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Number theory, the study of the properties of positive integers, is perhaps the purest form of mathematics. At first sight, it may seem far too abstract to apply to the natural world. In fact, the influential American number theorist Leonard Dickson wrote, “Thank God that number theory is unsullied by any application.”

And yet, again and again, number theory finds unexpected applications in science and engineering, from leaf angles that (almost) universally follow the Fibonacci sequence, to modern encryption techniques based on factoring prime numbers. Now, researchers have demonstrated an unexpected link between number theory and evolutionary genetics. Their work is published in the Journal of The Royal Society Interface.

Specifically, the team of researchers (from Oxford, Harvard, Cambridge, GUST, MIT, Imperial, and the Alan Turing Institute) have discovered a deep connection between the sums-of-digits function from number theory and a key quantity in genetics, the phenotype mutational robustness. This quality is defined as the average probability that a point mutation does not change a phenotype (a characteristic of an organism).

The discovery may have important implications for evolutionary genetics. Many genetic mutations are neutral, meaning that they can slowly accumulate over time without affecting the viability of the phenotype. These neutral mutations cause genome sequences to change at a steady rate over time. Because this rate is known, scientists can compare the percentage difference in the sequence between two organisms and infer when their latest common ancestor lived.

But the existence of these neutral mutations posed an important question: what fraction of mutations to a sequence are neutral? This property, called the phenotype mutational robustness, defines the average amount of mutations that can occur across all sequences without affecting the phenotype.

Professor Ard Louis from the University of Oxford, who led the study, said, “We have known for some time that many biological systems exhibit remarkably high phenotype robustness, without which evolution would not be possible. But we didn’t know what the absolute maximal robustness possible would be, or if there even was a maximum.”

It is precisely this question that the team has answered. They proved that the maximum robustness is proportional to the logarithm of the fraction of all possible sequences that map to a phenotype, with a correction which is given by the sums of digits function s_k(n), defined as the sum of the digits of a natural number n in base k. For example, for n = 123 in base 10, the digit sum would be s₁₀(123) = 1 + 2 + 3 = 6.

Another surprise was that the maximum robustness also turns out to be related to the famous Tagaki function, a bizarre function that is continuous everywhere, but differentiable nowhere. This fractal function is also called the blancmange curve, because it looks like the French dessert.

First author Dr. Vaibhav Mohanty (Harvard Medical School) added, “What is most surprising is that we found clear evidence in the mapping from sequences to RNA secondary structures that nature in some cases achieves the exact maximum robustness bound. It’s as if biology knows about the fractal sums-of-digits function.”

Professor Ard Louis added, “The beauty of number theory lies not only in the abstract relationships it uncovers between integers, but also in the deep mathematical structures it illuminates in our natural world. We believe that many intriguing new links between number theory and genetics will be found in the future.”

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Credit of the article given to University of Oxford

Mathematicians at The University of Manchester have answered the question: How many lottery tickets do you need to buy to guarantee wining something on the U.K. National Lottery?

Focusing on the National Lottery’s flagship game “Lotto,” which draws six random numbersfrom 1 to 59, Dr. David Stewart and Dr. David Cushing found that 27 is the lowest possible number of tickets needed to guarantee a win—although, importantly, with no guarantee of a profit.

They describe the solution using a mathematical system called finite geometry, which centers around a triangle-like structure called a Fano plane. Each point of the structure is plotted with pairs of numbers and connected with lines—each line generates a set of six numbers, which equates to one ticket.

It takes three Fano planes and two triangles to cover all 59 numbers and generate 27 sets of tickets.

Choosing tickets in this way guarantees that no matter which of the 45,057,474 possible draws occurs, at least one of the tickets will have at least two numbers in common. From any draw of six, two numbers must appear on one of the five geometric structures, which ensures they appear on at least one ticket.

But Dr. Stewart and Dr. Cushing say that the hard work is actually showing that achieving the same outcome with 26 tickets is not possible.

