Category: mathematics

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Mathematics is a “science which requires a great amount of imagination”, said the 19th-century Russian maths professor Sofya Kovalevskaya – a pioneering figure for women’s equality in this subject.

We all have an imagination, so I believe everyone has the ability to enjoy mathematics. It’s not just arithmetic but a magical mixture of logic, reasoning, pattern spotting and creative thinking.

Of course, more and more research also shows the benefits of doing puzzles like these for brain health and development. Canadian psychologist Donald Hebb’s theory of learning has come to be known as “when neurons fire together, they wire together” (which, by the way, is one of the guiding principles behind training large neural networks in AI). New pathways start to form which can build and maintain strong cognitive function.

What’s more, doing maths is often a collaborative endeavour – and can be a great source of fun and fulfilment when people work together on problems. Which brings me to these festive-themed puzzles, which can be tackled by the whole family. No formal training in maths is required, and no complicated formulas are needed to solve them.

I hope they bring you some moments of mindful relaxation this holiday season. You can read the answers (and my explanations for them) here.

Festive maths puzzlers

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Puzzle 1: You are given nine gold coins that look identical. You are told that one of them is fake, and that this coin weighs less than the real ones. You are also given a set of old-fashioned balance scales that weigh groups of objects and show which group is heavier.

Question: What is the smallest number of weighings you need to carry out to determine which is the fake coin?

Puzzle 2: You’ve been transported back in time to help cook Christmas dinner. Your job is to bake the Christmas pie, but there aren’t even any clocks in the kitchen, let alone mobile phones. All you’ve got is two egg-timers: one that times exactly four minutes, and one that times exactly seven minutes. The scary chef tells you to put the pie in the oven for exactly ten minutes and no longer.

Question: How can you time ten minutes exactly, and avoid getting told off by the chef?

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Puzzle 3: Having successfully cooked the Christmas pie, you are now entrusted with allocating the mulled wine – which is currently in two ten-litre barrels. The chef hands you one five-litre bottle and one four-litre bottle, both of which are empty. He orders you to fill the bottles with exactly three litres of wine each, without wasting a drop.

Question: How can you do this?

Puzzle 4: For the sake of this quiz, imagine there are not 12 but 100 days of Christmas. On the n-th day of Christmas, you receive £n as a gift, from £1 on the first day to £100 on the final day. In other words, far too many gifts for you to be able to count all the money!

Question: Can you calculate the total amount of money you have been given without laboriously adding all 100 numbers together?

(Note: a variation of this question was once posed to the German mathematician and astronomer Carl Friedrich Gauss in the 18th century.)

Puzzle 5: Here’s a Christmassy sequence of numbers. The first six in the sequence are: 9, 11, 10, 12, 9, 5 … (Note: the fifth number is 11 in some versions of this puzzle.)

Question: What is the next number in this sequence?

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Puzzle 6: Take a look at the following list of statements:

Exactly one statement in this list of statements is false.

Exactly two statements in this list are false.

Exactly three statements in this list are false.

… and so on until:

Exactly 99 statements in this list are false.

Exactly 100 statements in this list are false.

Question: Which of these 100 statements is the only true one?

Puzzle 7: You are in a room with two other people, Arthur and Bob, who both have impeccable logic. Each of you is wearing a Christmas hat which is either red or green. Nobody can see their own hat but you can all see the other two.

You can also see that both Arthur’s and Bob’s hats are red. Now you are all told that at least one of the hats is red. Arthur says: “I do not know what colour my hat is.” Then Bob says: “I do not know what colour my hat is.”

Question: Can you deduce what colour your Christmas hat is?

Puzzle 8: There are three boxes under your Christmas tree. One contains two small presents, one contains two pieces of coal, and one contains a small present and a piece of coal. Each box has a label on it that shows what’s inside – but the labels have got mixed up, so every box currently has the wrong label on it. You are now told that you can open one box.

Question: Which box should you open, in order to then be able to switch the labels so that every label correctly shows the contents of its box?

Puzzle 9: Just before Christmas dinner, naughty Jack comes into the kitchen where there is one-litre bottle of orange juice and a one-litre bottle of apple juice. He decides to put a tablespoon of orange juice into the bottle of apple juice, then stirs it around so it’s evenly mixed.

But naughty Jill has seen what he did. Now she comes in, and takes a tablespoon of liquid from the bottle of apple juice and puts it into the bottle of orange juice.

Question: Is there now more orange juice in the bottle of apple juice, or more apple juice in the bottle of orange juice?

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Puzzle 10: In Santa’s home town, all banknotes carry pictures of either Santa or Mrs Claus on one side, and pictures of either a present or a reindeer on the other. A young elf places four notes on a table showing the following pictures:

Santa   |   Mrs Claus   |   Present | Reindeer

Now an older, wiser elf tells him: “If Santa is on one side of the note, a present must be on the other.”

Question: Which notes must the young elf must turn over to confirm what the older elf says is true?

Bonus puzzle

If you need a festive tiebreaker, here’s a question that requires a little bit of algebra (and the formula “speed = distance/time”). It’s tempting to say this question can’t be solved because the distance is not known – but the magic of algebra should give you the answer.

Santa travels on his sleigh from Greenland to the North Pole at a speed of 30 miles per hour, and immediately returns from the North Pole to Greenland at a speed of 40 miles per hour

Tiebreaker: What is the average speed of Santa’s entire journey?

(Note: a non-Christmassy version of this question was posed by the American physicist Julius Sumner-Miller.)

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

*Credit for article given to Neil Saunders*

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OpenAI chief executive Sam Altman – perhaps the most prominent face of the artificial intelligence (AI) boom that accelerated with the launch of ChatGPT in 2022 – loves scaling laws.

