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Young people in Germany are less proficient in mathematics, reading and science as compared to 2018. This is revealed in a PISA study. About one-third of the 15-year-olds tested achieved only a very low level of proficiency in at least one of the three subjects. The results confirmed a downward trend already in evidence in the preceding PISA studies. The mathematics and reading scores of German students are only at OECD average levels. They remain above that level only in natural sciences.

The PISA studies are regular assessments of the ability of 15-year-old students to solve problems in mathematics, reading and science in real-world contexts as they approach the end of compulsory schooling. The current study, coordinated by the Organization for Economic Cooperation and Development (OECD) and conducted in Germany by the Center for International Student Assessment (ZIB) at the Technical University of Munich (TUM), was carried out in the spring of 2022.

In many OECD countries the average scores in mathematics and reading skills were lower as compared to the previous PISA study in 2018. Scores were also down in natural sciences, although to a lesser extent.

In Germany the decrease in scores was larger than average in all three subjects. As a result, Germany is now significantly above the OECD average (492 vs. 485 points) only in natural sciences. In mathematics (475 vs. 472 points) and reading (480 vs. 476 points), the results now match the OECD average, which has also fallen in both subjects.

After the first PISA study in 2000, Germany initially achieved improvements in its results and was able to maintain them at a high level. In the most recent PISA rounds, however, there were signs of a negative trend. The scores in mathematics and natural sciences are now below those of the PISA studies in the 2000s, when those subjects were assessed in detail for the first time (mathematics: PISA 2003; natural sciences: PISA 2006). The reading scores of the current study are around the same as in PISA 2000, when that subject was a focal point for the first time.

Only a few OECD countries were able to improve some of their results between 2018 and 2022, for example Japan in reading and Italy, Ireland and Latvia in science. In mathematics, students in Japan and Korea show the highest average performance. The top countries in reading are Ireland, Japan, Korea and Estonia. Japan, Korea, Estonia and Canada have the best results in natural sciences.

Focus of the eighth PISA study: Mathematics

In the eighth Program for International Student Assessment (PISA) study, the skills were assessed of a representative selection of approximately 6,100 15-year-old secondary school students attending around 260 schools of all types in Germany. Students also answered a questionnaire about learning conditions, attitudes and social background. School principals, teachers and parents answered questions on the lesson structure, classroom resources and the role of learning in the home. Approximately 690,000 students took part in the study worldwide. Every PISA cycle has a focus subject. This time it was mathematics.

The German part of the study was headed by the ZIB on behalf of the Standing Conference of the Ministers of Education and Cultural Affairs (KMK) and the Federal Ministry of Education and Research. Partners in the ZIB, alongside TUM, are the Leibniz Institute for Research and Information in Education (DIPF) and the Leibniz Institute for Pedagogy of Natural Sciences and Mathematics (IPN).

More students at low proficiency levels

According to the test scores, the study classifies the students in six proficiency levels. Students whose skills do not exceed proficiency level 1 require additional support in order to meet the demands of vocational training or further schooling and participate fully in society.

About one-third of the 15-year-olds achieved only these very low proficiency levels in at least one of the three tested subjects. Around one in six have significant deficiencies in all three subjects. The percentages of these particularly low-performing students have increased significantly since 2018. They now stand at 30% in mathematics, 26% in reading and 23% in science.

At the other end of the scale are the highly proficient students. In mathematics and reading they now represent only 9% and 8% of the total, respectively, while in natural sciences their share of the total remained unchanged, at around 10%.

The pandemic factor

The students’ answers to the questionnaires point to possible reasons for the lower scores: First, the researchers believe that school closures during the COVID pandemic negatively affected the ability to learn skills. In Germany schools made less use of digital media than the OECD average and relied more on materials mailed to students. “By comparison with other countries, Germany was not well prepared for distance learning in terms of digital hardware—but then caught up,” says study head Prof. Doris Lewalter, an educational researcher at TUM and managing director of the ZIB. Fewer than half of the low-proficiency students made use of available remedial options.