Dr. David Stewart, a Reader in Pure Mathematics at The University of Manchester, said, “Fundamentally there is a tension which comes from the fact that there are only 156 entries on 26 tickets. This means a lot of numbers can’t appear a lot of times. Eventually you see that you’ll be able to find six numbers that don’t appear on any ticket together. In graph theory terms, we end up proving the existence of an independent set of size six.”

Although guaranteed a win, the researchers say that the chances of making a profit are very unlikely and shouldn’t be used as a reason to gamble.

The 27 lottery tickets would set you back £54. And Peter Rowlett, a mathematician from The Aperiodical website, has shown that in almost 99% of cases, you wouldn’t make that money back.

When putting the theory to the test in the lottery draw on 1 July 2023; the researchers matched just two balls on three of the tickets, the reward being three lucky dip tries on a subsequent lottery, each of which came to nothing.

The researchers say that the finding is interesting from a computational point of view. They use a fifty-year-old programming language called Prolog, which they say makes it one of the oldest examples of real artificial intelligence.

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Credit of the article given to University of Manchester

In addition to describing biological interactions, evolutionary theory has also become a valuable tool to make sense of the dynamics of social norms. Social norms determine which behaviours should be regarded as positive, and how community members should act towards each other.

In a recent publication, published in PLOS Computational Biology, researchers from RIKEN, Japan, and the Max-Planck-Institute for Evolutionary Biology (MPI) describe a new class of social norms for cooperative interactions.

Social norms play an important role in people’s everyday lives. They govern how people should behave and how reputations are formed based on past behaviours.

In the last 25 years, there has been an effort to describe these dynamics of reputations more formally, using mathematical models borrowed from evolutionary game theory. These models describe how social norms evolve over time—how successful norms can spread in a society and how detrimental norms fade.

Most of these models assume that an individual’s reputation should only depend on what this person did in the past. However, everyday experience and experimental evidence suggest that additional external factors may as well influence a person’s reputation. People do not only earn a reputation for how they act, but also based on who they interact with, and how they are affected by those interactions.

For example, with a recent series of experiments, researchers from Harvard University have shown that victims of harmful actions are often regarded as more virtuous than they actually are. To explore such phenomena more formally, researchers at the MPI for Evolutionary Biology in Plön and RIKEN, Japan, have developed a new mathematical framework to describe social norms.

According to the new framework, when a person’s action affects the well-being of another community member, the reputations of both individuals may be updated. Using this general framework, the researchers explore which properties such norms ought have to support cooperative interactions. Surprisingly, some of these social norms indeed have the property observed in the earlier experiments: when one individual defects against another, the victim’s reputation should improve.

Moreover, the researchers also observe a fundamental trade-off. Norms that are particularly good in sustaining cooperation tend to be less robust with respect to noise (such as when reputations are shaped by third-party gossip).

Overall, this work is part of a bigger effort to understand key properties of social norms in a rigorous manner. These studies shed light on which ecological and social environments facilitate cooperation, and on the effects of social norms more generally.

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Credit of the article given to Max Planck Society

In the face of existential dilemmas that are shared by all of humanity, including the consequences of inequality or climate change, it is crucial to understand the conditions leading to cooperation. A new game theory model developed at the Institute of Science and Technology Austria (ISTA) based on 192 stochastic games and on some elegant algebra finds that both cases—available information and the lack thereof—can lead to cooperative outcomes.

The journal Nature Communications has published a new open-access paper on the role information plays in reaching a cooperative outcome. Working at ISTA with the Chatterjee group, research scholar Maria Kleshnina developed a framework of stochastic games, a tool to describe how people make decisions in changing environments. The new model finds that availability of information is intricately linked to cooperative outcomes.

“In this paper, we present a new model of games where a group’s environment changes, based on actions of group members who do not necessarily have all relevant information about their environment. We find that there are rich interactions between the availability of information and cooperative behaviour.

“Counter-intuitively there are instances where there is a benefit of ignorance, and we characterize when information helps in cooperation,” says Professor Krishnendu Chatterjee who leads the “Computer-Aided Verification, Game Theory” group at the Institute of Science and Technology Austria, where this work was done.