These widely admired rules of thumb linking the size of an AI model with its capabilities inform much of the headlong rush among the AI industry to buy up powerful computer chips, build unimaginably large data centres, and re-open shuttered nuclear plants.

As Altman argued in a blog post earlier this year, the thinking is that the “intelligence” of an AI model “roughly equals the log of the resources used to train and run it” – meaning you can steadily produce better performance by exponentially increasing the scale of data and computing power involved.

First observed in 2020 and further refined in 2022, the scaling laws for large language models (LLMs) come from drawing lines on charts of experimental data. For engineers, they give a simple formula that tells you how big to build the next model and what performance increase to expect.

Will the scaling laws keep on scaling as AI models get bigger and bigger? AI companies are betting hundreds of billions of dollars that they will – but history suggests it is not always so simple.

Scaling laws aren’t just for AI

Scaling laws can be wonderful. Modern aerodynamics is built on them, for example.

Using an elegant piece of mathematics called the Buckingham π theorem, engineers discovered how to compare small models in wind tunnels or test basins with full-scale planes and ships by making sure some key numbers matched up.

Those scaling ideas inform the design of almost everything that flies or floats, as well as industrial fans and pumps.

Another famous scaling idea underpinned the boom decades of the silicon chip revolution. Moore’s law – the idea that the number of the tiny switches called transistors on a microchip would double every two years or so – helped designers create the small, powerful computing technology we have today.

But there’s a catch: not all “scaling laws” are laws of nature. Some are purely mathematical and can hold indefinitely. Others are just lines fitted to data that work beautifully until you stray too far from the circumstances where they were measured or designed.

When scaling laws break down

History is littered with painful reminders of scaling laws that broke. A classic example is the collapse of the Tacoma Narrows Bridge in 1940.

The bridge was designed by scaling up what had worked for smaller bridges to something longer and slimmer. Engineers assumed the same scaling arguments would hold: if a certain ratio of stiffness to bridge length worked before, it should work again.

Instead, moderate winds set off an unexpected instability called aeroelastic flutter. The bridge deck tore itself apart, collapsing just four months after opening.

Likewise, even the “laws” of microchip manufacturing had an expiry date. For decades, Moore’s law (transistor counts doubling every couple of years) and Dennard scaling (a larger number of smaller transistors running faster while using the same amount of power) were astonishingly reliable guides for chip design and industry roadmaps.

As transistors became small enough to be measured in nanometres, however, those neat scaling rules began to collide with hard physical limits.

When transistor gates shrank to just a few atoms thick, they started leaking current and behaving unpredictably. The operating voltages could also no longer be reduced with being lost in background noise.

Eventually, shrinking was no longer the way forward. Chips have still grown more powerful, but now through new designs rather than just scaling down.

Laws of nature or rules of thumb?

The language-model scaling curves that Altman celebrates are real, and so far they’ve been extraordinarily useful.

They told researchers that models would keep getting better if you fed them enough data and computing power. They also showed earlier systems were not fundamentally limited – they just hadn’t had enough resources thrown at them.

But these are undoubtedly curves that have been fit to data. They are less like the derived mathematical scaling laws used in aerodynamics and more like the useful rules of thumb used in microchip design – and that means they likely won’t work forever.

The language model scaling rules don’t necessarily encode real-world problems such as limits to the availability of high-quality data for training, or the difficulty of getting AI to deal with novel tasks – let alone safety constraints or the economic difficulties of building data centres and power grids. There is no law of nature or theorem guaranteeing that “intelligence scales” forever.

Investing in the curves

So far, the scaling curves for AI look pretty smooth – but the financial curves are a different story.

Deutsche Bank recently warned of an AI “funding gap” based on Bain Capital estimates of a US$800 billion mismatch between projected AI revenues and the investment in chips, data centres and power that would be needed to keep current growth going.

JP Morgan, for their part, has estimated that the broader AI sector might need around US$650 billion in annual revenue just to earn a modest 10% return on the planned build-out of AI infrastructure.

We’re still finding out which kind of law governs frontier LLMs. The realities may keep playing along with the current scaling rules; or new bottlenecks – data, energy, users’ willingness to pay – may bend the curve.

Altman’s bet is that the LLM scaling laws will continue. If that’s so, it may be worth building enormous amounts of computing power because the gains are predictable. On the other hand, the banks’ growing unease is a reminder that some scaling stories can turn out to be Tacoma Narrows: beautiful curves in one context, hiding a nasty surprise in the next.

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

*Credit for article given to Nathan Garland*

 

Math teachers have to accommodate high school students’ different approaches to problem-solving. RJ Sangosti/MediaNews Group/The Denver Post via Getty Images

Among high school students and adults, girls and women are much more likely to use traditional, step-by-step algorithms to solve basic math problems – such as lining up numbers to add, starting with the ones place, and “carrying over” a number when needed. Boys and men are more likely to use alternative shortcuts, such as rounding both numbers, adding the rounded figures, and then adjusting to remove the rounding.

But those who use traditional methods on basic problems are less likely to solve more complex math problems correctly. These are the main findings of two studies our research team published in November 2025.

This new evidence may help explain an apparent contradiction in the existing research – girls do better at math in school, but boys do better on high-stakes math tests and are more likely to pursue math-intensive careers. Our research focuses not just on getting correct answers, but on the methods students use to arrive at them. We find that boys and girls approach math problems differently, in ways that persist into adulthood.

A possible paradox

In a 2016 study of U.S. elementary students, boys outnumbered girls 4 to 1 among the top 1% of scorers on a national math test. And over many decades, boys have been about twice as likely as girls to be among the top scorers on the SAT and AP math exams.

