Author: IMC

Breathe. As your lungs expand, air fills 500 million tiny alveoli, each a fraction of a millimeter across. As you exhale, these millions of tiny breaths merge effortlessly through larger and larger airways into one ultimate breath.

These airways are fractal.

The branches within lungs are an example of self-similarity. Brockhaus and Efron Encyclopedic Dictionary/Wikimedia

Fractals are a mathematical tool for describing objects with detail at every scale. Mathematicians and physicists like me use fractals and related concepts to understand how things change going from small to big.

You and I translate between vastly different scales when we think about how our choices affect the world. Is this latte contributing to climate change? Should I vote in this election?

These conceptual tools apply to the body as well as landscapes, natural disasters and society.

Fractals everywhere

In 1967, mathematician Benoit Mandelbrot asked, “How long is the coast of Britain?”

It’s a trick question. The answer depends on how you measure it. If you trace the outline on a map, you get one answer, but if you walk the coastline with a meter stick, the result is quite different. Anyone who has tried to estimate the length of a rugged hiking trail from a map knows the treachery of the large-scale picture.

Satellite image of Great Britain and Northern Ireland. NASA

That’s because lungs, the British coastline and hiking trails all have fractality: their length, number of branches or some other quantity depends on the scale or resolution you use to measure them.

The coastline is also self-similar – it’s made out of smaller copies of itself. Fern fronds, trees, snail shells, landscapes, the silhouettes of mountains and river networks all look like smaller versions of themselves.

That’s why, when you’re looking at an aerial photograph of a landscape, it’s often hard to tell whether the scale bar should be 50 km or 500 m.

Your lungs are self-similar, because the body finely calibrates each branch in exact proportions, making each branch a smaller replica of the previous. This modular design makes lungs efficient at any size. Think of a child and an adult, or a mouse, a whale. The only difference between small and large is in how many times the airways branch.

Self-similarity and fractality appear in art and architecture, in the arches within arches of Roman aqueducts and the spires of Gothic cathedrals that mirror the forest canopy. Even ancient Chinese calligraphers Huai Su and Yan Zhenqing prized the fractality of summer clouds, cracks in a wall and water stains in a leaking house in 722.

Scale invariance

Self-similar objects have a scale invariance. In other words, some property holds regardless of how big they get, such as the efficiency of lungs.

In effect, scale invariance describes what changes between scales by saying what doesn’t change.

A sketch from Leonardo da Vinci’s notes on tree branches. Fractal Foundation

Leonardo da Vinci observed that, as trees branch, the total cross-sectional area of all branches is preserved. In other words, going from trunk to twigs, the number of branches and their diameter change with each branching, but the total thickness of all branches bundled together stays the same.

Da Vinci’s observation implies a scale invariance: For every branch of a certain radius, there are four downstream branches with half that radius.

Earthquake frequency has a similar scale invariance, which was observed in the 1940s. The big ones come to mind – Lisbon 1755, San Francisco 1989 – but many small earthquakes occur in California every day. The Gutenberg-Richter law says that earthquake frequency depends on the size of the earthquake. The answer is surprisingly simple. A tenfold bigger earthquake occurs roughly one-tenth as often.

Society and the power law

A 19th-century economist Vilifredo Pareto – famous in business school for the 80/20 rule – observed that the number of families with a certain wealth is inversely proportional to their wealth, raised to some exponent. Pareto measured the exponent for different years and different countries and found that it was usually around 1.5.

Patterns in an oak’s branches. Schlegelfotos/shutterstock.com

Pareto’s wealth distribution came to be known as the power law, ostensibly because of the exponent or “power.”

Anything self-similar has a corresponding power law. In an April paper, my colleague and I describe the corresponding power law for lungs, blood vessels and trees. It differs from Pareto’s power law only by taking into account specific ratios between branches.

The sizes of fortunes then are akin to the sizes of tree twigs or blood vessels – a few trunks or large branches and exponentially more tiny twigs.

Pareto thought of his distribution of wealth as a natural law, but many different models of social organization give rise to a Pareto distribution and societies do vary in wealth inequality. The higher Pareto’s exponent, the more egalitarian the society.

From understanding how humans are made up of tiny cells to how we affect the planet, self-similarity, fractality and scale invariance often help translate from one level of organization to another.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Mitchell Newberry

Credit: Getty Images

The right mix of people who already know one another, of boys and girls–Ramsey numbers may hold the answer

Let’s say you’re planning your next party and agonizing over the guest list. To whom should you send invitations? What combination of friends and strangers is the right mix?

It turns out mathematicians have been working on a version of this problem for nearly a century. Depending on what you want, the answer can be complicated.

