Month: August 2025

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*