However, the analysis of the international data shows no systematic link between the decreases in proficiency between 2018 and 2022 and the duration of school closures. Some countries with relatively few school closure days have significantly lower scores than in 2018 while others, with higher numbers of days lost, show only small decreases or even slightly higher scores.

Language difficulties as a factor

A second possible factor to explain the results in the study’s focal area of mathematics: in Germany there is still a strong link between students’ proficiency in mathematics and the socio-economic status of their families and their immigrant background. Today’s 15-year-olds who themselves have immigrated to Germany are significantly less proficient at mathematics than the same group in 2012, when this question was last investigated. German is spoken less often in the homes of these students than in those of comparable students in 2012.

“This conclusion is only a partial explanation of the overall results, however,” says Prof. Lewalter. “The mathematical scores of non-immigrant students are also lower than in 2012—and even more so than for German-born children of immigrant parents.”

The factor of interest and motivation

To explain the long-term negative trend, the researchers therefore also take the students’ responses to the questionnaire regarding motivation, attitudes and classroom conditions. Compared to 2012, the students showed less enjoyment and interest in mathematics. The subject was also causing them more anxiety. In addition, the 15-year-olds see fewer potential benefits from learning mathematics.

“The results also show that the students feel less supported by their mathematics teachers. But this support is a key prerequisite for good instruction. In addition, the students have only limited awareness of teachers’ efforts to make lessons relevant to real-world contexts. This makes it more difficult for them to recognize the importance of mathematics in their lives—which can in turn decrease their motivation for the subject,” says Prof. Lewalter.

‘A big push’

As key conclusions from the PISA results, the educational researchers recommend:

• A systematic diagnosis and development of linguistic and reading proficiency from pre-school to the secondary level. “German language skills are the foundation of success at school,” says Prof. Lewalter.
• Ongoing development of instruction and the inclusion of digital media. “The students’ living realities are constantly changing and, along with them, the baseline conditions for applying mathematics, reading and science skills,” says Prof. Lewalter.
• Needs-oriented resource allocation to better equip schools with large numbers of students from disadvantaged families and with immigrant backgrounds.

“After the first PISA study in 2000, Germany was able to significantly improve students’ skills with effective support programs,” says Prof. Lewalter. “With a big push, combining the efforts of policy makers, schools and society, we can do it again.”

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Credit of the article given to Technical University Munich

Giving and receiving effective feedback is a pivotal tool across all levels of education and teaching. In mathematics education, constructive feedback enhances students’ understanding, builds their mathematical skills, and helps to foster a deeper connection between educators and their students.

This article explores what feedback is, how we can ensure the feedback we give our students is specific and actionable and, as teachers, how we, too, can constructively receive feedback.

What is feedback?

Feedback in maths goes beyond praise, criticism, reward or punishment. A mere ‘good job’ or ‘try again’ falls short of offering the constructive guidance required for improvement.

Effective feedback should:

• be clear and concise
• explain what the student did well, their strengths and what they understood
• point out and explain any misunderstandings
• suggest ways to improve or move forward.

In explicit teaching, feedback should be specific and actionable.

Specific: I know what needs to be done.

Actionable: I know what I need to do to move ahead.

Students should know precisely what needs improvement and how to achieve it. This not only clarifies the path to success, it also makes the feedback process more motivating for the individual student.

Why is it important?

Feedback in maths education serves various essential purposes:

• Promoting learning: It is a powerful catalyst for academic growth.
• Building motivation:Constructive feedback can inspire students to strive for excellence.
• Fostering relationships: Feedback provides an excellent opportunity for teachers to connect with students on a deeper level.

The human element: empathy and trust

Delivering feedback is not a formulaic, one-size-fits-all process. As with all elements of teaching, many factors influence how we interact with our students and how we establish what kind of feedback will work best for them.

Two key ingredients to delivering meaningful feedback are empathy and trust.

Without empathy and trust, feedback can feel invasive, critical and unwarranted. If we build a safe learning environment built on empathy and trust, our feedback will be more effective and our students more motivated for improvement.

