Author: IMC

How can we explain and predict human behaviour? Are mathematics and probability up to the task, or are humans too complex and irrational?

Often, people’s actions take us by surprise, particularly when they seem irrational. Take the COVID pandemic: one thing nobody saw coming was a rush on toilet paper that left supermarket shelves bare in many countries.

But by combining ideas from mathematics, economics and behavioural science, researchers were eventually able to make mathematical models of how panic spreads between people, which made sense of the toilet paper panic.

In new research published in the Journal of the Royal Society Interface, we have taken a similar approach to the spread of disease – and showed that human reactions to the spread of disease can be as important as the behaviour of the disease itself when it comes to determining how an outbreak develops.

The power of context

One thing we know is that context can shape people’s behaviour in surprising ways. A nightly example of this is the popular TV game show Deal or No Deal, in which contestants regularly turn down offers of free money because they hope they will get a larger sum later.

If you carry out a rational calculation of the probabilities, most of the time the contestant’s “best” move is to accept the offer. But in practice, people often turn down a reasonable offer and hold out for a tiny chance at the big bucks.

Would a person refuse $5,000 if they were offered it in any other context? In this situation, straightforward maths can’t predict how people will behave.

The science of irrationality

What if we go beyond maths? Behavioural science has much to say about what drives people to take specific actions.

In this case, it might suggest people behave more reasonably if they set a realistic goal (such as getting $5,000) and position the goal in a powerful motivational context (such as planning to use the money to pay for a holiday).

Yet time and again even people with clear, achievable goals are swept up by emotion and context. At the right time and place, they will believe that luck is with them and refuse a $5,000 offer in the hope of something bigger.

Nevertheless, researchers have found ways to understand the behaviour of Deal or No Deal contestants by combining ideas from mathematics, economics and the study of behaviour around risky choices.

In essence, the researchers found contestants’ decisions are “path-dependent”. This means their choice to accept a bank offer depends not only on their goal and the odds, but also the choices they have already made.

Group behaviours

Deal or No Deal, of course, is largely about individuals making decisions in a certain context. But when we’re trying to understand the spread of disease, we’re interested in how whole groups of people behave.

This is the realm of social psychology, where group behaviours and attitudes can influence individual actions. In some ways this makes groups easier to predict, and it’s where combining mathematics and behavioural science really starts to produce results.

Although some mass behaviours at the start of the COVID pandemic were highly visible – like panic-buying toilet paper – others were not. Mobility data from Google showed people were choosing to limit their own movement, for example, before any mandated restrictions were in place.

Feedback loops

Fear and perceived risk can promote self-preservation through positive mass behaviours. For example, as more sickness appears in the community, people are more likely to act to prevent themselves getting sick.

These actions in turn have a direct impact on the spread of the disease, which further affects human behaviour, and so on. Many mathematical models of how diseases spread have failed to take this feedback loop into account.

Our new study is a step toward combining population disease spread modelling with mass behaviour modelling, aimed at understanding the links between behaviour and infection.

Our framework accounts for dynamic and self-driven protective health behaviours in the presence of an infectious disease. This puts us in a better position to make informed choices and policy recommendations for future epidemics.

Notably, our approach allows us to understand how mass behaviours influence how great a burden the disease will impose on the population in the long term. There is still much work to develop in this area.

To better understand human behaviour from a mathematical perspective, we will need better data around human choices in the presence of an infectious disease. This lets us pick out patterns that can be used for prediction.

Predicting behaviour

So, to come back to the question: can we predict human behaviour? Well, it depends. Many factors contribute to our choices: emotion, context, risk perception, social observation, fear, excitement.

Understanding which of these factors to explore with mathematics is no easy feat. However, when society faces so many challenges related to changes in mass behaviour – from infectious diseases to climate change – using mathematics to describe and predict patterns is a powerful tool.

But no single discipline can provide the answer to global challenges which need changes in human behaviour at scale. We will need more interdisciplinary teams to achieve meaningful impacts.

