Author: IMC

All a_i lie on the same sheet of a cone with vertex x. The right-hand picture is not true to scale relative to the given numerical example. Credit: Advances in Applied Mathematics (2024). DOI: 10.1016/j.aam.2024.102741

The summer holidays are ending, which for many concludes with a long drive home and reliance on GPS devices to get safely home. But every now and then, GPS devices can suggest strange directions or get briefly confused about your location. But until now, no one knew for sure when the satellites were in a good enough position for the GPS system to give reliable direction.

TU/e’s Mireille Boutin and her co-worker Gregor Kemper at the Technical University of Munich have turned to mathematics to help determine when your GPS system has enough information to determine your location accurately. The research is published in the journal Advances in Applied Mathematics.

“In 200 meters, turn right.” This is a typical instruction that many have heard from their global positioning system (GPS).

Without a doubt, advancements in GPS technologies and mobile navigation apps have helped GPS play a major role in modern car journeys.

But, strictly adhering to instructions from GPS devices can lead to undesirable situations. Less serious might be turning left instead of right, while more serious could be driving your car into a harbor—just as two tourists did in Hawaii in 2023. The latter incident is very much an exception to the rule, and one might wonder: “How often does this happen and why?”

GPS and your visibility

“The core of the GPS system was developed in the mid-1960s. At the time, the theory behind it did not provide any guarantee that the location given would be correct,” says Boutin, professor at the Department of Mathematics and Computer Science.

It won’t come as a surprise then to learn that calculating an object’s position on Earth relies on some nifty mathematics. And they haven’t changed much since the early days. These are at the core of the GPS system we all use. And it deserved an update.

So, along with her colleague Gregor Kemper at the Technical University of Munich, Boutin turned to mathematics to expand on the theory behind the GPS system, and their finding has recently been published in the journal Advances in Applied Mathematics.

How does GPS work?

Before revealing Boutin and Kemper’s big finding, just how does GPS work?

Global positioning is all about determining the position of a device on Earth using signals sent by satellites. A signal sent by a satellite carries two key pieces of information—the position of the satellite in space and the time at which the position was sent by the satellite. By the way, the time is recorded by a very precise clock on board the satellite, which is usually an atomic clock.

Thanks to the atomic clock, satellites send very accurate times, but the big issue lies with the accuracy of the clock in the user’s device—whether it’s a GPS navigation device, a smartphone, or a running watch.

“In effect, GPS combines precise and imprecise information to figure out where a device is located,” says Boutin. “GPS might be widely used, but we could not find any theoretical basis to guarantee that the position obtained from the satellite signals is unique and accurate.”

Google says ‘four’

If you do a quick Google search for the minimum number of satellites needed for navigation with GPS, multiple sources report that you need at least four satellites.

But the question is not just how many satellites you can see, but also what arrangements can they form? For some arrangements, determining the user position is impossible. But what arrangements exactly? That’s what the researchers wanted to find out.

“We found conjectures in scientific papers that seem to be widely accepted, but we could not find any rigorous argument to support them anywhere. Therefore, we thought that, as mathematicians, we might be able to fill that knowledge gap,” Boutin says.

To solve the problem, Boutin and Kemper simplified the GPS problem to what works best in practice: equations that are linear in terms of the unknown variables.

“A set of linear equations is the simplest form of equations we could hope for. To be honest, we were surprised that this simple set of linear equations for the GPS problem wasn’t already known,” Boutin adds.

The problem of uniqueness

With their linear equations ready, Boutin and Kemper then looked closely at the solutions to the equations, paying special attention as to whether the equations gave a unique solution.

“A unique solution implies that the only solution to the equations is the actual position of the user,” notes Boutin.

If there is more than one solution to the equations, then only one is correct—that is, the true user position—but the GPS system would not know which one to pick and might return the wrong one.

The researchers found that nonunique solutions can emerge when the satellites lie in a special structure known as a “hyperboloid of revolution of two sheets.”

“It doesn’t matter how many satellites send a signal—if they all lie on one of these hyperboloids then it’s possible that the equations can have two solutions, so the one chosen by the GPS could be wrong,” says Boutin.

But what about the claim that you need at least four satellites to determine your position? “Having four satellites can work, but the solution is not always unique,” points out Boutin.

Why mathematics matters

For Boutin, this work demonstrates the power and application of mathematics.

“I personally love the fact that mathematics is a very powerful tool with lots of practical applications,” says Boutin. “I think people who are not mathematicians may not see the connections so easily, and so it is always nice to find clear and compelling examples of everyday problems where mathematics can make a difference.”