However, girls tend to be more diligent in elementary school and get better grades in math class throughout their schooling. And girls and boys across the grades tend to score similarly on state math tests, which tend to be more aligned with the school curriculum and have more familiar problems than the SAT or other national tests.

Beyond grades and test scores, the skills and confidence acquired in school carry far beyond, into the workforce. In lucrative STEM occupations, such as computer science and engineering, men outnumber women 3 to 1. Researchers have considered several explanations for this disparity, including differences in math confidence and occupational values, such as prioritizing helping others or making money. Our study suggests an additional factor to consider: gender differences in approaches to math problems.

When older adults think of math, they may recall memorizing times tables or doing the tedious, long-division algorithm. Memorization and rule-following can pay off on math tests focused on procedures taught in school. But rule-following has its limits and seems to provide more payoff among low-achieving than high-achieving students in classrooms.

More advanced math involves solving new, perplexing problems rather than following rules.

Math can be creative, not rote. AP Photo/Jacquelyn Martin

Differing strategies

In looking at earlier studies of young children, our research team was struck by findings that young boys use more inventive strategies on computation problems, whereas girls more often use standard algorithms or counting. We wondered whether these differences disappear after elementary school, or whether they persist and relate to gender disparities in more advanced math outcomes.

In an earlier study, we surveyed students from two high schools with different demographic characteristics to see whether they were what we called bold problem-solvers. We asked them to rate how much they agreed or disagreed with specific statements, such as “I like to think outside the box when I solve math problems.” Boys reported bolder problem-solving tendencies than girls did. Importantly, students who reported bolder problem-solving tendencies scored higher on a math problem-solving test we administered.

Our newer studies echo those earlier results but reveal more specifics about how boys and girls, and men and women, approach basic math problems.

Algorithms and teacher-pleasing

In the first study, we gave three questions to more than 200 high school students: “25 x 9 = ___,” “600 – 498 = ___,” and “19 + 47 + 31 = ___.” Each question could be solved with a traditional algorithm or with a mental shortcut, such as solving 25 x 9 by first multiplying 25 x 8 to get 200 and then adding the final 25 to get 225.

Regardless of their gender, students were equally likely to solve these basic computation items correctly. But there was a striking gender difference in how they arrived at that answer. Girls were almost three times as likely as boys – 52% versus 18% – to use a standard algorithm on all three items. Boys were far more likely than girls – 51% versus 15% – to never use an algorithm on the questions.

Girls were far more likely than boys to use an algorithm

When given three basic math problems, high school girls were three times more likely than boys to use a standard algorithm to solve all three. High school boys were nearly three times more likely than girls to use an alternative strategy for all three problems.

We suspected that girls’ tendency to use algorithms might stem from greater social pressure toward compliance, including complying with traditional teacher expectations.

So, we also asked all the students eight questions to probe how much they try to please their teachers. We also wanted to see whether algorithm use might relate to gender differences in more advanced problem-solving, so we gave students several complex math problems from national tests, including the SAT.

As we suspected, we found that girls were more likely to report a desire to please teachers, such as by completing work as directed. Those who said they did have that desire used the standard algorithm more often.

Also, the boys in our sample scored higher than the girls on the complex math problems. Importantly, even though students who used algorithms on the basic computation items were just as likely to compute these items correctly, algorithm users did worse on the more complex math problems.

Continuing into adulthood

In our second study, we gave 810 adults just one problem: “125 + 238 = ___.” We asked them to add mentally, which we expected would discourage them from using an algorithm. Again, there was no gender difference in answering correctly.

But 69% of women, compared to 46% of men, reported using the standard algorithm for their mental calculation, rather than using another strategy entirely.

We also gave the adults a more advanced problem-solving test, this time focused on probability-related reasoning, such as the chances that rolling a seven-sided die would result in an even number. Similar to our first study, women and those who used the standard algorithm on the computation problem performed worse on the reasoning test.

The importance of inventiveness

We identified some factors that may play a role in these gender differences, including spatial-thinking skills, which may help people develop alternate calculation approaches. Anxiety about taking tests and perfectionism, both more prevalent among women, may also be a factor.

We are also interested in the power of gender-specific social pressures on girls. National data has shown that young girls exhibit more studious behavior than do boys. And the high school girls we studied were more likely than boys to report they made a specific effort to meet teachers’ expectations.

More research definitely is needed to better understand this dynamic, but we hypothesize that the expectation some girls feel to be compliant and please others may drive teacher-pleasing tendencies that result in girls using algorithms more frequently than boys, who are more socialized to be risk-takers.

While compliant behavior and standard math methods often lead to correct answers and good grades in school, we believe schools should prepare all students – regardless of gender – for when they face unfamiliar problems that require inventive problem-solving skills, whether in daily life, on high-stakes tests or in math-intensive professions.

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

*Credit for article given to Sarah Lubienski, Colleen Ganley & Martha Makowski*

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As the artificial intelligence (AI) and quantum computing industries explode, trained STEM professionals are in high demand. Mathematics is foundational to these fields.

But mathematics is missing an important ingredient: people who are female or gender-diverse.

In New South Wales, for example, only one-third of high school graduates who complete mathematics at the highest level are female or gender-diverse. And when students choose university courses in December, a large proportion of these highly qualified people will step away from mathematics and STEM.

Australia cannot stay competitive by only accessing half of its young talent. By leaving mathematics early, young women and gender-diverse people limit their own career opportunities. Worse, the new technologies resulting from the current revolutions may not serve broader society well, if women and gender-diverse people are not involved in their development.

But at the University of Sydney over the past five years we have run a successful pilot program to reverse this trend – and to empower young women to make informed career choices. Better, the program is cheap to run and can be easily adopted elsewhere so mathematics – and the many industries it underpins – can be more diverse in ways that benefit everyone, regardless of their gender.

