Author: IMC

There’s a strange trend in mathematics education in England. Maths is the most popular subject at A-level since overtaking English in 2014. It’s taken by around 85,000 and 90,000 students a year.

But many universities – particularly lower-tariff institutions, which accept students with lower A-level grades – are recruiting far fewer students for maths degrees. There’s been a 50% drop in numbers of maths students at the lowest tariff universities over the five years between 2017 and 2021. As a result, some universities are struggling to keep their mathematics departments open.

The total number of students studying maths has remained largely static over the last decade. Prestigious Russell Group universities which require top A-level grades have increased their numbers of maths students.

This trend in degree-level mathematics education is worrying. It restricts the accessibility of maths degrees, especially to students from poorer backgrounds who are most likely to study at universities close to where they live. It perpetuates the myth that only those people who are unusually gifted at mathematics should study it – and that high-level maths skills are not necessary for everyone else.

Research carried out in 2019 by King’s College London and Ipsos found that half of the working age population had the numeracy skills expected of a child at primary school. Just as worrying was that despite this, 43% of those polled said “they would not like to improve their numeracy skills”. Nearly a quarter (23%) stated that “they couldn’t see how it would benefit them”.

Mathematics has been fundamental in recent technological developments such as quantum computing, information security and artificial intelligence. A pipeline of more mathematics graduates from more diverse backgrounds will be essential if the UK is to remain a science and technology powerhouse into the future.

But maths is also vital to a huge range of careers, including in business and government. In March 2024, campaign group Protect Pure Maths held a summit to bring together experts from industry, academia and government to discuss concerns about poor maths skills and the continuing importance of high-quality mathematics education.

Prior to the summit, the London Mathematical Society commissioned a survey of over 500 businesses to gauge their concerns about the potential lack of future graduates with strong mathematical skills.

They found that 72% of businesses agree they would benefit from more maths graduates entering the workforce. And 75% would worry if UK universities shrunk or closed their maths departments.

A 2023 report on MPs’ staff found that skills in Stem subjects (science, technology, engineering and mathematics) were particularly hard to find among those who worked in Westminster. As many as 90% of those who had taken an undergraduate degree had studied humanities or social sciences. While these subject backgrounds are valuable, the lack of specialised maths skills is stark.

Limited options

The mathematics department at Oxford Brookes has closed and other universities have seen recruitment reductions or other cuts. The resulting maths deserts will remove the opportunity for students to gain a high-quality mathematics education in their local area. Universities should do their best to keep these departments open.

This might be possible if the way that degrees are set up changes. For many degree courses in countries such as the US and Australia, students are able to take a broad selection of subjects, from science and maths subjects through to the humanities. Each are taught in their respective academic departments. This allows students to gain advanced knowledge and see how each field feeds into others.

This is scarcely possible in the UK, where students must choose a specialist and narrow degree programme at age 18.

Another possible solution would be to put core mathematics modules in degree disciplines that rely so heavily on it – such as engineering, economics, chemistry, physics, biology and computer science – and have them taught by specialist mathematicians. This would help keep mathematics departments open, while also ensuring that general mathematical literacy improves in the UK.

The relevance of mathematics and its vast range of applications would be abundantly clear, better equipping every student with the necessary mathematical skills the workforce needs.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Neil Saunders, The Conversation

What do the flow of cars on a highway and the movement of bacteria towards a food source have in common? In both cases, annoying traffic jams can form. Especially for cars, we might want to understand how to avoid them, but perhaps we’ve never thought of turning to statistical physics.

Alexandre Solon, a physicist from Sorbonne Université, and Eric Bertin, from the University of Grenoble, both working for the Centre national de la recherche scientifique (CNRS), have done just that. Their research, recently published in the Journal of Statistical Mechanics: Theory and Experiment, has developed a one-dimensional mathematical model that describes the movement of particles in situations similar to cars moving along a road or bacteria attracted to a nutrient source, which they then tested with computer simulations to observe what happened as parameters varied.

“The model is one-dimensional because the elements can only move in one direction, like on a one-lane one-way street,” explains Solon.

It’s an idealized situation, but not so different from what happens on many roads where you can find yourself stuck in rush hour traffic. The models from which this research is derived historically come from studying the behaviour of atoms and molecules: for example, those in a gas being heated or cooled. In the case of Bertin and Solon’s model, however, the behaviour of the individual elements is a bit more sophisticated than that of an atom.

