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Reduce wastage and enjoy deeply satisfying neat folds by applying a little geometry to your gift-wrapping, says Katie Steckles.

Wrapping gifts in paper involves converting a 2D shape into a 3D one, which presents plenty of geometrical challenges. Mathematics can help with this, in particular by making sure that you are using just the right amount of paper, with no wastage.

When you are dealing with a box-shaped gift, you might already wrap the paper around it to make a rectangular tube, then fold in the ends. With a little measuring, though, you can figure out precisely how much paper you will need to wrap a gift using this method, keeping the ends nice and neat.

For example, if your gift is a box with a square cross-section, you will need to measure the length of the long side, L, and the thickness, T, which is the length of one side of the square. Then, you will need a piece of paper measuring 4 × T (to wrap around the four sides with a small overlap) by L + T. Once wrapped around the shape, a bit of paper half the height of the square will stick out at each end, and if you push the four sides in carefully, you can create diagonal folds to make four points that meet neatly in the middle. The square ends of the gift make this possible (and deeply satisfying).

Similarly, if you are wrapping a cylindrical gift with diameter D (such as a candle), mathematics tells us you need your paper to be just more than π × D wide, and L + D long. This means the ends can be folded in – possibly less neatly – to also meet exactly in the middle (sticky bows are your friend here).

How about if your gift is an equilateral triangular prism? Here, the length of one side of the triangle gives the thickness T, and your paper should be a little over 3 × T wide and L + (2 × T) long. The extra length is needed because it is harder to fold the excess end bits to make the points meet in the middle. Instead, you can fold the paper to cover the end triangle exactly, by pushing it in from one side at a time and creating a three-layered triangle of paper that sits exactly over the end.

It is also possible to wrap large, flat, square-ish gifts using a diagonal method. If the diagonal of the top surface of your box is D, and the height is H, you can wrap it using a square piece of paper that measures a little over D + (√2 × H) along each side.

Place your gift in the centre of the paper, oriented diagonally, and bring the four corners to meet in the middle of your gift, securing it with one piece of tape or a sticky bow. This will cover all the faces exactly, and look pretty smart too.

For maximum mathematical satisfaction, what you want is to get the pattern on the paper to line up exactly. This is easier for a soft gift, where you can squash it to line up the pattern, but will only work with a box if the distance around it is exactly a multiple of the width of the repeat on the pattern. Otherwise, follow my example (above) and get your own custom wrapping paper printed!

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

*Credit for article given to Katie Steckles*

Want to get a slight edge during a coin toss? Check out which side is facing upwards before the coin is flipped –- then call that same side.

This tactic will win 50.8 percent of the time, according to researchers who conducted 350,757-coin flips.

For the preprint study, which was published on the arXiv database last week and has not yet been peer-reviewed, 48 people tossed coins of 46 different currencies.

They were told to flip the coins with their thumb and catch it in their hand—if the coins fell on a flat surface that could introduce other factors such as bouncing or spinning.

Frantisek Bartos, of the University of Amsterdam in the Netherlands, told AFP that the work was inspired by 2007 research led by Stanford University mathematician Persi Diaconis—who is also a former magician.

Diaconis’ model proposed that there was a “wobble” and a slight off-axis tilt that occurs when humans flip coins with their thumb, Bartos said.

Because of this bias, they proposed it would land on the side facing upwards when it was flipped 51 percent of the time—almost exactly the same figure borne out by Bartos’ research.

While that may not seem like a significant advantage, Bartos said it was more of an edge that casinos have against “optimal” blackjack players.

It does depend on the technique of the flipper. Some people had almost no bias while others had much more than 50.8 percent, Bartos said.

For people committed to choosing either heads or tails before every toss, there was no bias for either side, the researchers found.

None of the many different coins showed any sign of bias either.

Happily, achieving a fair coin flip is simple: just make sure the person calling heads or tails cannot see which side is facing up before the toss.

‘It’s fun to do stupid stuff’

Bartos first heard of the bias theory while studying Bayesian statistics during his master’s degree and decided to test it on a massive scale.

But there was a problem: he needed people willing to toss a lot of coins.

At first, he tried to persuade his friends to flip coins over the weekend while watching “Lord of the Rings”.

“But nobody was really down for that,” he said.

Eventually Bartos managed to convince some colleagues and students to flip coins whenever possible, during lunch breaks, even while on holiday.

“It will be terrible,” he told them. “But it’s fun to do some stupid stuff from time to time.”