Central to Boutin and Kemper’s research is the field of algebraic geometry in which abstract algebraic methods are used to solve geometrical, real-world problems.

“Algebraic geometry is an area of mathematics that is considered very abstract. I find it nice to be reminded that any piece of mathematics, however abstract it might be, may turn out to have practical applications at some point,” says Boutin.

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

Credit of the article to be given Eindhoven University of Technology

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

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

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

Why do schools stream?

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

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

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

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

[We stream] to push the better students forward.

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

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

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

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

The ‘glass ceiling effect’

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

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

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

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

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

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

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

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

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

What does other research say?

International research has also found streaming students is inequitable.

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

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

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

What should we do instead?

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

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

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

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

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

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

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

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

A pair of mathematicians studied the UK National Lottery and figured out a combination of 27 tickets that guarantees you will always win, but they tell New Scientist they don’t bother to play.

David Cushing and David Stewart calculate a winning solution

Earlier this year, two mathematicians revealed that it is possible to guarantee a win on the UK national lottery by buying just 27 tickets, despite there being 45,057,474 possible draw combinations. The pair were shocked to see their findings make headlines around the world and inspire numerous people to play these 27 tickets – with mixed results – and say they don’t bother to play themselves.

David Cushing and David Stewart at the University of Manchester, UK, used a mathematical field called finite geometry to prove that particular sets of 27 tickets would guarantee a win.

They placed each of the lottery numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then used these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets will cover every possible winning combination of two numbers, the minimum needed to win a prize. Each ticket costs £2.

It was an elegant and intuitive solution to a tricky problem, but also an irresistible headline that attracted newspapers, radio stations and television channels from around the world. And it also led many people to chance their luck – despite the researchers always pointing out that it was, statistically speaking, a very good way to lose money, as the winnings were in no way guaranteed to even cover the cost of the tickets.

Cushing says he has received numerous emails since the paper was released from people who cheerily announce that they have won tiny amounts, like two free lucky dips – essentially another free go on the lottery. “They were very happy to tell me how much they’d lost basically,” he says.

The pair did calculate that their method would have won them £1810 if they had played on one night during the writing of their research paper – 21 June. Both Cushing and Stewart had decided not to play the numbers themselves that night, but they have since found that a member of their research group “went rogue” and bought the right tickets – putting himself £1756 in profit.

“He said what convinced him to definitely put them on was that it was summer solstice. He said he had this feeling,” says Cushing, shaking his head as he speaks. “He’s a professional statistician. He is incredibly lucky with it; he claims he once found a lottery ticket in the street and it won £10.”

Cushing and Stewart say that while their winning colleague – who would prefer to remain nameless – has not even bought them lunch as a thank you for their efforts, he has continued to play the 27 lottery tickets. However, he now randomly permutes the tickets to alternative 27-ticket, guaranteed-win sets in case others have also been inspired by the set that was made public. Avoiding that set could avert a situation where a future jackpot win would be shared with dozens or even hundreds of mathematically-inclined players.

Stewart says there is no way to know how many people are doing the same because Camelot, which runs the lottery, doesn’t release that information. “If the jackpot comes up and it happens to match exactly one of the [set of] tickets and it gets split a thousand ways, that will be some indication,” he says.

Nonetheless, Cushing says that he no longer has any interest in playing the 27 tickets. “I came to the conclusion that whenever we were involved, they didn’t make any money, and then they made money when we decided not to put them on. That’s not very mathematical, but it seemed to be what was happening,” he says.

And Stewart is keen to stress that mathematics, no matter how neat a proof, can never make the UK lottery a wise investment. “If every single man, woman and child in the UK bought a separate ticket, we’d only have a quarter chance of someone winning the jackpot,” he says.

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

*Credit for article given to Matthew Sparkes*

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

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

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

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

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

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

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

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

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

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

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

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

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

*Credit for article given to Alex Wilkins*

Math experts have developed new ways to provide further evidence for human-caused global heating and predict how close Earth is to reaching dangerous climate tipping points.

Tipping points occur when lots of small changes in climate build up to create a sudden, large change. Passing these thresholds leads to accelerated warming and extreme weather events.

For example, in years to come, the Atlantic Meridional Overturning Circulation (AMOC), is at risk of collapsing. The AMOC is an ocean current transport system that brings warm waterto the North Atlantic and helps to regulate temperatures in Europe and North America. If it collapses, the regional and global climate may become more erratic.