Ignorance can be beneficial for cooperation too

In 2016, Štěpán Šimsa, one of the authors of the new paper was working with the Chatterjee group, when he ran some preliminary simulations to find that ignorance about the state of the game may benefit cooperation. This is counter-intuitive since the availability of information is generally thought to be universally beneficial. Christian Hilbe, then a postdoc with the Chatterjee group, along with Kleshnina, thought this to be a worthy research direction. The group then took on the task of investigating how information or the lack thereof affects the evolution of cooperation.

“We quantified in which games it is useful to have precise information about the environmental state. And we find that in most cases, around 80 to 90% it is indeed really good if players are aware of the environment’s state and which game they are playing right now. Yet, we also find some very interesting exceptional cases, where it’s actually optimal for cooperation if everyone is ignorant about the game they are playing,” says co-author Christian Hilbe, who now leads the research group Dynamics of Social Behaviour at the Max Planck Institute for Evolutionary Biology in Germany.

The researchers’ framework represents an idealized model for cooperation in changing environments. Therefore, the results cannot be directly transferred to real-world applications like solving climate change. For this, they say, a more extensive model would be required. Although, from the basic science model that she has built, Kleshnina is able to offer a qualitative direction.

“In a changing system, a benefit of ignorance is more likely to occur in systems that naturally punish non-cooperation. This could happen, for example, if the group’s environment quickly deteriorates if players no longer cooperate mutually. In such a system, individuals have strong incentives to cooperate today, if they want to avoid playing an unprofitable game tomorrow,” she says.

To illustrate the benefit of ignorance, Kleshnina says, “For example, we found that in informed populations, individuals can use their knowledge to employ more nuanced strategies. These nuanced strategies, however, can be less effective in sustaining cooperation. In such a case, there is indeed a small benefit of ignorance towards cooperation.”

A brilliant method

Game theory is, in its essence, a study of mathematical models set up within the framework of games or exchange of logical decisions being played between rational players. Its applications in understanding social and biological evolution have been welcomed by interdisciplinary researchers given its game-changing approach.

Within the context of evolutionary game theory, many models investigate cooperation but most assume that the same game is played over and over again, and also that the players are always perfectly aware of the game that they are playing and its state at any given moment. The new study weakens these general assumptions, first by allowing the simulated players to play different games over time. And second, by accounting for the impact of information.

“The beauty of this approach is that one can combine some elegant linear algebra with extensive computer simulations,” says Kleshnina.

The new framework opens up many new research directions. For instance, what is the role of asymmetric information? One player might know the exact game being played, but another may not. This is not something that the model currently covers. “In that sense, our paper has quite [a few] future applications within theory itself,” Hilbe adds.

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Credit of the article given to Institute of Science and Technology Austria

Everyone knows that 2 + 2 = 4, but why do we have arithmetic in the first place, and why is it true? Researchers at the University of Canterbury have recently answered these questions by “reverse engineering” arithmetic from a psychological perspective. To do this, they considered all possible ways that quantities could be combined, and proved (for the first time in mathematical terms) that addition and multiplication are the simplest.

Their proof is based on four assumptions—principles of perceptual organization—that shape how we and other animals experience the world. These assumptions eliminate all possibilities except arithmetic, like how a sculptor’s work reveals a statue hidden in a block of stone.

Monotonicity is the idea of “things changing in the same direction,” and helps us keep track of our place in the world, so that when we approach an object it looms larger but smaller when we move away. Convexity is grounded in intuitions of betweenness. For example, the four corners of a football pitch define the playing field even without boundary lines connecting them. Continuity describes the smoothness with which objects seem to move in space and time. Isomorphism is the idea of sameness or analogy. It’s what allows us to recognize that a cat is more similar to a dog than it is to a rock.

Taken together, these four principles structure our perception of the world so that our everyday experience is ordered and cognitively manageable.