Declining enrolments

Before 2020, female and gender-diverse enrolments in advanced mathematics at the University of Sydney were in decline.

In 2020 the incoming cohort was nearly 80% male. Non-STEM directions offer attractive and important career options, and some movement between specialisations is expected. But a nosedive from 35% female students at the end of high school to 22% at the start of university indicates a problem.

Over five years, a team I lead piloted an intervention which has increased the ratio of female and gender-diverse students in advanced first-year mathematics from 22% to 30% – nearly back to the high school levels.

Our program consists of two components:

information, personalised invitations, and enrolment advice for incoming female and gender-diverse students, and a mentoring program for female and gender-diverse students who enrol in advanced mathematics.

Targeting the problem from year one

Before the start of semester, we compare first year enrolments with students’ high school certificates and majors. Like in high school, mathematics at the university is offered at multiple parallel levels.

When students are enrolled at a lower level than their background and major would justify, we send personalised emails encouraging them to switch to the advanced level. We hold a welcome event and multiple drop-in sessions, offering tailored advice.

In the mentoring program we match female and gender diverse advanced maths students with groups of eight to twelve peers of mixed year levels. Matching is based on timetables.

Each group is mentored by a senior (Honours or PhD) student, and an academic – at least one of whom is female or gender-diverse. Student mentors bring invaluable insight to the program, as they had walked in the mentees’ shoes only a few years before.

Each year 50–80 students participate in the program, roughly two-thirds of whom are first-year students.

Mentoring groups meet weekly for an hour: sometimes with both mentors, sometimes with the student mentor alone. Meeting topics are loosely structured around academic advice and sharing experiences.

Many groups develop their own agendas organically. The program does not focus on tutoring, though students enjoy discussing key mathematical techniques and concepts.

Fostering community and belonging

At the heart of the program is the opportunity to build community with peers, away from the pressure of assessments. While student feedback on the program is overall enthusiastic, it is a puzzle to maintain engagement with mentoring as semesters get hectic. It is difficult for students to prioritise community building when marks are on the line elsewhere.

We suspected the large drop in female and gender diverse enrolments at the transition to university is at least partly explained by these students’ lack of confidence in their mathematical abilities.

Research shows such insecurities disproportionately affect women. General messaging is ineffective in the face of self-doubt, so we aimed for a personalised but scalable approach.

The mentoring component fosters community and belonging. This combats isolation, provides ongoing support and enables long-term retention.

A low-cost solution

Our program is a low-cost solution that can be implemented in most academic contexts.

The first year of university is a place to start, but it is too late to fully address Australia’s pipeline problem. We can’t expect to have women and gender-diverse students participating in STEM at university in higher numbers than they did at the end of high school.

Similar programs could be put in place in high schools, and personal invitations can even be used to bring more girls to elementary school enrichment programs. This would help boost diverse and equitable participation in STEM from the roots.

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

*Credit for article given to Zsuzsanna Dancso*

 

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What’s seven times nine? Quick, you’ve got six seconds to answer.

This June, over 600,000 children in England in year four, aged eight and nine, will be expected to answer questions like this. They will be sitting the multiplication tables check (MTC), a statutory assessment of their multiplication fact recall.

The MTC was introduced in 2022 with the aim of driving up standards in mathematics. It’s an online test that children take on a tablet or computer, made up of 25 questions with six seconds per question.

Being able to quickly recall multiplication facts is valuable. Not having to think about seven times nine, just knowing that it’s 63, frees up a child’s mental thinking space. This means they can focus on different aspects of the mathematics they are doing, such as completing multi-step problems or using reasoning to solve context-based problems.

Being able to quickly recall multiplication facts is also the foundation for more advanced mathematics topics that children will encounter at secondary school.

Our research shows that the MTC is an accurate reflection of children’s multiplication fact recall. But the learning they do for this test doesn’t necessarily help them apply this knowledge in other areas of mathematics. What’s more, focus on the MTC may be diverting teaching time away from other maths knowledge.

Since the multiplication tables check was introduced in 2022, the average score in the test has increased year-on-year from 19.8 in 2022 to 20.6 in 2024. This suggests that schools are placing more emphasis on children’s multiplication fact recall – and on preparing them for this test.

Teaching union the NAHT (National Association of Head Teachers) has suggested that the test is unnecessary, and that it places too much emphasis on fact recall at a cost to other areas of mathematics. The union has also expressed concerns that it disadvantages some children for reasons such as digital accessibility.

Our research has investigated whether the MTC is a good way of testing children’s recall of multiplication facts. We have found that children perform just as well on a more traditional paper-and-pencil timed fact test as on a computer test equivalent to the MTC. However, having a time limit per question – which is only possible with a computerised test – is essential to assess recall, rather than fast calculation.

Pupils taking part in the research project. Lisa Gilligan-Lee/University of Nottingham, Author provided (no reuse)

There was no evidence that any children were particularly disadvantaged by the computerised test. However, we did find that children’s attention skills and how quickly they could enter numbers into the tablet they were using did influence their scores.

This suggests that, for it to be a fair test, it is important that children are familiar with the technology they are using to complete the test. Given that there are stark differences in access to technology in schools, this may pose an issue for some children.

The purpose of introducing the MTC was to improve children’s broader mathematics attainment by improving their multiplication fact recall. But performance in the year six Sats tests, which assess a range of mathematical skills, shows little change.

Crucially, improving children’s multiplication fact recall through retrieval practice doesn’t equate to improving their ability to use the multiplication facts they know. If posed a question such as “Tara has seven books. Ravi has four times as many. How many books do they have altogether?” Children who can recall that 5 x 7 = 35 may still not be able to solve the problem.