Our book, “The Fascinating World of Graph Theory,” explores puzzles like these and shows how they can be solved through graphs. A question like this one might seem small, but it’s a beautiful demonstration of how graphs can be used to solve mathematical problems in such diverse fields as the sciences, communication and society.

A puzzle is born

While it’s well-known that Harvard is one of the top academic universities in the country, you might be surprised to learn that there was a time when Harvard had one of the nation’s best football teams. But in 1931, led by All–American quarterback Barry Wood, such was the case.

That season Harvard played Army. At halftime, unexpectedly, Army led Harvard 13–0. Clearly upset, Harvard’s president told Army’s commandant of cadets that while Army may be better than Harvard in football, Harvard was superior in a more scholarly competition.

Though Harvard came back to defeat Army 14-13, the commandant accepted the challenge to compete against Harvard in something more scholarly. It was agreed that the two would compete – in mathematics. This led to Army and Harvard selecting mathematics teams; the showdown occurred in West Point in 1933. To Harvard’s surprise, Army won.

The Harvard–Army competition eventually led to an annual mathematics competition for undergraduates in 1938, called the Putnam exam, named for William Lowell Putnam, a relative of Harvard’s president. This exam was designed to stimulate a healthy rivalry in mathematics in the United States and Canada. Over the years and continuing to this day, this exam has contained many interesting and often challenging problems – including the one we describe above.

Red and blue lines

The 1953 exam contained the following problem (reworded a bit): There are six points in the plane. Every point is connected to every other point by a line that’s either blue or red. Show that there are three of these points between which only lines of the same color are drawn.

In math, if there is a collection of points with lines drawn between some pairs of points, that structure is called a graph. The study of these graphs is called graph theory. In graph theory, however, the points are called vertices and the lines are called edges.

Graphs can be used to represent a wide variety of situations. For example, in this Putnam problem, a point can represent a person, a red line can mean the people are friends and a blue line means that they are strangers.

Show that there are three points connected by lines of the same color. Credit: richtom80 Wikimedia (CC BY-SA 3.0)

For example, let’s call the points A, B, C, D, E, F and select one of them, say A. Of the five lines drawn from A to the other five points, there must be three lines of the same color.

Say the lines from A to B, C, D are all red. If a line between any two of B, C, D is red, then there are three points with only red lines between them. If no line between any two of B, C, D is red, then they are all blue.

What if there were only five points? There may not be three points where all lines between them are colored the same. For example, the lines A–B, B–C, C–D, D–E, E–A may be red, with the others blue.

From what we saw, then, the smallest number of people who can be invited to a party (where every two people are either friends or strangers) such that there are three mutual friends or three mutual strangers is six.

What if we would like four people to be mutual friends or mutual strangers? What is the smallest number of people we must invite to a party to be certain of this? This question has been answered. It’s 18.

What if we would like five people to be mutual friends or mutual strangers? In this situation, the smallest number of people to invite to a party to be guaranteed of this is – unknown. Nobody knows. While this problem is easy to describe and perhaps sounds rather simple, it is notoriously difficult.

Ramsey numbers

What we have been discussing is a type of number in graph theory called a Ramsey number. These numbers are named for the British philosopher, economist and mathematician Frank Plumpton Ramsey.

Ramsey died at the age of 26 but obtained at his very early age a very curious theorem in mathematics, which gave rise to our question here. Say we have another plane full of points connected by red and blue lines. We pick two positive integers, named r and s. We want to have exactly r points where all lines between them are red or s points where all lines between them are blue. What’s the smallest number of points we can do this with? That’s called a Ramsey number.

For example, say we want our plane to have at least three points connected by all red lines and three points connected by all blue lines. The Ramsey number – the smallest number of points we need to make this happen – is six.

When mathematicians look at a problem, they often ask themselves: Does this suggest another question? This is what has happened with Ramsey numbers – and party problems.

For example, here’s one: Five girls are planning a party. They have decided to invite some boys to the party, whether they know the boys or not. How many boys do they need to invite to be certain that there will always be three boys among them such that three of the five girls are either friends with all three boys or are not acquainted with all three boys? It’s probably not easy to make a good guess at the answer. It’s 41!

Very few Ramsey numbers are known. However, this doesn’t stop mathematicians from trying to solve such problems. Often, failing to solve one problem can lead to an even more interesting problem. Such is the life of a mathematician.