“Among other things, a component of inertia has been inserted, which can be more or less pronounced, replicating for example the reactivity of a driver at the wheel. We can imagine a fresh and reactive driver, who brakes and accelerates at just the right moments, or another one at the end of the day, more tired and struggling to stay in sync with the rhythm of the flow of cars they are in,” Solon explains.

By conducting simulations with different values of certain parameters (the density of the elements, inertia, speed), Solon and Bertin were able to determine both situations in which traffic flowed smoothly, or on the contrary, became congested, as well as the type of jams that formed: large and centralized, or smaller and distributed along the route, akin to a “stop-and-go” pattern.

Borrowing language from statistical mechanics, Solon speaks of phase transitions: “Just as when the temperature changes water becomes ice, when the values of some parameters change, a smooth flow of cars becomes a congestion, a knot where no movement is possible.”

When the system reaches a critical density or when movement conditions favour accumulation rather than dispersion, the particles begin to form dense clusters, similar to traffic jams, while other areas may remain relatively empty. Traffic jams, therefore, can be seen as the dense phase in a system that has undergone a phase transition, characterized by low mobility and high localization of particles.

Solon and Bertin have thus identified conditions that can favour this congestion. Continuing with the metaphor of cars, contributing to the formation of traffic jams is the high density of vehicles, which reduces the space between one vehicle and another and increases the likelihood of interaction (and thus slowdown). Another condition is the frequent entries and exits from the flow: The addition of vehicles from the access ramp or attempts to change lanes in dense areas increase the risk of slowdowns, especially if vehicles try to merge without leaving sufficient space.

A third factor is the already-mentioned inertia in the behaviour of drivers, who—when they react with some delay to changes in the speed of the vehicles ahead of them—create a chain reaction of braking that can lead to the formation of a traffic jam. In contrast, the aggregation observed in bacterial colony happens in absence of any inertia, and bacteria can move in any direction contrary to cars that need to follow the direction of traffic.

As Bertin says, “It is thus interesting and surprising to find that both types of behaviours are connected and can be continuously transformed into one another.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to International School of Advanced Studies (SISSA)

A mathematical link between two key equations—one that deals with the very big and the other, the very small—has been developed by a young mathematician in China.

The mathematical discipline known as differential geometry is concerned with the geometry of smooth shapes and spaces. With roots going back to antiquity, the field flourished in the early 20th century, enabling Einstein to develop his general theory of relativity and other physicists to develop quantum field theory and the Standard Model of particle physics.

Gao Chen, a 29-year-old mathematician at the University of Science and Technology of China in Hefei, specializes in a branch known as complex differential geometry. Its complexity is not in dealing with complicated structures, but rather because it is based on complex numbers—a system of numbers that extends everyday numbers by including the square root of -1.

This area appeals to Chen because of its connections with other fields. “Complex differential geometry lies at the intersection of analysis, algebra, and mathematical physics,” he says. “Many tools can be used to study this area.”

Chen has now found a new link between two important equations in the field: the Kähler–Einstein equation, which describes how mass causes curvature in space–time in general relativity, and the Hermitian–Yang–Mills equation, which underpins the Standard Model of particle physics.

Chen was inspired by his Ph.D. supervisor Xiuxiong Chen of New York’s Stony Brook University, to take on the problem. “Finding solutions to the Hermitian–Yang–Mills and the Kähler–Einstein equations are considered the most important advances in complex differential geometry in previous decades,” says Gao Chen. “My results provide a connection between these two key results.”

“The Kähler –Einstein equation describes very large things, as large as the universe, whereas the Hermitian–Yang–Mills equation describes tiny things, as small as quantum phenomena,” explains Gao Chen. “I’ve built a bridge between these two equations.” Gao Chen notes that other bridges existed previously, but that he has found a new one.

“This bridge provides a new key, a new tool for theoretical research in this field,” Gao Chen adds. His paper describing this bridge was published in the journal Inventiones mathematicae in 2021.

In particular, the finding could find use in string theory—the leading contender of theories that researchers are developing in their quest to unite quantum physics and relativity. “The deformed Hermitian–Yang–Mills equation that I studied plays an important role in the study of string theory,” notes Gao Chen.