The implications, explained in a paper in Psychological Review, are far-reaching because arithmetic is fundamental for mathematics and science. They suggest arithmetic is biologically-based and a natural consequence of our perception. Mathematics is thus a realization in symbols of the fundamental nature of the mind, and as such both invented and discovered. The seemingly magical success of mathematics in the physical sciences hints that our mind and the world are not separate, but part of a common unity.

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Credit of the article given to University of Canterbury

You thought the maze looked fun, but now you can’t find your way out. Luckily, mathematics is here to help you escape, says Katie Steckles.

Getting lost in a maze is no fun, and on that rare occasion when you find yourself stuck in one without a map or a bird’s-eye view, it can be difficult to choose which way to go. Mathematics gives us a few tools we can use – in particular, topology, which concerns shapes and how they connect.

The most devious mazes are designed to be as confusing as possible, with dead ends and identical-looking junctions. But there is a stunningly simple rule that will always get you out of a maze, no matter how complicated: always turn right.

Any standard maze can be solved with this method (or its equivalent, the “always-turn-left” method). To do it, place one hand on the wall of the maze as you go in and keep it there. Each time you come to a junction, keep following the wall – if there is an opening on the side you are touching, take it; otherwise go straight. If you hit a dead end, turn around and carry on.

The reason this works is because the walls of any solvable maze will always have at least two distinct connected pieces: one to the left of the optimal solution path (shown in red), and one to the right. The section of wall next to the entrance is part of the same connected chunk of maze as the wall by the exit, and if you keep your hand on it, you will eventually walk along the whole length of the edge of this object – no matter how many twists and turns this involves – and reach the part at the exit.

While it is guaranteed to work, this certainly won’t be the most efficient path – you might find you traverse as much as half of the maze in the process, or even more depending on the layout. But at least it is easy to remember the rule.

Some mazes have more than two pieces. In these, disconnected sections of wall (shown in yellow) inside the maze create loops. In this case, if you start following the wall somewhere in the middle of the maze, there is a chance it could be part of an isolated section, which would leave you walking around a loop forever. But if you start from a wall that is connected to the outside, wall-following will still get you out.

It is reassuring to know that even if you are lost in a maze, you can always get out by following some variation on this rule: if you notice you have reached part of the maze you have been to before, you can detect loops, and switch to the opposite wall.

This is especially useful for mazes where the goal is to get to the centre: if the centre isn’t connected to the outside, wall-following won’t work, and you will need to switch walls to get onto the centre component. But as long as there are a finite number of pieces to the maze, and you keep trying different ones, you will eventually find a piece that is connected to your goal. You might, however, miss the bus home.

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*Credit for article given to Katie Steckles*

Board games based on numbers, like Monopoly, Othello and Chutes and Ladders, make young children better at math, according to a comprehensive review of research published on the topic over the last 23 years.

Board games are already known to enhance learning and development including reading and literacy.

Now this new study, published in the journal Early Years, finds, for 3- to 9-year-olds, the format of number-based board games helps to improve counting, addition, and the ability to recognize if a number is higher or lower than another.

The researchers say children benefit from programs—or interventions—where they play board games a few times a week supervised by a teacher or another trained adult.

“Board games enhance mathematical abilities for young children,” says lead author Dr. Jaime Balladares, from Pontificia Universidad Católica de Chile, in Santiago, Chile.

“Using board games can be considered a strategy with potential effects on basic and complex math skills.

“Board games can easily be adapted to include learning objectives related to mathematical skills or other domains.”

Games where players take turns to move pieces around a board differ from those involving specific skills or gambling.

Board game rules are fixed which limits a player’s activities, and the moves on the board usually determine the overall playing situation.

However, preschools rarely use board games. This study aimed to compile the available evidence of their effects on children.

The researchers set out to investigate the scale of the effects of physical board games in promoting learning in young children.

They based their findings on a review of 19 studies published from 2000 onwards involving children aged from 3 to 9 years. All except one study focused on the relationship between board games and mathematical skills.

All children participating in the studies received special board game sessions which took place on average twice a week for 20 minutes over one-and-a-half months. Teachers, therapists, or parents were among the adults who led these sessions.

