Author: IMC

Most people have heard the famous phrase “the enemy of my enemy is my friend.” Now, Northwestern University researchers have used statistical physics to confirm the theory that underlies this famous axiom. The study, “Proper network randomization is key to assessing social balance,” is published in the journal Science Advances.

In the 1940s, Austrian psychologist Fritz Heider introduced social balance theory, which explains how humans innately strive to find harmony in their social circles. According to the theory, four rules—an enemy of an enemy is a friend, a friend of a friend is a friend, a friend of an enemy is an enemy and, finally, an enemy of a friend is an enemy—lead to balanced relationships.

Although countless studies have tried to confirm this theory using network science and mathematics, their efforts have fallen short, as networks deviate from perfectly balanced relationships. Hence, the real question is whether social networks are more balanced than expected according to an adequate network model.

Most network models were too simplified to fully capture the complexities within human relationships that affect social balance, yielding inconsistent results on whether deviations observed from the network model expectations are in line with the theory of social balance.

The Northwestern team, however, successfully integrated the two key pieces that make Heider’s social framework work. In real life, not everyone knows each other, and some people are more positive than others. Researchers have long known that each factor influences social ties, but existing models could only account for one factor at a time.

By simultaneously incorporating both constraints, the researchers’ resulting network model finally confirmed the famous theory some 80 years after Heider first proposed it.

The useful new framework could help researchers better understand social dynamics, including political polarization and international relations, as well as any system that comprises a mixture of positive and negative interactions, such as neural networks or drug combinations.

“We have always thought this social intuition works, but we didn’t know why it worked,” said Northwestern’s István Kovács, the study’s senior author.

“All we needed was to figure out the math. If you look through the literature, there are many studies on the theory, but there’s no agreement among them. For decades, we kept getting it wrong. The reason is because real life is complicated. We realized that we needed to take into account both constraints simultaneously: who knows whom and that some people are just friendlier than others.”

“We can finally conclude that social networks align with expectations that were formed 80 years ago,” added Bingjie Hao, the study’s first author. “Our findings also have broad applications for future use. Our mathematics allows us to incorporate constraints on the connections and the preference of different entities in the system. That will be useful for modeling other systems beyond social networks.”

Kovács is an assistant professor of Physics and Astronomy at Northwestern’s Weinberg College of Arts and Sciences. Hao is a postdoctoral researcher in his laboratory.

What is social balance theory?

Using groups of three people, Heider’s social balance theory maintains the assumption that humans strive for comfortable, harmonious relationships.

In balanced relationships, all people like each other. Or, if one person dislikes two people, those two are friends. Imbalanced relationships exist when all three people dislike each other, or one person likes two people who dislike each other, leading to anxiety and tension.

Studying such frustrated systems led to the 2021 Nobel Prize in physics to Italian theoretical physicist Giorgio Parisi, who shared the prize with climate modelers Syukuro Manabe and Klaus Hasselmann.

“It seems very aligned with social intuition,” Kovács said. “You can see how this would lead to extreme polarization, which we do see today in terms of political polarization. If everyone you like also dislikes all the people you don’t like, then that results in two parties that hate each other.”

However, it has been challenging to collect large-scale data where not only friends but also enemies are listed. With the onset of Big Data in the early 2000s, researchers tried to see if such signed data from social networks could confirm Heider’s theory. When generating networks to test Heider’s rules, individual people serve as nodes. The edges connecting nodes represent the relationships among individuals.

If the nodes are not friends, then the edge between them is assigned a negative (or hostile) value. If the nodes are friends, then the edge is marked with a positive (or friendly) value. In previous models, edges were assigned positive or negative values at random, without respecting both constraints. None of those studies accurately captured the realities of social networks.

Finding success in constraints

To explore the problem, Kovács and Hao turned to four large-scale, publicly available signed network datasets previously curated by social scientists, including data from 1) user-rated comments on social news site Slashdot; 2) exchanges among Congressional members on the House floor; 3) interactions among Bitcoin traders; and 4) product reviews from consumer review site Epinions.

In their network model, Kovács and Hao did not assign truly random negative or positive values to the edges. For every interaction to be random, every node would need to have an equal chance of encountering one another. In real life, however, not everyone actually knows everyone else within a social network. For example, a person might not ever encounter their friend’s friend, who lives on the other side of the world.

To make their model more realistic, Kovács and Hao distributed positive or negative values based on a statistical model that describes the probability of assigning positive or negative signs to the interactions that exist. That kept the values random—but random within limits given by constraints of the network topology. In addition to who knows whom, the team took into account that some people in life are just friendlier than others. Friendly people are more likely to have more positive—and fewer hostile—interactions.