When delivering feedback, avoid emotional reactions and remember that the level of trust in your classroom will significantly impact how your feedback is received.

Integrating feedback into planning

To create space for feedback, when planning teaching educators should:

• Set clear goals and success criteria: Determine what you aim to achieve with your students.
• Anticipate misconceptions: Be prepared to address common misunderstandings.
• Deliberate noticing: Continuously assess your students and their understanding, both academically and emotionally.
• Establish empathy and trust: Deliberate noticing allows teachers to connect with their students on a deeper level.

Feedback goes both ways

Teachers are often very adept at giving feedback but are not always so comfortable on the receiving end of it. Be open to receiving feedback from students, both sought feedback and inferred feedback.

Sought feedback: Give your students regular opportunities to share with you what is and isn’t working. Encourage them to be specific.

Inferred feedback: Where teachers identify students may not be fully comprehending the material. Roving around the classroom will give you more opportunities to pick up on your students’ inferred feedback.

Asking your students for feedback is a great way to improve your practice and also provides an opportunity as the teacher to model how to receive feedback. Giving and receiving purposeful feedback will help to create a learning environment that encourages open communication.

Feedback in mathematics education is not a one-size-fits-all approach but a human-centred process rooted in empathy, trust and a genuine desire for improvement. By embracing these principles and making space for feedback in teaching, educators can create a supportive environment where every student can excel in mathematics.

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Credit of the article given to The Mathematics Hub

Matrix multiplication – where two grids of numbers are multiplied together – forms the basis of many computing tasks, and an improved technique discovered by an artificial intelligence could boost computation speeds by up to 20 per cent.

Multiplying numbers is a fundamental task for computers

An artificial intelligence created by the firm DeepMind has discovered a new way to multiply numbers, the first such advance in over 50 years. The find could boost some computation speeds by up to 20 per cent, as a range of software relies on carrying out the task at great scale.

Matrix multiplication – where two grids of numbers are multiplied together – is a fundamental computing task used in virtually all software to some extent, but particularly so in graphics, AI and scientific simulations. Even a small improvement in the efficiency of these algorithms could bring large performance gains, or significant energy savings.

For centuries, it was believed that the most efficient way of multiplying matrices would be proportional to the number of elements being multiplied, meaning that the task becomes proportionally harder for larger and larger matrices.

But the mathematician Volker Strassen proved in 1969 that multiplying a matrix of two rows of two numbers with another of the same size doesn’t necessarily involve eight multiplications and that, with a clever trick, it can be reduced to seven. This approach, called the Strassen algorithm, requires some extra addition, but this is acceptable because additions in a computer take far less time than multiplications.

The algorithm has stood as the most efficient approach on most matrix sizes for more than 50 years, although some slight improvements that aren’t easily adapted to computer code have been found. But DeepMind’s AI has now discovered a faster technique that works perfectly on current hardware. The company’s new AI, AlphaTensor, started with no knowledge of any solutions and was presented with the problem of creating a working algorithm that completed the task with the minimum number of steps.

It found an algorithm for multiplying two matrices of four rows of four numbers using just 47 multiplications, which outperforms Strassen’s 49 multiplications. It also developed improved techniques for multiplying matrices of other sizes, 70 in total.

AlphaTensor discovered thousands of functional algorithms for each size of matrix, including 14,000 for 4×4 matrices alone. But only a small minority were better than the state of the art. The research builds on AlphaZero, DeepMind’s game-playing model, and has been two years in the making.

Hussein Fawzi at DeepMind says the results are mathematically sound, but are far from intuitive for humans. “We don’t really know why the system came up with this, essentially,” he says. “Why is it the best way of multiplying matrices? It’s unclear.”

“Somehow, the neural networks get an intuition of what looks good and what looks bad. I honestly can’t tell you exactly how that works. I think there is some theoretical work to be done there on how exactly deep learning manages to do these kinds of things,” says Fawzi.

DeepMind found that the algorithms could boost computation speed by between 10 and 20 per cent on certain hardware such as an Nvidia V100 graphics processing unit (GPU) and a Google tensor processing unit (TPU) v2, but there is no guarantee that those gains would also be seen on common devices like a smartphone or laptop.