The flippers even held weekend-long events where they tossed coins from 9am to 9pm. A massage gun was deployed to soothe sore shoulders.

Countless decisions have been made by coin tosses throughout human history.

While writing his paper, Bartos visited the British Museum and learned that the Wright brothers used one to determine who would attempt the first plane flight.

Coin tosses have also decided numerous political races, including a tied 2013 mayoral election in the Philippines.

But they are probably most common in the field of sport. During the current Cricket World Cup, coin tosses decide which side gets to choose whether to bat or field first.

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Credit of the article given to Daniel Lawler

Students who are more digitally skilled also perform better in math. New research from Renae Loh and others at Radboud University shows that in countries with better availability of information and communication technology (ICT) in schools, math performance benefits greatly. It further suggests that improving the ICT environment in schools can reduce inequality in education between countries. The paper is published in European Educational Research Journal today.

For anyone growing up today, ICT skills play a tremendously important role. Today’s youth constantly come into contact with technology throughout their life, both in work and leisure. Though previous studies have shown the importance of ICT skills in students’ learning outcomes, a new study focuses specifically on its relevance to math and how that differs between countries.

“Both ICT and math rely on structural and logical thinking, which is why ICT skills overlap with and boosts math learning. But we were also curious to find out how much of that depends on a country’s ICT environment,” says Renae Loh, primary author of the paper and a sociologist at Radboud University.

Benefits of a strong ICT infrastructure

Loh and her colleagues used data from the 2018 PISA Study and compares 248,720 students aged 15 to 16 across 43 countries. Included in this data is information about the ICT skills of these students. They were asked whether they read new information on digital devices, and if they would try to solve problems with those devices themselves, among other questions. The more positively students responded to these questions, the more skilled in ICT the researchers judged these students to be.

Loh says, “What we found is that students get more educational benefit out of their digital skills in countries with a strong ICT infrastructure in education. This is likely because the more computers and other digital tools are available to them in their studies, the more they were able to put those skills to use, and the more valued these skills were. It is not a negligible difference either.”

“A strong ICT infrastructure in education could boost what math performance benefits students gain from their digital skills by about 60%. Differences in ICT infrastructure in education accounted for 25% of the differences between countries in how much math benefits students gain from their digital skills. It is also a better indicator than, for example, looking at a more general indicator of country wealth, because it is more pinpointed and more actionable.”

Reducing inequality

Especially notable to Loh and her colleagues was the difference that was apparent between countries with a strong ICT infrastructure, and countries without. “It was surprisingly straightforward, in some ways: the higher the computer-to-student ratio in a country, the stronger the math performance. This is consistent with the idea that these skills serve as a learning and signaling resource, at least for math, and students need opportunities to put these resources to use.”

Loh points out that there are limits to the insight offered by the data, however. “Our study doesn’t look at the process of how math is taught in these schools, specifically. Or how the ICT infrastructure is actually being used. Future research might also puzzle over how important math teachers themselves believe ICT skills to be, and if that belief and their subsequent teaching style influences the development of students, too.”

“There is still vast inequality in education around the world,” warns Loh. “And now there’s an added ICT dimension. Regardless of family background, gender, and so on, having limited access to ICT or a lack in digital skills is a disadvantage in schooling. What is clear is that the school environment is important here. More targeted investments in a robust ICT infrastructure in education would help in bridging the educational gap between countries and may also help to address inequalities in digital skills among students in those countries.”

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

Credit of the article given to Radboud University

Rob Eastaway and Brian Hobbs take over our maths column to reveal who solved their puzzle and won a copy of their New Scientist puzzle book, Headscratchers.

When solving problems in the real world, it is rare that the solution is purely mathematical, but maths is often a key ingredient. The puzzle we set a few weeks ago (New Scientist, 30 September, p 45) embraced this by encouraging readers to come up with ingenious solutions that didn’t have to be exclusively maths-based.

Here is a reminder of the problem: Prince Golightly found himself tied to a chair near the centre of a square room, in the dark, with chained monsters in the four corners and an escape door in the middle of one wall. With him, he had a broom, a dictionary, some duct tape, a kitchen clock and a bucket of water with a russet fish.

John Offord was one of several readers to spot an ambiguity in our wording. Four monsters in each corner? Did this mean 16 monsters? John suggested the dictionary might help the captors brush up on their grammar.