This article was originally published on The Conversation. Read the original article.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Gary Chartrand, Arthur Benjamin, Ping Zhang & The Conversation US

Alice in Wonderland enthusiasts recently celebrated the story’s anniversary with creative events like playing with puzzles and time — and future Alice exhibits are in the works. The original 1865 children’s book Alice’s Adventures in Wonderland, sprung from a mathematician’s imagination, continues to inspire exploration and fun.

But is a connection between math and creativity captured in schools? Much discussion across the western world from both experts and the public has emphasized the need to revitalize high school mathematics: critics say the experience is boring or not meaningful to most students. Experts concerned with the public interest and decision-making say students need skills in critical thinking, creativity, communication and collaboration.

Mathematicians, philosophers and educators are also concerned with the excitement and energy of creative expression, with invention, with wonder and even with what might be called the romance of learning.

Mathematics has all the attributes of the paragraph above, and so it seems to me that what’s missing from high school math is mathematics itself.

I am now working with colleagues at Queen’s University and the University of Ottawa to develop RabbitMath, a senior level high-school math curriculum designed to enable students to work together creatively with a high level of personal engagement. My preparation for this has been 40 years of working with teachers in high-school classrooms.

In partnership with grades 11 and 12 math teachers, we will be piloting this curriculum over the next few years.

Mathematical novels

When students study literature, drama or the creative arts in high school, the curriculum centres on what can be called sophisticated works of art, created in response to life’s struggles and triumphs.

But currently in school mathematics, this is rarely the case: students are not connected to the larger imaginative projects through which professional mathematicians confront the world’s problems or explore the world’s mysteries.

The author, Peter Taylor, right, at a Lisgar Collegiate Institute Grade 11 math classroom in Ottawa, 2018. (Ann Arden), Author provided

Mathematician Jo Boaler from the Stanford Graduate School of Education says that a “wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in school education.”

Of the subject of mathematics, Boaler notes that:

“Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that is an aesthetic, creative, and beautiful subject. Why are these descriptions so different?”

She points out the same gulf isn’t seen if people ask students and English-literature professors what literature is about.

In the process of constructing the RabbitMath curriculum, problems or activities are included when team members find them engaging and a challenge to their intellect and imagination. Following the analogy with literature, we call the models we are working with mathematical novels.

For example, one project invites students to work with ocean tides. It would hard to find a dramatic cycle as majestic as the effect of that sublime distant moon on the powerful tidal action in the Bay of Fundy.

Student engagement

In the 1970s, the extraordinary mathematician and computer scientist at Massachusetts Institute of Technology, Seymour Papert, noticed that in art class, students, just as mature artists, are involved in personally meaningful work. Papert’s objective was to be able to say the same of a mathematics student.

I had a parallel experience in 2013 when I was the internal reviewer for the Drama program at Queen’s. I marvelled at students’ creative passion as they prepared to stage a performance. And they weren’t all actors: they were singers, musicians, writers, composers, directors and technicians.

In Papert’s curriculum model, students with diverse abilities and interests work together on projects, whereby they collaborate on problems, strategies and outcomes.

As a pioneering computer scientist, Papert understood that students could directly access the processes of design and construction through digital technology. Papert used his computer system LOGO for this technical interface. LOGO was limited in its scope, but Papert’s idea was way ahead of its time.

Students in the RabbitMath classroom will work together using the programming language Python to construct diagrams and animations to better understand their experiments with springs and tires, mirrors and music. They will produce videos that can explain to their classmates the workings of a sophisticated structure.

Today, technology, the internet, computer algebra systems and mathematical programming provide possibilities for immediate engagement in processes of design and construction — exactly what Papert wanted. The platform for RabbitMath is the Jupyter Notebook, a direct descendant of LOGO.

Technical skill

For too many years, real progress in school mathematics education has been hamstrung by a ridiculous confrontation between so-called “traditional” and “discovery” math. The former is concerned with technical facility and the latter is about skills of inquiry and investigation.

There is no conflict between the two; in fact they support each other rather well. Every sophisticated human endeavour, from conducting a symphony orchestra to putting a satellite into orbit, understands the complementary nature of technical facility and creative investigation.

Stanford University Graduate School of Education mathematician Keith Devlin advises parents to ensure their child has mastery of what he calls number sense, “fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons.” But for students embarking on careers in science, technology or engineering, that is not enough, he says. They need a deep understanding of both those procedures and the concepts they rely on — the capacity to analyze and work with complex systems.

A high-school math class is a rich ecosystem of differing abilities, capacities, objectives and temperaments.