In some of the 19 studies, children were grouped into either the number board game or to a board game that did not focus on numeracy skills. In others, all children participated in number board games but were allocated different types (e.g., Dominoes).

All children were assessed on their math performance before and after the intervention sessions which were designed to encourage skills such as counting out loud.

The authors rated success according to four categories including basic numeric competency such as the ability to name numbers, and basic number comprehension (e.g., ‘nine is greater than three’).

The other categories were deepened number comprehension—where a child can accurately add and subtract—and interest in mathematics.

In some cases, parents attended a training session to learn arithmetic that they could then use in the games.

Results showed that math skills improved significantly after the sessions among children for more than half (52%) of the tasks analysed.

In nearly a third (32%) of cases, children in the intervention groups gained better results than those who did not take part in the board game intervention.

The results also show that from analysed studies to date, board games on the language or literacy areas, while implemented, did not include scientific evaluation (i.e. comparing control with intervention groups, or pre and post-intervention) to evaluate their impact on children.

Designing and implementing board games along with scientific procedures to evaluate their efficacy, therefore, are “urgent tasks to develop in the next few years,” Dr. Balladares, who was previously at UCL, argues.

And this, now, is the next project they are investigating.

Dr. Balladares concludes, “Future studies should be designed to explore the effects that these games could have on other cognitive and developmental skills.

“An interesting space for the development of intervention and assessment of board games should open up in the next few years, given the complexity of games and the need to design more and better games for educational purposes.”

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Credit of the article given to Taylor & Francis

Life, from the perspective of thermodynamics, is a system out of equilibrium, resisting tendencies towards increasing their levels of disorder. In such a state, the dynamics are irreversible over time. This link between the tendency toward disorder and irreversibility is expressed as the ‘arrow of time’ by the English physicist Arthur Eddington in 1927.

Now, an international team including researchers from Kyoto University, Hokkaido University, and the Basque Center for Applied Mathematics, has developed a solution for temporal asymmetry, furthering our understanding of the behaviour of biological systems, machine learning, and AI tools.

“The study offers, for the first time, an exact mathematical solution of the temporal asymmetry—also known as entropy production—of nonequilibrium disordered Ising networks,” says co-author Miguel Aguilera of the Basque Center for Applied Mathematics.

The researchers focused on a prototype of large-scale complex networks called the Ising model, a tool used to study recurrently connected neurons. When connections between neurons are symmetric, the Ising model is in a state of equilibrium and presents complex disordered states called spin glasses. The mathematical solution of this state led to the award of the 2021 Nobel Prize in physics to Giorgio Parisi.

Unlike in living systems, however, spin crystals are in equilibrium and their dynamics are time reversible. The researchers instead worked on the time-irreversible Ising dynamics caused by asymmetric connections between neurons.

The exact solutions obtained serve as benchmarks for developing approximate methods for learning artificial neural networks. The development of learning methods used in multiple phases may advance machine learning studies.

“The Ising model underpins recent advances in deep learning and generative artificial neural networks. So, understanding its behaviour offers critical insights into both biological and artificial intelligence in general,” added Hideaki Shimazaki at KyotoU’s Graduate School of Informatics.

“Our findings are the result of an exciting collaboration involving insights from physics, neuroscience and mathematical modeling,” remarked Aguilera. “The multidisciplinary approach has opened the door to novel ways to understand the organization of large-scale complex networks and perhaps decipher the thermodynamic arrow of time.”

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Credit of the article given to Kyoto University

On June 23, 1993, the mathematician Andrew Wiles gave the last of three lectures detailing his solution to Fermat’s last theorem, a problem that had remained unsolved for three and a half centuries. Wiles’ announcement caused a sensation, both within the mathematical community and in the media.

Beyond providing a satisfying resolution to a long-standing problem, Wiles’ work marks an important moment in the establishment of a bridge between two important, but seemingly very different, areas of mathematics.

History demonstrates that many of the greatest breakthroughs in math involve making connections between seemingly disparate branches of the subject. These bridges allow mathematicians, like the two of us, to transport problems from one branch to another and gain access to new tools, techniques and insights.