Gao Chen now has his eyes set on other important problems, including one of the seven Millennium Prize Problems. These are considered the most challenging in the field by mathematicians and carry a $1 million prize for a correct solution. “In the future, I hope to tackle a generalization of the Kähler–Einstein equation,” he says. “I also hope to work on other Millennium Prize problems, including the Hodge conjecture.”

Provided by University of Science and Technology of China

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of Science and Technology of China

Imagine a small village where every action someone takes, good or bad, is quietly followed by ever-attentive, nosy neighbours. An individual’s reputation is built through these actions and observations, which determines how others will treat them. They help a neighbour and are likely to receive help from others in return; they turn their back on a neighbour and find themselves isolated. But what happens when people make mistakes, when good deeds go unnoticed, or errors lead to unjust blame?

Here, the study of behaviour intersects with Bayesian and abductive reasoning, says Erol Akçay, a theoretical biologist at the University of Pennsylvania’s School of Arts & Sciences.

Bayesian reasoning refers to a method for assessing probability, in which individuals use prior knowledge paired with new evidence to update their beliefs or estimates about a certain condition, in this case the reputation of other villagers. While abductive reasoning involves a simple “what you see is what you get” approach to rationalizing and making a decision, Akçay says.

In two papers, one published in PLoS Computational Biology and the other in the Journal of Theoretical Biology, researchers from the Department of Biology explored how these reasoning strategies can be effectively modeled and applied to enhance biologists’ understanding of social dynamics.

Making the educated guess

The PLoS Computational Biology paper investigates how Bayesian statistical methods can be used to weigh the likelihood of errors and align the judgments of actors within a social network with a more nuanced understanding of reputation. “It’s something we may commonly do when we’re trying to offer up an explanation for some phenomena with no obvious, straightforward, or intuitive solution,” Akçay says.

Bryce Morsky, a co-author on both papers and now an assistant professor at Florida State University, began the work during his postdoctoral research in Akçay’s lab. He says that he initially believed that accounting for errors in judgment could substantially enhance the reward-and-punishment system that underpins cooperation and that he expected that a better understanding of these errors and incorporating them into the model would promote more effective cooperation.

“Essentially, the hypothesis was that reducing errors would lead to a more accurate assessment of reputations, which would in turn foster cooperation,” he says.

The team developed a mathematical model to simulate Bayesian reasoning. It involved a game-theoretical model where individuals interact within a framework of donation-based encounters. Other individuals in the simulation assess the reputations of actors based on their actions, influenced by several predefined social norms.

In the context of the village, this means judging each villager by their actions—whether helping another (good) or failing to do so (bad)—but also taking into account their historical reputation and the potential that you didn’t assess correctly.

“So, for example, if you observe someone behaving badly, but you thought they were good before, you keep an open mind that you perhaps didn’t see correctly. This allows for a nuanced calculation of reputation updates,” Morsky says. He and colleagues use this model to see how errors and reasoning would affect the villagers’ perception and social dynamics.

The five key social norms the study explores are: Scoring, Shunning, Simple Standing, Staying, and Stern Judging; each affects the reputation and subsequent behaviour of individuals differently, altering the evolutionary outcomes of cooperative strategies.

“In some scenarios, particularly under Scoring, Bayesian reasoning improved cooperation, Morsky says. “But under other norms, like Stern judging, it generally resulted in less cooperation due to stricter judgment criteria.”

Morsky explains that under Scoring a simple rule is applied: It is good to cooperate (give) and bad to defect (not give), regardless of the recipient’s reputation. Whereas under Stern judging not only are the actions of individuals considered, but their decisions are also critically evaluated based on the reputation of the recipient.

In the context of the nosy-neighbours scenario, if a villager decides to help another, this action is noted positively under Scoring, regardless of who receives the help or their standing in the village. Conversely, under Stern Judging if a villager chooses to help someone with a bad reputation it is noted negatively, the researchers say.

He adds that lack of cooperation was particularly evident in norms where Bayesian reasoning led to less tolerance for errors, which could exacerbate disagreements about reputations instead of resolving them. This, coupled with the knowledge that humans do not weigh all the relevant information prior to deciding who to work with, prompted Akçay and Morsky to investigate other modes of reasoning.

More than just a hunch

While working in Akçay’s lab, Morsky recruited Neel Pandula, then a sophomore in high school. “We met through the Penn Laboratory Experience in the Natural Sciences program,” Morsky says. “In light of the Bayesian reasoning model, Neel proposed abductive reasoning as another approach to modeling reasoning, and so we got to writing that paper for the Journal of Theoretical Biology, which he became first author of.”

