Month: September 2023

A major new study has revealed that American teenagers are more likely than any other nationality to brag about their math ability.

Research using data from 40,000 15-year-olds from nine English-speaking nations internationally found those in North America were the most likely to exaggerate their mathematical knowledge, while those in Ireland and Scotland were least likely to do so.

The study, published in Assessment in Education: Principles, Policy & Practice, used responses from the OECD Programme for International Student Assessment (PISA), in which participants took a two-hour math test alongside a 30-minute background questionnaire.

They were asked how familiar they were with each of 16 mathematical terms—but three of the terms were fake.

Further questions revealed those who claimed familiarity with non-existent mathematical concepts were also more likely to display overconfidence in their academic prowess, problem-solving skills and perseverance.

For instance, they claimed higher levels of competence in calculating a discount on a television and in finding their way to a destination. Two thirds of those most likely to overestimate their mathematical ability were confident they could work out the petrol consumption of a car, compared to just 40% of those least likely to do so.

Those likely to over-claim were also more likely to say if their mobile phone stopped sending texts they would consult a manual (41% versus 30%) while those less likely to do so tended to say they would react by pressing all the buttons (56% versus 49%).

Over-claimers were also more likely to say they were popular with their peers at school, although the evidence was less strong on this topic.

Overall, boys were more likely to overclaim than girls, and those from advantaged backgrounds were more likely to do so than those from less advantaged groups. In most countries, immigrants were more likely to do this than the native-born, particularly in Northern Ireland and New Zealand although not in the United States.

Three broad clusters of countries emerged, with the United States and Canada at the top of the rankings when it came to excessive claims on math knowledge, and with Ireland, Northern Ireland and Scotland at the bottom. In the middle were Australia, New Zealand, England and Wales.

The report’s lead author is John Jerrim, Professor of Education and Social Statistics at the UCL Institute of Education. “Our research provides important new insight into how those who over-claim about their math ability also exhibit high levels of over-confidence in other areas,” he said.

“Although ‘overclaiming’ may at first seem to be a negative social trait, we have previously found that overconfident individuals are more likely to land top-jobs. The fact that young men tend to overclaim their knowledge more than young women, and the rich are more likely to overclaim than the poor, could be related to the different labor market outcomes of these groups.”

Students were shown a list of 16 items and asked to indicate their knowledge of each on a five-point scale ranging from “never heard of it” to “know it well, understand the concept.” They were:

1. Exponential function
2. Divisor
3. Quadratic function
4. Proper number
5. Linear equation
6. Vectors
7. Complex number
8. Rational number
9. Radicals
10. Subjunctive scaling
11. Polygon
12. Declarative fraction
13. Congruent figure
14. Cosine
15. Arithmetic mean
16. Probability

Numbers 4, 10 and 12 were fake terms.

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Credit of the article given to Taylor & Francis

Richard Schwartz, a mathematician at Brown University has found a solution to the problem of how small a Möbius band can be made without intersecting itself—at least for a smooth piece of paper. The paper is published on the arXiv preprint server.

A Möbius strip (or band) is both a physical and mathematical object. A sample can be constructed by twisting a simple strip of paper one time and then taping the ends together. Since they were first discovered back in the mid-1800s, mathematicians have been scratching their heads trying to determine one simple constraint—what is the shortest strip necessary for making one? Back in the late 1970s, a pair of mathematicians, Charles Sidney Weaver and Benjamin Rigler Halpern, found that the problem could be made simpler by allowing self-intersections—that changed the problem to one that involved seeking the minimum amount of strip needed to avoid self-intersections.

Four years ago, Schwartz found himself intrigued by the problem and, as he describes in his paper, became “hooked” on finding a solution. Two years ago, he thought he had finally found it and published a proof showing his work—it involved breaking down the problem into multiple pieces and then using geometry principles to solve the puzzle as a whole.

Unfortunately, there turned out to be a major flaw in his work that he did not discover until much more recently. He found it by creating physical samples and cutting them in different ways to see how they worked on a deeper level. He discovered that the 2D strip was not shaped like a parallelogram as had been thought—instead, it was a trapezoid.

Inspired by his discovery, he went back to this original proof and corrected the error, and in doing so, found that the proof worked much better than it had originally. It was also much simpler. He also used the proof to work out the optimization problem, and got what he was hoping for: √3. He notes that while pleased with his own work, he has already turned his attention to another problem—to determine how short a band can be if it is twisted three times instead of once.\

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

Mathematics and engineering go hand in hand. Mathematics is an essential tool for engineers and plays a crucial role in helping students become better engineers. In this article, we will explore how math helps students become better engineers.

