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Exam Code: 8002 Practice exam 2022 by Killexams.com team
II- Mathematical Foundations of Risk Measurement
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Killexams : PRMIA Mathematical obtain - BingNews https://killexams.com/pass4sure/exam-detail/8002 Search results Killexams : PRMIA Mathematical obtain - BingNews https://killexams.com/pass4sure/exam-detail/8002 https://killexams.com/exam_list/PRMIA Killexams : Applied and Computational Mathematics

The course is a basic introductory graduate course in partial differential equations. syllabus include: Laplacian, properties of harmonic functions, boundary value problems, wave equation, heat equation, Schrodinger equation, hyperbolic conservation laws, Hamilton-Jacobi equations, Fokker-Planck equations, basic function spaces and inequalities, regularity theory for second order elliptic linear PDE, De Giorgi method, basic harmonic analysis methods, linear evolution equations, existence, uniqueness and regularity results for classes of nonlinear PDE with applications to equations of nonlinear and statistical physics.

Sun, 21 May 2017 14:06:00 -0500 en text/html https://www.princeton.edu/academics/area-of-study/applied-and-computational-mathematics
Killexams : How to use mathematical formulas in Google Sheets No result found, try new keyword!Equations are just math. You likely won't use something as simple as 2 + 2, but that's the basic idea. To input a formula into Google Sheets, preface what you type with the equal sign ... Sat, 03 Dec 2022 04:40:00 -0600 https://www.androidpolice.com/how-to-use-mathematical-formulas-in-google-sheets/ Killexams : Mathematical Sciences

Consult the Smith College Course Catalog for information on the current courses available in mathematics and statistics.

There are also several courses that are available for credit from other departments, including art, psychology and more. Consult the catalog.


What classes you should take depends a great deal on what you find most interesting and on what your goals are. Discuss your options with your adviser and also talk to the instructors of particular courses that interest you.

If you are interested in the sciences

The department offers a variety of courses to supply you a solid mathematical experience. Calculus III and Linear Algebra are fundamental courses. You may also want to consider taking one or more of the following: Intro to Probability and Statistics, Differential Equations, Differential Equations and Numerical Methods, Discrete Mathematics, Advanced syllabus in Continuous Applied Mathematics.

If you are interested in computer science

Consider taking some of these: Calculus III, Linear Algebra, Modern Algebra, Discrete Mathematics. Many of our students are double–majoring in mathematics and computer science.

If you are interested in economics

Calculus will supply you a good, basic experience. You may consider other courses as well, so be sure to discuss your options with your adviser. If you are contemplating graduate school in economics, the economics department recommends you to take MTH 211, 212, 280 and 281. Taking a solid course in statistics is also a good idea (any of MTH 220, 246, 290, 291 and 320 would do). Many economics majors want to take MTH 264 as well. Double–majoring in mathematics and economics is a good choice.

If you are interested in applied mathematics

The following courses work specifically with applications: MTH 205, 264, 353 and 364. Other courses that contain many applications and are important for anyone considering graduate school in applied mathematics are: MTH 220, 246, 254, 255, 280, 290, 291, and 320. 

If you are interested in theoretical mathematics

The following courses work with abstract structures: MTH 233, 238, 246, 254, 255, 280, 281, 333, 370, 381, and 382.

If you liked calculus

There are many reasons for liking calculus. If you delighted in the geometry, for example, you should consider MTH 270, 280, 370 and 382. If you enjoyed the power of calculus to describe and understand the world, you will want to take MTH 264. If you are fascinated with the ideas of limit and infinity and want to get to the bottom of them, you should take MTH 281.

If you liked linear algebra

You will like MTH 233 very much, and you will also like MTH 238 and 333.

If you liked discrete mathematics

The natural sequel to Discrete Mathematics is MTH 254 or 255 and then 353. In addition, you may be interested in MTH 246 and in CSC 252 (counts 2 credits toward the mathematics major).

If you are interested in graduate school in mathematics

Take a lot of courses, but be sure to take MTH 233, 254, and 281 and as many of MTH 264, 333, 370, 381, and 382 as possible. You should also consider taking a graduate course at the University of Massachusetts.

If you are interested in graduate school in statistics

The MST Mathematical Statistics joint Major between MTH and SDS is explicitly designed as a preparation for graduate school in Statistics. 

If you are interested in graduate school in operations research

Operations research is a relatively new subarea of mathematics, bringing together mathematical ideas and techniques that are applied to large organizations such as businesses, computers, and governments. You should take MTH 211 and at least some of the courses listed for statistics above, some combinatorics (MTH 254) and some computer science. Consider also syllabus in Applied Mathematics and Numerical Analysis.

If you want to be a teacher

Certification requirements vary widely from state–to–state. If you are interested in teaching in secondary school, a mathematics major plus practice teaching may be enough to get started. In Massachusetts, the major should include either MTH 233 or 238 and one of MTH 220 or 246. A course involving geometry, such as MTH 270 or MTH 370 is also helpful. You should also have some introduction to computers. For guidelines, look at the list of courses listed in the MAT program. Finally, while MTH 307 syllabus in Mathematics Education is rarely offered, something equivalent is taught as a special studies whenever there are MAT students.

