On July 14, Maryam Mirzakhani, Stanford professor of mathematics and the only female winner of the prestigious Fields Medal in Mathematics, died at the age of 40.

In just a few hours, her name, both in her native Farsi (#مریم میرزاخانی) and English (#maryammirzakhani), was trending on Twitter and Facebook. Most major news agencieswere covering the news of her death as well as recounting her many achievements.

The grief was especially hard-hitting for a generation of younger academics like me who have always held Maryam as a role model whose example is helping redefine women’s status in science and especially mathematics.

The irony was that Maryam always tried to avoid the media’s spotlight. Her modesty and simplicity despite being the only woman to gain such high status in the world of mathematics – winning what’s often called the “Nobel Prize of math” – stood out to those who knew her.

Unfortunately, I did not get the chance to meet Maryam personally. But like many of my Iranian peers in academia, I looked to her example as proof that the world would welcome us and our scientific contributions no matter our skin color, nationality or religion.

As people around the globe grieve the loss of this talented mathematician, Maryam’s life stands as an inspiration for young girls and boys from all walks of life the world over.

Despite her calm expression and warm smile, Maryam was a warrior. She and her family, alongside many other Iranians, lived through the hard economic and social transformations after the Iran revolution in 1979 and also survived the eight years of the Iran-Iraq war a few years after that.

Maryam originally wanted to be a writer, a passion of hers that never faded away even during her postgraduate studies. However, she found an even greater joy in how rewarding it felt to solve mathematical problems. As a student, she was the first female member of Iran’s national team to participate in the International Math Olympiad, and she won two gold medals in two consecutive years – still a record.

She received her bachelor’s degree from Sharif University of Technology in Iran and later a doctorate from Harvard. In 2014, Maryam was recognized with the Fields Medal, the highest-ranking award in mathematics, for her efforts in what’s known as hyperbolic geometry. Her work focused on curved surfaces – such as spheres or donut shapes – and how to understand their properties. Her achievements have applications in other fields of science including quantum field theory, engineering and material science, and could even influence theories around how our universe was born.

Maryam was a “hall of fame” all by herself. She modestly attributed her own success to her perseverance, hard work and patience. As she put it:

“The beauty of mathematics only shows itself to more patient followers.”

Unfortunately, when she was honored with the Fields Medal, she was already tackling her last challenge, the breast cancer that eventually killed her.

Maryam’s contributions to the field of mathematics will long be remembered. But just as important is her legacy as a role model.

Maryam was an Iranian, a woman and an immigrant to the United States. Unfortunately, these three words together raise red flags for some in Western countries, particularly in the U.S., in the time of Trump’s proposed travel ban.

Against all odds, Maryam’s talent was nurtured in Iran and later flourished in the U.S. Her successes discredit the xenophobic stereotypes that are encouraged by a politics of fear. Maryam defied expectations and rose above all the labels that make it easy to judge others who are not like “us.”

Maryam’s legend may continue to grow after her early death. Still only 20 percent of full-time math faculty at U.S. universities are women, according to a 2015 demographic surveyof 213 departments by the American Mathematical Society. Research shows that stereotyped role models can influence whether people “see themselves” in certain STEM careers. The example of a woman who rose to the top of this still very male field may help inspire math’s next generation.

In the same way people think of Marie Curie or Jane Goodall as scientific pioneers, Maryam Mirzakhani will go down in history as a trailblazer as well as a mathematical genius.

*The author of this article is Mehrdokht Pournader, Lecturer in Operations Management and Organizational Behavior, Macquarie Graduate School of Management. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may want to browse our portfolio of math books. When you purchase a print or e-copy, apply discount code **STC317** and save up to 30% off the list price and free global shipping. If you prefer to browse electronically, visit us on ScienceDirect.

In 1939, the fictional professor J. Abner Pediwell published a curious book called “The Saber-Tooth Curriculum.”

Through a series of satirical lectures, Pediwell (or the actual author, education professor Harold R. W. Benjamin) describes a Paleolithic curriculum that includes lessons in grabbing fish with your bare hands and scaring saber-toothed tigers with fire. Even after the invention of fishnets proves to be a far superior method of catching fish, teachers continued teaching the bare-hands method, claiming that it helps students develop “generalized agility.”

Pedwill showed how curricula can become entrenched and ritualistic, failing to respond to changes in the world around it. In math education, the problem is not quite so dire – but it’s time to start breaking a few of our own traditions. There’s a growing interest in emphasizing problem-solving and understanding concepts over skills and procedures. While memorized skills and procedures are useful, knowing the underlying meanings and understanding how they work builds problem-solving skills so that students may go beyond solving the standard book chapter problem.

**Chapter Download: I Hate Maths: Changing Primary School Teachers’ Relationship with Mathematics**

As education researchers, we see two different ways that educators can build alternative mathematics courses. These updated courses work better for all students by changing what they teach and how they teach it.

In math, the usual curricular pathway – or sequence of courses – starts with algebra in eighth or ninth grade. This is followed by geometry, second-year algebra and trigonometry, all the way up to calculus and differential equations in college.

This pathway still serves science, technology, engineering and mathematics (STEM) majors reasonably well. However, some educators are now concerned about students who may have other career goals or interests. These students are stuck on largely the same path, but many end up terminating their mathematics studies at an earlier point along the way.

In fact, students who struggle early with the traditional singular STEM pathway are more likely to fall out of the higher education pipeline entirely. Many institutions have identified college algebra courses as a key roadblock leading to students dropping out of college altogether.

Another issue is that there is a growing need for new quantitative skills and reasoning in a wide variety of careers – not just STEM careers. In the 21st century, workers across many fields need to know how to deal effectively with data (statistical reasoning), detect trends and patterns in huge amounts of information (“big data”), use computers to solve problems (computational thinking) and make predictions about the relationships between different components of a system (mathematical modeling).