Understanding and Applying Principles:

Engineering is all about applying scientific principles to solve real-world problems. Mathematics is the language of science, and without it, engineers would not be able to understand the fundamental principles that govern the world around us. By studying math, students learn how to analyze and solve complex problems, which is a critical skill for any engineer. Moreover, math helps students understand the fundamental concepts of physics, which is essential to many engineering fields.

Analyzing and Solving Problems:

Engineers are problem solvers, and math is an essential tool for problem-solving. Math helps students develop critical thinking skills and teaches them how to analyze and solve problems systematically. Engineers use mathematical concepts to create models, analyze data, and make predictions. These models and predictions help engineers design and build products that meet specific needs and requirements. One standard approach to building your maths skills is by participating in Olympiads such as the International Maths Olympiad Challenge.

Design and Optimization:

Designing and optimizing systems is another essential part of engineering. Math plays a critical role in helping engineers design and optimize systems. Mathematical models help engineers simulate and optimize systems to ensure that they meet specific requirements. By understanding mathematical concepts like calculus, optimization, and linear algebra, students can learn how to design and optimize complex systems.

Communication:

Engineers must be able to communicate complex technical concepts to non-technical stakeholders. Math helps students develop this skill by teaching them how to use graphs, charts, and other visual aids to communicate complex data and concepts. By using math to present data and findings, engineers can help non-technical stakeholders understand the technical aspects of their work.

Mathematics is an essential tool for engineers. By studying math, students can develop critical thinking skills, learn how to solve complex problems, and design and optimize systems. Moreover, math helps students communicate complex technical concepts to non-technical stakeholders, an essential skill for any engineer. Therefore, it is important for engineering students to have a strong foundation in mathematics. By doing so, they can become better engineers and contribute to solving the world’s complex problems.

For much of her teaching career, Carrie Stark relied on math games to engage her students, assuming they would pick up concepts like multiplication by seeing them in action. The kids had fun, but the lessons never stuck.

A few years ago, she shifted her approach, turning to more direct explanation after finding a website on a set of evidence-based practices known as the science of math.

“I could see how the game related to multiplication, but the kids weren’t making those connections,” said Stark, a math teacher in the suburbs of Kansas City. “You have to explicitly teach the content.”

As American schools work to turn around math scores that plunged during the pandemic, some researchers are pushing for more attention to a set of research-based practices for teaching math. The movement has passionate backers, but is still in its infancy, especially compared with the phonics-based “science of reading” that has inspired changes in how classrooms across the country approach literacy.

Experts say math research hasn’t gotten as much funding or attention, especially beyond the elementary level. Meanwhile, the math instruction schools are currently using doesn’t work all that well. The U.S. trails other high-income countries in math performance, and lately more students graduate high school with deficits in basic math skills.

Advocates say teaching practices supported by quantitative research could help, but they are still coming into focus.

“I don’t think the movement has caught on yet. I think it’s an idea,” said Matthew Burns, a professor of special education at the University of Florida who was among researchers who helped create a Science of Math website as a resource for teachers.

WHAT IS THE SCIENCE OF MATH?

There’s a debate over which evidence-based practices belong under the banner of the science of math, but researchers agree on some core ideas.

The foremost principle: Math instruction must be systematic and explicit. Teachers need to give clear and precise instructions and introduce new concepts in small chunks while building on older concepts. Such approaches have been endorsed by dozens of studieshighlighted by the Institute of Education Sciences, an arm of the U.S. Education Department that evaluates teaching practices.

That guidance contrasts with exploratory or inquiry-based models of education, where students explore and discover concepts on their own, with the teacher nudging them along. It’s unclear which approaches are used most widely in schools.

In some ways, the best practices for math parallel the science of reading, which emphasizes detailed, explicit instruction in phonics, instead of letting kids guess how to read a word based on pictures or context clues. After the science of reading gained prominence, 18 states in just three years have passed legislation mandating that classroom teachers use evidence-backed methods to teach reading.

Margie Howells, an elementary math teacher in Wheeling, West Virginia, first went researching best practices because there weren’t as many resources for dyscalculia, a math learning disability, as there were for dyslexia. After reading about the science of math movement, she became more explicit about things that she assumed students understood, like how the horizontal line in a fraction means the same thing as a division sign.