If you are interested in teaching elementary school, most of your required courses will be in the education department. In the mathematics department, our concern would be that you are comfortable with mathematics, have seen its variety, and most important, that you enjoy it. For all that, you should take the mathematics courses which appeal to you most. For education courses, the latest information is that you should take EDC 235, 238, 346, 347, 404 (practice teaching), and one elective to be certified. Note that during the semester when you take practice teaching EDC 404, you will likely be unable to take a math course. Plan ahead and consult the education department.

If you want to be a doctor

You are doing fine by majoring in mathematics. A course in statistics would be a very good idea. Other areas of mathematics that would be useful are differential equations and combinatorics.

If you want to be an actuary

Take MTH 246, 290, 291 and 320 and the actuarial exams that are offered periodically. Advancement as an actuary is achieved by passing of a series of examinations. Informal student study groups often form (ask around!).

If you want to get a good job when you graduate

A major in mathematics prepares you well, regardless of which courses you choose. Math majors learn to think on their feet; they aren't frightened of numbers and they're at home with abstract ideas. Often, this alone is what employers are looking for. That said, we should add that knowledge of computer programming is very useful, as is some familiarity with statistics.

If you want something Smith does not offer

If you are interested in a subject we do not offer, you should talk to professors whose fields of interest are closest to the subject, as a special studies. The arrangement must be approved by the department, but reasonable requests are not refused. If your interest is particularly strong, you might consider an honors project, or summer research work. You should also consider taking a course (or courses) at one of the consortium schools.

Sun, 04 Dec 2022 13:53:00 -0600 en text/html https://www.smith.edu/academics/mathematical-sciences
Killexams : How Math Became an Object of the Culture Wars

William Heard Kilpatrick, one of the most influential pedagogical figures of the early twentieth century, would have felt right at home in today’s educational culture wars. Back then, as now, the traditionalist defense of math education came from the idea that the subject created order and discipline in the minds of young students. The child who could solve a geometric proof, for example, would carry that logic and work ethic into his professional life, even if it did not entail any numbers at all. Kilpatrick, a popular reformer who was known as the “million-dollar professor,” not for his salary but for the huge tuition-paying crowds his lectures drew, dismissed that idea. Algebra and geometry, he believed, should not be widely taught in high schools because they were an “intellectual luxury,” and “harmful rather than helpful to the kind of thinking necessary for ordinary living.” Not everyone was going to need or even have the intelligence to complete an algebra course, Kilpatrick reasoned. Why bother teaching it to them?

In 1915, Kilpatrick chaired an influential National Education Association committee tasked with looking into the reform of math instruction in high school. He amplified his attack on the place of math in schools, as the committee’s report declared that nothing in mathematics should be taught unless “its (probable) value can be shown,” and recommended the traditional high-school-mathematics curriculum for only a select few.

What Do We Really Know About Teaching Kids Math?

Read Part II of Jay Caspian Kang’s series on math education.

Kilpatrick’s ideas were taken up by the progressivist movement in education, a powerful force in the early twentieth century inspired by the work of the philosopher John Dewey and guided by a set of principles that included “freedom for children to develop naturally,” “interest as the motive of all work,” and “teacher as guide, not taskmaster.” These ideas had their roots in the University of Chicago but ultimately went mainstream when they were championed by professors at the Columbia University Teachers College, where Kilpatrick and Dewey taught. The coalition of anti-math parents and academics had a steady influence on education policy for decades. From the start of the twentieth century to after the Second World War, the percentage of high-school students enrolled in algebra fell. In 1909, roughly fifty-seven per cent of high-school students were enrolled in algebra. By 1955, that number had been cut by more than half to about twenty-five per cent.

The decline in advanced math coincided with what can only be called a revolution in secondary schooling. In 1890, less than seven per cent of fourteen-year-olds were in high school, which was almost entirely the provenance of the wealthy and connected; by the late nineteen-thirties, nearly three out of four children between fourteen and seventeen were enrolled in some form of high school, which meant that schools and educators had a much broader and complicated problem than when they were just figuring out how to prepare the richest kids in the country.

So began the backlash to Kilpatrick and his coalition of progressive educators. Starting in the nineteen-fifties, something called “new math” was introduced into classrooms across the country. This movement promised an acceleration of math education and a new, scientific approach to how kids learned, but it didn’t really gain much traction until 1957, when the U.S.S.R. launched Sputnik. The ensuing panic about the U.S.’s ability to compete led to nationwide reforms that brought calculus into the high-school curriculum and, for a time, retired the progressivist movement in education.

This same fight has repeated itself on several different occasions since then. The open-education movement of the late nineteen-sixties and nineteen-seventies tried to eliminate curricula and standardized tests, and stop the separation of kids into different grades; some “open schools” even did away with walls and allowed the children to dictate what was taught. The backlash to this approach in the following decade culminated in the “math wars” of the nineteen-nineties, in which a new focus on ameliorating racial inequalities in math achievement came up against a panic about how American children measured up to those in Asia, and the potential economic ramifications of a math deficit. The reformers believed that it was vital to shrink achievement gaps in part because they saw—somewhat correctly—an emerging economy that would be dominated by math-related fields, whether computer science, engineering, or economics. A math gap in schools, in essence, was a preview for accelerating income inequality.