By introducing these two constraints, the resulting model showed that large-scale social networks consistently align with Heider’s social balance theory. The model also highlighted patterns beyond three nodes. It shows that social balance theory applies to larger graphlets, which involve four and possibly even more nodes.

“We know now that you need to take into account these two constraints,” Kovács said. “Without those, you cannot come up with the right mechanisms. It looks complicated, but it’s actually fairly simple mathematics.”

Insights into polarization and beyond

Kovács and Hao currently are exploring several future directions for this work. In one potential direction, the new model could be used to explore interventions aimed at reducing political polarization. But the researchers say the model could help better understand systems beyond social groups and connections among friends.

“We could look at excitatory and inhibitory connections between neurons in the brain or interactions representing different combinations of drugs to treat disease,” Kovács said. “The social network study was an ideal playground to explore, but our main interest is to go beyond investigating interactions among friends and look at other complex networks.”

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

Credit of the article given to Northwestern University

Malawi introduced free primary education in 1994. This has significantly improved access to schooling. However, the country—which is one of the poorest in the world—still faces a high learning poverty rate of 87%. Learning poverty is a measure of a child’s inability to meet minimum proficiency in reading, numeracy and other skills at the primary school level. Malawi’s rate means that 87% of children in standard 4, at age 10, are unable to read. Only 19% of children aged between 7 and 14 have foundational reading skills and 13% have foundational numeracy skills. This leads to social and financial dependency. It also limits the extent to which individuals can actively participate in society. Children become especially vulnerable to pernicious social issues such as forced marriage, female genital mutilation, and child labor.

The primary education sector also has many challenges. These include overcrowded classrooms, limited learning materials, and a shortage of trained teachers.

There is a pressing need for innovative, transformative approaches to providing foundational education to meet the goals envisioned in Malawi 2063, the country’s long-term national plan. To accomplish this, the government of Malawi is using scientific evidence to enable meaningful and effective learning happen at scale.

This evidence has been generated in parallel by researchers from the University of Nottingham in the UK and the NGO Imagine Worldwide in the US and Africa. We have been testing the efficacy of an interactive educational technology (EdTech) developed by UK-based non-profit onebillion to raise foundational education by different groups of learners in Malawi.

The EdTech delivers personalized, adaptive software that enables each child to learn reading, writing and numeracy at the right level. Children work on tablets through a carefully structured course made up of thousands of engaging activities, games and stories. Over the past 11 years, we have built a complementary and robust evidence base focusing on different aspects of the software and program.

In 2013, I conducted the first pupil-level randomized control trial at a state primary school in Malawi’s capital city, Lilongwe. Randomized controlled trials are prospective studies that measure the effectiveness of a new intervention compared to standard practice. They are considered the gold standard in effectiveness research. We wanted to test whether the EdTech could raise young children’s numeracy skills. The study showed that after eight weeks of using the EdTech for 30 minutes a day, learners in grades 1–3 (aged 6 to 9) made significant improvements in basic numeracy compared to standard classroom practice. Teachers were also able to put the EdTech to use with ease.

Now, after many studies, Malawi’s government, in collaboration with Imagine Worldwide, is embedding the EdTech program in all state primary schools nationwide. This will serve 3.8 million children per year in grades 1–4 across all 6,000 state primary schools in Malawi.

Rigorous testing

After our initial 2013 study, we kept testing the EdTech through rigorous studies. Oneshowed that the EdTech program significantly raised foundational numeracy and literacy skills of early grade learners. Our results showed similar learning gains for girls and boys with the EdTech. This equalizes foundational education across gender.

Another study showed that children with special educational needs and disabilities could interact and learn with the EdTech, albeit at a slower pace than mainstream peers.

The EdTech wasn’t just tested in Malawi. We wanted to see if it could address learning poverty in different contexts, thus equalizing all children’s opportunities, no matter where they live.

Research in the UK demonstrated that the same EdTech raised the basic numeracy skills of children in the early years of primary schools compared to standard classroom instruction. It was also found to support numeracy acquisition by developmentally young children, including those with Down syndrome.

It was also shown to be effective in a bilingual setting. Brazilian children’s basic numeracy skills improved compared to standard practice after instruction with the EdTech delivered in either English, their language of instruction, or their home language, Brazilian-Portuguese.

Alongside the research from the University of Nottingham, Imagine Worldwide undertook a series of studies in Malawi and other countries to investigate how this EdTech could raise foundational skills over longer periods of time and in different languages and contexts, including refugee camps.

Imagine Worldwide conducted six randomized control trials, including two of the longest over eight months and two years. They showed robust learning gains in literacy and numeracy. They also found that children’s excitement about school, their attendance, and their confidence as learners improved.

The EdTech program also mitigated against learning loss during school closures. During Imagine’s 2-year randomized control trial in Malawi, program delivery was interrupted for seven months by COVID-related closures. Yet, results showed that children who had participated in the EdTech program prior to schools closing returned to school with higher achievement levels than their peers who had received standard instruction only.