“I’m doing a lot more instruction in vocabulary and symbol explanations so that the students have that built-in understanding,” said Howells, who is working on developing a science-based tutoring program for students with dyscalculia and other learning differences.

THE SO-CALLED MATH WARS

Some elements of math instruction emphasize big-picture concepts. Others involve learning how to do calculations. Over the decades, clashes between schools of thought favouring one or another have been labeled the “math wars.” A key principle of the science of math movement is that both are important, and teachers need to foster procedural as well as conceptual understanding.

“We need to be doing all those simultaneously,” Stark said.

When Stark demonstrates a long division problem, she writes out the steps for calculating the answer while students use a chart or blocks to understand the problem conceptually.

Stark helps coach fellow teachers at her school to support struggling students—something she used to feel unequipped to do, despite 20 years of teaching experience. Most of the resources she found online just suggested different math games. So, she did research online and signed up for special trainings, and started focusing more on fundamentals.

For one fifth grader who was struggling with fractions, she explicitly re-taught equivalent fractions from third grade—why two-fourths are the same as one-half, for instance. He had been working with her for three years, but this was the first time she heard him say, “I totally get it now!”

“He was really feeling success. He was super proud of himself,” Stark said.

Still, skeptics of the science of math question the emphasis placed on learning algorithms, the step-by-step procedures for calculation. Proponents say they are necessary along with memorization of math facts (basic operations like 3×5 or 7+9) and regular timed practice—approaches often associated with mind-numbing drills and worksheets.

Math is “a creative, artistic, playful, reasoning-rich activity. And it’s very different than algorithms,” said Nick Wasserman, a professor of math education at Columbia University’s Teachers College.

Supporters argue mastering math facts unlocks creative problem-solving by freeing up working memory—and that inquiry, creativity and collaboration are still all crucial to student success.

“When we have this dichotomy, it creates an unnecessary divide and it creates a dangerous divide,” said Elizabeth Hughes, a professor of special education at Penn State and a leader in the science of math movement. People feel the need to choose sides between “Team Algorithms” and “Team Exploratory,” but “we really need both.”

A HIGHER IMPORTANCE ON READING?

Best practices are one thing. But some disagree such a thing as a “science of math” exists in the way it does for reading. There just isn’t the same volume of research, education researcher Tom Loveless said.

“Reading is a topic where we have a much larger amount of good, solid, causal research that can link instruction to student achievement,” he said.

To some, the less advanced state of research on math reflects societal values, and how many teachers themselves feel more invested in reading. Many elementary school teachers doubt their own math ability and struggle with anxiety around teaching it.

“Many of us will readily admit that we weren’t good at math,” said Daniel Ansari, a professor of cognitive neuroscience at Western University in Canada. “If I was illiterate, I wouldn’t tell a soul.”

Still, Ansari said, there is enough research out there to make a difference in the classroom.

“We do understand some of the things that really work,” he said, “and we know some of the things that are not worth spending time on.”

Correction note: This story has been corrected to reflect that Burns is now at the University of Florida, and not the University of Missouri.

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Credit of the article given to Sharon Lurye

Until recently, working out how three objects can stably orbit each other was nearly impossible, but now mathematicians have found a record number of solutions.

The motion of three objects is more complex than you might think

The question of how three objects can form a stable orbit around each other has troubled mathematicians for more than 300 years, but now researchers have found a record 12,000 orbital arrangements permitted by Isaac Newton’s laws of motion.

While mathematically describing the movement of two orbiting bodies and how each one’s gravity affects the other is relatively simple, the problem becomes vastly more complex once a third object is added. In 2017, researchers found 1223 new solutions to the three-body problem, doubling the number of possibilities then known. Now, Ivan Hristov at Sofia University in Bulgaria and his colleagues have unearthed more than 12,000 further orbits that work.

The team used a supercomputer to run an optimised version of the algorithm used in the 2017 work, discovering 12,392 new solutions. Hristov says that if he repeated the search with even more powerful hardware he could find “five times more”.

All the solutions found by the researchers start with all three bodies being stationary, before entering freefall as they are pulled towards each other by gravity. Their momentum then carries them past each other before they slow down, stop and are attracted together once more. The team found that, assuming there is no friction, this pattern would repeat infinitely.