What’s more, sophisticated computational tools provide us with mathematical capabilities far beyond arithmetic calculations. For example, large numerical data sets can be visually examined for patterns using computer graphing software. Other tools can derive predictive equations that would be impractical for anyone to compute with paper and pencil. What’s really needed are people who can make use of those tools productively, by posing the right questions and then interpreting the results sensibly.

The quest to improve student retention has led schools to consider other pathways that would provide students with the quantitative skills they need. For example, courses that use spreadsheets extensively for mathematical modeling and powerful statistical software packages have been developed as part of an alternative pathway designed for students with interests in business and economics.

The Carnegie Foundation for the Advancement of Teaching has created alternative math curricula called Quantway and Statway as examples of alternative pathways – used primarily in community colleges – that focus on quantitative reasoning and statistics/data analysis, respectively.

These alternative pathways involve activities that go beyond students writing examples down in their notebooks. Students might use software, build mathematical models or exercise other skills – all of which require flexible instruction.

Both new and old pathways can benefit from new and more flexible methods. In 2012, the President’s Council of Advisors on Science and Technology called for a 34 percent increasein the number of STEM graduates by 2020. Their report suggested current STEM teaching practices could improve through evidence-based approaches like active learning.

In a traditional classroom, students act as passive observers, watching an expert correctly work out problems. This approach doesn’t foster an environment where mistakes can be made and answers can be questioned. Without mistakes, students lack the opportunity to more deeply explore how and why things don’t work. They then tend to view mathematics as a series of isolated problems for which the solution is merely a prescribed formula.

Mathematician David Bressoud summarized this well:

*“No matter how engaging the lecturer may be, sitting still, listening to someone talk, and attempting to transcribe what they have said into a notebook is a very poor substitute for actively engaging with the material at hand, for doing mathematics.”*

Conversely, classrooms that incorporate active learning allow students to ask questions and explore. Active learning is not a specifically defined teaching technique. Rather, it’s a spectrum of instructional approaches, all of which involve students actively participating in lessons. For example, teachers could pose questions during class time for students to answer with an electronic clicker. Or, the class could skip the lecture entirely, leaving students to work on problems in groups.

While the idea of active learning has existed for decades, there has been a greater push for widespread adoption in recent years, as more scientific research has emerged. A 2014 analysis looked at 225 studies comparing active learning with traditional lecture in STEM courses. Their findings unequivocally support using active learning and question whether or not lecture should even continue in STEM classrooms. If this were a medical study in which active learning was the experimental drug, the authors write, trials would be “stopped for benefit” – because active learning is so clearly beneficial for students.

The studies in this analysis varied greatly in the level of active learning that took place. In other words, active learning, no matter how minimal, leads to greater student achievement than a traditional lecture classroom.

Regardless of pathway, all students can benefit from active engagement in the classroom. As mathematician Paul Halmos put it: “The best way to learn is to do; the worst way to teach is to talk.”

*The authors of this article are Mary E. Pilgrim, Assistant Professor of Mathematics Education, Colorado State University, and Thomas Dick, Professor of Mathematics, Oregon State University. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may also like a new book *called **Understanding Emotions in Mathematical Thinking and Learning**, *a multidisciplinary approach to the role of emotions in numerical cognition, mathematics education, learning sciences, and affective sciences.

We are pleased to offer you a look at the book by providing you with a chapter from the book called *“I Hate Maths”: Changing Primary School Teachers’ Relationship with Mathematics” *below. The book gives an overview of the current study of the relationship between emotions and mathematics. This chapter reports on the theory behind the design and implementation of the IHM workshops and on participants’ responses to them.

**Chapter Download: I Hate Maths: Changing Primary School Teachers’ Relationship with Mathematics**

If you would like to read additional chapters from the book, visit ScienceDirect. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny.

From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other.

Does art sometimes spur mathematical discovery? There’s no simple answer to this question, but in some instances it seems very likely.

Consider Islamic ornament, such as that found in the Alhambra in Granada, Spain.

In the 14th and 15th centuries, the Alhambra served as the palace and harem of the Berber monarchs. For many visitors, it’s a setting as close to paradise as anything on earth: a series of open courtyards with fountains, surrounded by arcades that provide shelter and shade. The ceilings are molded in elaborate geometric patterns that resemble stalactites. The crowning glory is the ornament in colorful tile on the surrounding walls, which dazzles the eye in a hypnotic way that’s strangely blissful. In a fashion akin to music, the patterns lift the onlooker into an almost out-of-body state, a sort of heavenly rapture.

It’s a triumph of art – and of mathematical reasoning. The ornament explores a branch of mathematics known as tiling, which seeks to fill a space completely with regular geometric patterns. Math shows that a flat surface can be regularly covered by symmetric shapes with three, four and six sides, but not with shapes of five sides.

It’s also possible to combine different shapes, using triangular, square and hexagonal tiles to fill a space completely. The Alhambra revels in elaborate combinations of this sort, which are hard to see as stable rather than in motion. They seem to spin before our eyes. They trigger our brain into action and, as we look, we arrange and rearrange their patterns in different configurations.

An emotional experience? Very much so. But what’s fascinating about such Islamic tilings is that the work of anonymous artists and craftsmen also displays a near-perfect mastery of mathematical logic. Mathematicians have identified 17 types of symmetry: bilateral symmetry, rotational symmetry and so forth. At least 16 appear in the tilework of the Alhambra, almost as if they were textbook diagrams.

The patterns are not merely beautiful, but mathematically rigorous as well. They explore the fundamental characteristics of symmetry in a surprisingly complete way. Mathematicians, however, did not come up with their analysis of the principles of symmetry until several centuries after the tiles of the Alhambra had been set in place.