James Knight at the University of Sussex, UK, says that a range of software run on supercomputers and powerful hardware, like AI research and weather simulation, is effectively large-scale matrix multiplication.
“If this type of approach was actually implemented there, then it could be a sort of universal speed-up,” he says. “If Nvidia implemented this in their CUDA library [a tool that allows GPUs to work together], it would knock some percentage off most deep-learning workloads, I’d say.”

Oded Lachish at Birkbeck, University of London, says the new algorithms could boost the efficiency of a wide range of software, because matrix multiplication is such a common problem – and more algorithms are likely to follow.

“I believe we’ll be seeing AI-generated results for other problems of a similar nature, albeit rarely something as central as matrix multiplication. There’s significant motivation for such technology, since fewer operations in an algorithm doesn’t just mean faster results, it also means less energy spent,” he says. If a task can be completed slightly more efficiently, then it can be run on less powerful, less power-intensive hardware, or on the same hardware in less time, using less energy.

But DeepMind’s advances don’t necessarily mean human coders are out of a job. “Should programmers be worried? Maybe in the far future. Automatic optimisation has been done for decades in the microchip design industry and this is just another important tool in the coder’s arsenal,” says Lachish.

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*Credit for article given to Matthew Sparkes*

When the NBA celebrated the start of its 75th season in fall 2021, it was clear that the 3-point shot adopted by the league in 1979-80 had transformed the sport.

The number of attempts beyond the arc had increased in each of the previous 10 seasons, from 22.2% in 2010-11 to 39.2% in 2020-21, and it had been nearly five years since a team won a game without making at least one 3-pointer (that streak is now up to eight years). Led by 3-point specialists Steph Curry and Klay Thompson, the Golden State Warriors had won three of the previous seven NBA titles and were about to win a fourth in 2022.

It appeared that the 3-point revolution would never end. But a recent study by Falk College of Sport and Human Dynamics sport analytics professor Shane Sanders and associate professor Justin Ehrlich shows that while the number of 3-point shots continues to increase, the average expected value of 3-pointers has become less than 2-pointers since the 2017-18 season.

“When taking fouled shots and made free throws into consideration, we found that what had long been a premium for the 3-point shot started to become a dispremium in the 2017-18 season and that trend is continuing,” Ehrlich says. “The implication of these findings is enormous in terms of potential impact on roster construction and offensive philosophies.”

The research preprint from Sanders and Ehrlich, “Estimating NBA Team Shot Selection Efficiency from Aggregations of True, Continuous Shot Charts: A Generalized Additive Model Approach,” is available through the Social Science Research Network website. Sanders and Ehrlich will present their paper as one of seven finalists in the research competition at the NBA-centric MIT Sloan Sports Analytics Conference March 1-2 in Boston, Massachusetts.

“In past conferences, there has been a lot of discussion among NBA executives about how basketball analytics created the 3-point ‘moneyball’ era of basketball and how this has impacted the popularity of the game,” Sanders says. “Perhaps ironically, our research uses basketball analytics, along with a fully specified team offensive objective function, to say there is now too much 3-point shooting for a point-maximizing offense.”

To conduct their research, Sanders and Ehrlich developed a new shot chart that uses a generalized additive model to estimate total shot proficiency continuously in the half-court. Their shot chart incorporates missed shots that draw a shooting foul—and shot-pursuant free throw scoring—to determine total scoring yield following a shot decision.

Current expected value formulas fall short by not including this additional information, which, when combined with the outcome of the initial shot attempt, results in what Sanders and Ehrlich call the “true point value” of a shot. For the 2022-23 NBA season:

• True Value from 2-point shot attempts=1.181
• True Value from 3-point shot attempts=1.094

And even when not factoring in free throws, the researchers found that the expected value from 3-point shots are now worth less than 2-point shots. For the 2022-23 NBA season:

• Expected value from 2P field goal attempt=2P% * 2 = .548 * 2= 1.096
• Expected value from 3P field goal attempt=3P% * 3 = .361 * 3= 1.083

The true value data can be found in this dashboard, and the graph above shows the expected and true values of 2- and 3-point shots from 2016-22.

