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Sunday, July 22, 2018

Where did maths mastery come from? | Opinion - Schools Week

The roots of some ‘Asian’ teaching approaches lie closer to home than you might think, observes Mark Boylan, Professor at the Sheffield Institute of Education, Sheffield Hallam University.

Photo: Schools Week

Read or hear the word “maths” in England right now, and you’re likely to see or hear the word “mastery”, too. Teaching for mastery is both the name of the government’s preferred teaching approach in maths and the name for the programme of funds and initiatives to encourage schools to adopt it. Publishers, consultants and websites with maths resources are all adopting the mastery brand.

Given this, a new teacher could be forgiven for thinking that talk of mastery has been around for a long time. However, a quick search in the Schools Week archive reveals that in 2015 there were only passing references, and generally they were not linked to maths.

Teachers who have been around for a little longer probably have some inkling that this talk of mastery has something to do with maths teaching in East Asia, and particularly Singapore and Shanghai.

But teachers in Singapore and Shanghai don’t tend to talk about mastery in maths. They don’t do teaching for mastery – they do teaching maths. So where did this idea of mastery and mathematics come from? Fortunately, the modern adoption of the term is well documented.

The idea of mastery learning goes back to the 1960s, when Benjamin Bloom had a novel idea: if learners don’t get something the first time, then teach them again and in different ways until they do. More recently, the Ark academy chain began to develop a maths curriculum influenced by Singapore.

They got funding from the Education Endowment Foundation for further development and for trials of the new approaches. In 2011 the term “mathematics mastery” was adopted.

Over the next few years, Mathematics Mastery developed as a curriculum and professional development programme that eventually separated from Ark to become an independent not-for-profit organisation. In 2014, Helen Drury, the founder of Mathematics Mastery, published a book explaining the approach...

Singapore’s maths teaching and curriculum was strongly influenced by the 1982 Cockroft Report – a report by an HMI in England. The idea of sequencing material as concrete-pictorial-abstract is a Singaporean version of Jerome Bruner’s ideas about learning.
Read more... 

Related link

How To Teach Mathematics for Mastery
How To Teach Mathematics for Mastery by Helen Drury, leading educator and founder of the school improvement programme, Mathematics Mastery.

Source: Schools Week 


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Dorothea Rockburne: Painting by the Numbers | Discoveries - Robb Report

An exhibition at Dia:Beacon showcases the pioneering artist’s mathematically grounded work, as Robb Report reports.

Dorothea Rockburne; installation view. 
Photo: Dorothea Rockburne/Artists Rights Society (ARS), New York. Courtesy Dorothea Rockburne Studio. Bill Jacobson Studio.
“Although I am a painter, I also have a doctorate in mathematics and so the structure of my work is mathematical,” says Dorothea Rockburne, in describing her oeuvre. “When you get into higher math, it’s thrilling. It’s living in another world. The way that mathematics always uses the elegant solution, my work has an elegant solution aspect to it. But explain it, I can’t. That’s why I do it, because it’s not explainable in language.”

With their myriad folds and recognizable ratios, her torn paper and rough metal assemblages, particularly those from the 1960s and early 1970, reflect her numerical grounding—the Montreal-born artist, now 85, having tutored under noted German mathematician Max Dehn during her years at Black Mountain College, near Asheville, N.C. A half-dozen of Rockburne’s seminal works are now on view at Dia:Beacon in Beacon, N.Y., where they are to be joined by a suite of more recent works in 2019.

“At Black Mountain College, Rockburne began to pursue a new language born from the understanding of the presence of geometry in nature as well as human-made surroundings,” says curator Courtney Martin, adding that the exhibition includes the artist’s first “topological painting,” Tropical Tan (1967–68). The work consists of four sheets of pig iron that have been creased along the diagonal axes and partially coated with a wrinkle-finish paint. Rockburne, she says, has referred to this body of work as “visual equations,” underscoring its aesthetic and analytical aspects. In time, her painting practice would come to draw on ancient systems of proportion as well as astronomical phenomena.
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Source: Robb Report


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Saturday, July 21, 2018

The Peculiar Math That Could Underlie the Laws of Nature | Mathematics - Quanta Magazine

"New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”" argues Natalie Wolchover, senior writer and editor at Quanta Magazine covering the physical sciences.