At the heart of the nineties math wars was a relatively progressive report published, in 1980, by the National Council of Teachers of Mathematics (N.C.T.M.), which advocated for group learning and the use of calculators—in favor of “problem solving”—before students had even mastered the basics of arithmetic. Starting in 1989, the report was codified into national math standards that were eventually protested by angry parents—including two particularly heated groups from Princeton, New Jersey and Palo Alto, California—who demanded more advanced math offerings. The conflict lasted years, during which time anti-reform math traditionalists, decrying what they referred to as “fuzzy math,” had organized themselves into a political force that called for changes in textbooks and lobbied national politicians.

In some ways, the cyclical nature of these arguments makes sense: each generation of parents believes their children are facing problems the world has never seen before. As was true in the nineteen-nineties, today’s fights about math are not entirely about what kids actually learn in their classrooms, or how well those lessons prepare the U.S. for competition against a geopolitical enemy. I’d argue that the spectre haunting today’s discord over math education is a growing suspicion of the equity-based movement of educators who, as in the early twentieth century, largely come from graduate education programs like the ones at Stanford and Columbia’s Teachers College.

Parents tend to express their misgivings with the declaration “I just want my kids to learn math.” I have heard this phrase countless times: on social media, in viral clips of school board meetings, and in my own interactions with fellow-parents. I admit that I have even said it myself. The tone differs, but the message is more or less the same: math is the justification, both implicitly and explicitly, for the reaction against progressive curriculum reform in American schools. At an abstract, but deeply felt level, math is the ideal foil for so-called “wokeness” and the infusion of social justice into schools because it is supposed to exist outside of humans and their frivolous emotions. Pi is pi everywhere; two plus two, as they say, equals four.

A accurate iteration of the math-backlash cycle was a controversy over the California Mathematics Framework (C.M.F.), the state’s math-curriculum guidelines, which get updated about once a decade. The current proposal, which Rivka Galchen outlined in these pages in September, will be up for adoption in 2023. It suggests that California address achievement gaps in math by finding alternatives to “tracked” math classes for advanced kids in early education and, in a Kilpatrick-esque turn, creating a data-science track for the high-school students who might not have an interest in calculus. An early draft of the proposal even recommended eliminating algebra instruction at the middle-school level in order to keep the playing field level.

The draft framework was criticized by the usual suspects (math professors, and parents who worry that their kids will go to U.C. Santa Cruz instead of Stanford) but also many equity-focussed educators who worry that the program may be seen as a slackening of expectations for minority and low-income students.

A central figure in the debate was Jo Boaler, a professor at Stanford’s Graduate School of Education and an author of the C.M.F. who has conducted research pointing to the harmful effects of tracked classes. In a study published in 2008, Boaler and her co-author examined three California high schools, including one that de-tracked all of its math courses. Students at that school, she found, “learned more, enjoyed mathematics more and progressed to higher mathematics levels.” These findings have been supported by other studies in the field, which show that “low-ability” students performed better in “heterogeneous” settings, where they learn math with students of differing abilities, than when they were placed in a tracked group. Middle- and high-performing students performed equally as well in both environments.

The question for Boaler, then, centers on who usually gets put in lower tracks. “There is evidence that tracking results in severe racial stratification,” Boaler wrote in an e-mail. “Students of color, with the same achievement, end up in lower track classes.” Underrepresented students in science and math fields, Boaler and her C.M.F. co-authors write, “experience significantly more academic barriers (lack of academic exposure) in middle grades and below, and these barriers are negatively associated with math achievement in high school.” In addition to suggesting de-tracked math offerings, the draft framework also proposes to incorporate social-justice lessons into math, and to orient lessons toward seeing how math could advance social justice.

Predictably, the social-justice element fuelled a lot of the outrage around the C.M.F. Brian Lindaman, a professor at California State University, Chico, who chaired the committee of experts, academics, and former math teachers who wrote the C.M.F., said the creators of the framework were “asked in the guidelines to address equity.” He described this to me as a “challenge,” because of the ways in which it would almost automatically be seen as a political statement.

Lindaman did not exactly welcome the controversy—he told me he was a bit taken aback by its ferocity—but he is a true believer in the equity ethos. For him, it comes down to figuring out how Black, Latino, and other children who are underrepresented in STEM fields can start to think of themselves as “math people.” “What we would hope to see is that students find the joy and the beauty of math early,” Lindaman said. “Many of them do, it’s just that somewhere along the line that gets taken out and they stop seeing the beauty in mathematics. And we think that has something to do with the way it gets taught.” Lindaman also acknowledged that the framework’s emphasis on problem-solving questions and math scenarios rooted in social-justice concerns was “controversial,” but he defended it by stressing the importance of children seeing real-world applications of what they were doing, in a way that made sense to them.