According to this research, the expected value from average 2-point field goal attempts (FGA) is now worth 0.013 points more than average 3-point FGA, even before factoring in shot-pursuant free throw scoring. In other words, if you multiply the probability of making a 3-point FGA times the value of a 3-point FGA, it’s worth less than if you multiple a 2-point FGA times the value of a 2-point FGA.

When discussing true point value, the researchers use the term “shot attempts” instead of “field goal attempts” because their formula includes missed shots when a player is fouled, which is not included in standard field-goal attempt statistics. So, when including made and missed free throws, the disparity based on this new true value metric is even greater as average 2-point shot attempts are now worth 0.087 more points than 3-point shot attempts.

Officials from NBA teams and the league have discussed moving the 3-point line back from its current distance of 23 feet, 9 inches (22 feet in the corners). But as this study shows, the value of a 3-pointer is decreasing at the current distance, and teams are already starting to alter their shot selection to emphasize more high-percentage 2-point shots.

“These research findings do not coincide completely with the unresearched musings of NBA analysts Charles Barkley and Shaquille O’Neal,” Sanders says.

“For example, our findings do not suggest that such perimeter stars as Stephen Curry or Damian Lillard should not shoot a lot of threes. It means marginal stretch fours and other marginal outside shooters should not pull up for a 3 as often and that some marginal outside shooters should not extend their range to 25-26 feet or more. Players can still achieve the offensive spacing benefits of positioning on the perimeter without some players shooting from there quite as often.”

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Credit of the article given to Matt Michael, Syracuse University

Mathematicians who have studied the most efficient way to pack spheres in eight-dimensional space and the spacing of prime numbers are among this year’s recipients of the highest award in mathematics, the Fields medal.

Mathematicians who have studied the most efficient way to pack spheres in eight-dimensional space and the spacing of prime numbers are among this year’s recipients of the highest award in mathematics, the Fields medal.

The winners for 2022 are James Maynard at the University of Oxford; Maryna Viazovska at the Swiss Federal Institute of Technology in Lausanne (EPFL); Hugo Duminil-Copin at the University of Geneva, Switzerland; and June Huh at Princeton University in New Jersey.

Kyiv-born Viazovska is only the second female recipient among the 64 mathematicians to have received the award.

“Sphere packing is a very natural geometric problem. You have a big box, and you have an infinite collection of equal balls, and you’re trying to put as many balls into the box as you can,” says Viazovska. Her contribution was to provide an explicit formula to prove the most efficient stacking pattern for spheres in eight dimensions – a problem she says took 13 years to solve.

Maynard’s work involved understanding the gaps between prime numbers, while Duminil-Copin’s contribution was in the theory of phase transitions – such as water turning to ice, or evaporating into steam – in statistical physics.

June Huh, who dropped out of high school aged 16 to become a poet, was recognised for a range of work including the innovative use of geometry in the field of combinatorics, the mathematics of counting and arranging.

The medal, which is considered to be as prestigious as the Nobel prize, is given to two, three or four mathematicians under the age of 40 every four years.

The awards were first given out in 1936 and are named in honour of Canadian mathematician John Charles Fields. This year’s awards were due to be presented at the International Congress of Mathematicians in Saint Petersburg, Russia, but the ceremony was relocated to Helsinki, Finland.

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*Credit for article given to Matthew Sparkes*

It seems there’s a five-letter word describing what many players of the wildly popular Wordle puzzle do daily as they struggle to find a target word within six tries.

According to one mathematics expert, that word is “cheat.”

James P. Dilger, who by day is professor emeritus at Stony Brook University in New York specializing in the mechanisms of anesthetic action, and by night is a Wordle junkie, says the numbers behind published Wordle success rates don’t quite add up.

