of discovering new large primes increased. In 1952, Raphael M. Robinson identified five new Mersenne primes using a Standard Western Automatic Computer to carry out the Lucas-Lehmer prime number tests.

As computers improved, the list of Mersenne primes grew, especially with the Cray supercomputer’s arrival in 1964. Although there are infinitely many primes, researchers are unsure how many fit the type (2p – 1) and are Mersenne primes.

By the early 1980s, researchers had accumulated enough data to confidently believe that infinitely many Mersenne primes exist. They could even guess how often these prime numbers appear, on average. Mathematicians have not found proof so far, but new data continues to support these guesses.

George Woltman, a computer scientist, founded the Great Internet Mersenne Prime Search, or GIMPS, in 1996. Through this collaborative program, anyone can download freely available software from the GIMPS website to search for Mersenne prime numbers on their personal computers. The website contains specific instructions on how to participate.

GIMPS has now identified 18 Mersenne primes, primarily on personal computers using Intel chips. The program averages a new discovery about every one to two years.

The largest known prime

Luke Durant, a retired programmer, discovered the current record for the largest known prime, (2136,279,841 – 1), in October 2024

Referred to as M136279841, this 41,024,320-digit number was the 52nd Mersenne prime identified and was found by running GIMPS on a publicly available cloud-based computing network.

This network used Nvidia chips and ran across 17 countries and 24 data centers. These advanced chips provide faster computing by handling thousands of calculations simultaneously. The result is shorter run times for algorithms such as prime number testing.

New and increasingly powerful computer chips have allowed prime-number hunters to find increasingly larger primes. Fritzchens Fritz/Flickr

The Electronic Frontier Foundation is a civil liberty group that offers cash prizes for identifying large primes. It awarded prizes in 2000 and 2009 for the first verified 1 million-digit and 10 million-digit prime numbers.

Large prime number enthusiasts’ next two challenges are to identify the first 100 million-digit and 1 billion-digit primes. EFF prizes of US$150,000 and $250,000, respectively, await the first successful individual or group.

Eight of the 10 largest known prime numbers are Mersenne primes, so GIMPS and cloud computing are poised to play a prominent role in the search for record-breaking large prime numbers.

Large prime numbers have a vital role in many encryption methods in cybersecurity, so every internet user stands to benefit from the search for large prime numbers. These searches help keep digital communications and sensitive information safe.

This story was updated on May 30, 2025 to correct the name of the Greek mathematician Euclid and to correct the factors of 2047.

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

*Credit for article given to Jeremiah Bartz*