A lot of this isn’t worth arguing over. The claim that pedagogy should incorporate examples from the daily lives of all kinds of students is a reasonable one. The idea that math instruction should aim to create as many math people as possible isn’t what infuriates the parents of Palo Alto and Princeton. The problem is that the C.M.F., which, as noted before, was based on explicit instructions to address equity concerns, framed the problem in a much broader, almost philosophical way. It’s a credit to the authors that they don’t treat STEM subjects as hard, objective endeavors that separate exceptional children from their competitors, but rather as lifelong pursuits that should include as many different types of people as possible.

Mon, 14 Nov 2022 10:00:00 -0600 en-US text/html https://www.newyorker.com/news/our-columnists/how-math-became-an-object-of-the-culture-wars
Killexams : What You Need to Know About Becoming a Mathematics Major No result found, try new keyword!Mathematics majors study the relationships between numbers, structures and patterns. Their classes range from algebra to statistics, and the concepts build on one another. Students learn skills ... Tue, 05 May 2020 08:08:00 -0500 text/html https://www.usnews.com/education/best-colleges/mathematics-major-overview Killexams : Novel mathematical technique enables better modeling of 'multiphase' fluids

Researchers have developed a mathematical technique that radically reduces the enormous computational costs of trying to model fluids that combine both liquid and gas phases, especially within rocket engines. The computational burdens of this sort of modeling have long challenged researchers to accurately describe how shockwaves in such multiphase fluids produce wear and tear in machinery.

The technique is described in a paper that appeared in the Journal of Computational Physics.

Calculations for —the branch of physics concerned with the motion of liquids, gases, and anything else that acts like a fluid—can run into a computational challenge when considering the flow of materials in more than one phase or state of matter, such as a water and a gas flowing together simultaneously.

This sort of flow, termed "multiphase flow," which also covers the flow of two different liquids such as oil and water, is extremely widespread. It happens anywhere there might be a breakup of liquid droplets, and their phase changes evaporation, boiling, condensation and cavitation (the forming of small vapor-filled cavities within a liquid, i.e., bubbles).

In the oil and gas industry, for example a well is the source of a flow not just of petroleum but also natural gas and water. The flow inside a rocket engine that uses a is another example of a multiphase flow, again of gaseous and liquid phases, where some of the liquid oxygen evaporates and then together with the liquid remainder is set alight in a combustion chamber.

Study of multiphase flow is vital within such industries not least for how machine wear and tear occur. When the bubble vapor cavities of the earlier mentioned phenomenon of cavitation confront higher pressures, these bubbles can collapse. That collapse in turn produces a shock wave that can weather and damage machines or infrastructure.

Modeling of this is thus extremely important to industrial activities throughout much of the modern world. It is straightforward enough modeling a fluid in a single phase, but few fluids in the real world ever remain just one phase. The computational challenge here comes from quantifying the distribution of velocities of the different phases, including how velocities change at the interface between the two (or more) phases.

There are two main ways of modeling multiphase flows, the Euler-Lagrange method and the Euler-Euler method. Both are extremely computationally expensive, and both have certain drawbacks. In the Eulerian-Lagrange approach, for example, the liquid phase is expressed as a collection of particles, and, as a result, gas-liquid interfaces and primary atomization (the process of change from a bulk liquid to droplets) cannot be analyzed.

It was not until the 1990s, that increases in computing power permitted more realistic modeling of multiphase flow. Researchers for the first time were able to move up from simplified one-dimensional representations of multiphase flow to more realistic three-dimensional models.

But even with such computing power advances, multiphase model work remains computationally expensive (which means, in some really tricky cases, financially expensive too). This is especially problematic for the high degree of multiphase-flow complexity within rocket engines. The multiphase liquid and gaseous oxygen fuel combustion can be accompanied by instabilities such as resonance or chugging in the engine from fluctuations in the rate of heat release.

To address such problems, multiphase flow modeling is required that involves mathematical solutions describing both sound waves and compressible flows (in other words involving cavitation). In rocket engine research, compressible multiphase flow computation—and how to reduce its computational cost—has increasingly become an important subject of research.

Conventional methods of modeling compressible multiphase flow call for the use of a computationally expensive "exact Riemann solver" to describe interactions between air bubbles and water shockwaves (at the interface between liquid and gas phases).

"An exact Riemann solver is a method of approximation in to describe the flux across such discontinuities," said Junya Aono, a computational fluid dynamicist with the Department of Mechanical Engineering and Material Science at Yokohama National University (YNU). "But it incurs high computation costs and also struggles with what's called the carbuncle problem in which the captured shock waves are distorted, potentially affecting heat transfer to the chamber containing the multiphase flow."

So the researchers developed a way to model compressible multiphase flow that no longer needs Riemann solvers.

"This involves a tweaking of the Simple Low-dissipation Advection Upstream Splitting Method," said Keiichi Kitamura, co-author of the paper and who is also with YNU. "This is a used elsewhere but can be extended to complex physical interactions, such as multiphase flows."