Wordle was developed by a software engineerto pass the time during the early days of COVID restrictions. Players must determine a target five-letter word in six or fewer attempts. With each guess, the player is provided with three bits of information: correct letters in the correct position are displayed in green, correct letters placed in incorrect spots are displayed in yellow, and incorrect letters are displayed in black.

In the beginning, Wordle was played mainly among family and friends of the developer, Josh Wardle. Wordle’s popularity soared, reaching 3 million users after The New York Times purchased the game in January 2022. Today, some 2 million play Wordle daily. It is recreated in 50 languages globally.

Dilger’s suspicions arose while studying the game’s statistics published daily by The Times.

“I noticed one day an awful lot of people answered with one guess and thought, ‘that’s strange,'” Dilger said. “And then I paid attention to it and it was happening day after day. Well, I’m a science nerd and wanted to know what’s going on.”

Dilger imported statistics covering four months of user guesses into an Excel spreadsheet. His report, “Wordle: A Microcosm of Life. Luck, Skill, Cheating, Loyalty, and Influence!” appeared in the preprint server arXiv Sept. 6.

The game has a data bank containing 2,315 words, good for five years of play. (There actually are more than 12,000 five-letter words in the English language, but The Times weeded out the most obscure ones.)

Dilger calculated that the odds of randomly guessing the day’s word at 0.043%, totaling 860 players. Yet, Times statistics show that the number of players making correct first guesses in each game never dipped below 4,000.

“Do I mean to tell you that never, not once, was the share percent of the first guess less than 0.2%? Yup!” Dilger asserted.

He went further. His numbers are based on the 2,315-word master list compiled by The Times, but 800 of those words have already been used. Most players are not likely to know that detail, but if they did, and they excluded words already played, their odds of guessing the correct word would rise slightly. Yet, according to Dilger, their odds would still be a low 0.066%.

“Yet, it happens consistently every day,” Dilger said. “Some days it’s as high as 0.5%,” which would be 10,000 players.

He also noted how unlikely it would be that a user would correctly guess such poor first-choice candidates as “nanny” and “igloo.” Players gain maximum advantage when they surmise words with non-repeating characters and as many vowels as possible. “Nanny” repeats one letter three times and uses only two vowels. “Igloo” not only is a relatively rare word, but contains only two vowels, repeating one of them.

“What shall we call these people?” He asked. “‘Cheaters’ comes to mind, so that’s what I call ’em.”

Dilger did not offer any explanation for such nefarious behaviour, other than to say that many players “became frustrated at some point in the game and then felt joy or relief after having surpassed the hurdle with a cheat.”

“We are baffled as to how first-word cheaters actually have fun playing,” Dinger said, “but that does not diminish our enjoyment of the game.”

He might have quoted former wrestler, actor, philosopher and governor of Minnesota Jesse Ventura, who once suggested, “Winners never cheat, and cheaters never win.” Except maybe in Wordle.

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Credit of the article given to Peter Grad, Phys.org

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.”

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Credit of the article given to Northwestern University

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

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

Imagine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching “out-of-field”. This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is maths.

A 2021 report on Australia’s teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both maths and maths education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in maths education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the maths education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in maths had markedly higher results, compared with those taught by out-of-field teachers.

We also know maths skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Maths has also been identified as a national skill shortages priority area.

There are worrying signs students are not receiving the maths education they need. Aaron Lefler/ Unsplash, CC BY

What do we do about this?

There have been repeated efforts to address teacher shortages,including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries. There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers: upgrading current teachers’ maths and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in maths will mean they can move from being out-of-field to “in-field”.

How to give teachers this training

A new report commissioned by mathematics and statistics organisations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in maths.

It identified 12 different courses to give existing teachers maths teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach maths in the junior and middle years of high school. Some taught university-level maths and others taught school-level maths. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be “in-field” for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time maths teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don’t know exactly how many extra teachers need to be trained in maths. One of the key recommendations of the report is for accurate national data of every teacher’s content specialisations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be maths teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in maths.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field maths teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial maths lessons every day by teachers who are not trained to teach maths, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

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Credit of the article given to Monstera Production/ Pexels , CC BY

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