In July 2023, a study suggested the AMOC could collapse any time between 2025 and 2095. Another recent study, from October 2023, indicates that we are in dangerous proximity of the collapse of the Greenland ice sheet, yet with some hope of avoiding it if proper action is taken.

Now, a study published in the journal Nature Reviews Physics combines mathematics and physics to study climate change across time scales. It will help scientists refine their predictions for AMOC’s collapse as well as help them understand the proximity to various other climate tipping points.

Lead author Professor Valerio Lucarini, Professor of Statistical Mechanics at the University of Reading, said, “Our study provides academics and policymakers with rigorous mathematical tools needed to understand and predict climate change, and, specifically to detect and avoid climate tipping points resulting from human activities. It gives us a unified perspective linking climate variability and climate change. Our approach allows one to seamlessly study gradual climate changes and tipping points.

“We now have a robust framework for capturing the richness of climate dynamics across timescales. The findings provide tools for building up further evidence for human-caused global heating.”

Innovations and collaborations

The work will significantly advance the development of climate models and theories to generate more accurate predictions of climate change and evaluation of tipping point proximity, helping to inform mitigation policies and climate adaptation strategies.

The research critically investigated 2021 Nobel Prize in Physics winner Klaus Hasselmann’s stochastic modeling approach, which revolutionized climate change analysis. The new study refined Hasselmann’s techniques using recent tools developed within the mathematical and physical scientific literature.

The scientists noted how their methodology allows to better understand the nearing of the collapse of the AMOC. Observations show the AMOC is weakening. In addition to AMOC, the new method could be used to detect proximity to tipping points for:

• The collapse of ecosystems
• The melting of Greenland ice sheets
• The dieback of the Amazon rainforest

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

Credit of the article given to University of Reading

Chinese ice-ray lattice, or “binglie” as it is called in Chinese, is an intricate pattern that looks like cracked ice and is a common decorative element used in traditional Chinese window designs.

Originally inspired by fragmented patterns on ice or crackle-glazed ceramic surfaces, the design represents the melting of the ice and the beginning of a thriving spring.

When Dr. Iasef Md Rian, now an Associate Professor at Xi’an Jiaotong-Liverpool University’s Department of Architecture, arrived in China for the first time in 2019, he was immediately captivated by the latticed window designs in the classical gardens of Suzhou.

“Classical gardens in China strike me as very different from the Western ones, which are more symmetrical and organized,” he says. “Chinese gardens, however, have a more natural formation in their layout and design. The ice-ray window design is one of the manifestations.”

Having focused on fractal geometry in architectural design for many years, Dr. Rian felt an urge to explore the beauty in the patterns.

“My mind is always looking for this kind of inspiration source, so I was motivated right away to study the underlying geometric principles of the ice-ray patterns, he says.”

Revealing the underlying rule

Dr. Rian finds that the rule of creating ice-ray patterns is actually very simple.

He explains, “Take Type 1 as an example; a square is first divided into two quadrilaterals, and then each quadrilateral is further divided into two quadrilaterals. In each step, the proportions of the subdivided quadrilaterals are different, and this is how the random pattern is created using a simple rule.

“Through this configuration, Chinese craftsmen might have intended to increase its firmness so it can function as a window fence to provide protection. The random configuration of ice-ray lattices provides multi-angled connections, which transform the window into a collection of resultant forces and uniform stress distribution, in turn achieving a unique degree of stiffness.

“The microstructure of trabecular bone tissue in our own bodies serves as an excellent natural example of the potential of random lattices. It balances high stiffness, which contributes to strength, with a surprisingly lightweight structure.”

Dr. Rian recently published a paper in Frontiers of Architectural Research that explores the geometric qualities of ice-ray patterns and expands the possibilities of integrating random patterns into structural designs, especially the lattice shell design, which is often used in spherical domes and curved structures.

“In my research, I developed an algorithm to model the ice-ray patterns for lattice shell designs and assessed their feasibility and effectiveness compared to conventional gridshells. These gridshells, made from regular grids, contrast with continuous shells.

“While regular gridshells perform well under uniform loads, the ice-ray lattice offers strength under asymmetric loads. Some ice-ray patterns, resulting from optimization, surprisingly provide better strength than regular gridshells under self-weight. There is also an additional aesthetic advantage when applying the ice-ray pattern to a lattice shell design.

“I extend the application of this pattern to curved surfaces, which helps to unlock its potential in the geometric, structural, and constructional aspects of lattice shell design,” he says.