Applying the evidence to policy

Malawi’s government was pleased with the early results and the program was expanded to about 150 schools, with the help of UK non-profit Voluntary Service Overseas. A national steering committee was established by Malawi’s government to monitor the program and review additional emerging research. In 2022 the Education Ministry formally launched the program through which the EdTech will be rolled out; it was introduced in 500 new schools at the start of the 2023/2024 school year, in September 2023.

To achieve the promise of the early research, ongoing implementation research and monitoring is helping to ensure program quality and impacts are sustained as it rolls out nationwide.

Strong evidence

Basic literacy and numeracy are the keys to unlocking a child’s potential—improving their health, wealth and social outcomes. Our combined research has shown that child-directed EdTech can deliver high-quality education for millions of marginalized children worldwide. The evidence is strong, diverse and replicable. Now governments need to follow the lead of Malawi to abolish learning poverty and make foundational education a reality for all children, everywhere.

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

Credit of the article given to Nicola Pitchford and Dr. Karen Levesque, The Conversation

A centuries-old maths problem asks what shape a circle traces out as it rolls along a line. The answer, dubbed a “cycloid”, turns out to have applications in a variety of scientific fields.

Light reflecting off the round rim creates a mathematically significant shape in this coffee cup

Sarah Hart

The artist Paul Klee famously described drawing as “taking a line for a walk” – but why stop there? Mathematicians have been wondering for five centuries what happens when you take circles and other curves for a walk. Let me tell you about this fascinating story…

A wheel rolling along a road will trace out a series of arches

Imagine a wheel rolling along a road – or, more mathematically, a circle rolling along a line. If you follow the path of a point on that circle, it traces out a series of arches. What exactly is their shape? The first person to give the question serious thought seems to have been Galileo Galilei, who gave the arch-like curve a name – the cycloid. He was fascinated by cycloids, and part of their intriguing mystery was that it seemed impossible to answer the most basic questions we ask about a curve – how long is it and what area does it contain? In this case, what’s the area between the straight line and the arch? Galileo even constructed a cycloid on a sheet of metal, so he could weigh it to get an estimate of the area, but he never managed to solve the problem mathematically.

Within a few years, it seemed like every mathematician in Europe was obsessed with the cycloid. Pierre de Fermat, René Descartes, Marin Mersenne, Isaac Newton and Gottfried Wilhelm Leibniz all studied it. It even brought Blaise Pascal back to mathematics, after he had sworn off it in favour of theology. One night, he had a terrible toothache and, to distract himself from the pain, decided to think about cycloids. It worked – the toothache miraculously disappeared, and naturally Pascal concluded that God must approve of him doing mathematics. He never gave it up again. The statue of Pascal in the Louvre Museum in Paris even shows him with a diagram of a cycloid. The curve became so well known, in fact, that it made its way into several classic works of literature – it gets name-checked in Gulliver’s Travels, Tristram Shandy and Moby-Dick.

The question of the cycloid’s area was first solved in the mid-17th century by Gilles de Roberval, and the answer turned out to be delightfully simple – exactly three times the area of the rolling circle. The first person to determine the length of the cycloid was Christopher Wren, who was an extremely good mathematician, though I hear he also dabbled in architecture. It’s another beautifully simple formula: the length is exactly four times the diameter of the generating circle. The beguiling cycloid was so appealing to mathematicians that it was nicknamed “the Helen of Geometry”.

But its beauty wasn’t the only reason for the name. It was responsible for many bitter arguments. When mathematician Evangelista Torricelli independently found the area under the cycloid, Roberval accused him of stealing his work. “Team Roberval” even claimed that Torricelli had died of shame after being unmasked as a plagiarist (though the typhoid he had at the time may have been a contributing factor). Descartes dismissed Fermat’s work on the cycloid as “ridiculous gibberish”. And in response to a challenge from Johann Bernoulli, Isaac Newton grumpily complained about being “teased by foreigners about mathematics”.

An amazing property of the cycloid was discovered by Christiaan Huygens, who designed the first pendulum clock. Pendulums are good for timekeeping because the period of their motion – the time taken for one full swing of the pendulum – is constant, no matter what the angle of release. But in fact, that’s only approximately true – the period does vary slightly. Huygens wondered if he could do better. The end of a pendulum string moves along the arc of a circle, but is there a curved path it could follow so that the bob would reach the bottom of the curve in the same time no matter where it was released? This became known as the “tautochrone problem”. And guess which curve is the solution? An added bonus is its link to the “brachistochrone problem” of finding the curve between any two points along which a particle moving under gravity will descend in the shortest time. There’s no reason at all to think that the same curve could answer both problems, but it does. The solution is the cycloid. It’s a delightful surprise to find it cropping up in situations seemingly so unrelated to where we first encountered it.