Solutions to the three-body problem are of interest to astronomers, as they can describe how any three celestial objects – be they stars, planets or moons – can maintain a stable orbit. But it remains to be seen how stable the new solutions are when the tiny influences of additional, distant bodies and other real-world noise are taken into account.

“Their physical and astronomical relevance will be better known after the study of stability – it’s very important,” says Hristov. “But, nevertheless – stable or unstable – they are of great theoretical interest. They have a very beautiful spatial and temporal structure.”

Juhan Frank at Louisiana State University says that finding so many solutions in a precise set of conditions will be of interest to mathematicians, but of limited application in the real world.

“Most, if not all, require such precise initial conditions that they are probably never realised in nature,” says Frank. “After a complex and yet predictable orbital interaction, such three-body systems tend to break into a binary and an escaping third body, usually the least massive of the three.”

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*Credit for article given to Matthew Sparkes *

Getting in sync can be exhilarating when you’re dancing in rhythm with other people or clapping along in an audience. Fireflies too know the joy of synchronization, timing their flashes together to create a larger display to attract mates.

Synchronization is important at a more basic level in our bodies, too. Our heart cells all beat together (at least when things are going well) and synchronized electrical waves can help coordinate brain regions—but too much synchronization of brain cells is what happens in an epileptic seizure.

Sync most often emerges spontaneously rather than through following the lead of some central timekeeper. How does this happen? What is it about a system that determines whether sync will emerge, and how strong it will be?

In new research published in Proceedings of the National Academy of Sciences, we show how the strength of synchronization in a network depends on the structure of the connections between its members—whether they be brain cells, fireflies, or groups of dancers.

The science of sync

Scientists originally became interested in sync to understand the inner workings of natural systems. We have also become interested in designing sync as a desired behaviour in human-made systems such as power grids (to keep them in phase).

Mathematicians can analyse sync by treating the individuals in the system as “coupled oscillators.” An oscillator is something that periodically repeats the same pattern of activity, like the sequence of steps in a repetitive dance, and coupled oscillators are ones that can influence each other’s behaviour.

It can be useful to measure whether a system of oscillators can synchronize their actions, and how strong that synchronization would be. Strength of synchronization means how well the sync can recover from disturbances.

Take a group dance, for example. A disturbance might be one person starting to get some steps wrong. The person might quickly recover by watching their friends, they might throw their friends off for a few steps before everyone recovers, or in the worst case it might just cause chaos.

Synced systems are strong but hard to unravel

Two factors make it difficult to determine how strong the synchronization in a set of coupled oscillators could be.

First, it’s rare for a single oscillator to be in charge and telling everyone else what to do. In our dance example, that means there’s neither music nor lead dancers to set the tempo.

And second, usually each oscillator is only connected to a few others in the system. So each dancer can only see and react to a few others, and everyone is taking their cues from a completely different set of dancers.

This is the case in the brain, for example, where there is a complex network structure of connections between different regions.

Real complex systems like this, where there is no central guiding signal and oscillators are connected in a complex network, are very robust to damage and adaptable to change, and can more easily scale to different sizes.

Stronger sync comes from more wandering walks

One drawback of such complicated systems is for scientists, as they are mathematically difficult to come to grips with. However, our new research has made a significant advance on this front.

We have shown how the network structure connecting a set of oscillators controls how well they can synchronize. The quality of sync depends on “walks” on a network, which are sequences of hops between connected oscillators or nodes.

Our math examines what are called “paired walks.” If you start at one node and take two walks with randomly chosen next hops for a specific number of hops, the two walks might end up at the same node (these are convergent walks) or at different nodes (divergent walks).

We found that the more often paired walks on a network were convergent rather than divergent, the worse the synchronization on the network would be.

When more paired walks are convergent, disturbances tend to be reinforced.

In our dancing example, one person making the wrong steps might lead some neighbours astray, who may then lead some of their neighbours astray and so on.

These chains of potential disturbances are like walks on the network. When those disturbances propagate through multiple neighbours and then converge on one person, that person is going to be much more likely to copy the out-of-sync moves than if only one of their neighbours was offbeat.

Social networks, power grids and beyond

So networks with many convergent walks are prone to poorer synchronization. This is good news for the brain avoiding epilepsy, as its highly modular structure brings a high proportion of convergent walks.

We can see this reflected in the echo chamber phenomenon in social media. Tightly coupled subgroups reinforcing their own messages can synchronize themselves well, but may fall far out of step with the wider population.