The russet fish was deliberately inserted as a red herring (geddit?), but we loved that some readers found ways to incorporate it, either as a way of distracting the monsters or as a source of valuable protein for a hungry prince. Dave Wilson contrived a delightful monster detector, while Glenn Reid composed a limerick with the solution of turning off the computer game and going to bed.

And so to more practical solutions. Arlo Harding and Ed Schulz both suggested ways of creating a torch by igniting assorted materials with an electric spark from the light cable. But Ben Haller and Chris Armstrong had the cleverest mathematical approach. After locating the light fitting in the room’s centre with the broom, they used duct tape and rope to circle the centre, increasing the radius until they touched the wall at what must be its centre, and then continued circling to each wall till they found the escape door. Meanwhile, the duo of Denise and Emory (aged 11) used Pythagoras’s theorem to confirm the monsters in the corners would be safely beyond reach. They, plus Ben and Chris, win a copy of our New Scientist puzzle book Headscratchers.

It is unlikely you will ever have to escape monsters in this way, but spiral search patterns when visibility is limited are employed in various real-world scenarios: rescuers probing for survivors in avalanches, divers performing underwater searches and detectives examining crime scenes, for example. Some telescopes have automated spiral search algorithms that help locate celestial objects. These patterns allow for thorough searches while ensuring you don’t stray too far from your starting point.

Of course, like all real-world problems, mathematical nous isn’t enough. As our readers have displayed, lateral thinking and the ability to improvise are human skills that help us find the creative solutions an algorithm would miss.

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

*Credit for article given to Rob Eastaway and Brian Hobbs*

Economists from RUDN University have created a methodology based on mathematical modeling to manage production effectively with rapidly emerging innovations. The resultswere published in Sustainability.

Innovative products emerge so rapidly that enterprises have to change and adjust their management policies. Moreover, new management methods must ensure sustainable development. RUDN economists used mathematical modeling methods to develop a management methodology for enterprises that produce innovative products.

“An effective methodological tool for managing economic processes that ensure sustainable economic development of enterprises has not yet been developed. We set a goal to create such a tool based on mathematical modeling. Based on this tool, decision-makers could regulate processes, allocating additional resources and reducing risks,” said Zhanna Chupina, Ph.D., Associate Professor of the Department of Customs Affairs at RUDN University.

Economists have developed a tool that will allow, in modern conditions, to manage economic processes in such a way as to ensure sustainable development. To do this, the authors used mathematical modeling—they identified the main characteristics of the creation of innovative products and the activities of companies and described the dependence of profit on them. Economists included, for example, technical superiority over competitors, scientific and technological achievements based on the product, and price compliance with the capabilities of the buyer as the main values.

RUDN economists concluded that any innovation in a product is associated with the use or modernization of means of production. The authors determined the values of the technological level and the moments at which modernization should be carried out. It turned out that this needs to be done before the break-even point occurs, that is, the sales volume at which the proceeds from the sale of goods are equal to the costs of their production.

The general algorithm that economists have proposed is based on the fact that the properties of innovative products are involved in management. After management actions, it is necessary to evaluate the characteristics of the product and answer whether it is innovative. If so, then such an enterprise is considered capable of achieving global competitive superiority in the market. If not, then the characteristics need to be changed and return to the previous step.

“We have shown that for an organization to be in constant sustainable economic development, it is necessary to carry out updates to already produced innovative products even before the break-even point. We proposed a methodological toolkit to effectively manage the creation and production of innovative products and their renewal,” said Zhanna Chupina, Ph.D., associate professor of the Department of Customs Affairs at RUDN University.

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Credit of the article given to Russian Foundation for Basic Research

Politicians in the UK have maths on the mind. The Conservatives intend to extend compulsory maths education for young people until 18.

And at the Labour party conference, shadow education secretary Bridget Phillipson announced the opposition’s plans to improve maths skills across the country: a focus on primary school and pre-school education rather than post-16, with an emphasis on children learning the maths they will need for everyday life.

Paying attention to young children’s maths is a good idea. Evidence from the UK and beyond shows that children start primary school with varying levels of mathematical skills – and disadvantage gaps are already evident at this point, meaning that children from poorer backgrounds may not have skills at the same level as their more well-off peers.

The differences between children’s maths skills then remain remarkably stable over time. Children who start primary school with mathematical abilities behind the level of their peers will typically remain behind their peers throughout school.

To reduce these gaps, we need to act early. But positive change won’t be achieved simply by adding more content to the primary or early years mathematics curriculums. Neither is it helpful to push children to learn more complex mathematics earlier. These approaches might lead to children learning maths in a superficial and rote manner, rather than understanding the underlying ideas.