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Credit of the article given to The Conversation

In the 18th century, the philosopher David Hume observed that induction—inferring the future based on what’s happened in the past—can never be reliable. In 1997, SFI Professor David Wolpert with his colleague Bill Macready made Hume’s observation mathematically precise, showing that it’s impossible for any inference algorithm (such as machine learning or genetic algorithms) to be consistently better than any other for every possible real-world situation.

Over the next decade, the pair proved a series of theorems about this that were dubbed the “no-free-lunch” theorems. These proved that one algorithm could, in fact, be a bit better than another in most circumstances—but only at the cost of being far worse in the remaining circumstances.

These theorems have been extremely controversial since their inception, since they punctured the claims of many researchers that the algorithms they had developed were superior to other algorithms. As part of the controversy, in 2019, the philosopher Gerhard Schulz wrote a book wrestling with the implications of Hume’s and Wolpert’s work.

A special issue of the Journal for General Philosophy of Science published in March 2023 is devoted to Schulz’s book, and includes an article by Wolpert himself, in which he reviews the “no-free-lunch” theorems, pointing out that there are also many “free-lunch” theorems.

He states that the meta-induction algorithms that Schurz advocates as a “solution to Hume’s problem” are simply examples of such a free lunch based on correlations among the generalization errors of induction algorithms. Wolpert concludes that the prior algorithms that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference.

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Credit of the article given to Santa Fe Institute

Finding bats is hard. They are small, fast and they primarily fly at night.  But our new research could improve the way conservationists find bat roosts. We’ve developed a new algorithm that significantly reduces the area that needs to be searched, which could save time and cut labour cost.

Of course, you may wonder why we would want to find bats in the first place. But these flying mammals are natural pest controllers and pollinators, and they help disperse seeds. So they are extremely useful in contributing to the health of our environment.

Despite their importance though, bat habitats are threatened by human activities such as increased lighting, noise and land use. To ensure that we can study and enhance the health of our bat population, we need to locate their roosts. But finding bat roosts is a bit like finding a needle in a haystack.

Our previous work measured and modelled the motion of greater horseshoe bats in flight. Having such a model means we can predict where bats will be, depending on their roost position. But the position of the roost is something we often don’t know.

Our new research combines our previous mathematical model of bat motion with data gathered from acoustic recorders known as “bat detectors”. These bat detectors are placed around the environment and left there for several nights.

Seeing with sound

Bats use echolocation, which allows them to “see with sound” when they’re flying. If these ultrasonic calls are made within ten to 15 metres of a bat detector, the device is triggered to make a recording, providing an accurate record of where and when a bat was present.

The sound recordings also provide clues about the identity of the species. Greater horseshoe bats make a very distinctive “warbling” call at almost exactly 82kHz in frequency, so we can easily tell whether the species is present or not.

Assuming that a bat detector’s batteries last for a few nights, its memory card is not full, and the units are not stolen or vandalised, then we can use the bat call data to generate a map that shows the proportion of bat calls at each detector location.

Our model can also be used to predict the proportion of bat calls based on a given roost location. So, we split the environment up into a grid and simulate bats flying from each grid square. The grid square, or squares, whose simulations best reproduce the bat detector data will then be the most likely locations of the roost.

This simple algorithm can then be applied to whole terrains, meaning that we can create a map of likely roost locations. Cutting out the regions that are least likely to contain the roost can mean we shrink the search space to less than 1% of the initially surveyed area. Simplifying the process of finding bat roosts allows more of an ecologist’s time to be spent on conservation projects, rather than laborious searching.

In 2022, we developed an app that uses publicly available data to predict bat flight lines. At the moment the app can help ecologists, developers or local authority planners, know how the environment is used by bats. However, it needs a roost location to be specified first, and this information is not always known. Our new research removes this barrier, making the app easier to use.

Our work offers a way of identifying likely roost locations. These estimates can then be verified either by directly observing particular features, or by capturing bats at a nearby location and following them back home, using radiotracking.

Over the past two decades, bat detectors have gone from simple hand-held machines to high-performance devices that can collect data for days at a time. Yet they are usually deployed only to identify bat species. We have shown they can be used to identify the areas most likely to contain bat roosts, uncovering critical information about these most secretive of animals.

We hope that this will provide further tools for ecologists to optimise the initial microphone detector locations, thereby providing a holistic way of detecting bat roosts.