Cohl Furey, a mathematical physicist at the University of Cambridge, is finding links between the Standard Model of particle physics and the octonions, numbers whose multiplication rules are encoded in a triangular diagram called the Fano plane.
Photo: Susannah Ireland for Quanta Magazine

In 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours south to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin. Furey had figured out how to build on a finding of Günaydin’s from 40 years earlier — a largely forgotten result that supported a powerful suspicion about fundamental physics and its relationship to pure math.
The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight-dimensional numbers called “octonions.”
As numbers go, the familiar real numbers — those found on the number line, like 1, π and -83.777 — just get things started. Real numbers can be paired up in a particular way to form “complex numbers,” first studied in 16th-century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D “quaternions,” discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin’s Broome Bridge. John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. The first three of these “division algebras” would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein’s special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?...

Peculiar Numbers
I met Furey in June, in the porter’s lodge through which one enters Trinity Hall on the bank of the River Cam. Petite, muscular, and wearing a sleeveless black T-shirt (that revealed bruises from mixed martial arts), rolled-up jeans, socks with cartoon aliens on them and Vegetarian Shoes–brand sneakers, in person she was more Vancouverite than the otherworldly figure in her lecture videos. We ambled around the college lawns, ducking through medieval doorways in and out of the hot sun. On a different day I might have seen her doing physics on a purple yoga mat on the grass. 

Read more...

Recommended Reading

Photo: James O’Brien for Quanta Magazine

Quantum Questions Inspire New Math by Robbert Dijkgraaf, director and Leon Levy Professor at the Institute for Advanced Study in Princeton, New Jersey.
"In order to fully understand the quantum world, we may have to develop a new realm of mathematics."

Source: Quanta Magazine


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Taking Risks in Your Teaching | Faculty Development - Faculty Focus

Reprinted from The Teaching Professor, 28.3 (2014): 7. © Magna Publications. All rights reserved.


Maryellen Weimer, PhD, Author at Faculty Focus reports, "Often in workshops when I’m speaking about the process of implementing change—deciding what to change and how to change it or considering whether to add a new instructional strategy—the question of risk lurks in the choices being considered."
 
Photo: Faculty Focus
When attending a workshop or program that offers a range of instructional possibilities, teachers typically respond to some favorably. I see it—they write down the idea, nod, or maybe ask a follow-up question to be sure they understand the details. Not all the ideas presented get this favorable response. Occasionally, the response is overtly negative. But more often there is no response. The idea doesn’t resonate.

When I ask participants to look over their notes (I love teaching faculty because they do take notes) and share what criteria they used to select the new ideas they’re considering implementing, the responses are pretty nonspecific: “I liked it.” “It’s something I think I can do.” “I can use it when I’m teaching X.” I think they are really saying, “This approach fits comfortably with who I am and how I teach.” We first gravitate toward instructional changes that mesh with current practices and the content we teach. We choose them because we can see ourselves doing them...

Are some instructional approaches just too risky? Can teachers take on something they really shouldn’t be trying? Of course.
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Source: Faculty Focus


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MTSU renews academic partnership with Ningbo University in China | Daily News Journal

"Middle Tennessee State University renewed Friday an academic partnership that allows actuarial science students enrolled at a Chinese university to finish their classwork and earn their degree at the Murfreesboro campus" inform MTSU, Contributed.

MTSU President Sidney A. McPhee, left, and Shao Qianium, vice president of Ningbo University in China, sign papers that renew the academic partnership between the two institutions. Behind them is a portrait of the late Chinese leader Deng Xiaoping, inscribing on a scroll the name of Ningbo University at its founding in 1986.
Photo: Andrew Oppmann

MTSU President Sidney A. McPhee and Shao Qianium, vice president of Ningbo University in China, agreed to extend the Joint Mathematics and Applied Mathematics Program, established in the fall of 2012, for 10 more years.

Ningbo students selected for the program first attend classes taught by MTSU faculty on the Chinese university campus, then progress to Murfreesboro to finish their studies. Successful students receive degrees from both MTSU and Ningbo University.

The program can accept up to 40 students each academic year.

“This partnership with Ningbo University is a model for effective and productive international academic cooperation,” McPhee said. “It leverages our respective strengths to provide unique opportunities for exceptional students.”

Replied Shao, quoting the renowned Chinese philosopher Confucius, “Isn’t it great when friends visit from distant places?”