When you roll a circle along a line, you get a cycloid. But what happens when you roll a line along a circle? This is an instance of a curve called an involute. To make one, you take a point at the end of a line segment and roll that line along the curve so it’s always just touching it (in other words, it’s a tangent). The involute is the curve traced out by that point. For the involute of a circle, imagine unspooling a thread from a cotton reel and following the end of the thread as it moves. The result is a spiralling curve emerging from the circle’s circumference.

When a line rolls along a circle, it produces a curve called an involute

Huygens was the first person to ask about involutes, as part of his attempts to make more accurate clocks. It’s all very well knowing the cycloid is the perfect tautochrone, but how do you get your string to follow a cycloidal path? You need to find a curve whose involute is a cycloid. The miraculous cycloid, it turns out, has the beautiful property that it is its own involute! But those lovely spiralling circle involutes turn out to be extremely useful too.

A circle with many involutes

My favourite application is one Huygens definitely couldn’t have predicted: in the design of a nuclear reactor that produces high-mass elements for scientific research. This is done by smashing neutrons at high speed into lighter elements, to create heavier ones. Within the cylindrical reactor cores, the uranium oxide fuel is sandwiched in thin layers between strips of aluminium, which are then curved to fit into the cylindrical shape. The heat produced by a quadrillion neutrons hurtling around every square centimetre is considerable, so coolant runs between these strips. It’s vital that they must be a constant distance apart all the way along their curved surfaces, to prevent hotspots. That’s where a useful property of circle involutes comes in. If you draw a set of circle involutes starting at equally spaced points on the circumference of a circle, then the distances between them remain constant along the whole of each curve. So, they are the perfect choice for the fuel strips in the reactor core. What’s more, the circle involute is the only curve for which this is true! I just love that a curve first studied in the context of pendulum clocks turns out to solve a key design question for nuclear reactors.

We’ve rolled circles along lines and lines along circles. Clearly the next step is to roll circles along circles. What happens? Here, we have some choices. What size is the rolling circle? And are we rolling along the inside or the outside of the stationary one? The curve made by a circle rolling along inside of the circle is called a hypocycloid; rolling it along the outside gives you an epicycloid. If you’ve ever played with a Spirograph toy, you’ll almost have drawn hypocycloids. Because your pen is not quite at the rim of the rolling circle, technically you are creating what are called hypotrochoids.

A cardioid (left) and nephroid (right)

Of the epicycloids, the most interesting is the cardioid: the heart-shaped curve resulting when the rolling circle has the same radius as the fixed one. Meanwhile, the kidney-shaped nephroid is produced by a rolling circle half the radius of the fixed one. Cardioids crop up in the most fascinating places. The central region of the Mandelbrot set, a famous fractal, is a cardioid. Sound engineers will be familiar with cardioid microphones, which pick up sound in a cardioid-shaped region. You might also find cardioid-like curves in the light patterns created in coffee cups in some kinds of lighting. If light rays from a fixed source are reflected off a curved mirror, the curve to which each of those reflected rays are tangent will be visible as a concentrated region of light, called a caustic. It turns out that a light source on the circumference of a perfectly circular mirror will result precisely in a cardioid!

Of course, in our coffee cup example, usually the light source isn’t exactly on the rim of the cup, but some way away. If it were very far away, we could assume that the light rays hitting the rim of the cup are parallel. In that situation, it can be shown that the caustic is actually not a cardioid but another epicycloid: the nephroid. Since a strong overhead light is somewhere between these two extremes, the curve we get is usually going to be somewhere between a cardioid and a nephroid. The mathematician Alfréd Rényi once defined a mathematician as “a device for turning coffee into theorems”. That process is nowhere more clearly seen than with our wonderful epicycloids. Check them out if you’re reading this with your morning cuppa!

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

*Credit for article given to Sarah Hart*

An analysis of a mathematical economic game suggests that even learning from past mistakes will almost never help us optimise our decision-making – with implications for our ability to make the biggest financial gains.

When people trade stocks, they don’t always learn from experience

Even when we learn from past mistakes, we may never become optimal decision-makers. The finding comes from an analysis of a mathematical game that simulates a large economy, and suggests we may need to rethink some of the common assumptions built into existing economic theories.

In such theories, people are typically represented as rational agents who learn from past experiences to optimise their performance, eventually reaching a stable state in which they know how to maximise their earnings. This assumption surprised Jérôme Garnier-Brun at École Polytechnique in France because, as a physicist, he knew that interactions in nature – such as those between atoms – often result in chaos rather than stability. He and his colleagues mathematically tested whether economists are correct to assume that learning from the past can help people avoid chaos.