Time pressure

What’s more, because the MTC is a timed test, teachers and parents may use similar time-pressured approaches to prepare children and help them improve their multiplication fact recall. But our research showed that while practice with a computerised game can support children’s fact recall, the benefits to learning are the same whether or not children are encouraged to answer as quickly as possible.

In research not yet published in a peer-reviewed journal, we found that children who were anxious about mathematics learnt less when practising with time pressure compared to children without mathematics anxiety. Without time pressure, anxiety levels were not related to the amount of learning. Doing some regular multiplication fact retrieval practice is more important than the type of practice, for all learners.

Even though the MTC is a timed assessment, it doesn’t mean that children only need to do timed practice to prepare for this. Some children may benefit more from less time pressure when practising.

Multiplication fact recall is just one element of mathematics and so having a good balance is important. Fact recall and testing should go hand in hand with other areas of mathematics learning such as understanding concepts, choosing strategies and solving applied problems.

Recalling multiplication facts doesn’t automatically help children to apply their knowledge. So, although working towards the multiplication tables check can support fact recall, children will need extra support in knowing how to use and apply these facts.

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

*Credit for article given to Camilla Gilmore, Lucy Cragg & Natasha Guy*

 

Sydney Bourne/ AAP

If you ask a small child a simple maths question, such as 4+2, they may count on their fingers to work it out.

Should we encourage young children to do this? This seemingly simple question is surprisingly complex to answer.

Some teachers and parents might say, yes, it seems to help young children learn about numbers. Others might discourage finger counting, arguing it might slow the development of mental strategies.

A new Swiss study, released on Friday, shows kids who use finger counting from a young age perform better at addition than those who do not.

What does the research say?

There is a rich debate among researchers about the value of kids using their fingers to count.

Education psychologists say finger counting helps children think through strategies without overloading their working memory (how our brains hold pieces of information for short time while we work something out), until more abstract strategies are mastered.

Researchers in embodied cognition (learning through actions) argue associating fingers and numbers is “doing what comes naturally” and so, should be encouraged. Neuroscientists might also note similar parts of your brain activate when you move your fingers and think about numbers, which helps memory.

Several previous classroom studies have shown children who use finger strategies to solve maths questions perform better than children who do not, until around seven when the opposite becomes true.

So, before age seven, finger-counters are better. After seven, non-finger-counters are better.

Why does this happen? What does this mean for mathematics education? This has been a point of debate for several years.

A new study followed 200 kids

A new University of Lausane study has taken an important step in settling this debate.

The researchers say previous studies have left us with two possible explanations for the apparent change in the benefits of finger counting at about seven.

One interpretation is finger strategies become inefficient when maths questions become more complex (for example 13 + 9 is harder than 1 + 3), so children who use finger strategies don’t perform as well.

The other possibility is the children who are not using finger strategies at seven (and performing better than those who do) were previously finger-users, who have transitioned to more advanced mental strategies.

To untangle these contrasting explanations, the researchers followed almost 200 children from age 4.5 to 7.5 and assessed their addition skills and finger use every six months.

Notably, they tracked if and when the children started and stopped using their fingers. So, at each assessment point, it was noted whether children were non-finger users, new finger-users (newly started), continuing finger-users, or ex-finger users (had stopped).

What did the study find?

The study found that by 6.5 years most of the non-finger users were indeed ex-finger users. These ex-finger users were also the highest performers in the addition questions and were still improving a year later. The significance of this finding is that in previous studies, these high performing children had only been identified as non-finger users, not as former users of finger-based strategies.

In the new Swiss study, only 12 children never used their fingers over the years, and they were the lowest performing group.

Additionally, the study showed the “late starters” with finger-counting strategies, who were still using finger strategies at the age of 6.5 to 7.5 years, did not perform as well as the ex-finger users.

What does this mean?

The findings from this unique longitudinal study are powerful. It seems reasonable to conclude both teachers and parents should encourage finger counting development from preschool through the first couple of years of school.

However, the Swiss study focused on predominantly white European children from middle to high socioeconomic backgrounds. Would we find such clear outcomes in the average multicultural public school in Australia? We suspect that we might.

Our own 2025 study found a wide variety of finger counting methods in such schools, but when teachers paid attention to the development of finger counting strategies it supported children’s number skills.

What can parents do?

Parents can show preschoolers how they can use their fingers to represent numbers, such as holding up three fingers and saying “three”.

Help them practice counting from one to ten, matching one finger at a time. Once they get started, the rest should come naturally. There is no need to discourage finger counting at any time. Children naturally stop using their fingers when they no longer need them.

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

*Credit for article given to Jennifer Way & Katherin Cartwright*

Despite using numerical bases on a daily basis, few of us have reflected on the nature of these cognitive tools. (Getty Images/Unsplash+)

Most of us have little trouble working out how many millilitres are in 2.4 litres of water (it’s 2,400). But the same can’t be said when we’re asked how many minutes are in 2.4 hours (it’s 144).

That’s because the Indo-Arabic numerals we often use to represent numbers are base-10, while the system we often use to measure time is base-60.

Expressing time in decimal notation leads to an interaction between these two bases, which can have implications at both the cognitive and cultural level.

Such base interactions and their consequences are among the important topics covered in a new issue of the Philosophical Transactions of the Royal Society journal, which I co-edited with colleagues Andrea Bender (University of Bergen), Mary Walworth (French National Centre for Scientific Research) and Simon J. Greenhill (University of Auckland).

The themed issue brings together work from anthropology, linguistics, philosophy and psychology to examine how humans conceptualize numbers and the numeral systems we build around them.

What are bases, and why do they matter?

Despite using numeral bases on a daily basis, few of us have reflected on the nature of these cognitive tools. As I explain in my contribution to the issue, bases are special numbers in the numeral systems we use.