Dr. Rian has also integrated ice-ray patterns and complex geometries into his teaching. In 2022, he organized a workshop for students to design ice-ray lattice roofs.

He explains that learning the concept of fractal geometry can really push the students’ ideas toward a unique design.

“This is very different from what they’ve learned in high school. In learning to create this geometry system, they will also learn computational modeling and simulations. In the end, they’ll get comprehensive knowledge of advanced architectural and digital design,” he says.

Rediscovering traditional designs

To extend the research in this field, Dr. Rian is investigating the effectiveness of complex geometry in various aspects like micro-scale material design and structural design.

He says, “For instance, in facade design, we usually use conventional or parametric geometry to design regular shapes. However, the random shapes designed with complex geometry can offer a more natural impression and daylight penetration.”

He encourages design students and researchers to learn from the past.

“Any traditional design has a hidden rule in it. We can now use digital technologies and advanced tools to extend and expand the knowledge of traditional craftsmanship for contemporary design.

“There are many inspirations behind the traditional designs, and those principles can really inspire us designers to make innovative designs for the future,” he says.

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

Credit of the article given to Yi Qian, Xi’an jiaotong-Liverpool University

Putting off college math could improve the likelihood that students remain in college. But that may only be true as long as students don’t procrastinate more than one year. This is what colleagues and I found in a study published in 2023 of 1,119 students at a public university for whom no remedial coursework was required during their first year.

Enrolling in a math course during the first semester of college resulted in students being four times more likely to drop out. Although delayed enrollment in a math course had benefits in the first year, its advantages vanished by the end of the second year. In our study, almost 40% of students who postponed the course beyond a year did not attempt it at all and failed to obtain a degree within six years.

Why it matters

Nearly 1.7 million students who recently graduated from high school will immediately enroll in college. Math is a requirement for most degrees, but students aren’t always ready to do college-level math. By putting off college math for a year, it gives students time to adjust to college and prepare for more challenging coursework.

Approximately 40% of four-year college students must first take a remedial math course. This can extend the time it takes to graduate and increase the likelihood of dropping out. Our study did not apply to students who need remedial math.

For students who do not require remedial courses, some delay can be beneficial, but students’ past experiences in math can lead to avoidance of math courses. Many students experience math anxiety. Procrastination can be an avoidance strategy for managing fears about math. The fear of math for students may be a more significant barrier than their performance.

It is estimated that at least 17% of the population will likely experience high levels of math anxiety. Math anxiety can lead to a drop in math performance. It can also lead to avoiding majors and career paths involving math.

Our study fills the void in research on the effects of how soon students take college-level math courses. It also supports prior evidence that students benefit from a mix of coursework that is challenging yet not overwhelming as they transition to college.

What still isn’t known

We believe colleges need to better promote student confidence in math by examining how student success courses can reduce math anxiety. Student success courses provide students with study skills, note-taking skills, goal setting, time management and stress management, as well as career and financial decision making to support the transition to college. Although student success courses are a proven practice that help students stick with college, rarely do these courses address students’ fear of math.

Students are at the greatest risk of dropping out of college during their first year. Advisors play a crucial role in providing students with resources for success. This includes recommendations on what courses to take and when to take them. More research is also needed about how advisors can effectively communicate the impact of when math is taken by students.

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

Credit of the article given to Forrest Lane, The Conversation

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

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

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

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

Data shows homeschooled students fare well

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

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

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

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

Areas of concern abound as homeschool growth accelerates

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

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

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

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

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

3 ways to improve homeschooling

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

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

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

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

Credit of the article given to The Conversation

Australian teenagers have more disruptive maths classrooms and experience bullying at greater levels than the OECD average, a new report shows.

But in better news, Australian students report high levels of curiosity, which is important for both enjoyment and achievement at school.

The report, by the Australian Council for Educational Research (ACER) analysed questionnaire responses from more than 13,430 Australian students and 743 principals, to understand how their school experiences impact on maths performance.

What is the research?

This is the second report exploring Australian data from the 2022 Programme for International Student Assessment (PISA).

Australian teenagers have more disruptive maths classrooms and experience bullying at greater levels than the OECD average, a new report shows.

But in better news, Australian students report high levels of curiosity, which is important for both enjoyment and achievement at school.

The report, by the Australian Council for Educational Research (ACER) analysed questionnaire responses from more than 13,430 Australian students and 743 principals, to understand how their school experiences impact on maths performance.

What is the research?