The educator’s goal must be to enable a diverse mix of students to work together in a math class as creatively and intensely as students in the drama program, or to bring the same personal passion as they might to writing fiction.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Peter Taylor

Abstract algebra class gave me the kick in the rear I needed to focus on the relationships between notes

During my senior year of college, I decided I wanted to expand my musical horizons, so I joined the early music ensemble. I had entered college with a viola scholarship, so I had played in the orchestra throughout my time there as well as doing chamber music and working on solo viola pieces, but I had always enjoyed early music and wanted to try something new. In the ensemble, I played Baroque violin, and a lot of my technique as a modern violist translated well. I held the instrument and bow a little differently, but on the whole, I was able to pick it up quickly. Reading the music was another story.

Many people are exposed to treble and bass clefs in music classes. I was very young when my mom started to teach me how to read music, but I still remember the feeling of accomplishment I had when I mastered the idea that the position of a spot on an array of lines and spaces corresponded to a particular key on a piano or a particular pitch I could sing. The treble clef is based on the G above middle C. The bass clef uses the F below middle C.

A C major scale written across bass (bottom) and treble (top) clefs. The two dots on the bass clef indicate the F below middle C, and the treble clef circles the G above middle C. Credit: Martin Marte-Singer Wikimedia (CC BY-SA 4.0)

Many instruments’ ranges fit comfortably onto either the treble or bass clef, perhaps adjusted by an octave when necessary. Piano music, of course, uses both clefs. But the viola is a little too low to use treble clef all the time and a little too high to use bass clef all the time. When I started to play viola, I learned to read alto clef, which has middle C smack dab in the middle of the staff, and eventually I was the music-reading equivalent of trilingual.

A C major scale written in alto clef, starting on middle C. Credit: Hyacinth Wikimedia

My multilingualism had its limits, though. I could read all three clefs, but if I wanted to play music originally written for cello an octave higher or music originally written for flute or violin an octave lower—tasks that would have been trivial on a piano—I would struggle to read bass clef up an octave or treble clef down an octave, respectively. Tenor clef, which is like alto clef but with the C one line higher, flummoxed me entirely. I wasn’t as fluent as I wanted to be.

Early music ensemble pushed my limits. Music from the Baroque era and before was not always notated using the small number of clefs we tend to use now. I was reading music in French violin clef (ooh-la-la, this one looks like treble clef but has the G on the bottom line instead of the line above the bottom), soprano clef (a C clef like alto and tenor clefs with the C on the bottom line), and other currently unusual clefs. It was overwhelming. I made a lot of mistakes in rehearsal, despite the many note names I had to write in my music.

The same semester I started playing with the early music ensemble, I took an abstract algebra class. Abstract algebra looks at structures of sets of numbers and symmetries. It encourages people to see connections between sometimes very different mathematical objects and transformations and to view the relationships between objects as fundamental to understanding those objects.

At some point in the semester, a switch flipped in my brain, and my early music clef struggles virtually disappeared. At the beginning of a piece, I would look at the clef to get my bearings, and I could see the rest of the notes as representing relationships between one pitch and the next. I read intervals, not pitches. I was not perfect, but I felt like almost overnight I had unlocked a new music-reading level.

I have always felt like my journey into more abstract algebra and my new clef fluency were related, but I have struggled to put that connection into words. I feel like the structural and relational aspects of abstract algebra helped me to see clefs as descriptions of relationships between notes rather than as absolute pitches, but I can’t point to a particular theorem or insight in abstract algebra that would apply explicitly.

Last year, I learned about the Yoneda lemma, an important theorem in the mathematical field of category theory. (According to our My Favorite Theorem guest Emily Riehl, it’s every category theorist’s favorite theorem.) I am no category theorist, but I found Tai-Danae Bradley’s description of the Yoneda lemma helpful, particularly the big idea she shared in this post on the Yoneda perspective. She writes that the punchline of the Yoneda lemma, or at least two of its corollaries, is “mathematical objects are completely determined by their relationships to other objects.”

It has taken me a while to make the connection explicitly, but I think the “Yoneda perspective” describes the mental shift I made in early music ensemble. It’s not the exact notes that matter when you’re reading music written in an unfamiliar clef but the relationships between them. Since having this shift in perspective, it’s been easier for me to transpose music into different keys and read treble and bass clefs in whatever octaves I need to.