They devised a mathematical model for a game featuring hundreds of players. Each of these theoretical players can choose between two actions, like buying or selling a stock. They also interact with each other, and each player’s decision-making is influenced by what they have done before – meaning each player can learn from experience. The researchers could adjust the precise extent to which a player’s past experiences influenced their subsequent decision-making. They could also control the interactions between the players to make them either cooperate or compete with each other more.

With all these control knobs available to them, Garnier-Brun and his colleagues used methods from statistical physics to simulate different game scenarios on a computer. The researchers expected that in some scenarios the game would always result in chaos, with players unable to learn how to optimise their performance. Economic theory would also suggest that, given the right set of parameters, the virtual players would settle into a stable state where they have mastered the game – but the researchers found that this wasn’t really the case. The most likely outcome was a state that never settled.

Jean-Philippe Bouchaud at École Polytechnique, who worked on the project, says that in the absence of one centralised, omniscient, god-like player that could coordinate everyone, regular players could only learn how to reach “satisficing” states. That is, they could reach a level that satisfied minimum expectations, but not much more. Players gained more than they would have done by playing at random, so learning was not useless, but they still gained less than they would have if past experience had allowed them to truly optimise their performance.

“This work is such a powerful new way of looking at the problem of learning complex games and these questions are fundamental to the construction of models of economic decision-making,” says Tobias Galla at the Institute for Cross-Disciplinary Physics and Complex Systems in Spain. He says the finding that learning typically does not lead to outcomes better than satisficing could also be important for processes like foraging decisions by animals or for some machine learning applications.

Bouchaud says his team’s game model is too simple to be immediately adopted for making predictions about the real world, but he sees the study as a challenge to economists to drop many assumptions that currently go into theorising processes like merchants choosing suppliers or banks setting interest rates.

“The idea that people are always making complicated economic computations and learn how to become the most rational agents, our paper invites everyone to move on [from that],” he says.

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

*Credit for article given to Karmela Padavic-Callaghan*

French director Justine Triet’s “Anatomy of a Fall,” winner of the 2023 Oscar for best original script, reconstructs a fatal fall in order to dissect the collapse of the romantic relationship between the film’s leading couple, Sandra Voyter and Samuel Maleski.

Far from an exception, breakups of the sort depicted in the film are commonplace: global data shows high levels of marriage failure, with a marked increase towards the end of the last century.

In some Western countries, as many as 50% of marriages do not make it past 25 years, giving rise to the popular maxim “half of all marriages end in divorce.”

According to Triet, “the strange thing is for a relationship to work. The majority are hellish, and the film aims to go deep into that hell.”

Importantly, divorce statistics do not account for the number of relationships that are unhappy. Perhaps the majority are indeed hellish, but some marriages today are long-lasting, and seem stronger and more loving than any that came before. This dichotomy—widespread failure or exceptional success– seems to summarize the current state of marriage in the West. This has been dubbed the “all or nothing” marriage.

Supplying relationship energy

Scientific studies have established that romantic relationships tend to drop off, meaning that, on average, satisfaction levels reduce over time. Successful couples are able to arrest this fall, finding a satisfying level that can last indefinitely. Many others, however, gradually decline to the point where breaking up is only a matter of time.

Relationship psychology shows that love alone is not enough to keep a couple together –it requires effort. Relationship scientist John Gottman likens this to the second law of thermodynamics, whereby a closed system—such as a marriage– degenerates unless energy is supplied. As he puts it, “if you do nothing to make things get better in your marriage but do not do anything wrong, the marriage will still tend to get worse over time.”

The “all or nothing” theory therefore suggests that successful relationships require a significant investment of time and energy. Couples who make this commitment will be rewarded with a high level of satisfaction, while those who fail to do so, like Samuel and Sandra in Triet’s film, are destined to fail.

But why do some couples manage to stop this fall and stay happy? Like Samuel and Sandra, all couples start out in love, and want to be happy together forever. If we assume that they are compatible and willing to make the effort together, they form what some call an “Adam and Eve” relationship—the Biblical archetype of a harmonious, lasting union.

Analysing the ‘Adam and Eve’ relationship

Using dynamic systems to analyse this relationship model confirms the “all or nothing” theory.

Dynamic systems are a mathematical tool for understanding the evolution of a variable over time. In the case of romantic relationships, we are interested in the “feeling” of love in a couple. Because effort is needed to sustain the relationship, it becomes a dynamic system controlled by effort: effort regulates “feeling,” with the objective of making the “feeling” last forever.

By applying this effort control theory, our research has found that a successful relationship requires effort beyond the partners’ preferred level, and that this effort gap is difficult to sustain over time.

The mathematical anatomy of a fall

As Sandra Voyter says in Triet’s film, there are times when a relationship is chaotic, others when you fight alone, sometimes alongside your partner, and sometimes against your partner.