This is the second report exploring Australian data from the 2022 Programme for International Student Assessment (PISA).

Author provided (no reuse)

The advantage gap

ACER’s first PISA 2022 report showed students from disadvantaged socioeconomic backgrounds were six times more likely to be low performers in maths than advantaged students.

It also showed the achievement gap between these two groups had grown by 19 points (or about one year of learning) since 2018.

This second report provides more insight into the challenges faced by disadvantaged students.

It shows a greater proportion of this group report learning in a less favourable disciplinary climate, experience lower levels of teacher support and feel less safe at school than their more advantaged peers.

Girls are more worried than boys

In last year’s report, the mean score for maths performance across OECD countries was nine points lower for girls than it was for boys. In Australia, the difference was 12 points.

The new report also showed differences in wellbeing. In 2022, a greater number of girls reported they panicked easily (58% compared to 23% of boys), got nervous easily (71% compared to 39%) and felt nervous about approaching exams (75% compared 49%).

Almost double the percentage of girls reported feeling anxious when they didn’t have their “digital device” near them (20% compared to 11%). Whether this was a phone, tablet or computer was not specified.

Overall, students who reported feeling anxious when they did not have their device near them scored 37 points lower on the maths test than those who reported never feeling this way or feeling it “half the time”.

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Curiosity a strong marker for performance

Curiosity was measured for the first time in PISA 2022. This included student behaviours such as asking questions, developing hypotheses, knowing how things work, learning new things and boredom.

Students in Singapore, the highest performing country in PISA 2022, showed the greatest levels of curiosity, followed by Korea and Canada. These were the only comparison countries to have a significantly higher curiosity score than Australia, with the Netherlands showing the lowest curiosity score overall.

As ACER researchers note: “curiosity is associated with greater psychological wellbeing” and “leads to more enjoyment and participation in school and higher academic achievement”.

They found Australia’s foreign-born students reported being more curious than Australian-born students, with 74% compared to 66% reporting that they liked learning new things.

What next?

Their findings highlight concerns for Australian education, such as persistently poor outcomes for disadvantaged students and higher stress levels experienced by girls. We need to better understand why this is happening.

But they also identify behaviours and conditions – such as high levels of curiosity – that contribute to a good maths performance and can be used by schools and policymakers to plan for better outcomes.

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

Credit of the article given to The Conversation

Luis Caffarelli has been awarded the most prestigious prize in mathematics for his work on nonlinear partial differential equations, which have many applications in the real world.

Luis Caffarelli has won the 2023 Abel prize, unofficially called the Nobel prize for mathematics, for his work on a class of equations that describe many real-world physical systems, from melting ice to jet engines.

Caffarelli was having breakfast with his wife when he found out the news. “The breakfast was better all of a sudden,” he says. “My wife was happy, I was happy — it was an emotional moment.”

Based at the University of Texas at Austin, Caffarelli started work on partial differential equations (PDEs) in the late 1970s and has contributed to hundreds of papers since. He is known for making connections between seemingly distant mathematical concepts, such as how a theory describing the smallest possible areas that surfaces can occupy can be used to describe PDEs in extreme cases.

PDEs have been studied for hundreds of years and describe almost every sort of physical process, ranging from fluids to combustion engines to financial models. Caffarelli’s most important work concerned nonlinear PDEs, which describe complex relationships between several variables. These equations are more difficult to solve than other PDEs, and often produce solutions that don’t make sense in the physical world.

Caffarelli helped tackle these problems with regularity theory, which sets out how to deal with problematic solutions by borrowing ideas from geometry. His approach carefully elucidated the troublesome parts of the equations, solving a wide range of problems over his more than four-decade career.

“Forty years after these papers appeared, we have digested them and we know how to do some of these things more efficiently,” says Francesco Maggi at the University of Texas at Austin. “But when they appeared back in the day, in the 80s, these were alien mathematics.”

Many of the nonlinear PDEs that Caffarelli helped describe were so-called free boundary problems, which describe physical scenarios where two objects in contact share a changing surface, like ice melting into water or water seeping through a filter.

“He has used insights that combined ingenuity, and sometimes methods that are not ultra-complicated, but which are used in a manner that others could not see — and he has done that time and time again,” says Thomas Chen at the University of Texas at Austin.

These insights have also helped other researchers translate equations so that they can be solved on supercomputers. “He has been one of the most prominent people in bringing this theory to a point where it’s really useful for applications,” says Maggi.

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

*Credit for article given to Alex Wilkins*