Pandula, now a first-year student in the College of Arts and Sciences, explains that he and Morsky used Dempster-Shafer Theory—a probabilistic framework to infer best explanations—to form the basis of their approach.

“What’s key here is that Dempter-Shafer Theory allows for a bit of flexibility in handling uncertainty and allows for integrating new evidence into existing belief systems without fully committing to a single hypothesis unless the evidence is strong,” Pandula says.

For instance, the researchers explain, in a village, seeing a good person help another good person aligns with social norms and is readily accepted by observers. However, if a villager known as bad is seen helping a good person, it contradicts these norms, leading observers to question the reputations involved or the accuracy of their observation. Then they use the rules of abductive reasoning, specifically the Dempster-Shafer theory, considering error rates and typical behaviours to determine the most likely truth behind the unexpected action.

The team anticipated that abductive reasoning would handle errors in reputationassessments more effectively, especially in public settings in which individuals may be pressured one way or another resulting in discrepancies and errors. Under Scoring and the other norms, they found that abductive reasoning could better foster cooperation than Bayesian in public settings.

Akçay says that it came as a bit of a surprise to see that in navigating social networks, such a simple “cognitively ‘cheap, lazy’ reasoning mechanism proves this effective at dealing with the challenges associated with indirect reciprocity.”

Morsky notes that in both models the researchers chose not to factor in any cost of a cognitive burden. “You’d hope that performing a demanding task like remembering which individuals did what and using that to inform you on what they’re likely to do next would yield some positive, prosocial outcome. Yet even if you make this effort costless, under Bayesian reasoning, it generally undermines cooperation.”

As a follow up, the researchers are interested in exploring how low-cost reasoning methods, like abductive reasoning, can be evolutionarily favoured in larger, more complex social circles. And they are interested in applying these reasoning methods to other social systems.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Nathi Magubane, University of Pennsylvania

For all the students wondering why they would ever need to use the Pythagorean theorem, Katie Steckles is delighted to report on a real-world encounter.

Recently, I was building a flat-pack wardrobe when I noticed something odd in the instructions. Before you assembled the wardrobe, they said, you needed to measure the height of the ceiling in the room you were going to put it in. If it was less than 244 centimetres high, there was a different set of directions to follow.

These separate instructions asked you to build the wardrobe in a vertical orientation, holding the side panels upright while you attached them to the base. The first set of directions gave you a much easier job, building the wardrobe flat on the floor before lifting it up into place. I was intrigued by the value of 244 cm: this wasn’t the same as the height of the wardrobe, or any other dimension on the package, and I briefly wondered where that number had come from. Then I realised: Pythagoras.

The wardrobe was 236 cm high and 60 cm deep. Looking at it side-on, the length of the diagonal line from corner to corner can be calculated using Pythagoras’s theorem. The vertical and horizontal sides meet at a right angle, meaning if we square the length of each then add them together, we get the well-known “square of the hypotenuse”. Taking the square root of this number gives the length of the diagonal.

In this case, we get a diagonal length a shade under 244 cm. If you wanted to build the wardrobe flat and then stand it up, you would need that full diagonal length to fit between the floor and the ceiling to make sure it wouldn’t crash into the ceiling as it swung past – so 244 cm is the safe ceiling height. It is a victory for maths in the real world, and vindication for maths teachers everywhere being asked, “When am I going to use this?”

This isn’t the only way we can connect Pythagoras to daily tasks. If you have ever needed to construct something that is a right angle – like a corner in joinery, or when laying out cones to delineate the boundaries of a sports pitch – you can use the Pythagorean theorem in reverse. This takes advantage of the fact that a right-angled triangle with sides of length 3 and 4 has a hypotenuse of 5 – a so-called 3-4-5 triangle.

If you measure 3 units along one side from the corner, and 4 along the other, and join them with a diagonal, the diagonal’s length will be precisely 5 units, if the corner is an exact right angle. Ancient cultures used loops of string with knots spaced 3, 4 and 5 units apart – when held out in a triangle shape, with a knot at each vertex, they would have a right angle at one corner. This technique is still used as a spot check by builders today.

Engineers, artists and scientists might use geometrical thinking all the time, but my satisfaction in building a wardrobe, and finding the maths checked out perfectly, is hard to beat.

Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also puzzle adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks

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

*Credit for article given to Peter Rowlett*

My intention with this article is to give an intuitive and non-technical introduction to the field of evolutionary algorithms, particularly with regards to optimisation.

If I get you interested, I think you’re ready to go down the rabbit hole and simulate evolution on your own computer. If not … well, I’m sure we can still be friends.

Survival of the fittest

According to Charles Darwin, the great evolutionary biologist, the human race owes its existence to the phenomenon of survival of the fittest. And being the fittest doesn’t necessarily mean the biggest physical presence.

Once in high school, my lunchbox was targeted by swooping eagles, and I was reduced to a hapless onlooker. The eagle, though smaller in form, was fitter than me because it could take my lunch and fly away – it knew I couldn’t chase it.

As harsh as it sounds, look around you and you will see many examples of the rule of the jungle – the fitter survive while the rest gradually vanish.

The research area, now broadly referred to as Evolutionary Algorithms, simulates this behaviour on a computer to find the fittest solutions to a number of different classes of problems in science, engineering and economics.

The area in which this area is perhaps most widely used is known as “optimisation”.

Optimisation is everywhere

Your high school maths teacher probably told you the shortest way to go from point A to point B was along the straight-line joining A and B. Your mum told you that you should always get the right amount of sleep.

And, if you have lived on your own for any length of time, you’ll be familiar with the ever-increasing cost of living versus the constant income – you always strive to minimise the expenditures, while ensuring you are not malnourished.

Whenever you undertake an activity that seeks to minimise or maximise a well-defined quantity such as distance or the vague notion of the right amount of sleep, you are optimising.

Look around you right now and you’ll see optimisation in play – your Coke can is shaped like that for a reason, a water droplet is spherical for a reason, you wash all your dishes together in the dishwasher for a reason.

Each of these strives to save on something: volume of material of the Coke can, and energy and water, respectively, in the above cases.

So, we can safely say optimisation is the act of minimising or maximising a quantity. But that definition misses an important detail: there is always a notion of subject to or satisfying some conditions.

You must get the right amount of sleep, but you also must do your studies and go for your music lessons. Such conditions, which you also have to adhere to, are known as “constraints”. Optimisation with constraints is then collectively termed “constrained optimisation”.

After constraints comes the notion of “multi-objective optimisation”. You’ll usually have more than one thing to worry about (you must keep your supervisor happy with your work and keep yourself happy and also ensure that you are working on your other projects). In many cases these multiple objectives can be in conflict.

Evolutionary algorithms and optimisation

Imagine your local walking group has arranged a weekend trip for its members and one of the activities is a hill climbing exercise. The problem assigned to your group leader is to identify who among you will reach the hill in the shortest time.

There are two approaches he or she could take to complete this task: ask only one of you to climb up the hill at a time and measure the time needed or ask all of you to run all at once and see who reaches first.

That second method is known as the “population approach” of solving optimisation problems – and that’s how evolutionary algorithms work. The “population” of solutions are evolved over a number of iterations, with only the fittest solutions making it to the next.

This is analogous to the champion girl from your school making to the next round which was contested among champions from other schools in your state, then your country, and finally winning among all the countries.

Or, in our above scenario, finding who in the walking group reaches the hill top fastest, who would then be denoted as the fittest.

In engineering, optimisation needs are faced at almost every step, so it’s not surprising evolutionary algorithms have been successful in that domain.

Design optimisation of scramjets

At the Multi-disciplinary Design Optimisation Group at the University of New South Wales, my colleagues and I are involved in the design optimisation of scramjets, as part of the SCRAMSPACE program. In this, we’re working with colleagues from the University of Queensland.

Our evolutionary algorithms-based optimisation procedures have been successfully used to obtain the optimal configuration of various components of a scramjet.

There are, at the risk of sounding over-zealous, no limits to the application of evolutionary algorithms.

Has this whetted your appetite? Have you learnt something new today?

If so, I’m glad. May the force be with you!

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Amit Saha

A mathematical model of a particle that remembers its past so that it never travels the same path twice produces stunningly complex patterns.

A beautiful and surprisingly complex pattern produced by ‘mathematical billiards’

Albers et al. PRL 2024

In a mathematical version of billiards, particles that avoid retracing their paths get trapped in intricate and hard-to-predict patterns – which might eventually help us understand the complex movement patterns of living organisms.