Stunning as they are, the decorations of the Alhambra may have been surpassed by a masterpiece in Persia. There, in 1453, anonymous craftsmen at the Darbi-I Imam shrine in Isfahan discovered quasicrystalline patterns. These patterns have complex and mysterious mathematical properties that were not analyzed by mathematicians until the discovery of Penrose tilings in the 1970s.

Such patterns fill a space completely with regular shapes, but in a configuration which never repeats itself – indeed, is infinitely nonrepeated – although the mathematical constant known as the Golden Section occurs over and over again.

Daniel Schectman won the 2001 Nobel Prize for the discovery of quasicrystals, which obey this law of organization. This breakthrough forced scientists to reconsider their conception of the very nature of matter.

In 2005, Harvard physicist Peter James Lu showed that it’s possible to generate such quasicrystalline patterns relatively easily using girih tiles. Girih tiles combine several pure geometric shapes into five patterns: a regular decagon, an irregular hexagon, a bow tie, a rhombus and a regular pentagon.

Whatever the method, it’s clear that the quasicrystalline patterns at Darbi-I Imam were created by craftsmen without advanced training in mathematics. It took several more centuries for mathematicians to analyze and articulate what they were doing. In other words, intuition preceded full understanding.

Geometric perspective made it possible to portray the visible world with a new verisimilitude and accuracy, creating an artistic revolution in the Italian Renaissance. One could argue that perspective also led to a major reexamination of the fundamental laws of mathematics.

According to Euclidian mathematics, two parallel lines will remain parallel into infinity and never meet. In the world of Renaissance perspective, however, parallel lines eventually do meet in the far distance at the so-called “vanishing point.” In other words, Renaissance perspective present a geometry which follows regular mathematical laws, but is non-Euclidian.

When mathematicians first devised non-Euclidian mathematics in the early 19th century, they imagined a world in which parallel lines meet at infinity. The geometry they explored was, in many ways, similar to that of Renaissance perspective.

Non-Euclidian mathematics has since moved on to explore space which has 12 or 13 dimensions, far outside the world of Renaissance perspective. But it’s worth asking whether Renaissance art may have made easier to make that initial leap.

An interesting modern case of art that broke traditional boundaries – and that has suggestive parallels with recent developments in mathematics – is that of the paintings of Jackson Pollock.

To those who first encountered them, the paintings of Pollock seemed chaotic and senseless. With time, however, we’ve come to see that they have elements of order, though not a traditional sort. Their shapes are simultaneously predictable and unpredictable, in a fashion similar to the pattern of dripping water from a faucet. There’s no way to predict the exact effect of the next drip. But, if we chart the pattern of drips, we find that they fall within a zone that has a clear shape and boundaries.

Such unpredictability was once out of bounds for mathematicians. But, in recent years, it has become one of the hottest areas of mathematical exploration. For example, chaos theory explores patterns that are not predictable but fall within a definable range of possibilities, while fractal analysis studies shapes that are similar but not identical.

Pollock himself had no particular interest in mathematics, and little known talent in that arena. His fascination with these forms was intuitive and subjective.

Intriguingly, mathematicians have not been able to accurately describe what Pollock was doing in his paintings. For example, there have been attempts to use fractal analysis to create a numerical “signature” of his style, but so far the method has not worked – we can’t mathematically distinguish Pollock’s autograph work from bad imitations. Even the notion that Pollock employed fractal thoughts is probably incorrect.

Nonetheless, Pollock’s simultaneously chaotic and orderly patterns have suggested a fruitful direction for mathematics. At some point, it may well be possible to describe what Pollock was doing with mathematical tools, and artists will have to move on and mark out a new frontier to explore.

*The author of this article is Henry Adams, Ruth Coulter Heede Professor of Art History, Case Western Reserve University. Read the original article here.*

If you found this article interesting, you may also like a new book *called **Understanding Emotions in Mathematical Thinking and Learning**, *a multidisciplinary approach to the role of emotions in numerical cognition, mathematics education, learning sciences, and affective sciences.

We are pleased to offer you a look at the book by providing you with a chapter from the book called *“An Overview of the Growth and Trends of Current Research on Emotions and Mathematics” *that gives an overview of the current study of the relationship between emotions and mathematics below:

If you would like to read additional chapters from the book, visit ScienceDirect. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

Once upon a time, mathematicians imagined their job was to discover new mathematics and then let others explain it.

Today, digital tools like 3-D printing, animation and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.

Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.

**Chapter Download: Basic Finance**

The atmosphere was electric. Talks ran the gamut, from wildly creative thinkers who apply mathematics in the world of design to examples of pure mathematical results discovered through computer experimentation and visualization. It shed light on how powerful visualization has become for studying and sharing mathematics.

Visualization plays a growing role in mathematical research. According to John Sullivan at the Technical University of Berlin, mathematical thinking styles can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.

Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles.

Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity.

The “bubble” version of the configuration, shown below, is minimal, in the sense that it is the shortest possible shape where the tubes around the rings do not overlap. It’s as if you were to blow a soap bubble around each of the rings in the configuration. Techniques for proving that configurations like this are optimal often involve concepts of flow: If a given configuration is not the best, there are often ways to tell it to move in a direction that will make it better. This topic has great potential for visualization.

At the workshop, Sullivan dazzled us with a video of the three bands flowing into their optimal position. This animation allowed the researchers to see their ideas in action. It would never be considered as a substitute for a proof, but if an animation showed the wrong thing happening, people would realize that they must have made an error in their mathematics.

Visualization tools have helped mathematicians share their work in creative and surprising ways – even to rethink what the job of a mathematician might entail.

Take mathematician Fabienne Serrière, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.

Edmund Harriss of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”

Mathematical algorithms power the products made by Nervous System, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology.