Because our memories aren’t unlimited, we can’t represent each number with its own unique label. Instead, we use a small set of numerals to build larger ones, like “three hundred forty-two.”

The degree to which numeral systems transparently reflect their bases has all sorts of implications. (Pablo Merchán Montes/Unsplash+)

That’s why most numeral systems are structured around a compositional anchor — a special number with a name that serves as a building block to form names for other numbers. Bases are anchors that exploit powers of a special number to form complex numerical expressions.

The English language, for example, uses a decimal system, meaning it uses the powers of 10 to compose numerals. So we compose “three hundred and forty-two” using three times the second power of 10 (100), four times the first power of 10 (10) and two times the zeroth power of 10 (one).

This base structure allows us to represent numbers of all sizes without overloading our cognitive resources.

Languages affect how we count

Despite the abstract nature of numbers, the degree to which numeral systems transparently reflect their bases has very concrete implications — and not just when we tell time. Languages with less transparent rules will take longer to learn, longer to process and can lead to more calculation and dictation errors.

Take French numerals, for example. While languages like French, English and Mandarin all share the same base of 10, most dialects of French have what could politely be called a quirky way of representing numbers in the 70-99 range.

Seventy is soixante-dix in French, meaning “six times 10 plus 10,” while 80 uses 20 as an anchor and becomes quatre-vingts, meaning “four twenties” (or “four twenty,” depending on the context). And 90 is quatre vingt dix, meaning “four twenty ten.”

French is far from being alone in being quirky with its numerals. In German, numbers from 13 to 99 are expressed with the ones before the tens, but numbers over 100 switch back to saying the largest unit first.

Even in English, the fact that “twelve” is said instead of “ten two” hides the decimal rules at play. Such irregularities spread far beyond languages.

How bases shape learning and thought

Base-related oddities are spread out across the globe and have very real implications for how easily children learn what numbers are and how they interact with objects such as blocks, and for how efficiently adults manipulate notations.

For example, one study found that lack of base transparency slows down the acquisition of some numerical abilities in children, while another found similar negative effects on how quickly they learn how to count.

A young boy learns counting on an abacus at a school in Allahabad, India, in 2015. (AP Photo/Rajesh Kumar Singh)

Another study found that children from base-transparent languages were quicker to use large blocks worth 10 units to represent larger numbers (for example, expressing 32 using three large blocs and two small ones) than children with base-related irregularities.

While Mandarin’s perfectly transparent decimal structure can simplify learning, a new research method suggests that children may find it easier to learn what numbers are if they are exposed to systems with compositional anchors that are smaller than 10.

In general, how we represent bases has very concrete cognitive implications, including how easily we can learn number systems and which types of systems will tend to be used in which contexts.

Technicians lower the Mars Climate Orbiter onto its work stand in the Spacecraft Assembly and Encapsulation Facility-2 in 1998. (NASA)

At a cultural level, base representation influences our ability to collaborate with scientists across disciplines and across cultures. This was starkly illustrated by the infamous Mars Climate Orbiter incident, when a mix-up between metric and imperial units caused a $327 million spacecraft to crash into Mars in 1999.

Why understanding bases matters

Numeracy — the ability to understand and use numbers — is a crucial part of our modern lives. It has implications for our quality of life and for our ability to make informed decisions in domains like health and finances.

For example, being more familiar with numbers will influence how easily we can choose between retirement plans, how we consider trade-offs between side-effects and benefits when choosing between medications or how well we understand how probabilities apply to our investments.

And yet many struggle to learn what numbers are, with millions suffering from math anxiety. Developing better methods for helping people learn how to manipulate numbers can therefore help millions of people improve their lives.

Research on the cognitive and cultural implications of bases collected in the Philosophical Transactions of the Royal Society journal can help make progress towards our understanding of how we think about numbers, marking an important step towards making numbers more accessible to everyone.

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

*Credit for article given to Jean-Charles Pelland*

 

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If the recent news of a new mathematics and statistics curriculum for years 0–10 felt familiar, that’s because it was.

In term four last year, the Ministry of Education released a previous new maths (and English) curriculum for Years 0–8, to be implemented from term one this year.

Schools must use the latest new curriculum from term one next year. This will be the third curriculum for primary and intermediate schools in less than three years.

Despite claims that the most recent curriculum is only an “update”, the changes are bigger than teachers might have expected.

The new curriculum is more difficult and more full. There is now a longer list of maths procedures and vocabulary to be memorised at each year of school.

For example, year 3 children should learn there are 366 days in a leap year and that leap years happen every four years. Year 5 students should know what acute, obtuse and reflex angles are.

Some concepts have been moved into earlier years. Year 6 children will learn calculations with rational numbers (such as “75% is 24, find the whole amount”), whereas previously this would have been taught at year 8. (If you’re wondering, the whole amount is 32.)

Cubes and cube roots have been moved a year earlier. A lot of statistics, a traditional area of strength for New Zealand in international tests, has been stripped out.

Much of the “effective maths teaching” material about clearly explaining concepts and planning for challenging problem solving has been removed. Also gone are the “teaching considerations” that helped guide teachers on appropriate ways to teach the content.

The maths children should learn was previously broken up into what they needed to “understand, know and do” – the UKD model. But this has changed to “knowledge” and “practices”.

In short, there are new things to teach, things to teach in different years, and the whole curriculum is harder and structured differently. It is effectively a new curriculum.

Not just a document

Most teachers now have about eight school weeks to plan for the changes, alongside teaching, planning, marking, reporting, pastoral support and extracurricular activities.

For busy schools heading into the end of the school year, the timeline is unrealistic, some say a “nightmare”.

For secondary teachers, who will be making changes in years 9 and 10, this is the first major curriculum change since 2007.