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Credit of the article given to Thomas Woolley and Fiona Mathews, The Conversation

“We wanted to investigate how second pillar pension funds react to financial crises and how to protect them from the crises,” says Kaunas University of Technology (KTU) professor Dr. Audrius Kabašinskas, who, together with his team, discovered a way to achieve this goal. The discovery in question is the development of stress tests for pension funds. Lithuanian researchers were the first in the world to come up with such an adaptation of the stress tests.

Stress tests are usually carried out on banks or other financial institutions to allow market regulators to determine and assess their ability to withstand adverse economic conditions.

According to the professor at KTU Faculty of Mathematics and Natural Sciences, this innovative pension fund stress testing approach will benefit both regulators and pension fund managers.

“Making sure your pension fund is resilient to harsh financial market conditions will help you sleep better, save more, and have increased trust in your funds and the pension system itself,” Kabašinskas adds.

Results based on two major crises

First, the study needed to collect data from previous periods. “Two major events that shocked the whole world—COVID-19 and the first year of Russian invasion of Ukraine—just happened to occur during the project. This allowed us to gather a lot of relevant information and data on changes in the performance of pension funds,” says Kabašinskas.

The Hidden Markov Model (HMM), which, according to a professor at KTU Department of Mathematical Modelling, is quite simple in its principle of operation, helped to forecast future market conditions in this study.

The paper is published in the journal Annals of Operations Research.

“The observation of air temperature could be an analogy for it. All year round, without looking at the calendar, we observe the temperature outside and, based on the temperature level, we decide what time of the year it is. Of course, 15 degrees can occur in winter and sometimes it snows in May but these are random events. The state of the next day depends only on today,” he explains vividly.

According to the KTU researcher, this describes the idea of the Hidden Markov Model: by observing the changes in value, one can judge the state of global markets and try to forecast the future.

“In our study, we observed two well-known investment funds from 2019 to 2022. Collected information helped us identify that global markets at any given moment are in one of four states: no shock regime, a state of shock in stock markets, a state of shock in bond markets, and a state of global financial shock—a global crisis,” says Kabašinskas.

Using certain methods, the research team led by a professor Miloš Kopa representing KTU and Charles University in Prague found that these periods were aligned with the global events in question. Once the transition probabilities between the states were identified, it was possible to link the data of pension funds to these periods and simulate the future evolution of the pension funds’ value.

That’s where the innovation of stress testing came in. The purpose of this test is to determine whether a particular pension fund can deliver positive growth in the future when faced with a shock in the financial markets.

“In our study, we applied several scenarios, extending financial crises and modeling the evolution of fund values over the next 5 years,” says a KTU researcher.

This methodology can be applied not only to pension funds but also to other investments.

Example of Lithuanian pension funds

The research and the new stress tests were carried out on Lithuanian pension funds.

Kabašinskas says that the study revealed several interesting things. Firstly, on average, Lithuanian second pillar pension funds can withstand crises that are twice as long.

“However, the results show that some Lithuanian funds struggle to cope with inflation, while others, the most conservative funds for citizens who are likely to retire within next few years or who have already retired, are very slow in recovering after negative shocks,” adds the KTU expert.

This can be explained by regulatory aspects and the related investment strategy, as stock markets recover several times faster than bond markets, and the above-mentioned funds invest more than 90% in bonds and other less risky instruments.

A complementary study has also been carried out to show how pension funds should change their investment strategy to avoid the drastic negative consequences of various financial crises and shocks.

“Funds that invest heavily in stocks and other risky instruments should increase the number of risk-free instruments slightly, up to 10%, before or after the financial crisis hits. Meanwhile, funds investing mainly in bonds should increase the number of stocks in their holdings. In both cases, the end of the crisis should be followed by a slow return to the typical strategy,” advises a mathematician.

Although the survey did not aim to increase people’s confidence in pension funds, the results showed that Lithuania’s second pillar pension funds are resilient to crisis and are worth trust. Historically they have delivered long-term growth, some have even outperformed inflation and price increases.