Their first 3-D printed dress consists of thousands of interlocking pieces designed to fit a particular model. In order to print the dress, the designers folded up their virtual version, using protein-folding algorithms. A selective laser sintering process fused together parts of a block of powder to make the dress, then let all the unwanted powder fall away to reveal its shape.

Meanwhile, a delightful collection called Geometry Games can help everyone, from elementary school students to professional mathematicians, explore the concept of space. The project was founded by mathematician Jeff Weeks, one of the rock stars of the mathematical world. The iOS version of his “Torus Games” teaches children about multiply-connected spaces through interactive animation. According to Weeks, the app is verging on one million downloads.

My own work, described in my book “Creating Symmetry: The Artful Mathematics of Wallpaper Patterns,” starts with a visualization technique called the domain coloring algorithm.

I developed this algorithm in the 1990s to visualize mathematical ideas that have one dimension too many to see in 3-D space. The algorithm offers a way to use color to visualize something seemingly impossible to visualize in one diagram: a complex-valued function in the plane. This is a formula that takes one complex number (an expression of the form *a*+_b_i, which has two coordinates) and returns another. Seeing both the 2-D input and the 2-D output is one dimension more than ordinary eyes can see, hence the need for my algorithm. Now, I use it to create patterns and mathematical art.

My main pattern-making strategy relies on a branch of mathematics called Fourier theory, which involves the superposition of waves. Many people are familiar with the idea that the sound of a violin string can be broken down into its fundamental frequencies. My “wallpaper functions” break down plane patterns in just the same way.

My book starts with a lesson in making symmetric curves. Taking the same idea into a new dimension, I figured out how to weave polyhedral solids – think cube, dodecahedron, and so on – from symmetric bands made from these waves. I staged three of these new shapes, using Photoshop’s 3-D ray-tracing capacity, in the “Platonic Regatta” shown below. The three windsails display the symmetries of Platonic solids: the icosahedron/dodecahedron, cube/octahedron and tetrahedron.

About an hour after I spoke at the workshop, mathematician Mikael Vejdemo-Johansson had posted a Twitter bot to animate a new set of curves every day!

5-fold symmetry, seed: 701035, complex Fourier coefficients pic.twitter.com/HsLUZuVhIw

— Symmetric Curves (@symmetric_curve) March 19, 2017

Mathematics in the 21st century has entered a new phase. Whether you want to crack an unsolved problem, teach known results to students, design unique apparel or just make beautiful art, new tools for visualization can help you do it better.

*The author of this article is Frank A. Farris, Associate Professor of Mathematics, Santa Clara University. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may also like a new book out called *The Joy of Finite Mathematics: The Language and Art of Math**. *This book teaches step-by-step procedures, and clearly defined formulae, to help readers learn to apply math to subjects ranging from reason (logic) to finance (personal budget), making this interactive and engaging book appropriate for non-science, undergraduate students in the liberal arts, social sciences, finance, economics, and other humanities areas.

We are pleased to offer you a look at the book by providing you with a chapter from the book called *“Basic Finance” *that covers basic financing including sinking funds and amortization, various savings situations, and comparison shopping (credit versus cash, leasing versus purchasing, and renting versus owning).

There is also a section on personal finance (how to create a monthly budget), insurance (what every homeowner should know), and your credit report.

If you would like to read additional chapters from the book, visit ScienceDirect here. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

In 2013, a meeting of academics specialising in teaching first year undergraduate mathematics (known as the FYiMaths network) identified that the broad removal of mathematics prerequisites for many undergraduate degrees had created the biggest challenge they faced in teaching.

Many individuals had made attempts to pass this message up the management line at their universities. But at that time, staff believed that reintroducing prerequisites would never happen.

However, earlier this year The University of Sydney announced it would do exactly that, by requiring students studying science, engineering, commerce and IT to have completed at least intermediate level mathematics in high school.

The Australian Academy of Science’s Decadal Plan for the Mathematical Sciences, launched in Canberra yesterday, continues this push. One of its key recommendations is the reinstatement of mathematics prerequisites for science, engineering and commerce degrees.

But will it improve the level of maths education? Will it bolster mathematics skills in those studying science, engineering and commerce?

**Chapter Download: Equations and Functions**

A prerequisite study for entry to a degree is considered to be essential background knowledge that students need in order to be successful in that degree. A student cannot be selected into the degree if they do not have the stated prerequisite or an equivalent to it.

Over the past two decades, most universities have moved away from mathematics prerequisites, replacing them with assumed knowledge statements. This means that students can be selected without verifying that they have in fact completed this background study.

So what’s wrong with that?

In most cases, the assumed knowledge statements are unclear and often difficult to find, so students may not be aware of the assumed requirements. The removal of mathematics prerequisites also grossly underplays the level of mathematical facility required for these courses and trivialises the learning and skill development required to acquire it.

It places the burden on students to decide what should or should not be known in order to succeed in a course, and to assume the risk of those decisions, even though they are in no position to know what the risks are.

As a consequence, large numbers of students have been enrolling in mathematics-dependent courses without the assumed knowledge.

Over the last decade or more, numbers of students studying intermediate and advanced level mathematics in school has been in steady decline. Students have been free to make subject choices based on maximising their ATAR score rather than choosing the subjects that will best prepare them for their chosen career.

Since intermediate and advanced mathematics subjects are seen as hard and deemed not necessary for entry, students have been allowed – in some cases even encouraged – to opt out.

On the other side of the enrolment gate, consequences for students include being required to undertake bridging courses (some at extra cost) and having limited pathways through their degrees. Students do not generally know this at the end of Year 10 when they decide on which subjects they will choose for their Year 12.

Neither do they know that these choices may impact on their ability to succeed in their tertiary studies. Failure and attrition rates are generally high in first-year STEM subjects. And lack of the requisite background in mathematics plays a significant part in this.