Actuarial science trains students to apply mathematical skills and statistical techniques to manage risks and solve problems in insurance and pension programs.

MTSU, through its College of Basic and Applied Sciences, is the only Tennessee university offering actuarial science coursework for both undergraduate and graduate degrees.

Students take courses in mathematics, statistics, economics, and insurance designed to help prepare for preliminary examinations from the actuarial professional societies and to prepare them for success in the field. 
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Source: Daily News Journal


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DC126 - Actuarial Mathematics - DCU | CareersPortal

Actuarial Mathematics involves applying the science of chance - probability - to complicated problems encountered every day in insurance and high finance. It is a career suited to those who excel in mathematics and problem solving.

 Photo: The Faculty of Science and Health - Dublin City University

The actuarial profession helps people to manage their exposure to risk and its impact on their lives, property, health or future by using mathematics.

Finance encompasses topics such as economics, accounting, statistics, and investment, all with the focus on managing money efficiently.
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Source: CareersPortal


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The 10 Highest Paid Degrees | Business - Perfectly Plain

"While a bachelor’s degree isn’t the only avenue to obtain a job that gives us financial independence, it is a path that many teenagers and parents feel is an investment to transition into adulthood" according to Pawan Naidu, journalist from California.

Photo: Perfectly Plain

However, when it comes to money, not all degrees are built the same. Your major largely determines your earning potential, and Payscale recently ranked majors based on alumni salaries.

Let’s take an in-depth look at the top 10 degrees, and whether the salary is going to be worth it.

(Note: Early Career Pay is the median salary for alumni with said degree with zero to five years of experience. Mid-Career Pay is the median salary for alumni with said degree with 10 plus years of experience.)

7. Economics and Mathematics
I’m not exactly sure if we’re ever going to need to trigonometry in our jobs or day-to-day life like are teachers told us, but having an advanced understanding of math helps the bank account. A person can expect to make around $60,000 a year soon after they graduate, and see a pay increase of $122,900 during the middle of their career. However, it needs to be mentioned that only 36 percent of alumni feel like they’re making a difference in the world.

According to Wake Forest, they have a program that combines these two majors and it allows students to apply mathematical methods to the development of economic theory, models and quantitative analysis...

3. Actuarial Science  
This major might not reap immediate benefits compared to previous ones on this list, but it makes up for it later on. Students majoring in actuarial science can expect to make $61,200 a year after graduation, and see more than double in pay increase in the middle of their career with $130,800. A little over 40 percent of alumni say they are doing something good in the world.

Actuarial science is statistical analysis involving mathematics, statistics and financial economics.

2. Actuarial Mathematics 
Actuarial mathematics is very similar to actuarial science. There is a difference in what they look at though. There is a slight pay decrease for starting salaries after graduation, but an increase when a person is in the middle of their career.
A recent graduate can expect to make $56,400 a year while receiving a significant pay increase to $131,700 later on. Almost half of the alumni feel their work is providing good in the world.

According to College Grad, actuaries working with math analyze the financial costs of risk and uncertainty. They use mathematics, statistics and financial theory to assess the risk of potential events, and they help businesses and clients develop policies that minimize the cost of that risk. Actuarial mathematicians’ work is essential to the insurance industry.
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Source: Perfectly Plain


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How 'big-picture thinking' and IB study go hand in glove | Education - The Sydney Morning Herald

"The International Baccalaureate alternative in high school is growing" summarizes Fran Molloy, Journalist.
 

IB students, from left, Alexandra Doubleday, Kate Addison and Sophia Kennaugh with Tara's head of Senior School Scott Baker.
Photo: Supplied

Students at Tara Anglican School for Girls will now learn their years 7 to 10 coursework via the International Baccalaureate (IB) Middle Years Program, a framework for teaching that the school introduced more than two years ago when students now in year 9 began their secondary schooling.

Head of Senior School Scott Baker says the content follows the Australian curriculum, but is taught differently. 

"The framework through which the material is taught is best practice," he says.

The IB middle years program (MYP) gives broad coverage in creative arts and design, language acquisition, literature, maths, physical education, societies and science...