Our results bring a new understanding to how synchronization functions in different natural network structures. It opens new opportunities in terms of designing network structures or interventions on networks, either to aid synchronization (in power grids, say) or to avoid synchronization (say in the brain).

More widely, it represents a major step forward in our understanding of how the structure of complex networks affects their behaviour and capabilities.

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Credit of the article given to Joseph Lizier, The Conversation

Knots are used in all sorts of ways, every day. They ensure safety both indoors and for outdoor activities such as boating or sailing, are used as surgical sutures, as decorations, and they can even be found at nanoscales in nature, for example in DNA molecules.

Elastic knots are those that bounce back into their original shape in the absence of friction. There are open elastic knots tied with a single length of wire with two ends, which revert to being a straight line, and closed elastic knots in which the ends of the wire used to tie them have been attached together. These tend to spring back into a curved shape.

With a focus on closed knots, researchers in the Ecole Polytechnique Federale de Lausanne Geometric Computing Laboratory, led by Professor Mark Pauly, along with colleagues in Canada and the United States, have discovered thousands of new transformable knots including three novel shapes that the humble figure-eight knot can assume, doubling the number documented to date in scientific literature.

The findings are published in the journal ACM Transactions on Graphics.

To make these discoveries, the team first developed a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find the stable equilibrium states of elastic knots. Leveraging results from knot theory, they ran their pipeline on thousands of different topological knot types to create an extensive data set of multistable knots.

“By applying a series of filters to this data, we discovered new transformable knots with interesting physical properties and beautiful geometric forms,” explained doctoral assistant Michele Vidulis, the lead author of the paper “Computational Exploration of Multistable Elastic Knots.”

“This rich set of fascinating shapes can be created simply by knotting an elastic wire, and we noticed how such seemingly simple objects can sometimes exhibit tens or even hundreds of different stable shapes. The novel geometric patterns we identified were at times surprising. For example, we found that most—but not all—the preferred shapes of elastic knots are flat and planar, while few of them assume three-dimensional shapes,” Vidulis continued.

The team conducted further analysis across knot types that revealed new geometric and topological patterns with constructive principles not seen in previously tabulated knot types, showing how multistable elastic knots might be used to design new structures.

“As a result of our research, we can see elastic knots being used in the design process of self-deployable structures, like pop-up tents or lightweight emergency shelters. New metamaterials can be designed that combine several elastic knotted elements to build a network with complex mechanical behaviour,” Vidulis explained.

The team also created engaging recreational puzzles with the challenge to deform an elastic knot and manually find some of the interesting geometric shapes that they have computed with their algorithms.

As satisfying as these new discoveries are, Vidulis and the team believe that the work opens the way to several other potential new research directions.

“We want to explore the design of self-deployable structures and consider coupling elastic rods with fabric materials. As well, despite simulating thousands of different knots, our exploration only scratched the surface of the millions of knots that are known. We also plan to study more complex ensembles of knotted systems, in which new mechanical properties might emerge from the way in which the individual components are intertwined to each other,” he concluded.

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Credit of the article given to anya Petersen, Ecole Polytechnique Federale de Lausanne

Exciting a brain region using electrical noise stimulation can help improve mathematical learning in those who struggle with the subject, according to a new study from the Universities of Surrey and Oxford, Loughborough University, and Radboud University in The Netherlands.

During this unique study, published in PLOS Biology, researchers investigated the impact of neurostimulation on learning. Despite the growing interest in this non-invasive technique, little is known about the neurophysiological changes induced and the effect it has on learning.

Researchers found that electrical noise stimulation over the frontal part of the brain improved the mathematical ability of people whose brain was less excited (by mathematics) before the application of stimulation. No improvement in mathematical scores was identified in those who had a high level of brain excitation during the initial assessment or in the placebo groups. Researchers believe that electrical noise stimulation acts on the sodium channels in the brain, interfering with the cell membrane of the neurons, which increases cortical excitability.

Professor Roi Cohen Kadosh, Professor of Cognitive Neuroscience and Head of the School of Psychology at the University of Surrey who led this project, said, “Learning is key to everything we do in life—from developing new skills, such as driving a car, to learning how to code. Our brains are constantly absorbing and acquiring new knowledge.

“Previously, we have shown that a person’s ability to learn is associated with neuronal excitation in their brains. What we wanted to discover in this case is if our novel stimulation protocol could boost, in other words excite, this activity and improve mathematical skills.”