Students who enter university without the assumed knowledge in mathematics also generally have lower success rates than students who have the assumed knowledge from school, even after they have completed bridging courses. In consumer terms, this buyer beware approach is not working.

So, where does that leave us?

Universities have a responsibility to determine what minimum background knowledge students require to be successful in a course. Once that determination is made, they should be required to ensure that the students they accept have that required knowledge.

Reintroducing appropriate mathematics prerequisites should increase participation in intermediate and advanced level mathematics at school. It has to.

We want students to take full advantage of the excellent education that is available to them through our secondary school system rather than trying to play catchup for years later.

Engaging students in the study of mathematics at school needs to be addressed on many levels. Certainly, making strong statements about prerequisites is one piece of the puzzle, but not the only one.

The Decadal Plan also calls for an urgent increase in the provision of professional development for teachers, especially those teaching mathematics out-of-field. It is essential that we support our teachers at all levels of education, so that we can give students the best possible education in mathematics that we must.

*The authors of this article are Deborah King, Associate Professor in Mathematics, University of Melbourne, and John Rice, Honorary Professor, University of Sydney. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may also like a new book out called *Engineering Mathematics with Examples and Applications* that provides a compact and concise primer in the field, starting with the foundations, and then gradually developing to the advanced level of mathematics that is necessary for all engineering disciplines.

If you would like to read additional chapters from the book, visit ScienceDirect here. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

In 2014, I read in *Mathematics Today* about a competition that was taking place called the Mathematical Competitive Game 2014-2015. It concerned GPS positioning and one had to try to estimate the uncertainty in the position of a GPS receiver from actual data. In due course, I submitted an entry called “Uncertainties in GPS Positioning.”

The intended readership for the entry was individuals interested in GPS, such as university students. The reader will benefit from being able to: understand how a GPS receiver calculates its position; understand why the calculated position is only an approximation to the true position; gain some appreciation of the factors which contribute to the difficulties in calculating an approximation of the true position; gain some appreciation of the mathematical steps that are employed in order to reduce errors in the approximation.

I was fortunate enough to achieve some success by winning the Joint First prize, Individual Category.

As a result of my research into the topic of GPS positioning, I submitted a book proposal to Elsevier called *Uncertainties in GPS Positioning: A Mathematical Discourse**, *it was accepted, and the book published in January, 2017.

Today, we all know what a GPS receiver is: it communicates with a satellite system and lets you know where you are on a map. A receiver receives signals from several orbiting satellites and processes them. The receiver has a built-in map. Uncertainties in GPS Positioning: A Mathematical Discourse describes the calculations performed by a GPS receiver and describes the problem associated with making sure that the estimated location is in close agreement with the actual location.

** Uncertainties in GPS Positioning: A Mathematical Discourse** provides a brief introduction to positioning and navigation systems, followed by the main topics that cover an introduction to GPS, basic GPS principles, signals from satellites to receiver for GPS, GPS modernization, signals from satellites to receiver for other satellite navigation systems, the solution of an idealized problem, and sources of inaccuracy. An example positioning problem with estimated inaccuracies is presented in detail, including a step-by-step mathematical solution. For each topic, background information is provided to aid the reader comprehend the subject matter. The future of satellite navigation systems is also discussed.

Dr. Alan Oxley is a tutor in the Faculty of Engineering, Design, and Information & Communications Technology (EDICT) at Bahrain Polytechnic, Kingdom of Bahrain. He and his former postgraduate students have published a number of researcher papers. In 2014-2015 the Mathematical Competitive Game took place with the topic ‘Uncertainties in GPS Positioning.’ Dr. Oxley’s entry received First Prize ex-aequo. His research interests are wide-ranging in both mathematics and computer science.

We are pleased to offer a free chapter of his book, “*Introduction to GPS.”*

If you would like to view more chapters, you can access the book on ScienceDirect. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** for up to 30% off the list price and free global shipping.

Being stuck in miles of halted traffic is not a relaxing way to start or finish a summer holiday. And as we crawl along the road, our views blocked by by slow-moving roofboxes and caravans, many of us will fantasise about a future free of traffic jams.

As a mathematician and motorist, I view traffic as a complex system, consisting of many interacting agents including cars, lorries, cyclists and pedestrians. Sometimes these agents interact in a free-flowing way and at other (infuriating) times they simply grind to a halt. All scenarios can be examined – and hopefully improved – using mathematical modelling, a way of describing the world in the language of maths.

Mathematical models tell us for instance that if drivers kept within the variable speed limits sometimes displayed on a motorway, traffic would flow consistently at, say, 50mph. Instead we tend to drive more aggressively, accelerating as soon as the opportunity arises – and being forced to brake moments later. The result is greater fuel consumption and a longer overall journey time. Cooperative driving seems to go against human nature when we get behind the wheel. But could this change if our roads were taken over by driverless cars?

Incorporating driverless cars into mathematical traffic models will prove key to improving traffic flow and assessing the various conditions in which traffic reaches a traffic jam threshold, or “jamming density”. The chances of reaching this point are affected by changes such as road layout, traffic volume and traffic light systems. And crucially, they are affected by whoever is in control of the vehicles.

In mathematical analysis, dense traffic can be treated as a flow and modelled using differential equations which describe the movement of fluids. Queuing models consider individual vehicles on a network of roads and the expected time they spend both in motion and waiting at junctions.

Another type of model consists of a grid in which cars’ positions are updated, according to certain rules, from one grid cell to the next. These rules can be based on their current velocity, acceleration and deceleration due to other vehicles and random events. This random deceleration is included to account for situations caused by something other than other vehicles – a pedestrian crossing the road for example, or a driver distracted by a passenger.

Adaptations to such models can take into account factors such as traffic light synchronisation or road closures, and they will need to be adapted further to take into account the movement of driverless cars.