A key part of IB study is learning how to take notes, how to reference and research. That's music to the ears of universities, some of whom have begun making unconditional early entry offers to IB students based on their predicted results.
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Source: The Sydney Morning Herald 


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Op-Ed: How Africa can embrace an artificial intelligence enabled future | CNBCAfrica.com

"Currently, no African country is among the top 10 countries expected to benefit most from AI and automation. But, the continent has the potential to catch up with the rest of world if we act fast" says Zoaib Hoosen, Microsoft Managing Director.


t’s no longer news that Artificial Intelligence (AI) will be a driving force behind the Fourth Industrial Revolution, with the global economic returns of this revolution expected to be in the region of about $16 trillion.

Along with these returns, AI is also expected to create 2.3 million new jobs by 2020, according to Gartner.

However, if we look at previous revolutions, history shows us that these revolutions have always been accompanied by a brief transition of temporary job loss followed by a period of recovery where job creation moves into more positive territory.

This means that we all need to take steps now to prepare AI in the future...

Combining STEM with the arts
Young people have a leg up on those already in the working world because they can easily develop the necessary skills for these new roles. It’s therefore essential that our education system constantly evolves to equip youth with the right skills and way of thinking to be successful in jobs that may not even exist yet.

As the division of tasks between man and machine changes, we must re-evaluate the type of knowledge and skills imparted to future generations.

For example, technical skills will be required to design and implement AI systems, but interpersonal skills, creativity and emotional intelligence will also become crucial in giving humans an advantage over machines.

“At one level, AI will require that even more people specialise in digital skills and data science. But skilling-up for an AI-powered world involves more than science, technology, engineering and math. As computers behave more like humans, the social sciences and humanities will become even more important. Languages, art, history, economics, ethics, philosophy, psychology and human development courses can teach critical, philosophical and ethics-based skills that will be instrumental in the development and management of AI solutions.” This is according to Microsoft president, Brad Smith, and EVP of AI and research, Harry Shum, who recently authored the book “The Future Computed”, which primarily deals with AI and its role in society.

Interestingly, institutions like Stanford University are already implementing this forward-thinking approach. The university offers a programme called CS+X, which integrates its computer science degree with humanities degrees, resulting in a Bachelor of Arts and Science qualification.
Read more... 

Recommended Reading

The Future Computed is available here 
The Future Computed: Artificial Intelligence and its role in society by Brad Smith, President and Chief Legal Officer and Harry Shum, Executive Vice President of Microsoft AI and Research Group. 

Source: CNBCAfrica.com


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Want to change the world? Pursue research in the humanities… | University - Study International News

“The calling of the humanities is to make us truly human in the best sense of the word.” J. Irwin Miller


"From age-old sketches across cave walls to time-worn folklore tales, the humanities have long been mankind’s means of decoding a complex world" argues Study International News.
 
Photo: School of Humanities, Nanyang Technological University Singapore

 
Our ancient ancestors recorded thoughts and feelings, created magnificent art, told fantastical stories and developed philosophical theories. Detailed research and analysis of these achievements reveal the intricate twists and turns of our past, offering a foundation that helps us understand and predict current and future trends.

When the relevance and credibility of this discipline comes under fire, it serves us well to take stock of its powerful legacy. Advancements in the humanities through the ages have come to define the very fabric of our existence. And since they have shaped the world today, they will greatly factor into our tomorrow.

But the world in which they were first conceived is fundamentally different to the one they’re used in now.

The current digital age is one centred on disruptive technologies, thriving on change and rapid-pace development. In this world, failure to adapt breeds the very real danger of becoming obsolete.

As the basis of all common knowledge we’ve come to discover, research in the humanities must continue to evolve to meet the challenges of life in the twenty-first century.

“Tackling today’s biggest social and technological challenges requires the ability to think critically about their human context, which is something that humanities graduates happen to be best trained to do,” explains Vivek Wadhwa of The Washington Post.

Research in the humanities has driven advances in diverse fields, from healthcare to robotics. Humanities graduates influence every major sector and tackle the greatest issues in our world, while leaders increasingly look for support from those who understand every layer of the human condition.

The masterminds who forge our path through the intricate turns of these disciplines are empowered by higher education. Statistics have revealed the humanities to hold the most popular subjects studied among global leaders.

Known for research excellence and technological innovation, Nanyang Technological University‘s (NTU) standing at number 12 in the QS World Ranking positions it among the global academic elite. Topping the QS Top 50 Under 50 chart for the fifth year in a row, NTU’s research is known and respected across all corners of the globe.
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Source: Study International News


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