“Although short-term changes can be drastic, long-term growth is clearly visible,” says KTU professor Dr. Kabašinskas. “Lithuania, by the way, has a better system than many European countries,” he adds.

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April 2023 marks the 100th anniversary of the death of mathematician and philosopher John Venn. You may well be familiar with Venn diagrams—the ubiquitous pictures of typically two or three intersecting circles, illustrating the relationships between two or three collections of things.

For example, during the pandemic, Venn diagrams helped to illustrate symptoms of COVID-19 that are distinct from seasonal allergies. They are also often taught to school children and are typically part of the early curriculum for logic and databases in higher education.

Venn was born in Hull, UK, in 1834. His early life in Hull was influenced by his father, an Anglican priest—it was expected John would follow in his footstep. He did initially begin a career in the Anglican church, but later moved into academia at the University of Cambridge.

One of Venn’s major achievements was to find a way to visualize a mathematical area called set theory. Set theory is an area of mathematics which can help to formally describe properties of collections of objects.

For example, we could have a set of cars, C. Within this set, there could be subsets such as the set of electric cars, E, the set of petrol based cars, say P, and the set of diesel powered cars, D. Given these, we can operate on them, for example, to apply car charges to the sets P and D, and a discount to the set E.

These sorts of operations form the basis of databases, as well as being used in many fundamental areas of science. Other major works of Venn’s include probability theory and symbolic logic. Venn had initially used diagrams developed by the Swiss mathematician Leonard Euler to show some relationships between sets, which he then developed into his famous Venn diagrams.

Venn used the diagrams to prove a form of logical statement known as a categorical syllogism. This can be used to model reasoning. Here’s an example: “All computers need power. All AI systems are computers.” We can chain these together to the conclusion that “all AI systems need power.”

Today, we are familiar with such reasoning to illustrate how different collections relate to each other. For example, the SmartArt tool in Microsoft products lets you create a Venn diagram to illustrate the relationships between different sets. In our earlier car example, we could have a diagram showing electric cars, E, and petrol powered cars, P. The set of hybrid cars that have a petrol engine would be in the intersection of P and E.

Logic and computing

The visualization of sets (and databases) is helpful, but the importance of Venn’s work then—and now—is the way they allowed proof of George Boole’s ideas of logic as a formal science.

Venn used his diagrams to illustrate and explore such “symbolic logic”—defending and extending it. Symbolic logic underpins modern computing, and Boolean logic is a key part of the design of modern computer systems—making his work relevant today.

Venn’s work was also crucial to the work of philosopher Bertrand Russell, showing that there are problems that are unsolvable. We can express such problems with sets, in which each is an unsolvable problem. One such unsolvable problem can be expressed with the “Barber paradox.” Suppose we had an article in Wikipedia containing all the articles that don’t contain themselves—a set. Is this new article itself in that set?

Luckily we can visualize that with a Venn diagram with two circles, where one circle is the set of entries that don’t include themselves, A, and the other circle is the set of entries that do include themselves, B.

We can then ask the question: where do we put the article that contains all the articles that don’t contain themselves? Have a think about it, then see where you would put it.

The problem is that it cannot be on the left, as it would contain itself, and would therefore be inconsistent. And it cannot be on the right, as then it would be missing, or incomplete. And it can’t be in both. It must be in one or the other. This paradox illustrates how unsolvable statements can arise—they are valid in terms of expressing them within the logical system, but ultimately unanswerable. We could possibly extend our system to solve this, but then we would end up with another unanswerable question.

Venn’s diagrams were crucial in understanding this. And this area of science is still important, for example when considering the limitations of machine learning and AI, where we may ask questions that cannot be answered.

Venn also had an interest in building mechanical machines—including a bowling machinewhich proved so effective it was able to bowl out some top Australian batsmen of the day.

Following his abstract work on logic, he developed the concept of a logical-diagram machine with a lot of processing power: though this brilliant idea from 1881 would take many decades to appear as modern computers.

We remember Venn here in Hull, with a bridge close to his birthplace decorated with Venn circle inspired artwork. At the University of Hull’s main administration building, there’s an intersection of management and academia which is called the Venn building.