The tweaking—which involves careful design to take into account numerical dissipation, or the way that a simulated fluid can exhibit higher rate of diffusion that the medium does in the real world—allows accurate description of interactions of the captured shock waves and other discontinuities in multiphase flows. Crucially, all of this can be easily coded.

Any potential users of the technique can now easily produce compressible multiphase flow simulations without huge computational burdens or having to take special care in selecting the parameters for the model.

The researchers now want to apply their mathematical technique to practical 3D compressible multi-phase flow simulations, in particular with respect to .

More information: Junya Aono et al, An appropriate numerical dissipation for SLAU2 towards shock-stable compressible multiphase flow simulations, Journal of Computational Physics (2022). DOI: 10.1016/j.jcp.2022.111256

Provided by Yokohama National University

Citation: Novel mathematical technique enables better modeling of 'multiphase' fluids (2022, December 8) retrieved 13 December 2022 from https://phys.org/news/2022-12-mathematical-technique-enables-multiphase-fluids.html

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Thu, 08 Dec 2022 05:27:00 -0600 en text/html https://phys.org/news/2022-12-mathematical-technique-enables-multiphase-fluids.html
Killexams : Flickering fireflies pulse to a beat that mathematicians have long been seeking

This article was originally featured on The Conversation.

Imagine an old-growth forest in the fading light of a summer evening. As the last of the sun’s rays disappear beneath the horizon, a tiny flash catches your eye.

You turn around, hold your breath; it blinks again, hovering 2 feet above the leaf litter. Across the dusky glade, a fleeting response. Then another one, and another, and within minutes flickering fireflies spread all over the quiet woods.

At first they seem disorganized. But soon a few coordinated pairs appear, little tandems flashing on the same tempo twice a second. Pairs coalesce into triads, quintuplets, and suddenly the entire forest is pulsating with a common, glittering beat. The swarm has reached synchrony.

Firefly congregations are sprawling speed-dating events. Flashes convey a courtship dialogue between advertising males and selective females. Shaped by the interplay of competition and cooperation among thousands of fireflies in interaction, collective light patterns emerge, twinkling analogs to the murmurations of bird flocks swooping together. The mystifying phenomenon of some fireflies’ flash synchronization has puzzled scientists for over a century.

Synchrony is ubiquitous throughout the universe, from electron clouds to biological cycles and planetary orbits. But synchrony is a complex concept with many ramifications. It encompasses various shapes and forms, usually revealed by mathematics and later explored in nature.

Take the firefly swarm. Wait a little longer and among the illuminated chorus, something else appears: Some discordant flashers secede and continue off-beat. They blink at the same pace but keep a resolute delay with their conformist peers. Could this be evidence of a phenomenon predicted by mathematical equations but never seen in nature before?

Synchrony, with a twist

Twenty years ago, while digging deeper into the equations that form the framework of synchrony, physicists Dorjsuren Battogtokh and Yoshiki Kuramoto noticed something peculiar. Under specific circumstances, their mathematical solutions would describe an ambivalent ensemble, showing widespread synchrony interspersed with some erratic, free-floating constituents.

Their model relied on a collection of abstract clocks, called oscillators, that have a tendency to align with their neighbors. The nonuniform state was surprising, because the equations assumed all oscillators were perfectly identical and similarly connected to others.

Spontaneous breaking of underlying symmetry is something that typically bothers physicists. We cherish the idea that some order in the fabric of a system should translate into similar order in its large-scale dynamics. If oscillators are indistinguishable, they should either all get in sync, or all remain chaotic – not show differentiated behaviors.

It piqued the curiosity of many, including mathematicians Daniel Abrams and Steven Strogatz, who named the phenomenon “chimera.” In Greek mythology, the Chimera was a hybrid monster made of parts of incongruous animals – so a fitting name for a hodgepodge of mismatched clusters of oscillators.

At first, chimeras were rare in mathematical models, requiring a very specific set of parameters to materialize. Over time, learning where to scout, theorists began to uncover them in many variations of these models, dubbing them “breathing,” “twisted,” “multiheaded” and other eerie epithets. Still, it remained mysterious whether these theoretical chimeras were also possible in the physical world  – or merely a mathematical myth.

A decade later, a few ingenious experiments set up in physics laboratories yielded the elusive chimeras. They involved finely tuned networks of interactions between sophisticated oscillators. While proving that engineering the coexistence of coherence and incoherence was a delicate, but possible, venture, they left the deeper question unanswered: Could mathematical chimeras also exist within the natural world?

It turned out it would take a tiny luminescent insect to shed light on them.

Chimera amid the fireflies’ blinking chorus

As a postdoc in the Peleg Lab at the University of Colorado, I work on deciphering the inner workings of firefly swarms. Our approach builds on the foundations of a little-known niche within modern physics: animal collective behavior. Simply put, the overarching objective is to reveal and characterize spontaneous, unsupervised large-scale patterns in the dynamics of groups of animals. We then investigate how these self-organized patterns emerge from individual interactions.