In theory, autonomous cars will typically drive within the speed limits, have faster reaction times allowing them to drive closer together and will behave less randomly than humans, who tend to overreact in certain situations. On a tactical level, choosing the optimum route, accounting for obstacles and traffic density, driverless cars will behave in a more rational way, as they can communicate with other cars and quickly change route or driving behaviour.

So driverless cars may well make the mathematician’s job easier. Randomness is often introduced into models in order to incorporate unpredictable human behaviour. A system of driverless cars should be simpler to model than the equivalent human-driven traffic because there is less uncertainty. We could predict exactly how individual vehicles respond to events.

In a world with only driverless cars on the roads, computers would have full control of traffic. But for the time being, to avoid traffic jams we need to understand how autonomous and human-driven vehicles will interact together.

Of course, even with the best modelling, cooperative behaviour from driverless cars is not guaranteed. Different manufacturers might compete to come up with the best traffic-controlling software to ensure their cars get from A to B faster than their rivals. And, like the behaviour of individual human drivers, this could negatively affect everyone’s journey time.

But even supposing we managed to implement rules that optimised traffic flow for everyone, we could still get to the point where there are simply too many cars on the road, and jamming density is reached. Yet there is still potential for self-driving cars to help in this scenario.

Some car makers expect that eventually we will stop viewing cars as possessions and instead simply treat them as a transport service. Again, by applying mathematical techniques and modelling, we could optimise how this shared autonomous vehicle service could operate most efficiently, reducing the overall number of cars on the road. So while driverless cars alone might not rid us of traffic jams completely by themselves, an injection of mathematics into future policy could help navigate a smoother journey ahead.

*The author of this article is Lorna Wilson, Commercial Research Associate, University of Bath. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

*On-Road Intelligent Vehicles*, provides a comprehensive account of the technology of autonomous vehicles, with a special focus on the navigation and planning aspects, including information on the use of different sensors to perceive the environment, the problem of motion planning, and the macroscopic concepts related to Intelligent Transportation Systems. Essential reading for Postgraduate students, researchers and practitioners, working in the areas of Intelligent Vehicles, Intelligent Transportation Systems, Autonomous Vehicles, Robot Motion Planning, Special Topics in Robotics, Cooperative Systems, Planning and Navigation.

*Intelligent Vehicular Networks and Communications* examines cognitive radio, big data, and the cloud in vehicular communications and the current and future evolution of today’s transportation system. Chapters examine how intelligent transportation systems make more efficient transportation in urban environments and outline next generation vehicular networks technology. Vehicular and Wireless Network researchers, instructors, students, designers, and engineers will find the book particularly useful.

These books are also available in print on the Elsevier Store at 30% off the list price and with free global shipping. Apply discount code **STC215** at checkout.

The film The Man Who Knew Infinity tells the gripping story of Srinivasa Ramanujan, an exceptionally talented, self-taught Indian mathematician. While in India, he was able to develop his own ideas on summing geometric and arithmetic series without any formal training. Eventually, his raw talent was recognised and he got a post at the University of Cambridge. There, he worked with Professor G.H. Hardy until his untimely death at the age of 32 in 1920.

Despite his short life, Ramanujan made substantial contributions to number theory, elliptic functions, infinite series and continued fractions. The story seems to suggest that mathematical ability is something at least partly innate. But what does the evidence say?

There are many different theories about what mathematical ability is. One is that it is closely tied to the capacity for understanding and building language. Just over a decade ago, a study examined members of an Amazonian tribe whose counting system comprised words only for “one”, “two” and “many”. The researchers found that the tribe were exceptionally poor at performing numerical thinking with quantities greater than three. They argued this suggests language is a prerequisite for mathematical ability.

But does that mean that a mathematical genius should be better at language than the average person? There is some evidence for this. In 2007, researchers scanned the brains of 25 adult students while they were solving multiplication problems. The study found that individuals with higher mathematical competence appeared to rely more strongly on language-mediated processes, associated with brain circuits in the parietal lobe.

However, recent findings have challenged this. One study looked at the brain scans of participants, including professional mathematicians, while they evaluated mathematical and non-mathematical statements. They found that instead of the left hemisphere regions of the brain typically involved during language processing and verbal semantics, high level mathematical reasoning was linked with activation of a bilateral network of brain circuits associated with processing numbers and space.

In fact, the brain activation in professional mathematicians in particular showed minimal use of language areas. The researchers argue their results support previous studies that have found that knowledge of numbers and space during early childhood can predict mathematical achievement.

For example, a recent study of 77 eight- to 10-year-old children demonstrates that visuo-spatial skills (the capacity to identify visual and spatial relationships among objects) have an important role in mathematical achievement. As part of the study, they took part in a “number line estimation task”, in which they had to position a series of numbers at appropriate places on a line where only the start and end numbers of a scale (such as 0 and 10) were given.

The study also looked at the children’s overall mathematical ability, visuospatial skills and visuomotor integration (for example, copying increasingly complex images using pencil and paper). It found that children’s scores on visuospatial skill and visuomotor integration strongly predicted how well they would do on number line estimation and mathematics.

An alternative definition of mathematical ability is that it represents the capacity to recognise and exploit hidden structures in data. This may account for an observed overlap between mathematical and musical ability. Similarly, it could also explain why training in chess can benefit children’s ability to solve mathematical problems. Albert Einstein famously claimed that images, feelings and musical structures formed the basis of his reasoning rather than logical symbols or mathematical equations.

However, the extent to which mathematical ability relies on innate or environmental factors remains controversial. A recent large scale twin and genome-wide analysis of 12-year-old children found that genetics could explain around half of the observed correlation between mathematical and reading ability. Although this is quite substantial, it still means that the learning environment has an important role to play.