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Credit of the article given to Neil Gordon, The Conversation

How can you figure out which points lie on a certain curve? And how many possible curves do you count by a given number of points? These are the kinds of questions Pim Spelier of the Mathematical Institute studied during his Ph.D. research. Spelier received his doctorate with distinction on June 12.

What does counting curves mean on an average day? “A lot of sitting and gazing,” Spelier replies. “When I’m asked what exactly do, can’t always answer that easily. Usually give the example about the particle traveling through time.”

All possible curves

Imagine a particle moving through space and you follow the path the particle makes through time. That path is a curve, a geometric object. How many possible paths can the particle follow, if we assume certain properties? For example, a straight line can only pass through two points in one way. But how many paths are possible for the particle if we look at more difficult curves? And how do you study that?

By looking at all possible curves at the same time. For example, all possible directions from a given point form with each other a circle, and that is called a modulspace. And that circle is itself a geometric object.

The mathematical magic can happen because this set of all curves itself has geometrical properties, Spelier says, to which you can apply geometrical tricks. Next, you can make that far more complicated with even more complex curves and spaces. So not counting in three but, for example, in eleven dimensions.

Spelier tries to find patterns that always apply to the curves he studies. His approach? Breaking up complicated spaces into small, easy spaces. You can also break curves into partial curves. That way, the spaces in which you’re counting are easier. But the curves sometimes get complicated properties, because you have to be able to glue them back together.

Spelier says, “The goal is to find enough principles to determine the number of curves exactly.”

In addition to curves, Spelier also counted points on curves. He studied the question: how many solutions does a given mathematical equation have?

These are equations that are a bit more complicated than the a² + b² = c² of the Pythagorean theorem. That equation is about the lengths of the sides of a right triangle. If you replace the squares with higher powers, it is more difficult to investigate solutions. Spelier studied solutions in whole numbers, for example, 3² + 4² = 5².

Meanwhile, there is a method to find those solutions. Professor of Mathematics Bas Edixhoven, who died in 2022, and his Ph.D. student Guido Lido developed an alternative approach to the same problem. But to what extent the two methods match and differ was still unclear. During his Ph.D. research, Spelier developed an algorithm to investigate this.

The first person with an answer

Developing that algorithm is necessary to implement the method. If you want to do it by hand, you get pages and pages of equations. Edixhoven’s method uses algebraic geometry. Through clever geometric tricks, you can calculate exactly the whole number points of a given curve. Spelier proved that the Edixhoven-Lido method is better than the old one.

David Holmes, professor of Pure Mathematics and supervisor of Spelier, praises the proof provided. “When you’re the first person to answer a question that everyone in our community wants an answer to, that’s very impressive. Pim proves that these two methods for finding rational points are similar, an issue that really kept mathematicians busy.”

Doing math together

The best part of his Ph.D.? The meetings with his supervisor. After the first year, it was more collaboration than supervision, both for Spelier and Holmes. Spelier says, “Doing math together is still more fun than doing it alone.”

Spelier starts in September as a postdoc in Utrecht and is apparently not yet done with counting. After counting points and curves, he will soon start counting surfaces.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Leiden University.

With mass shootings happening randomly every year in the United States, it may seem that there is no way to predict where the next horrific event is most likely to occur. In a new study published by the journal Risk Analysis, scientists at Iowa State University calculate the annual probability of a mass shooting in every state and at public places such as shopping malls and schools.

Their new method for quantifying the risk of a mass shooting in specific places could help security officials make informed decisions when planning for emergency events.

For their analysis, Iowa State associate professor Cameron MacKenzie and his doctoral student Xue Lei applied statistical methods and computer simulations to a database of mass shootings recorded from 1966 to 2020 by the Violence Project. The Violence Project defines a mass shooting as an incident with four or more victims killed by a firearm in a public place.

According to the Violence Project, the U.S. has experienced 173 public mass shootings from 1966 to 2020—with at least one mass shooting every year since 1966.

After they generated a probability distribution of annual mass shootings in the U.S., the scientists used two different models to simulate the annual number of mass shootings in each state. The results were used to calculate the expected number of mass shootings and the probability that at least one mass shooting would occur in each state in one year.