Advised by knowledgeable firefly experts, my colleagues and I drove across the country to Congaree National Park in South Carolina to chase Photuris frontalis, one of few North American species known to synchronize. We set up our cameras in a small forest clearing among the loblolly pines. Soon after the first flickers poked through the twilight, we observed a very rhythmic, precise synchrony, apparently as clean as predicted by equations.

This was an enchanting experience, yet one that left me reflective. I panic that this display was too orderly to let us infer anything from it. Physicists learn about things by looking at their natural fluctuations. Here, there seemed to be little variability to investigate.

Synchrony manifests itself in the data in the form of sharp spikes in the graph of the number of flashes over time. These peaks indicate that most flashes occur at the same instant. When they don’t, the trace looks irregular, like scribbles. In our plots, I first saw nothing but the flawless comblike pattern of impeccable synchrony.

It turned out the chimera was hiding in plain sight, but I had to roam further along the data to encounter it. There, in between the spikes of the light chorus, some shorter peaks indicated smaller factions in sync among themselves but not with the main group. I called them “characters.” Together with the synchronized chorus, these incongruous characters make up the chimera.

Like in the ancient Greek theater, the chorus sets the background while characters create the action. The two groups are intertwined, roaming the same stage, as we revealed from the three-dimensional reconstruction of the swarm. Despite the split in their rhythm, their spatial dynamics appear indistinguishable. Characters don’t seem to congregate or follow one another.

This unexpectedly intermingled self-organization raises even more questions. Do characters among the swarm consciously decide to break away, maybe to signal their emancipation? Or do they spontaneously find themselves trapped off-beat? Can mathematical insights enlighten the social dynamics at play among luminous beetles?

Unlike abstract oscillators in math equations, fireflies are cognitive beings. They incorporate complex sensory information and process it through a decision-making pipeline. They are also constantly in motion, forming and breaking visual bonds with their peers. Streamlined mathematical models don’t yet capture these intricacies.

In the quiet woods, the synchronized flashes and their dissonant counterparts may have illuminated a trove of new chimeras for mathematicians and physicists to chase.

Sun, 20 Nov 2022 21:23:00 -0600 Raphael Sarfati / The Conversation en-US text/html https://www.popsci.com/environment/firefly-synchrony-mathematics/
Killexams : Dyscalculia: how to support your child if they have mathematical learning difficulties

A good grasp of maths has been linked to greater success in employment and better health. But a large proportion of us – up to 22% – have mathematical learning difficulties. What’s more, around 6% of children in primary schools may have dyscalculia, a mathematical learning disability.

Developmental dyscalculia is a persistent difficulty in understanding numbers which can affect anyone, regardless of age or ability.

If 6% of children have dyscalculia, that would mean one or two children in each primary school class of 30 – about as many children as have been estimated to have dyslexia. But dyscalculia is less well known, by both the general public and teachers. It is also less well researched in comparison to other learning difficulties.

Children with dyscalculia may struggle to learn foundational mathematical skills and concepts, such as simple counting, adding, subtracting and simple multiplication as well as times tables. Later, they may have difficulty with more advanced mathematical facts and procedures, such as borrowing and carrying over but also understanding fractions and ratios, for instance. Dyscalculia not only affects children during maths lessons: it can have an impact on all areas of the curriculum.

These persistent difficulties cannot be explained by a general below-average ability level, or other developmental disorders. Nevertheless, children with dyscalculia may also experience other learning difficulties, such as dyslexia and ADHD.

Here are some practical tips to support children with mathematical learning difficulties.

Use props

Children with dyscalculia can find additional practical supports useful when working out even simple sums and maths problems. They may often need to use practical aids, such as their fingers or an abacus. They can benefit from using counters and beads to make sets or groups, as well as using number lines to work out answers to maths problems.

Older children may find it helpful to keep crib sheets handy, which make information such as the times tables or certain formulas easily accessible. Inclusive teaching methods like these are likely to benefit all learners, not just those with dyscalculia.

Break the problem down

Research shows that metacognition can have a positive effect on maths learning. Metacognition is “thinking about thinking” – for example, thinking about the information you do and don’t know, or self awareness about the strategies you have to work out problems.

Teaching children strategies to identify where to start on a problem and how to break mathematical problems down could be a good starting point. For example, parents and teachers could encourage children to use songs and mnemonics to help them remember strategies to solve particular problems.

For example, the mnemonic DRAW provides students with a strategy for solving addition, subtraction, multiplication, & division problems:

D: discover the sign – the student finds, circles, and says the name of the operator (+,-, x or /).

R: read the problem – the student reads the equation.

A: answer – the student draws tallies or circles to find the answer, and checks it over.

W: write the answer – the student writes out the answer to the problem.

Find out where help is needed

Children with mathematical learning difficulties often get stuck with maths problems and may quickly supply up. Teachers and parents should ask children what they find difficult – even young children can explain this – and provide explicit instruction to support them with what they find difficult.

Focus on one thing at a time

As mathematical problems can be confusing for young people with mathematical difficulties, make sure to only work on one problem at the time. This could mean covering other maths questions on the page, and removing irrelevant pictures. Provide immediate feedback on both correct and incorrect answers. This will help children learn from their practice and understand the difference between correct and incorrect problem-solving strategies.