For the study, 102 participants were recruited, and their mathematical skills were assessed through a series of multiplication problems. Participants were then split into four groups including a learning group exposed to high-frequency random electrical noise stimulation and an overlearning group in which participants practiced the multiplication beyond the point of mastery with high-frequency random electrical noise stimulation.

The remaining two groups consisted of a learning and overlearning group but they were exposed to a sham (i.e., placebo) condition, an experience akin to real stimulation without applying significant electrical currents. EEG recordings were taken at the beginning and at the end of the stimulation to measure brain activity.

Dr. Nienke van Bueren, from Radboud University, who led this work under Professor Cohen Kadosh’s supervision, said, “These findings highlight that individuals with lower brain excitability may be more receptive to noise stimulation, leading to enhanced learning outcomes, while those with high brain excitability might not experience the same benefits in their mathematical abilities.”

Professor Cohen Kadosh adds, “What we have found is how this promising neurostimulation works and under which conditions the stimulation protocol is most effective. This discovery could not only pave the way for a more tailored approach in a person’s learning journey but also shed light on the optimal timing and duration of its application.”

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Credit of the article given to University of Surrey

Statistical methods can evaluate whether pivotal military events, like the Battle of Jutland, American involvement in the Vietnam war or the nuclear arms race, could’ve turned out otherwise, according to a new book.

Military historical narratives and statistical modeling bring fresh perspectives to the fore in a ground-breaking text, “Quantifying Counterfactual Military History,” by Brennen Fagan, Ian Horwood, Niall MacKay, Christopher Price and Jamie Wood, a team of historians and mathematicians.

The authors explain, “In writing history, it must always be remembered that a historical fact is simply one of numberless possibilities until the historical actor moves or an event occurs, at which point it becomes real. To understand the one-time possibility that became evidence we must also understand the possibilities that remained unrealized.”

Re-examining the battlefield

Midway through the First World War, Britain and Germany were locked in a technologically driven arms race which culminated in the Battle of Jutland in 1916. By that time, both nations had built over 50 dreadnoughts—speedy, heavily-armored, turbine-powered “all-big-gun” warships.

In the context of a battle that Britain couldn’t afford to lose, Winston Churchill was quoted as saying that the British commander John Jellicoe was “the only man on either side who could lose the war in an afternoon.”

Both nations, at various points in time, have claimed victory in the Battle of Jutland, and there is no consensus on who “won.” Using mathematical modeling, “Quantifying Counterfactual Military History” probes whether the Germans could have achieved a decisive victory.

The five scholars note, “This reconstructive battling enables us to put some level of statistical insight into multiple realizations of a key phase of Jutland. The model is crude and laden with assumptions—as are all wargames—but, unlike in a wargame, our goal is simply to understand what is plausible and what is not.”

Understanding nuclear deterrence

Counterfactual reasoning is positioned centrally when it comes to the fraught history of nuclear deterrence. By the 1980s, the nuclear arms race between the US and the Soviet Union had already spanned three decades and 1983 would bring a crisis less well-known than the Cuban missile crisis of 1962.

The authors draw attention to the peak of intensity in November 1983 during the so-called “Second Cold War.” A NATO “command post” exercise in Western Europe—known as Able Archer—was created to test communications in the event of nuclear war.

The Soviets, however—likely the result of faulty intelligence gathering—believed that an attack was imminent with the NATO exercise interpreted as the first phase. “Quantifying Counterfactual Military History” highlights the example as one where each side placed themselves in dangerous counterfactual mindsets.

“Mutual misapprehension in 1983 continued a long tradition of misunderstanding which had always created catastrophic potential for war, now based consciously and unconsciously on game theory and its erroneous assumption that rational actors were guided by accurate information,” the authors explain. “In this way they stumbled towards a war that neither had willed.”

Embrace the alternative

“Quantifying Counterfactual Military History” uses case studies of Jutland, Able Archer, the Battle of Britain and the Vietnam War to appraise long-established narratives around military events and examine the probabilities of the events that took place alongside the potential for alternative outcomes.

The book’s authors, however, take a restrained approach to counterfactual theory, one that acknowledges and considers why some events—including the actions of individuals or the rise of institutions—are more important than others and can be considered “critical junctures.” They understand this as very different from the arbitrary, loosely substantiated suppositions made by “exuberant” counterfactuals.

They say, “We can never be certain of the existence of critical junctures, or of the grounds of their criticality, but ‘restrained’ counterfactuals, if done with multiple perspectives and sufficient thoroughness, can surely make a distinctive contribution to the literature.”