What is Fermat’s last theorem?

Fermat’s last theorem is similar to the Pythagorean theorem, which states that the sides of any right triangle give a solution to the equation x² + y² = z² .

Every differently sized triangle gives a different solution, and in fact there are infinitely many solutions where all three of x, y and z are whole numbers—the smallest example is x=3, y=4 and z=5.

Fermat’s last theorem is about what happens if the exponent changes to something greater than 2. Are there whole-number solutions to x³ + y³ = z³ ? What if the exponent is 10, or 50, or 30 million? Or, most generally, what about any positive number bigger than 2?

Around the year 1637, Pierre de Fermat claimed that the answer was no, there are no three positive whole numbers that are a solution to xⁿ + yⁿ = zⁿ for any n bigger than 2. The French mathematician scribbled this claim into the margins of his copy of a math textbook from ancient Greece, declaring that he had a marvelous proof that the margin was “too narrow to contain.”

Fermat’s purported proof was never found, and his “last theorem” from the margins, published posthumously by his son, went on to plague mathematicians for centuries.

Searching for a solution

For the next 356 years, no one could find Fermat’s missing proof, but no one could prove him wrong either—not even Homer Simpson. The theorem quickly gained a reputation for being incredibly difficult or even impossible to prove, with thousands of incorrect proofs put forward. The theorem even earned a spot in the Guinness World Records as the “most difficult math problem.”

That is not to say that there was no progress. Fermat himself had proved it for n=3 and n=4. Many other mathematicians, including the trailblazer Sophie Germain, contributed proofs for individual values of n, inspired by Fermat’s methods.

But knowing Fermat’s last theorem is true for certain numbers isn’t enough for mathematicians—we need to know it’s true for infinitely many of them. Mathematicians wanted a proof that would work for all numbers bigger than 2 at once, but for centuries it seemed as though no such proof could be found.

However, toward the end of the 20th century, a growing body of work suggested Fermat’s last theorem should be true. At the heart of this work was something called the modularity conjecture, also known as the Taniyama-Shimura conjecture.

A bridge between two worlds

The modularity conjecture proposed a connection between two seemingly unrelated mathematical objects: elliptic curves and modular forms.

Elliptic curves are neither ellipses nor curves. They are doughnut-shaped spaces of solutions to cubic equations, like y² = x³—3x + 1.

A modular form is a kind of function which takes in certain complex numbers—numbers with two parts: a real part and an imaginary part—and outputs another complex number. What makes these functions special is that they are highly symmetrical, meaning there are lots of conditions on what they can look like.

There is no reason to expect that those two concepts are related, but that is what the modularity conjecture implied.

Finally, a proof

The modularity conjecture doesn’t appear to say anything about equations like xⁿ + yⁿ = zⁿ . But work by mathematicians in the 1980s showed a link between these new ideas and Fermat’s old theorem.

First, in 1985, Gerhard Frey realized that if Fermat was wrong and there could be a solution to xⁿ + yⁿ = zⁿ for some n bigger than 2, that solution would produce a peculiar elliptic curve. Then Kenneth Ribet showed in 1986 that such a curve could not exist in a universe where the modularity conjecture was also true.

Their work implied that if mathematicians could prove the modularity conjecture, then Fermat’s last theorem had to be true. For many mathematicians, including Andrew Wiles, working on the modularity conjecture became a path to proving Fermat’s last theorem.

Wiles worked for seven years, mostly in secret, trying to prove this difficult conjecture. By 1993, he was close to having a proof of a special case of the modularity conjecture—which was all he needed to prove Fermat’s last theorem.

He presented his work in a series of lectures at the Isaac Newton Institute in June 1993. Though subsequent peer review found a gap in Wiles’ proof, Wiles and his former student Richard Taylor worked for another year to fill in that gap and cement Fermat’s last theorem as a mathematical truth.