Primary focus

Labour has raised the prospect of creating a “phonics for maths”. Phonics is a method of learning to read that teaches children the sounds that letters and combinations of letters make. It is required in primary schools, and pupils take a phonics screening check in year one to assess their progress.

Although not universally supported, phonics has been linked to improvements in reading levels among children in England.

However, phonics is a specific technique for teaching word reading, while mathematics is incredibly broad. It involves multiple skills as well as different types of knowledge and understanding.

Even in early primary school, mathematics is complex. Children need to understand quantities and their relationships, to recognise digits and understand place value, to carry out arithmetic procedures, to identify patterns in numbers and shapes, and much more. It is unlikely that a single technique, as phonics is, can underpin this breadth of knowledge and understanding.

But in another sense, the parallel with phonics is encouraging. The phonics revolution was informed by research and developed from a better understanding of how children learn to read. This can and should be emulated for mathematics. Research evidence on the early stages of learning maths can help build a solid approach to teaching mathematical skills to young children.

Another feature of Labour’s plans is their aim to “bring maths to life” by using real-world examples: budgeting, exchange rates, sports league tables.

A desire to give meaning to numbers and mathematics by building on children’s experiences is a good ambition. This can be achieved through play-based and hands-on activities, which involve children manipulating objects such as counters and cards to better understand mathematical ideas and relationships. It is also important to help children see numbers and mathematical patterns in the world around them: the number of red cars on the street or the shapes of windows and doors, for instance.

These approaches may provide a stronger foundation for future learning than focusing on using written digits or learning mathematical facts (such as 2 + 3 = 5) too early.

Taking care

But care is needed to ensure that bringing maths to life truly reflects children’s experiences and doesn’t become a gimmick. It could even increase disadvantage gaps due to differences in children’s experiences, for example, for children from families who lack access to bank accounts or have never had the experience of travelling abroad and using different currency.

There are already good examples out there of how to teach in this way – such as the Mastering Number programme. Any curriculum changes need to be properly funded and developed in collaboration with experts in the field.

Giving children better mathematical foundations through engaging and meaningful activities can set them up for success throughout school and beyond. This would not only positively affect children’s achievement but could also change attitudes to mathematics for the better.

Changing attitudes to mathematics from the foundations upwards can help children and young people feel confident and engaged with the subject and see its value in their life, leading to more wanting to study the subject.

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Credit of the article given to NadyaEugene/Shutterstock

Five secret phrases used to create the encryption algorithms that secure everything from online banking to email have been lost to history – but now cryptographers are offering a bounty to rediscover them.

Could you solve a cryptography mystery?

Secret phrases that lie at the heart of modern data encryption standards were accidentally forgotten decades ago – but now cryptographers are offering a cash bounty for anyone who can figure them out. While this won’t allow anyone to break these encryption methods, it could solve a long-standing puzzle in the history of cryptography.

“This thing is used everywhere, and it’s an interesting question; what’s the full story? Where did they come from?” says cryptographer Filippo Valsorda. “Let’s help the trust in this important tool of cryptography, and let’s fill out this page of history that got torn off.”

The tool in question is a set of widely-used encryption algorithms that rely on mathematical objects called elliptic curves. In theory, any of an infinite number of curves can be used in the algorithms, but in the late 1990s the US National Security Agency (NSA), which is devoted to protecting domestic communications and cracking foreign transmissions, chose five specific curves it recommended for use. These were then included in official US encryption standards laid down in 2000, which are still used worldwide today.

Exactly why the NSA chose these particular curves is unclear, with the agency saying only that they were chosen at random. This led some people to believe that the NSA had secretly selected curves that were weak in some way, allowing the agency to crack them. Although there is no evidence that the elliptic curves in use today have been cracked, the story persists.

In the intervening years, it has been confirmed that the curves were chosen by an NSA cryptographer named Jerry Solinas, who died earlier this year. Anonymous sources have suggested that Solinas chose the curves by transforming English phrases into a string of numbers, or hashes, that served as a parameter in the curves.

It is thought the phrases were along the lines of “Jerry deserves a raise”. But rumours suggest Solinas’s computer was replaced shortly after making the choice, and keeping no record of them, he couldn’t figure out the specific phrases that produced the hashes used in the curves. Turning a phrase into a hash is a one-way process, meaning that recovering them was impossible with the computing power available at the time.