So what does all this tell us about geniuses like Ramanujan? If mathematical ability does stem from a core non-linguistic capacity to reason with spatial and numerical representation, this can help explain how a prodigious talent could blossom in the absence of training. While language might still play a role, the nature of the numerical representations being manipulated could be crucial.

The fact that genetics seems to be involved also helps shed light on the case – Ramanujan could have simply inherited the ability. Nevertheless, we should not forget the important contribution of environment and education. While Ramanujan’s raw talent was sufficient to attract attention to his remarkable ability, it was the later provision of more formal mathematical training in India and England that allowed him to reach his full potential.

*The author of this article is David Pearson, Reader of Cognitive Psychology, Anglia Ruskin University. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

A new book by the power team of Daniel Berch, David Geary, and Kathleen Mann Koepke, is a review of how contemporary brain imaging techniques and genetic methods can inform our understanding of mathematical cognitive development and ways to improve it. *Development of Mathematical Cognition* is available in print or as an e-book, and is also available online via ScienceDirect.

To read more from the book, you can visit ScienceDirect, or you can order a print or e-book via the Elsevier Store. Apply discount code STC215 for up to 30% off the list price and free shipping worldwide! Prefer to read it online, access it via ScienceDirect today.

]]>

A bewildering physics problem has apparently been solved by researchers, in a study which provides a mathematical basis for understanding issues ranging from predicting the formation of deserts, to making artificial intelligence more efficient.

In research carried out at the University of Cambridge, a team developed a computer program that can answer this mind-bending puzzle: Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

The answer, it turns out, is something like 10^{250} (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

Being able to calculate configurational entropy would, in theory, eventually enable us to answer a host of seemingly impossible problems – such as predicting the movement of avalanches, or anticipating how the shifting sand dunes in a desert will reshape themselves over time.

These questions belong to a field called granular physics, which deals with the behaviour of materials such as snow, soil or sand. Different versions of the same problem, however, exist in numerous other fields, such as string theory, cosmology, machine learning, and various branches of mathematics. The research shows how questions across all of those disciplines might one day be addressed.

Stefano Martiniani, a Gates Scholar at St John’s College, University of Cambridge, who carried out the study with colleagues in the Department of Chemistry, explained: “The problem is completely general. Granular materials themselves are the second most processed kind of material in the world after water and even the shape of the surface of the Earth is defined by how they behave.”

“Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that.”

At the heart of these problems is the idea of entropy – a term which describes how disordered the particles in a system are. In physics, a “system” refers to any collection of particles that we want to study, so for example it could mean all the water in a lake, or all the water molecules in a single ice cube.

When a system changes, for example because of a shift in temperature, the arrangement of these particles also changes. For example, if an ice cube is heated until it becomes a pool of water, its molecules become more disordered. Therefore, the ice cube, which has a tighter structure, is said to have lower entropy than the more disordered pool of water.

At a molecular level, where everything is constantly vibrating, it is often possible to observe and measure this quite clearly. In fact, many molecular processes involve a spontaneous increase in entropy until they reach a steady equilibrium.

The brute force way of doing this would be to keep changing the system and recording the configurations. Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store them, because there isn’t enough matter in the universe. – Stefano Martiniani

In granular physics, however, which tends to involve materials large enough to be seen with the naked eye, change does not happen in the same way. A sand dune in the desert will not spontaneously change the arrangement of its particles (the grains of sand). It needs an external factor, like the wind, for this to happen.

This means that while we can predict what will happen in many molecular processes, we cannot easily make equivalent predictions about how systems will behave in granular physics. Doing so would require us to be able to measure changes in the structural disorder of all of the particles in a system – its configurational entropy.

To do that, however, scientists need to know how many different ways a system can be structured in the first place. The calculations involved in this are so complicated that they have been dismissed as hopeless for any system involving more than about 20 particles. Yet the Cambridge study defied this by carrying out exactly this type of calculation for a system, modelled on a computer, in which the particles were 128 soft spheres, like tennis balls.

“The brute force way of doing this would be to keep changing the system and recording the configurations,” Martiniani said. “Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store the configurations, because there isn’t enough matter in the universe with which to do it.”

Instead, the researchers created a solution which involved taking a small sample of all possible configurations and working out the probability of them occurring, or the number of arrangements that would lead to those particular configurations appearing.

Based on these samples, it was possible to extrapolate not only in how many ways the entire system could therefore be arranged, but also how ordered one state was compared with the next – in other words, its overall configurational entropy.

Martiniani added that the team’s problem-solving technique could be used to address all sorts of problems in physics and maths. He himself is, for example, currently carrying out research into machine learning, where one of the problems is knowing how many different ways a system can be wired to process information efficiently.

“Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one,” he said. “By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find.”

The paper, Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings, is published in the journal, Physical Review E.

The text in this work is licensed under a Creative Commons Attribution 4.0 International License and first appeared on The University of Cambridge website. Click here for the original article.

If you want more on the subject, then try these books at up to 30% off the list price and free global shipping. Apply discount code STC215 at checkout.

*Introduction to Statistical Machine Learning*– Bridges the gap between theory and practice by providing a general introduction to machine learning that covers a wide range of topics concisely, and includes accompanying MATLAB/Octave programs to provide the practical skills needed to accomplish a wide range of data analysis tasks.*Machine Learning: A Bayesian and Optimization Perspective*– Gain an in-depth understanding of all the main machine learning methods, including sparse modeling, online and convex optimization, Bayesian inference, graphical models, deep networks, learning in RKH spaces, dimensionality reduction and dictionary learning.*Thermal Physics: Thermodynamics and Statistical Mechanics for Scientists and Engineers*–A comprehensive exposition of the fundamental laws of thermodynamics are stated precisely as postulates and subsequently connected to historical context and developed mathematically. It includes appendices to handle background and technical details.*Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation*– Building from first principles, this book gives a transparent explanation of the physical behaviour of equilibrium thermodynamic systems, presents a comprehensive, self-contained account of the modern mathematical and computational techniques of statistical mechanics.