Focus on one subject or problem at a time. Ground Picture/Shutterstock

It may also help to provide plenty of repetition and revisiting, teach short and frequent sessions, and make sure learners know what they should do if they get stuck, such as ask an adult for help.

Use the right vocabulary

Mathematical language and symbols can also be confusing. For example, a negative number carries a minus sign, but a minus sign can also be used to define an operation such as subtraction. We often use the word “minus” for both – for instance, saying “14 minus minus 9” (14 – –9). This can be difficult to interpret. Various different words, such as subtract, minus and take away, can describe the same concept.

It is important to use clear language (for instance, “14 take away negative 9”). Helping children expand their maths vocabulary, as well as checking their understanding, will also be useful.

Play games

Mathematics is everywhere around us in the environment and what is learned in the classroom also applies to our daily lives. Our own research has shown that young children benefit from playing short mathematical games using the tools and materials around them.

Counting and collecting sets of items can be done in any place: at the dining table, in the bath, or when out and about. Practice-based educational apps can also help children master foundational maths skills.

Be positive

Finally, it is crucial to promote positive feelings towards mathematics. This might include not voicing your own concerns and negative feelings about maths. Rather, foster an interest in maths that will help children persevere and overcome their difficulties.

Tue, 29 Nov 2022 03:20:00 -0600 en text/html https://theconversation.com/dyscalculia-how-to-support-your-child-if-they-have-mathematical-learning-difficulties-194304
Killexams : Squeeze slowly and REMOVE the cap: Scientists create a mathematical model to help avoid spattering squeezy sauce bottles

Squeeze slowly and REMOVE the cap: Scientists create a mathematical model to help avoid spattering squeezy sauce bottles

  • University of Oxford mathematical model avoids spattering sauce bottles
  • Key is buying a smaller bottle and unscrewing cap to pour out the final remnants
  • It's also important to remember to squeeze more gently and slowly

It is one of life's enduring frustrations, trying to squeeze the last drops out of a ketchup bottle only to end up splattered in tiny flecks of sauce.

Luckily scientists now have some tips on how to stop it happening. 

A team from the University of Oxford has created a mathematical model to help avoid spattering squeezy sauce bottles. 

Their findings suggest that the key is buying a smaller ketchup bottle, unscrewing the cap to pour out the final remnants, or remembering to squeeze more gently and slowly.

It is one of life's enduring frustrations, trying to squeeze the last drops out of a ketchup bottle only to end up splattered in tiny flecks of sauce

How to avoid the dreaded spatter 

  1. Buy a smaller ketchup bottle
  2. Unscrew the cap to pour out the final remnants
  3. Remember to squeeze more gently and slowly
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To understand why, it is first important to know that squeezing a sauce bottle forces the air inside into a smaller space.

The compressed air forces sauce out onto someone's chips to create more space for itself again.

To stop sauce splattering everywhere normally, the forceful air inside the bottle is kept in check by the sauce itself.

A thick sauce becomes almost 'stuck' to the sides of its container, due to the laws of physics, and this resistance slows down the flow as it leaves the bottle.

But this happy balance between the sauce and the air in the bottle is lost when nearly all the sauce is gone, and it therefore provides less resistance.

Then, if the air pushes the sauce out more forcefully than resistance can slow it down, sauce will come out fast and haphazardly, splattering a clean, white shirt within seconds.

This point at which air in a bottle is more forceful than sauce resistance is the 'sauce splatter threshold'.

Luckily the scientists who identified it have some helpful ideas on how to avoid a ketchup disaster.

They say unscrewing the cap of a sauce bottle for the last few uses gives the sauce a larger hole than the usual thin nozzle, reducing the air pressure needed to force it out, so the splatter threshold is avoided and the sauce comes out smoothly.

Buying a smaller ketchup bottle may also work, as a smaller bottle contains less air, meaning less total air pressure, so lower odds of the sauce making a mess.

Another tip is to squeeze more gently, as an impatient hard squeeze to a sauce bottle compresses the air inside faster, building up more energy which forces the sauce out in a fast, uncontrolled manner

Another tip is to squeeze more gently, as an impatient hard squeeze to a sauce bottle compresses the air inside faster, building up more energy which forces the sauce out in a fast, uncontrolled manner.

Researchers worked out the sauce threshold using a series of experiments, injecting air from a syringe into a thin tube filled with oil, which provides similar resistance to ketchup, mayonnaise, brown sauce or mustard, but is simpler to study.

Dr Callum Cuttle, a co-author the study from the Department of Engineering Science at the University of Oxford, said: 'The problem of being splattered with sauce is a universal one, and a source of immense frustration at barbecues, especially when you've just put a perfect swirl of sauce on your burger, only for it to be ruined.

'No one wants their clothes to be ruined by ketchup, and these findings do provide some insights on unscrewing the cap of ketchup bottles, or squeezing them more slowly and softly, which people might want to experiment with at home.

'Our analysis reveals that the splattering of a ketchup bottle can come down to the finest of margins - squeezing even slightly too hard will produce a splatter rather than a steady stream of liquid.'