The book is underpinned by an inter-disciplinary method which combines historical narrative and statistical data and analysis, offering both quantitative and qualitative rigor.

They explain, “This study has taken us in directions which are not common in academic collaboration, but which we hope demonstrate that collaborative research exploring what had been dead ground between the sciences and the humanities is long overdue.”

Rather than attempt to merely reinvent the past, “Quantifying Counterfactual Military History” calls attention to the dynamism inherent in historical practice and offers another tool for understanding historical actors, the decisions they made and the futures they shaped.

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Credit of the article given to Taylor & Francis

Calculus is the study of change. Calculus teaching methods, however, have changed little in recent decades. Now, FIU research shows a new model could improve calculus instruction nationwide.

A study published in Science shows a reimagined, innovative active learning approach to calculus instruction benefits all students. The model, developed at FIU, focuses on mastering different ways of thinking and solving problems—skills that are important beyond the classroom.

Rote memorization and large lecture halls have been replaced by active learning classrooms where students work collaboratively to solve problems. The result is greater learning outcomes and an understanding of calculus concepts, as well as better grades than their peers in traditional, lecture-based classes, according to the research.

“This large-scale study shows us what we’ve been seeing at FIU: If you put students in an interactive, active learning environment, they can and do learn significantly more, developing the ‘habits of mind’ they’ll use for a long time and throughout their careers,” said Laird Kramer, the study’s lead author and founding director of FIU’s STEM Transformation Institute.

Kramer and a team from the STEM Transformation Institute followed 811 FIU undergraduates enrolled in different sections of the same Calculus I course with two very different teaching methods—half of the sections were traditional lecture-based classes and the other half employed the evidence-based active learning model developed at FIU.

To see which group retained more information and better understood calculus concepts, the students were tested at the end of the course. Active learning classes had a higher average pass rate of 11%. Apply that to the roughly 300,000 students taking calculus each year in the U.S. and it could mean an additional 33,000 students passing calculus and getting closer to a STEM degree and career.

The active learning group’s learning gains cut across majors and academic paths and included underrepresented groups in STEM. This finding is significant since less than half of students entering universities as STEM majors actually graduate with a STEM degree. Failing calculus is a major reason.

“Calculus remains a critical step on the pathway to numerous STEM careers in engineering and the sciences,” said Michael J. Ferrara, Program Director at NSF Directorate of STEM Education. “This study makes a rigorous and compelling argument that active, student-centered calculus courses result in significantly greater learning and success outcomes when compared to more traditional approaches.

“These benefits are particularly profound for students from populations that have traditionally been underrepresented in the STEM workforce, which underscores how a more modern approach to teaching mathematics is critical as we look to nurture the full spectrum of STEM talent across the nation.”

Improving teaching methods in calculus means students are more likely to stay on track and stick with a STEM program. That, in turn, helps graduate more STEM professionals.

“Student success is FIU’s priority, as demonstrated by the development and successful implementation of active learning strategies in our calculus courses,” FIU Executive Vice President and Provost Elizabeth M. Béjar said. “This research builds on years of studying the positive impacts of active learning in the classroom to ensure students have the knowledge and skills they need to move through their STEM courses with confidence.”

FIU has led collaborative initiatives through its nationally recognized STEM Transformation Institute to improve learning. Research has informed the development and introduction of innovative instructional strategies for calculus, mathematics and other sciences and engineering. That’s led to significant increases in four-year graduation rates for STEM majors at FIU.

FIU’s active learning model is just as challenging and rigorous as a traditional lecture style, but more effective for the often incredibly complex process of learning and provides opportunities for different parts of the brain to engage and store information. Students learn by doing. Class time is collaboration time. In small groups, students work face-to-face developing and testing hypotheses. The goal is to learn to ask the right questions and look at problems in new ways—meaning, they think and act like mathematicians, engineers, scientists, doctors.

“This research began as an experiment to see if we could identify new ways of teaching coursework and give students an alternative way of learning the rigorous content in calculus,” said Mike Heithaus, executive dean of the College of Arts, Sciences & Education. “Based on prior research, we felt confident the active learning method would be effective. But even we were surprised at how much better students did in the active learning sections versus the traditional. As soon as we got these results, we began implementing these methods throughout our entire math curriculum.”

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Credit of the article given to Angela Nicoletti, Florida International University