Lasting consequences

The impacts of Fermat’s last theorem and its solution continue to reverberate through the world of mathematics. In 2001, a group of researchers, including Taylor, gave a full proof of the modularity conjecture in a series of papers that were inspired by Wiles’ work. This completed bridge between elliptic curves and modular forms has been—and will continue to be—foundational to understanding mathematics, even beyond Fermat’s last theorem.

Wiles’ work is cited as beginning “a new era in number theory” and is central to important pieces of modern math, including a widely used encryption technique and a huge research effort known as the Langlands Program that aims to build a bridge between two fundamental areas of mathematics: algebraic number theory and harmonic analysis.

Although Wiles worked mostly in isolation, he ultimately needed help from his peers to identify and fill in the gap in his original proof. Increasingly, mathematics today is a collaborative endeavor, as witnessed by what it took to finish proving the modularity conjecture. The problems are large and complex and often require a variety of expertise.

So, finally, did Fermat really have a proof of his last theorem, as he claimed? Knowing what mathematicians know now, many of us today don’t believe he did. Although Fermat was brilliant, he was sometimes wrong. Mathematicians can accept that he believed he had a proof, but it’s unlikely that his proof would stand up to modern scrutiny.

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Credit of the article given to Maxine Calle and David Bressoud, The Conversation

Children do better at math when music is a key part of their lessons, an analysis of almost 50 years of research on the topic has revealed.

It is thought that music can make math more enjoyable, keep students engaged and help many ease fear or anxiety they have about math. Motivation may be increased and pupils may appreciate math more, the peer-reviewed article in Educational Studies details.

Techniques for integrating music into math lessons range from clapping to pieces with different rhythms when learning numbers and fractions, to using math to design musical instruments.

Previous research has shown that children who are better at music also do better at math. But whether teaching music to youngsters actually improves their math has been less clear.

To find out more, Turkish researcher Dr. Ayça Akın, from the Department of Software Engineering, Antalya Belek University, searched academic databases for research on the topic published between 1975 and 2022.

She then combined the results of 55 studies from around the world, involving almost 78,000 young people from kindergarten pupils to university students, to come up with an answer.

Three types of musical intervention were included the meta-analysis: standardized music interventions (typical music lessons, in which children sing and listen to, and compose, music), instrumental musical interventions (lessons in which children learn how to play instruments, either individually or as part of a band) and music-math integrated interventions, in which music is integrated into math lessons.

Students took math tests before and after taking part in the intervention and the change in their scores was compared with that of youngsters who didn’t take part in an intervention.

The use of music, whether in separate lessons or as part of math classes, was associated with greater improvement in math over time.

The integrated lessons had the biggest effect, with around 73% of students who had integrated lessons doing significantly better than youngsters who didn’t have any type of musical intervention.

Some 69% of students who learned how to play instruments and 58% of students who had normal music lessons improved more than pupils with no musical intervention.

The results also indicate that music helps more with learning arithmetic than other types of math and has a bigger impact on younger pupils and those learning more basic mathematical concepts.

Dr. Akin, who carried out the research while at Turkey’s National Ministry of Education and Antalya Belek University, points out that math and music have much in common, such as the use of symbols symmetry. Both subjects also require abstract thought and quantitative reasoning.

Arithmetic may lend itself particularly well to being taught through music because core concepts, such as fractions and ratios, are also fundamental to music. For example, musical notes of different lengths can be represented as fractions and added together to create several bars of music.

Integrated lessons may be especially effective because they allow pupils to build connections between math and music and provide extra opportunities to explore, interpret and understand math.

Plus, if they are more enjoyable than traditional math lessons, any anxiety students feel about math may be eased.

Limitations of the analysis include the relatively small number of studies available for inclusion. This meant it wasn’t possible to look at the effect of factors such as gender, socio-economic status and length of musical instruction on the results.

Dr. Akin, who is now based at Antalya Belek University, concludes that while musical instruction overall has a small to moderate effect on achievement in math, integrated lessons have a large impact.

She adds, “Encouraging mathematics and music teachers to plan lessons together could help ease students’ anxiety about mathematics, while also boosting achievement.”

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Credit of the article given to Taylor & Francis