When searching for food, animals including ants and slime moulds leave chemical trails in their environment, which helps them avoid accidentally retracing their steps. This behaviour is not uncommon in biology, but when Maziyar Jalaal at the University of Amsterdam in the Netherlands and his colleagues modelled it as a simple mathematical problem, they uncovered an unexpected amount of complexity and chaos.

They used the framework of mathematical billiards, where an infinitely small particle bounces between the edges of a polygonal “table” without friction. Additionally, they gave the particle “spatial memory” – if it reached a point where it had already been before, it would reflect off it as if there was a wall there.

The researchers derived equations describing the motion of the particle and then used them to simulate this motion on a computer. They ran over 200 million simulations to see the path the particle would take inside different polygons – like a triangle and a hexagon – over time. Jalaal says that though the model was simple, idealised and deterministic, what they found was extremely intricate.

Within each polygon, the team identified regions where the particle was likely to become trapped after bouncing around for a long time due to its “remembering” its past trajectories, but zooming in on those regions revealed yet more patterns of motion.

“So, the patterns that you see if you keep zooming in, there is no end to them. And they don’t repeat, they’re not like fractals,” says Jalaal.

Katherine Newhall at the University of North Carolina at Chapel Hill says the study is an “interesting mental exercise” but would have to include more detail to accurately represent organisms and objects that have spatial memory in the real world. For instance, she says that a realistic particle would eventually travel in an imperfectly straight line or experience friction, which could radically change or even eradicate the patterns that the researchers found.

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

*Credit for article given to Karmela Padavic-Callaghan*

It is very common in Australian schools to “stream” students for subjects such as English, science and maths. This means students are grouped into different classes based on their previous academic attainment, or in some cases, just a perception of their level of ability.

Students can also be streamed as early as primary school. Yet there are no national or state policies on this. This means school principals are free to decide what will happen in their schools.

Why are students streamed in Australians schools? And is this a good idea? Our research on streaming maths classes shows we need to think much more carefully about this very common practice.

Why do schools stream?

At a maths teacher conference in Sydney in late 2023, WEdid a live survey about school approaches to streaming.

This survey was done via interactive software while WEwas giving a presentation. There were 338 responses from head teachers in maths in either high schools or schools that go all the way from Kindergarten to Year 12. Most of the teachers were from public schools.

In a sign of how widespread streaming is, 95% of head teachers said they streamed maths classes in their schools.

Respondents said one of the main reasons is to help high-achieving students and make sure they are appropriately challenged. As one teacher said:

[We stream] to push the better students forward.

But almost half the respondents said they believed all students were benefiting from this system.

We also heard how streaming is seen as a way to cope with the teacher shortage and specific lack of qualified maths teachers. These qualifications include skills in both maths and maths teaching. More than half (65%) of respondents said streaming can “aid differentiation [and] support targeted student learning interventions”. In other words, streaming is a way to cope with different levels of ability in the classrooms and make the job of teaching a class more straightforward. One respondent said:

[we stream because] it’s easier to differentiate with a class of students that have similar perceived ability.

Teachers said they streamed classes to push the best students ‘forward’.

The ‘glass ceiling effect’

But while many schools and teachers assume streaming is good for students, this is not what the research says.

Our 2020 study, on streaming was based on interviews with 85 students and 22 teachers from 11 government schools.

This found streaming creates a “glass ceiling effect” – in other words, students cannot progress out of the stream they are initially assigned to without significant remedial work to catch them up.

As one teacher told us, students in lower-ability classes were then placed at a “massive disadvantage”. This is because they can miss out on segments of the curriculum because the class may progress more slowly or is deliberately not taught certain sections deemed too complex.

Often students in our study were unaware of this missed content until Year 10 and thinking about their options for the final years of school and beyond. They may not be able to do higher-level maths in Year 11 and 12 because they are too far behind. As one teacher explained:

they didn’t have enough of that advanced background for them to be able to study it. It was too difficult for them to begin with.

This comes as fewer students are completing advanced (calculus-based) maths.

If students do not study senior maths, they do not have the background for studying for engineering and other STEM careers, which we know are in very high demand.

On top of this, students may also be stigmatised as “low ability” in maths. While classes are not labelled as such, students are well aware of who is in the top classes and who is not. This can have an impact on students’ confidence about maths.