Dustin Moody at the US National Institute of Standards and Technology, which sets US encryption standards, confirmed the stories to New Scientist: “I asked Jerry Solinas once, and he said he didn’t remember what they were. Jerry did seem to wish he remembered, as he could tell it would be useful for people to know exactly how the generation had gone. I think that when they were created, nobody [thought] that the provenance was a big deal.”

Now, Valsorda and other backers have offered a $12,288 bounty for cracking these five hashes – which will be tripled if the recipient chooses to donate it to charity. Half of the sum will go to the person who finds the first seed phrase, and the other half to whoever can find the remaining four.

Valsorda says that finding the hashes won’t weaken elliptic curve cryptography – because it is the nature of the curves that protects data, not the mathematical description of those curves – but that doing so will “help fill in a page of cryptographic history”. He believes that nobody in the 1990s considered that the phrases would be of interest in the future, and that the NSA couldn’t have released them anyway once they discovered that they were jokey phrases about one of their staff wanting a raise.

There are two main ways someone could claim the prize. The first is brute force – simply trying vast numbers of possible seeds, and checking the values created by hashing them against the known curves, which is more feasible than in the 1990s because of advances in computing power. 

But Valsorda says someone may already have the phrases written down. “Some of the people who did this work, or were in the same office as the people who did this work, probably are still around and remember some details,” he says. “The people who are involved in history sometimes don’t realise the importance of what they remember. But I’m not actually suggesting anybody, like, goes stalking NSA analysts.”

Keith Martin at Royal Holloway, University of London, says that the NSA itself would be best-equipped to crack the problem, but probably has other priorities, and anybody else will struggle to find the resources.

“I would be surprised if they’re successful,” he says. “But on the other hand, I can’t say for sure what hardware is out there and what hardware will be devoted to this problem. If someone does find the [phrases], what would be really interesting is how did they do it, rather than that they’ve done it.”

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*Credit for article given to Matthew Sparkes*

College students learn more calculus in an active learning course in which students solve problems during class than in a traditional lecture-based course. That’s according to a peer-reviewed study my colleagues and I published in science. We also found that college students better understood complex calculus concepts and earned better grades in the active learning course.

The findings held across racial and ethnic groups, genders and college majors, and for both first-time college and transfer students—thus, promoting success for all students. Students in the active learning course had an associated 11% higher pass rate.

If you apply that rate to the current 300,000students taking calculus each year in the U.S., it could mean an additional 33,000 pass their class.

Our experimental trial ran over three semesters—fall 2018 through fall 2019—and involved 811 undergraduate students at a public university that has been designated as a Hispanic-serving institution. The study evaluated the impact of an engagement-focused active learning calculus teaching method by randomly placing students into either a traditional lecture-based class or the active learning calculus class.

The active learning intervention promoted development of calculus understanding during class, with students working through exercises designed to build calculus knowledge and with faculty monitoring and guiding the process.

This differs from the lecture setting where students passively listen to the instructor and develop their understanding outside of class, often on their own.

An active learning approach allows students to work together to solve problems and explain ideas to each other. Active learning is about understanding the “why” behind a subject versus merely trying to memorize it.

Along the way, students experiment with their ideas, learn from their mistakes and ultimately make sense of calculus. In this way, they replicate the practices of mathematicians, including making and testing educated guesses, sense-making and explaining their reasoning to colleagues. Faculty are a critical part of the process. They guide the process through probing questions, demonstrating mathematical strategies, monitoring group progress and adapting pace and activities to foster student learning.

Florida International University made a short video to accompany a research paper on how active learning improves outcomes for calculus students.

Why it matters

Calculus is a foundational discipline for science, technology, engineering and mathematics, as it provides the skills for designing systems as well as for studying and predicting change.

But historically it’s been a barrier that has ended the opportunity for many students to achieve their goal of a STEM career. Only 40% of undergraduate students intending to earn a STEM degree complete their degree, and calculus plays a role in that loss. The reasons vary depending on the student. Failing calculus can be a final straw for some.

And it is particularly concerning for historically underrepresented groups. The odds of female students leaving a STEM major after calculus is 1.5 times higher than it is for men. And Hispanic and Black students have a 50% higher failure rate than white students in calculus. These losses deprive the individual students of STEM aspirations, career dreams and financial security. And it deprives society of their potentially innovative contributions to solving challenging problems, such as climate resilience, energy independence, infrastructure and more.