Primary and intermediate teachers, who have worked hard this year getting up to speed with a new curriculum that will soon expire, might legitimately ask why they bothered.

A curriculum change is a big deal in a school, something that normally happens once in a decade or more. A curriculum is not just a document, it is used every day for planning, teaching and assessment. Any change requires more lead time than this.

 

When England launched a new National Curriculum in 2013, teachers had it 12 months ahead of implementation. Singapore, a country whose education system Education Minister Erica Stanford paints as exemplary, gave teachers two years to prepare for the secondary maths curriculum change in 2020.

Expecting teachers to prepare for major curriculum changes in eight weeks is not only unnecessarily rushed and stressful – it is also a risk to children’s learning.

Time to slow down

Term one next year also marks the implementation of the new “student monitoring, assessment and reporting tool” (SMART) which teachers have not yet seen.

Children in Years 3–10 will take maths tests twice a year and will be described as emerging, developing, consolidating, proficient or exceeding. Children in the top three categories (during the year) or top two categories (at the end of year) are “on track”.

For the rest, the curriculum says “teachers will need to adjust classroom practice, develop individualised responses, or trigger additional learning support”.

The original curriculum rewrite shifted the goalposts – only 22% of year 8 students would be at the “expectation” level, compared with 42% previously – and this curriculum shifts those goalposts further.

The inevitably poorer results from testing against a more challenging curriculum risk damaging children’s self confidence, disappointing parents and placing blame on teachers.

Test results may improve in later years, compared to those produced in the first year of assessment against a harder curriculum that will take time to embed. But that will not necessarily be evidence the change was justified.

Pausing this latest curriculum change for at least 12 months would give time for adequate consultation and preparation. That would be more consistent with the change processes of education systems internationally.

According to a recent report from the Education Review Office, teachers have mostly demonstrated professionalism in their conscientious adoption of the previous curriculum.

In our view, the most recent changes will severely test that goodwill.

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

*Credit for article given to David Pomeroy & Lisa Darragh*

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Mathematics has been the broccoli of school subjects for generations of Australian teenagers.

Often pushed aside, dreaded, or even feared, nearly one third of students opt out of any senior maths courses.

This has serious implications for Australia’s future. As an Australian Academy of Science report warned on Thursday, we need people with maths skills to support a whole range of careers in science. This includes agricultural science, artificial intelligence, data science, biotechnology and climate science.

The skills we gain during school mathematics – problem-solving, pattern-finding, reasoning logically, and computational thinking – are essential to the work of many STEM careers.

The challenge is turning maths from broccoli to the ingredient every student wants on their plate for their future. So, what can we do?

What has been happening with high school maths?

Across Australia, there has been a decline in students studying maths in years 11 and 12 since the 1990s. Today, only 8.4% of Australian high school students study the most difficult level of maths.

There are diverse reasons explaining why students opt out of maths during school.

Many students struggle to see the relevance of the maths they are learning for their future. Others have low self-confidence and avoid maths, believing they are not capable. An increasing range of senior subjects has also led to students being drawn to more enticing alternatives.

What can parents do?

Research shows parents’ attitudes towards maths can predict the attitudes their children will have towards the subject.

This means we need to be careful as parents. If we have negative attitudes towards maths due to our own anxieties or past struggles, this can affect our children’s attitudes and performance too.

Instead, parents should try to focus on the positive aspects of maths.

For example, this is a subject where you learn about the mechanics of the world, rather than a subject to be endured before moving to the “fun” stuff. Maths can come alive once we notice how we use it in sports, art, cooking, travel, money management and games.

Parents can also be curious co-learners with their children – we never need to have all the answers ourselves. But showing interest, having a growth mindset (a belief you can improve your abilities through effort), and asking questions can support students’ positive attitudes and performance in maths.

You can also talk to your child about why mastering maths is central to a wide range of occupations, from coding to trades, retail, nursing, animation and architecture.

What should schools do?

Research suggests 20% of 15-year-old boys and 33% of 15-year-old girls do not think maths will be relevant to their future.

So we need a new approach to careers advice in schools. Students need adequate support from informed adults to make accurate judgements about career pathways – emphasising how maths can help.

On top of this, schools could consider the ways in which mathematics is celebrated and promoted in schools. While music, drama, and sport days are regular features of the school calendar, maths is rarely included. Exciting maths competitions and maths days are prime opportunities to show students how important maths is in our world.

What about teachers?

Some of us may remember maths lessons as rather dry with a focus on lots of questions and whether something was “wrong” or “right”.

So teachers who make maths engaging for students and maximise opportunities for success are crucial.

This involves making abstract mathematics real (how does this concept apply to something physical in the real world?).

Teachers should also provide step-by-step support to students (what educators call “scaffolding”), so young people experience a sense of achievement and success with maths. Success builds motivation, creating an upward spiral of positive maths experiences.

What can governments do?

The alarm bells over maths participation have been raised for 30 years, with government funding supporting research into this phenomenon.

Despite this, the declines persist, and gender gaps in maths have widened, with more boys doing maths and more boys achieving higher marks.

So while governments should continue to support research into this matter, they should prioritise translating it into practical strategies for schools and teachers.

Some evidence-based approaches include:

high-expectation teaching, where teachers set ambitious goals, create supportive classrooms, and believe all students can achieve

relevance interventions, where teachers show students the practical implications of their learning

mindset interventions, which help students believe in their abilities.

Getting kids back into maths

Maths participation is both a national concern and something we should all be personally attuned to.

The lifestyles of future generations will be dependent on our capacity to be STEM innovators.

At an individual level, when students opt-out of mathematics, they are potentially closing many doors in their lives and career.

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

*Credit for article given to Bronwyn Reid O’Connor & Ben Zunica*

 

When Grade 5 teacher Jared Bremner joined First Baptist Christian School in the Cayman Islands, Mathletics was already woven into their math curriculum.