What does other research say?

International research has also found streaming students is inequitable.

As a 2018 UK study showed, students from disadvantaged backgrounds are more likely to be put in lower streamed classes.

A 2009 review of research studies found that not streaming students was better for low-ability student achievement and had no effect on average and high-ability student achievement.

Streaming is also seen as a way to cope with teachers shortages, and teachers teaching out of their field of expertise.

What should we do instead?

Amid concerns about Australian students’ maths performance in national and international tests, schools need to stop assuming streaming is the best approach for students.

The research indicates it would be better if students were taught in mixed-ability classes – as long as teachers are supported and class sizes are small enough.

This means all students have the opportunity to be taught all of the curriculum, giving them the option of doing senior maths if they want to in Year 11 and Year 12.

It also means students are not stigmatised as “poor at maths” from a young age.

But to do so, teachers and schools must be given more teaching resources and support. And some of this support needs to begin in primary school, rather than waiting until high school to try and catch students up.

Students also need adequate career advice, so they are aware of how maths could help future careers and what they need to do to get there.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Karolina Grabowska/Pexels, CC BY

Here’s a game: Ask a friend to give you any number and you’ll return one that’s bigger. Just add “1” to whatever number they come up with and you’re sure to win.

The reason is that numbers go on forever. There is no highest number. But why? As a professor of mathematics, WEcan help you find an answer.

First, you need to understand what numbers are and where they come from. You learned about numbers because they enabled you to count. Early humans had similar needs – whether to count animals killed in a hunt or keep track of how many days had passed. That’s why they invented numbers.

But back then, numbers were quite limited and had a very simple form. Often, the “numbers” were just notches on a bone, going up to a couple hundred at most.

When numbers got bigger

As time went on, people’s needs grew. Herds of livestock had to be counted, goods and services traded, and measurements made for buildings and navigation. This led to the invention of larger numbers and better ways of representing them.

About 5,000 years ago, the Egyptians began using symbols for various numbers, with a final symbol for one million. Since they didn’t usually encounter bigger quantities, they also used this same final symbol to depict “many.”

The Greeks, starting with Pythagoras, were the first to study numbers for their own sake, rather than viewing them as just counting tools. As someone who’s written a book on the importance of numbers, WEcan’t emphasize enough how crucial this step was for humanity.

By 500 BCE, Pythagoras and his disciples had not only realized that the counting numbers – 1, 2, 3 and so on – were endless, but also that they could be used to explain cool stuff like the sounds made when you pluck a taut string.

Zero is a critical number

But there was a problem. Although the Greeks could mentally think of very large numbers, they had difficulty writing them down. This was because they did not know about the number 0.

Think of how important zero is in expressing big numbers. You can start with 1, then add more and more zeroes at the end to quickly get numbers like a million – 1,000,000, or 1 followed by six zeros – or a billion, with nine zeros, or a trillion, 12 zeros.

It was only around 1200 CE that zero, invented centuries earlier in India, came to Europe. This led to the way we write numbers today.

This brief history makes clear that numbers were developed over thousands of years. And though the Egyptians didn’t have much use for a million, we certainly do. Economists will tell you that government expenditures are commonly measured in millions of dollars.

Also, science has taken us to a point where we need even larger numbers. For instance, there are about 100 billion stars in our galaxy – or 100,000,000,000 – and the number of atoms in our universe may be as high as 1 followed by 82 zeros.

Don’t worry if you find it hard to picture such big numbers. It’s fine to just think of them as “many,” much like the Egyptians treated numbers over a million. These examples point to one reason why numbers must continue endlessly. If we had a maximum, some new use or discovery would surely make us exceed it.

Exceptions to the rule

But under certain circumstances, sometimes numbers do have a maximum because people design them that way for a practical purpose.

A good example is a clock – or clock arithmetic, where we use only the numbers 1 through 12. There is no 13 o’clock, because after 12 o’clock we just go back to 1 o’clock again. If you played the “bigger number” game with a friend in clock arithmetic, you’d lose if they chose the number 12.

Since numbers are a human invention, how do we construct them so they continue without end? Mathematicians started looking at this question starting in the early 1900s. What they came up with was based on two assumptions: that 0 is the starting number, and when you add 1 to any number you always get a new number.

These assumptions immediately give us the list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, and so on, a progression that continues without end.