What still isn’t known

A vexing challenge in calculus instruction—and across the STEM disciplines—is broad adoption of active learning strategies that work. We started this research to provide compelling evidence to show that this model works and to drive further change. The next step is addressing the barriers, including lack of time, questions about effectiveness and institutional policies that don’t provide an incentive for faculty to bring active learning to their classrooms.

A crucial next step is improving the evidence-based instructional change strategies that will promote adoption of active learning instruction in the classroom.

What’s next

Our latest results are motivating our team to further delve into the underlying instructional strategies that drive student understanding in calculus. We’re also looking for opportunities to replicate the experiment at a variety of institutions, including high schools, which will provide more insight into how to expand adoption across the nation.

We hope that this paper increases the rate of change of all faculty adopting active learning in their classrooms.

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Credit of the article given to Laird Kramer, The Conversation

A large team of researchers affiliated with multiple institutions across Europe, has found evidence backing up work by Persi Diaconis in 2007 in which he suggested tossed coins are more likely to land on the same side they started on, rather than on the reverse. The team conducted experiments designed to test the randomness of coin flipping and posted their results on the arXiv preprint server.

For many years, the coin toss (or flip) has represented a fair way to choose between two options—which side of a team goes first, for example, who wins a tied election, or gets to eat the last brownie. Over the years, many people have tested the randomness of coin tossing and most have found it to be as fair as expected—provided a fair coin is used.

But, Diaconis noted, such tests have only tested the likelihood that a fair coin, once flipped, has an equal chance of landing on heads or tails. They have not tested the likelihood of a fair coin landing with the same side up as that when it was flipped. He suggested that due to precession, a coin flipped into the air spends more time there with its initial side facing up, making it more likely to end up that way, as well. He suggested that the difference would be slight, however—just 1%. In this new effort, the research team tested Diaconis’ ideas.

The experiment involved 48 people flipping coins minted in 46 countries (to prevent design bias) for a total of 350,757-coin flips. Each time, the participants noted whether the coin landed with the same side up as when it was launched. The researchers found that Diaconis was right—there was a slight bias. They found the coin landed with the same side up as when it was launched 50.8% of the time. They also found there was some slight variation in percentages between different individuals tossing coins.

The team concludes that while the bias they found is slight, it could be meaningful if multiple coin tosses are used to determine an outcome—for example, flipping a quarter 1,000 times and betting $1 each time (with winnings of 0 or 2$ each round) should result in an average overall win of $19.

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

Researchers Jayne Spiller and Camilla Gilmore at the Center for Mathematical Cognition, University of Loughborough, U.K., have investigated the intersection of sleep and mathematical memory, finding that sleep after learning improves recall.

In their paper, “Positive impact of sleep on recall of multiplication facts,” published in Royal Society Open Science, the duo investigated whether learning complex multiplication problems before sleep would benefit recall compared to learning them during wakefulness to understand how sleep affects the memory of mathematical facts, specifically multiplication tables.

The study involved 77 adult participants aged 18 to 40 from the U.K. Each participant learned complex multiplication problems in two conditions: before sleep (sleep learning) and in the morning (wake learning). Participants completed online sessions where they learned new complex multiplication problems or were tested on previously learned material. Learning sessions included both untimed and timed trials.

Participants had better recall in the sleep learning condition than in the wake learning condition, with a moderate effect size. Even when participants had varying learning abilities, the sleep learning condition showed a beneficial effect on recall, with a smaller effect size.

Mathematical proficiency of the participants, as measured by accuracy in simple multiplication problems, was associated with learning scores but not with the extent of sleep-related benefit for recall.

The study highlights the potential educational implications of leveraging sleep-related benefits for learning. The positive impact of sleep on the recall of complex multiplication problems could be particularly useful for children learning multiplication tables or other math memorization skills, though it would be interesting to see how well a bedtime math lesson would be received.

While the authors suggest that sleep conferred the additional benefit on recall compared with learning during the daytime, the mechanisms by which encoding takes place are possibly enforced by a lack of continued external inputs. The authors point out this limitation of a lack of other comparative stimuli with a similar complexity of encoding to conclusively demonstrate in their study the specificity of sleep-related benefits on recall.

Asleep, the brain may be locking in the new learning because it has no other competition.

In contrast, an awake brain may be confronted with conversations, media reading or viewing and even other classes packed with learning material. This competition for memory encoding in the waking brain could be the cause of the memory differences seen in the study, though outside of recommending multi-hour meditation sessions between classes the likelihood of finding an alternative to sleep on memory may be limited.

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Credit of the article given to P by Justin Jackson, Phys.org