Six years later, he can’t imagine teaching without it.

First Baptist’s approach was different from the start: they were intentional about implementation. What started as simply adding another digital resource has become a complete shift in how students experience math from Kindergarten through Grade 8.

Mathletics, our ESSA-certified online math program for ages 5–16, delivers personalized learning through explicit and systematic instruction, engaging activities, gamified challenges and immediate feedback. But at First Baptist, it’s become much more than a supplement: it’s central to how they teach.

Bremner now oversees implementation across the entire school, ensuring every new teacher successfully integrates the program into their lessons.

His personal journey, from newcomer to passionate champion, reflects the school’s own evolution in digital math education.

From ‘just another resource’ to essential instruction

Six years ago, First Baptist had a clear goal: find a math platform with diverse resources that teachers could actually use to improve instruction through technology.

“While it started off as just a resource they wanted to use,” Bremner reflects, “is now an important part of our instruction.” What made the difference? “We find Mathletics to be very encompassing. It’s a very layered product,” Bremner explains, “from working through problem-solving questions to assessment to fun games.”

This comprehensive nature meant teachers weren’t juggling multiple tools: they had everything in one platform. The school committed to systematic implementation and real teacher training, instead of just handing teachers another digital tool and hoping for the best. Their approach? Start with the youngest learners and build up.

Students learn the platform progressively from Kindergarten through Grade 8, teachers get ongoing support (not just a one-time training session) and most importantly, Mathletics becomes woven into daily instruction rather than treated as an add-on.

“We’ve been intentional about using Mathletics,” Bremner explains, “and so from Kindergarten up to Grade 8, we’re really training our children how to understand the platform, how to use it and how they can benefit from the resources.

When learning becomes play (without students realizing it)

Ask Bremner about gamification and he lights up. He’s watched something remarkable happen in his classroom over the years: students so absorbed in math challenges they forget they’re actually learning.

“Making things in a gamified way allows them to enjoy their learning and learning almost becomes secondary,” he shares. “And so, through that gamification, they learn that they can progress, and they can make mistakes, but they can still improve.”

For today’s digital-native students, this isn’t just nice to have – it’s speaking their language.

And nowhere is this more evident than with Live Mathletics, the program’s real-time math competitions where students can compete with peers from around the world.

 

These live challenges test students’ math fluency skills and reflexes, allowing whole schools, classes and individual learners to go head-to-head.

For Bremner’s class at this international school, it means they’re not just competing with classmates: they’re facing off against students from India, Pakistan, South Africa and Canada.

“[They] find ways of how they can get their answers in quicker and how they can compete with children around the world,” Bremner observes. “And so, it grows a global mindset not just within the classroom but on the platform that we’re using.”

The competitive element has genuinely engaged students. They enjoy customizing their avatars, working to beat their personal bests in timed challenges and tracking their progress against peers worldwide.

The immediate feedback and global competition keep them motivated to practice more, proving that when learning feels like play, students naturally want to keep improving.

The teacher’s game-changer: Data that actually helps

The engagement is just the beginning. For Bremner, the real value comes from how Mathletics helps him meet every student where they are.

Using the platform’s reporting features, Bremner tracks how students perform on activities and quests, gauging their responses against the school’s grading scale. But he doesn’t stop at assessment.

“What we do is build from that,” he explains, “build their understanding and make sure they’re [not just] working on topics being taught in class at that moment… so it’s not just about current learning, it’s about identifying students that need to be challenged and also children that need to be supported.”

This means advanced learners get extension activities that push them beyond the current curriculum, while struggling students receive targeted support exactly where they need it.

The school reinforces this differentiated approach with Mathletics’ printable booklets in their Response to Intervention (RTI) program.

The result? Every student gets what they need (challenge or support) based on real data, not guesswork.

Leadership that values what works

At First Baptist, administrative support comes with accountability. The principal’s approach is clear: she’s committed to providing Mathletics but expects to see it actively used in classrooms.

“She wants to make sure the teachers are intentional in its use,” Bremner explains, “and that it’s not just another resource that we just add to the list.”

This results-focused leadership means the school maintains accountability through:

• Usage monitoring via administrative accounts
• Activity tracking to measure engagement levels
• Regular review of minutes used and activities assigned
• Data-driven decisions about program effectiveness.

“I have an admin account and I look at how many activities are being assigned and how many minutes are being used,” Bremner shares. “We make sure that it’s actually being used for its true value.”

From one teacher’s journey to school-wide success

When Bremner first arrived at First Baptist, integrating a new platform while adapting to a new country and school felt challenging. Yet his perspective shifted dramatically through experience:

“And over time of using it, I grew to love it!” he reflects.

His transformation, from newcomer to the teacher who now guides colleagues through Mathletics implementation, mirrors First Baptist’s own six-year evolution.

Through Bremner’s experience, we see the key elements that drive success:

• Comprehensive training that unlocks features teachers didn’t know existed
• Engaged students participating because they want to, not because they must
• Meaningful data that informs teaching decisions and student support
• Committed leadership that backs investment with accountability and clear usage expectations
• Systematic progression building platform fluency from kindergarten through eighth grade
• Sustained support offering professional development beyond one-time training sessions.

Today, Bremner monitors usage data and provides the support he once needed himself, embodying the long-term approach First Baptist has built.

“When it comes to Mathletics, because we’ve embedded it within our culture of numeracy in the school, it’s allowed our teachers to feel confident in how they use it and our students to feel comfortable and confident with the platform.”

Bremner’s six-year journey shows exactly how sustainable maths success can happen – one teacher, one classroom, one school at a time!

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

*Credit for article given to Kristina Gobetti*