Deciding which teams to pick in your NCAA basketball pool? Then you’re faced with a classic decision problem – and here, science can help.

On one hand, you want to pick good teams, the “favorites,” because those teams seem more likely to win. On the other hand, you want to pick some weaker teams, the “underdogs,” so your bracket will stand out from the rest and win the pool. These two opposing forces make for an interesting math problem, because somewhere in the middle is an optimal solution.

In my heart, I always know which teams will win, or at least which teams I want to root for. As an academic, though, I’d rather squeeze all the fun out of it by overanalyzing the situation. Let’s do that here!

To find the best way to build our own brackets, we need to first build a mathematical model for simulating the tournament.

Suppose we model the tournament by replacing basketball games with coin flips, except with coins that don’t land evenly heads or tails but rather are weighted to reflect each game’s actual odds. For example, when Baylor plays Yale on Thursday, instead of playing the game, we just flip a coin that gives the higher-seeded Baylor a greater chance of winning. We’d need to flip one of these coins for every first-round game, every potential second-round game, and for each possible matchup in the tournament. Each coin must be weighted in a way that models the actual game, so its probabilities must be determined by the specific matchup.

Where should we get these probabilities? The NCAA provides you with a handy little number next to each team, the team’s seed. For the first few rounds, each game has a favorite, and that choice was made by people with a tremendous amount of basketball knowledge. You could look back over history and observe that when a #5 seed plays a #12 seed, the #5 seed wins 65 percent of the time.

But there are plenty of other methods: Las Vegas betting odds give a point spread for each game, and based on those teams’ scoring averages, you can convert the point spread into a probability of winning. Computer rating systems abound, and you can convert these ratings into probabilities by considering the ratings difference between two teams – a method known as the Bradley-Terry model. Some more sophisticated systems can even produce a probability custom fit to the two teams in the game.

So, pick your favorite method. Even then, things aren’t as simple as they seem. The most likely outcome of the tournament is not necessarily that all favorites win. Look at this example:

Imagine a four-team tournament with teams A, B, C and D as shown. Assume that A always beats B, and C beats D with probability 0.6. Finally, A always beats D, but has only 0.5 probability of beating C. The only possible outcomes are: A wins over C (probability 0.3), C wins over A (probability 0.3) and A wins over D (probability 0.4). The most likely outcome contains the upset D beats C.

Further complicating the situation, the rules of your office or friends’ pool probably mean that picking correctly in later-round games earns more points than early-round picks. How do you pick a bracket that gets you those crucial late-round points?

In one of the first analytic papers on this subject, Kaplan and Garstka gave an algorithm for deciding which picks are expected to score the highest. Their method builds a list of 64 brackets backwards, round-by-round, starting each one with a different team as the winner. For example, Duke’s bracket starts with just Duke, and adds one round at a time, doubling in size but always keeping Duke as the winner. In the end, the algorithm selects the best from each of the 64 team-specific brackets.

This doesn’t sound like something a human would do, and in fact it is best implemented by a computer. The brackets produced tend to be “chalk” – in which higher-ranked teams are most likely to win – but do not always select the higher seed. And Kaplan and Garstka did observe that their algorithm did better than just automatically picking the high seeds.

To this point our model is ignoring an important fact: the goal of picking your bracket is not to achieve a high score, but to win a pool against other people. And people behave irrationally.

In a psychological experiment, McCrea and Hirt found evidence that pool participants pursue “probability matching”: if a collection of games (say, the 5-12 matchups) has historically produced an upset one-third of the time, people will attempt to predict upsets in about one-third of those games in their brackets. In fact, people do no better than random chance at making such predictions, and so hurt their overall chances in the pool.

On the other hand, when choosing the tournament winner, people flock to the favorites. Every year, ESPN Tournament Challenge publishes data on its 11 million entries. Last year, 48 percent of their players had selected prohibitive favorite Kentucky as champion. Picking the correct champion is important, but if everyone else has the same opinion then you need to pick a bunch of other games well, too.

This brings us back to what makes this problem interesting: you need to pick teams that win, but not the same teams as everyone else – so you come out on top in your pool.

To improve your odds in your pool, you need to model the other players you’re up against. Each year, large, free, Internet pools publish data on player behavior, and they publish it before your brackets are due on Thursday morning.

Let’s assume people make their picks the same way we modeled the games, by flipping biased coins for each game in the bracket. The national Internet pools give exactly the data you need to properly bias the coins. Nobody I know actually picks their bracket this way, but it turns out that real (human-picked) brackets and randomized brackets have nearly the same score distribution.

In my own research, we used this model to calculate optimal picks. The brackets produced tend to be very conservative in the first two rounds, include one or two surprises in the Final Four, and a strong but not heavily favored champion. They never, ever, pick an upset in a 5-12 game. According to the computers, these picks increase the chances of winning a big Internet pool by a factor of 100 to 1,000.

This sounds great. It is great! But there’s a catch: the NCAA basketball tournament happens only once a year. And your probability of winning is very low indeed – even with a boost from math and computer analytics. It will likely take thousands of years before the strategy pays off.

And that’s the beautiful thing about scientific studies of the NCAA tournament. Serious modeling and data analysis quail before the absurdity of predicting such a notoriously unpredictable event. After a decade of study, the only things we really know are that the tournament is madness and that your friend whose picks are based on mascots will probably win your pool.

Don’t want to just pick your favorite team and depend on luck? Here’s a chapter from the book *The Joy of Finite Mathematics* called *“Game Theory” *to help increase your odds:

If the print book is what you need, then for a limited time, you order it at up to 30% off the list price and free global shipping via the Elsevier Store. Enter discount code STC215 at checkout.

]]>