The Violence Project also provides the percentage of mass shootings in different types of locations. MacKenzie and Lei used that data to calculate the probability of a mass shooting in nine different types of public locations (including a restaurant, school, workplace, or house of worship) in the states of California and Iowa and also at the two largest high schools in each of those states.

Their findings include the following:

• The states with the greatest risk of a mass shooting are the most populous states: California, Texas, Florida, New York, and Pennsylvania. Together they account for almost 50% of all mass shootings.
• Some states, such as Iowa and Delaware, have never experienced a mass shooting.
• The annual risk of a mass shooting at the largest California high school is about 10 times greater than the risk at the largest Iowa high school.
• The number of mass shootings in the U.S. has increased by about one shooting every 10 years since the 1970s.

Importantly, MacKenzie points out that the probability of a mass shooting at a specific location depends on the definition of a mass shooting. In contrast to the Violence Project, the Gun Violence Archive defines a mass shooting as four or more individuals shot, injured or killed, in any location, not necessarily a public location. As a result, The Gun Violence Archive has collected data on shootings that occur in both public and private locations as well as targeted shootings (i.e., a gang shooting).

When the researchers applied data from The Gun Violence Archive to their models, the predicted number of annual mass shootings was nearly 100 times greater than the forecast based on The Violence Project’s data. The models predicted 639 mass shootings in 2022 with a 95% chance that the U.S. would experience between 567 and 722 mass shootings in that same year.

MacKenzie points out that “most media appear to use this broader definition of mass shootings.” Because of this, he urges that journalists explain how they are defining a mass shooting when reporting the statistical data.

With regard to the danger posed to children at school, MacKenzie explains, that “our results show that it is very, very unlikely that a specific student will attend a K-12 school and experience a mass shooting. But to parents of a child at a school that has experienced a mass shooting, explaining that the school was extremely unlucky provides no comfort.”

While it is important to take precautions, he adds that “we should not live in fear that our children will experience such a horrific event. Mass shootings are very low probability but very high consequence events.”

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Credit of the article given to Society for Risk Analysis

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

Researchers at The University of Western Australia’s ARC Industrial Transformation Research Hub for Transforming Energy Infrastructure through Digital Engineering (TIDE) have made a significant mathematical breakthrough that could help transform ocean research and technology.

Research Fellow Dr. Lachlan Astfalck, from UWA’s School of Physics, Mathematics and Computing, and his team developed a new method for spectral density estimation, addressing long-standing biases and paving the way for more accurate oceanographic studies.

The study was published in the journal Biometrika, known for its emphasis on original methodological and theoretical contributions of direct or potential value in applications.

“Understanding the ocean is crucial for numerous fields, including offshore engineering, climate assessment and modeling, renewable technologies, defense and transport,” Dr. Astfalck said.

“Our new method allows researchers and industry professionals to advance ocean technologies with greater confidence and accuracy.”

Spectral density estimation is a mathematical technique used to measure the energy contribution of oscillatory signals, such as waves and currents, by identifying which frequencies carry the most energy.

“Traditionally, Welch’s estimator has been the go-to method for this analysis due to its ease of use and widespread citation, however this method has an inherent risk of bias, which can distort the expected estimates based on the model’s assumption, a problem often overlooked,” Dr. Astfalck said.

The TIDE team developed the debiased Welch estimator, which uses non-parametric statistical learning to remove these biases.

“Our method improves the accuracy and reliability of spectral calculations without requiring specific assumptions about the data’s shape or distribution, which is particularly useful when dealing with complex data that doesn’t follow known analytical patterns, such as internal tides in oceanic shelf regions,” Dr. Astfalck said.

The new method was recently applied in a TIDE research project by Senior Lecturer at UWA’s Oceans Graduate School and TIDE collaborator, Dr. Matt Rayson, to look at complex non-linear ocean processes.

“The ocean is difficult to measure and understand and the work we are doing is all about uncovering some of those mysteries,” Dr. Rayson said.

“The new method means we can better understand ocean processes, climate models, ocean currents and sediment transport, bringing us closer to developing the next generation of numerical ocean models.

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Credit of the article given to University of Western Australia.

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