Samuel and Sandra’s relationship has elements in common with any other couple’s relationship. The starting point is very high: “feeling” is at its peak, and there is a shared belief that it will never end. Both are willing to contribute to the happiness of the relationship by making their own individual efforts, and both know that some kind of shock or external event will eventually alter this state.

Generally speaking, couples with the same socioeconomic, cultural, or religious background—known as homogamous couples— are more stable. Many couples, however, are heterogamous, meaning they differ in one or more of these regards.

Heterogamy can extend beyond an individual’s circumstances: on its most elemental level, it can boil down to a mismatch or imbalance in how efficient one member of a couple is in transforming effort into “feeling” or happiness. Such a disparity may lead to asymmetrical levels of effort being dedicated to making the relationship successful, which are already higher than those both members would prefer to make.

This is the case in Samuel and Sandra’s relationship: at one point in the film Samuel highlights this imbalance, and Sandra replies that she does not believe a couple should make equal efforts, saying she finds the idea depressing.

Who contributes more?

Our latest computational models for assessing the dynamics of imbalanced effort levels in couples allow us to simulate the evolution of happiness in a relationship, both in predictable environments and with varying levels of uncertainty. Our simulations suggest that Sandra is right: each partner does not have to make the same level of effort.

One of the film’s scenes—where Sandra and Samuel reproach each other for the efforts made or not made to sustain the relationship—displays typical negative couple dynamics, where each has a bone to pick. The film also implies that Samuel has made or is making more effort than Sandra in their relationship. Our analysis shows, perhaps surprisingly, that the more emotionally efficient partner has to make a greater effort to sustain the relationship. In the film, it appears that this is Samuel.

External events play a big part

Our analysis also shows that when the couple is subjected to a stressful episode, both partners need to increase their effort levels if the relationship is to survive. However, the more efficient partner’s effort level has to increase more. In the film, Sandra and Samuel’s relationship is subjected to a tremendous misfortune, which has a prolonged and pronounced effect on its narrative arc. This is why Samuel feels much more stressed than Sandra.

Mathematics offers an outcome in line with the film’s plot: the continuous overexertion of the most emotionally efficient partner—amplified by a prolonged period of crisis—leads to the relationship falling apart. In the case of the film, this also leads to Samuel’s fall.

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

Credit of the article given to José-Manuel Rey and Jorge Herrera de la Cruz

Will a certain tritium atom decay by a certain time? According to our current science, this question concerning physical phenomena should be answered by sampling from a probability distribution, a process not unlike spinning a roulette wheel or rolling dice. However, a paper in Foundations of Physics suggests that the same could be true of a question concerning mathematical phenomena, even one as prosaic as “what is 2+2?

SFI Professor David Wolpert, a physicist, and former SFI Omidyar Fellow David Kinney, a philosopher and cognitive scientist, propose a unified, probabilistic framework to describe math, the physical universe, and even describe how humans reason about both.

In their framework, mathematics and science are both represented as a process of asking and answering questions. How a given mathematician or scientist answers a question will depend on the probabilities that they assign to different answers. In an extension of the basic formalism, these same probabilities also determine how correctly that mathematician or scientist answers those questions.

To illustrate, say you answer a mathematics question now, and in the faraway future, some mathematicians answer the same question. The correctness of your answer depends on how your probabilities match theirs. (In physics, your correctness would involve physical experiments, not future physicists.)

Randomness pervades every part of the question-answer process, from question selection to the answer given. This results in an important benefit of the proposed framework: a novel, formal justification for two common-sense shortcuts in science and math—believing more strongly in ideas supported by multiple lines of evidence, and believing more strongly in ideas that best explain something you already believe.

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

Credit of the article given to Santa Fe Institute

Algorithms have become integral to our lives. From social media apps to Netflix, algorithms learn your preferences and prioritise the content you are shown. Google Maps and artificial intelligence are nothing without algorithms.

So, we’ve all heard of them, but where does the word “algorithm” even come from?

Over 1,000 years before the internet and smartphone apps, Persian scientist and polymath Muhammad ibn Mūsā al-Khwārizmī invented the concept of algorithms.

In fact, the word itself comes from the Latinised version of his name, “algorithmi”. And, as you might suspect, it’s also related to algebra.

Largely lost to time

Al-Khwārizmī lived from 780 to 850 CE, during the Islamic Golden Age. He is considered the “father of algebra”, and for some, the “grandfather of computer science”.

Yet, few details are known about his life. Many of his original works in Arabic have been lost to time.

It is believed al-Khwārizmī was born in the Khwarazm regionsouth of the Aral Sea in present-day Uzbekistan. He lived during the Abbasid Caliphate, which was a time of remarkable scientific progress in the Islamic Empire.