You might wonder why these two rules are assumptions. The reason for the first one is that we don’t really know how to define the number 0. For example: Is “0” the same as “nothing,” and if so, what exactly is meant by “nothing”?

The second might seem even more strange. After all, we can easily show that adding 1 to 2 gives us the new number 3, just like adding 1 to 2002 gives us the new number 2003.

But notice that we’re saying this has to hold for any number. We can’t very well verify this for every single case, since there are going to be an endless number of cases. As humans who can perform only a limited number of steps, we have to be careful anytime we make claims about an endless process. And mathematicians, in particular, refuse to take anything for granted.

Here, then, is the answer to why numbers don’t end: It’s because of the way in which we define them.

Now, the negative numbers

How do the negative numbers -1, -2, -3 and more fit into all this? Historically, people were very suspicious about such numbers, since it’s hard to picture a “minus one” apple or orange. As late as 1796, math textbooks warned against using negatives.

The negatives were created to address a calculation issue. The positive numbers are fine when you’re adding them together. But when you get to subtraction, they can’t handle differences like 1 minus 2, or 2 minus 4. If you want to be able to subtract numbers at will, you need negative numbers too.

A simple way to create negatives is to imagine all the numbers – 0, 1, 2, 3 and the rest – drawn equally spaced on a straight line. Now imagine a mirror placed at 0. Then define -1 to be the reflection of +1 on the line, -2 to be the reflection of +2, and so on. You’ll end up with all the negative numbers this way.

As a bonus, you’ll also know that since there are just as many negatives as there are positives, the negative numbers must also go on without end!

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Conversation

Avi Wigderson has won the 2023 Turing award for his work on understanding how randomness can shape and improve computer algorithms.

The mathematician Avi Wigderson has won the 2023 Turing award, often referred to as the Nobel prize for computing, for his work on understanding how randomness can shape and improve computer algorithms.

Wigderson, who also won the prestigious Abel prize in 2021 for his mathematical contributions to computer science, was taken aback by the award. “The [Turing] committee fooled me into believing that we were going to have some conversation about collaborating,” he says. “When I zoomed in, the whole committee was there and they told me. I was excited, surprised and happy.”

Computers work in a predictable way at the hardware level, but this can make it difficult for them to model real-world problems, which often have elements of randomness and unpredictability. Wigderson, at the Institute for Advanced Study in Princeton, New Jersey, has shown over a decades-long career that computers can also harness randomness in the algorithms that they run.

In the 1980s, Wigderson and his colleagues discovered that by inserting randomness into some algorithms, they could make them easier and faster to solve, but it was unclear how general this technique was. “We were wondering whether this randomness is essential, or maybe you can always get rid of it somehow if you’re clever enough,” he says.

One of Wigderson’s most important discoveries was making clear the relationship between types of problems, in terms of their difficulty to solve, and randomness. He also showed that certain algorithms that contained randomness and were hard to run could be made deterministic, or non-random, and easier to run.

These findings helped computer scientists better understand one of the most famous unproven conjectures in computer science, called “P ≠ NP”, which proposes that easy and hard problems for a computer to solve are fundamentally different. Using randomness, Wigderson discovered special cases where the two classes of problem were the same.

Wigderson first started exploring the relationship between randomness and computers in the 1980s, before the internet existed, and was attracted to the ideas he worked on by intellectual curiosity, rather than how they might be used. “I’m a very impractical person,” he says. “I’m not really motivated by applications.”

However, his ideas have become important for a wide swath of modern computing applications, from cryptography to cloud computing. “Avi’s impact on the theory of computation in the last 40 years is second to none,” says Oded Goldreich at the Weizmann Institute of Science in Israel. “The diversity of the areas to which he has contributed is stunning.”

One of the unexpected ways in which Wigderson’s ideas are now widely used was his work, with Goldreich and others, on zero-knowledge proofs, which detail ways of verifying information without revealing the information itself. These methods are fundamental for cryptocurrencies and blockchains today as a way to establish trust between different users.

Although great strides in the theory of computation have been made over Wigderson’s career, he says that the field is still full of interesting and unsolved problems. “You can’t imagine how happy I am that I am where I am, in the field that I’m in,” he says. “It’s bursting with intellectual questions.”

Wigderson will receive a $1 million prize as part of the Turing award.

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

*Credit for article given to Alex Wilkins*