The sauce splatter threshold, worked out using a branch of mathematics known as nonlinear dynamics, has not yet been reviewed by other scientists but will be presented at the American Physical Society's Division of Fluid Dynamics conference this weekend.

The findings could have important implications for other activities which involve displacing a fluid with a gas, including storing captured carbon dioxide, reinflating collapsed lungs, and designing better fuel cells.

The best way to eat a Chocolate Digestive biscuit, according to science 

1. Remove the biscuit from the packet at room temperature

2. Bring it towards your mouth with the chocolate side facing up

3. As you go to take a bite, turn the biscuit over so the chocolate is facing down

4. Take a substantial bite with the chocolate directly hitting the tongue first

5. If you’re by yourself, eat initially with the mouth open to hear the crunching of the biscuit which makes the taste sensation more memorable

6. Chew slowly after the first few mouthfuls to maximise the full taste experience

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Wed, 16 Nov 2022 02:11:00 -0600 text/html https://www.dailymail.co.uk/sciencetech/article-11435573/Scientists-create-mathematical-model-help-avoid-spattering-squeezy-sauce-bottles.html
Killexams : A former math teacher explains why some students are "good" at math, and others lag behind

When Frances E. Anderson saw the latest math scores for America's fourth-and eighth-graders, she was hardly surprised that they had dropped. Until recently — including the period of remote instruction during the pandemic — Anderson taught high school math to students at all levels. Now she is a researcher seeking to change how people understand children's math ability. In the following Q&A, Anderson explains what makes some kids "good" at math and what it will take to catch up those who have fallen behind.

What was the hardest part about teaching during the pandemic?

Seeing students who already struggled not be able to get what they needed during that time. Before the pandemic, I could work with students one on one, have students work in pairs, or have students in more advanced classes come tutor students in entry-level classes. During the pandemic, all of this was taken away because we didn't share the room with our students and — at least in the initial stages of the pandemic — many of us didn't have the skills to use comparable teaching strategies online.

How do you explain the accurate drop in math scores?

Once schools shifted to remote learning during the pandemic, teachers didn't have as many ways to keep students engaged. It was difficult to do hands-on activities and project-based learning, which are better for students who struggle in math.

Math teachers had to tell students what to do in mathematics, but this kind of direct instruction works for only about 20% of students. A lot of teaching math is visual. You need so much more space than just one screen. Teachers might use their words, hand gestures, whiteboards, graphs, diagrams, objects, physical movements, student work examples and more. These actions and items build a comprehensive experience and build more of the skills that math students need since the students can look at several of these teaching aids at once. Online, the teacher is limited only to what can be seen on their screen or on one student's screen at a time, which is vastly different.

In addition to being visual, teaching math is a lot about what is said during class. In fact, one of the most important functions of effective math teaching is how the teacher engages in conversations with students about mathematics. This conversation, known as classroom discourse, has great power to help students learn. When every student is muted so that they can hear the teacher, it's impossible to hear the students speak about mathematics.

Why are some students 'good' at math and others can't solve basic problems?

It's not true that some kids are good at math and others aren't. It comes down to what kind of exposure and experiences children have early in their lives. Some parents see to it that their kids do more with numbers than others. They do more at home, they do more in social events, and they do more in school. These routine exposures make them appear good at math. It isn't that they're good so much as it is that they had more time to work with mathematics.

Why did you leave teaching?

I still teach today, only I teach a different set of students: future teachers. As a schoolteacher, my impact was limited to the 180 students that I had each year. But now I am in a position where I can impact about 100 future teachers every year. That means each of those 100 future teachers can turn around and impact 180 students themselves every single year. In my position now, I can help so many more students in education than I ever could being a classroom teacher.

What is the focus of your research?

The purpose of my research is to change how people think about math ability and inability, which means a lot of my time is spent memorizing about mathematics teaching and learning. One of the most compelling articles that I've read explained that the brains of math experts, such as people who are mathematicians, compared to nonexperts are no different. Then, watching Jo Boaler, a widely respected math education researcher, explain how plastic the brain is, even through adulthood, has made me realize that math is not an innate ability; it is a learned skill, just like a lot of things. The goal of my research is to find enough convincing evidence for everyone to believe this as well.

What's the best way for students to catch up?

More time.

Students who have fallen behind should have twice as much instruction to engage in grade-level mathematics. And the time spent in math should be organic, rich, task-based teaching and learning. What this means is meaningful, personal experiences need to happen every day in math class. For example, a hands-on activity in math class, a story problem that is relevant to every student, or the students creating their own story problem with a teacher asking different types of questions to challenge the learners. All students need to see themselves as mathematicians so that they develop a personal connection to mathematics learning.


Frances E. Anderson, Faculty Member in Teacher Education, University of Nebraska Omaha

This article is republished from The Conversation under a Creative Commons license. Read the original article.

Tue, 22 Nov 2022 02:05:00 -0600 en text/html https://www.salon.com/2022/11/22/a-former-math-teacher-explains-why-some-students-are-good-at-math-and-others-lag-behind_partner/
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