Al-Khwārizmī made important contributions to mathematics, geography, astronomy and trigonometry. To help provide a more accurate world map, he corrected Alexandrian polymath Ptolemy’s classic cartography book, Geographia.

He produced calculations for tracking the movement of the Sun, Moon and planets. He also wrote about trigonometric functions and produced the first table of tangents.

Al-Khwārizmī was a scholar in the House of Wisdom (Bayt al-Hikmah) in Baghdad. At this intellectual hub, scholars were translating knowledge from around the world into Arabic, synthesising it to make meaningful progress in a range of disciplines. This included mathematics, a field deeply connected to Islam.

There are no images of what al-Khwārizmī looked like, but in 1983 the Soviet Union issued a stamp in honour of his 1,200th birthday. Wikimedia Commons

The ‘father of algebra’

Al-Khwārizmī was a polymath and a religious man. His scientific writings started with dedications to Allah and the Prophet Muhammad. And one of the major projects Islamic mathematicians undertook at the House of Wisdom was to develop algebra.

Around 830 CE, Caliph al-Ma’mun encouraged al-Khwārizmī to write a treatise on algebra, Al-Jabr (or The Compendious Book on Calculation by Completion and Balancing). This became his most important work.

A page from The Compendious Book on Calculation by Completion and Balancing. World Digital Library

At this point, “algebra” had been around for hundreds of years, but al-Khwārizmī was the first to write a definitive book on it. His work was meant to be a practical teaching tool. Its Latin translation was the basis for algebra textbooks in European universities until the 16th century.

In the first part, he introduced the concepts and rules of algebra, and methods for calculating the volumes and areas of shapes. In the second part he provided real-life problems and worked out solutions, such as inheritance cases, the partition of land and calculations for trade.

Al-Khwārizmī didn’t use modern-day mathematical notation with numbers and symbols. Instead, he wrote in simple prose and employed geometric diagrams:

Four roots are equal to twenty, then one root is equal to five, and the square to be formed of it is twenty-five.

In modern-day notation we’d write that like so:

4x = 20, x = 5, x² = 25

Grandfather of computer science

Al-Khwārizmī’s mathematical writings introduced the Hindu-Arabic numerals to Western mathematicians. These are the ten symbols we all use today: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The Hindu-Arabic numerals are important to the history of computing because they use the number zero and a base-ten decimal system. Importantly, this is the numeral system that underpins modern computing technology.

Al-Khwārizmī’s art of calculating mathematical problems laid the foundation for the concept of algorithms. He provided the first detailed explanations for using decimal notation to perform the four basic operations (addition, subtraction, multiplication, division) and computing fractions.

The contrast between algorithmic computations and abacus computations, as shown in Margarita Philosophica (1517). The Bavarian State Library

This was a more efficient computation method than using the abacus. To solve a mathematical equation, al-Khwārizmī systematically moved through a sequence of steps to find the answer. This is the underlying concept of an algorithm.

Algorism, a Medieval Latin term named after al-Khwārizmī, refers to the rules for performing arithmetic using the Hindu-Arabic numeral system. Translated to Latin, al-Khwārizmī’s book on Hindu numerals was titled Algorithmi de Numero Indorum.

In the early 20th century, the word algorithm came into its current definition and usage: “a procedure for solving a mathematical problem in a finite number of steps; a step-by-step procedure for solving a problem”.

Muhammad ibn Mūsā al-Khwārizmī played a central role in the development of mathematics and computer science as we know them today.

The next time you use any digital technology – from your social media feed to your online bank account to your Spotify app – remember that none of it would be possible without the pioneering work of an ancient Persian polymath.

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

Credit of the article given to The Conversation

Homeschooled children don’t always get a well-rounded curriculum. miniseries via Getty Images

Homeschooling is the fastest-growing education setting in the United States. More than 3 million students were educated at home in the 2021-22 school year, up from 2.5 million in the spring of 2019. Current estimates from the U.S. Census Bureau indicate that there are 3.62 million students homeschooled in the United States. That’s a meteoric increase from about 1 million in 1997.

Some experts, including Harvard law professor Elizabeth Bartholet, find the increase a cause to call for greater regulation. University of Washington education policy professor David Knight agrees, citing a lack of accountability and measures of student progress. Knight also worries about an absence of certain disciplines such as social studies that public schools are required to teach.

For those of us who have researched the homeschool movement and studied its past, these are not new concerns. So, what do we know about homeschooling and preparedness for college?

Data shows homeschooled students fare well

In 2020, we reviewed the evidence about how well homeschooling prepares people for college, career and life and published what we learned in a book chapter titled: “Life after Homeschool.”

We found evidence that homeschooled students are just as prepared academically for higher education as traditionally schooled peers. In one study, researchers drew a sample of 825,672 students – including 732 students who had been homeschooled – and found the homeschooled group scored higher on several measures of college preparedness, including the SAT and first-year GPA in college.

Ave Maria University education professor Marc Snyder came to similar results in a 2013 study. Snyder compared homeschooled and traditional students at his Catholic university in Florida to find the average ACT scores for homeschooled students was 26. Public school students averaged 24.22, and students who attended Catholic schools averaged 24.53.

Snyder’s study reinforces data from the ACT itself. The testing outlet reported that from 2001-2019, the average ACT scores for homeschooled students trended up, while public school students’ scores declined slightly. In 2023, the national average on the ACT was 19.9; the average for homeschoolers was 22.8.

Areas of concern abound as homeschool growth accelerates

Still, calls for regulation persist because of a host of challenges homeschooled students present. The Coalition for Responsible Home Education wants states to require minimal qualifications of a high school diploma or GED for the parent providing primary instruction, instruction for students in the same subjects as in public schools, and annual standardized assessments. In some states, they note, parents don’t even have to tell their local school district of their intent to homeschool.

The pro-regulation side points to studies showing homeschooled students feel less prepared for college and are four times less likely to go to college after high school. Homeschool students also take an average of one fewer math and science course than traditional peers.

Homeschooled students also often lack resources and guidance provided in traditional high schools for college prep. And social challenges abound when these students transition; a study of seven homeschooled graduates in Pennsylvania found students struggling to maintain their existing moral beliefs related to drinking, drug use and sexual norms, with the majority admitting they changed some beliefs and practices.

There’s also data that shows homeschooled students find the more structured academic environment on university campuses to be difficult to adjust to after a more lax experience learning at home.

Still, efforts to regulate homeschooling face opposition from parents as well as advocacy groups such as the Home School Legal Defense Association. In March 2024, for example, these forces defeated an attempt in New Hampshire to require homeschool students to take a statewide exam.

3 ways to improve homeschooling

To help homeschooled students transition to college, we recommend parents take three steps to better prepare their kids.

• Prioritize math and scienceto help address the math and science gap. Parents can use online courses offered through virtual high schools or employ tutors.
• Enroll in dual-credit or community college coursesto provide a taste of the structure of college life and to interact with peers from diverse backgrounds.
• Talk to children about the diversity of perspectivesthey will encounter at college. This can help prepare them for how to negotiate and respect the opinions of others.

Homeschooled students can successfully transition to college and compete with their peers. The challenges they face are entirely foreseeable, which means they can be addressed easily.

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

Credit of the article given to The Conversation

A small team of political scientists, statisticians and data scientists from Harvard University, New York University, and Yale University, has found that by switching to a new method to better protect privacy, the U.S. Census Department has introduced factors that reduce accuracy in some cases.

In their paper published in the journal Science Advances, the group describes how they analysed a file provided by Census officials to measure accuracy in publicly available census data and their results.

Prior to the 2020 U.S. census, officials with the U.S. Census Bureau worried about the privacy of the people who provide answers to the census, opted to change the method by which they ensured data security.

The old method was called, “swapping.” It involved swapping data from people living in one block of a city with people in another block, thereby preventing people from being identified based on their data. The new method is called “differential privacy” and it involves adding what the Bureau describes as “noise” to each piece of data that is collected.

In this new effort, the research team could find no instance of an outside entity conducting research to determine if the new method did indeed provide more privacy or if the processed data was more or less accurate than had been the case when swapping was used. So, they began one of their own.

The study began with the research team asking the Census Bureau to give them access to what is called the noisy measurement file (NMF)—the one used for the 2020 census. The Bureau denied the request, which led the team to sue them. Eventually, the lawsuit was dropped when the Bureau agreed to give the team the NMF associated with the much smaller 2010 census—one that was carried out as a way to test the new method, and involved both swapping and differentiating.

The researchers then analysed that file as a way to study the impact on accuracy of changing to the new system. In so doing, they found that overall, the two systems provided roughly equal accuracy on a broad scale. But they also found evidence of a reduction in accuracy at the block level of a type that could adversely impact minorities and multiracial populations.

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

Credit of the article given to Bob Yirka , Phys.org

There’s a strange trend in mathematics education in England. Math 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 math degrees. There’s been a 50% drop in numbers of math 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 math has remained largely static over the last decade. Prestigious Russell Group universities which require top A-level grades have increased their numbers of math students.

This trend in degree-level mathematics education is worrying. It restricts the accessibility of math 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 math 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 math is also vital to a huge range of careers, including in business and government. In March 2024, campaign group Protect Pure Math held a summit to bring together experts from industry, academia and government to discuss concerns about poor math 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 math graduates entering the workforce. And 75% would worry if UK universities shrunk or closed their math 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 specialized math 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 math 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 math 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 program at aged 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 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