Some organists and pianists can transpose music seemingly effortlessly to accommodate the needs of their church choirs or musical theater performers, and I think it’s because they’ve already shifted to the Yoneda perspective, even if that’s not how they would describe it. They didn’t necessarily get there via advanced mathematics classes, but for me, I think abstract algebra class gave me the kick in the rear I needed not to be tied to any one set of exact pitches but to focus on the relationships between them. I won’t claim that studying abstract algebra or category theory will improve your music-reading skills—making music, not studying a math book, is usually the best way to get better at making music—but pondering these connections enriches my experience of both math and music, and I hope it can do the same for you.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Evelyn Lamb

Bias in, bias out: many algorithms have inherent design problems. Vintage Tone/Shutterstock

Could this be the first line of a “Hippocratic Oath” for mathematicians and data scientists? Hannah Fry, Associate Professor in the mathematics of cities at University College London, argues that mathematicians and data scientists need such an oath, just like medical doctors who swear to act only in their patients’ best interests.

“In medicine, you learn about ethics from day one. In mathematics, it’s a bolt-on at best. It has to be there from day one and at the forefront of your mind in every step you take,” Fry argued.

But is a tech version of the Hippocratic Oath really required? In medicine, these oaths vary between institutions, and have evolved greatly in the nearly 2,500 years of their history. Indeed, there is some debate around whether the oath remains relevant to practising doctors, particularly as it is the law, rather than a set of ancient Greek principles, by which they must ultimately abide.

How has data science reached the point at which an ethical pledge is deemed necessary? There are certainly numerous examples of algorithms doing harm – criminal sentencing algorithms, for instance, have been shown to disproportionately recommend that low-income and minority people are sent to jail.

Similar crises have led to proposals for ethical pledges before. In the aftermath of the 2008 global financial crisis, a manifesto by financial engineers Emanuel Derman and Paul Wilmott beseeched economic modellers to swear not to “give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.”

Just as prejudices can be learned as a child, the biases of these algorithms are a result of their training. A common feature of these algorithms is the use of black-box (often proprietary) algorithms, many of which are trained using statistically biased data.

In the case of criminal justice, the algorithm’s unjust outcome stems from the fact that historically, minorities are overrepresented in prison populations (most likely as a result of long-held human biases). This bias is therefore replicated and likely exacerbated by the algorithm.

Machine learning algorithms are trained on data, and can only be expected to produce predictions that are limited to those data. Bias in, bias out.

Promises, promises

Would taking an ethical pledge have helped the designers of these algorithms? Perhaps, but greater awareness of statistical biases might have been enough. Issues of unbiased representation in sampling have long been a cornerstone of statistics, and training in these topics may have led the designers to step back and question the validity of their predictions.

Fry herself has commented on this issue in the past, saying it’s necessary for people to be “paying attention to how biases you have in data can end up feeding through to the analyses you’re doing”.

But while issues of unbiased representation are not new in statistics, the growing use of high-powered algorithms in contentious areas make “data literacy” more relevant than ever.

Part of the issue is the ease with which machine learning algorithms can be applied, making data literacy no longer particular to mathematical and computer scientists, but to the public at large. Widespread basic statistical and data literacy would aid awareness of the issues with statistical biases, and are a first step towards guarding against inappropriate use of algorithms.

Nobody is perfect, and while improved data literacy will help, unintended biases can still be overlooked. Algorithms might also have errors. One easy (to describe) way to guard against such issues is to make them publicly available. Such open source code can allow joint responsibility for bias and error checking.

Efforts of this sort are beginning to emerge, for example the Web Transparency and Accountability Project at Princeton University. Of course, many proprietary algorithms are commercial in confidence, which makes transparency difficult. Regulatory frameworks are hence likely to become important and necessary in this area. But a precondition is for practitioners, politicians, lawyers, and others to understand the issues around the widespread applicability of models, and their inherent statistical biases.

Ethics is undoubtedly important, and in a perfect world would form part of any education. But university degrees are finite. We argue that data and statistical literacy is an even more pressing concern, and could help guard against the appearance of more “unethical algorithms” in the future.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Lewis Mitchell, Joshua Ross

Credit: Getty Images

Processing high-level math concepts uses the same neural networks as the basic math skills a child is born with

Alan Turing, Albert Einstein, Stephen Hawking, John Nash—these “beautiful” minds never fail to enchant the public, but they also remain somewhat elusive. How do some people progress from being able to perform basic arithmetic to grasping advanced mathematical concepts and thinking at levels of abstraction that baffle the rest of the population? Neuroscience has now begun to pin down whether the brain of a math wiz somehow takes conceptual thinking to another level.

Specifically, scientists have long debated whether the basis of high-level mathematical thought is tied to the brain’s language-processing centers—that thinking at such a level of abstraction requires linguistic representation and an understanding of syntax—or to independent regions associated with number and spatial reasoning. In a study published this week in Proceedings of the National Academy of Sciences, a pair of researchers at the INSERM–CEA Cognitive Neuroimaging Unit in France reported that the brain areas involved in math are different from those engaged in equally complex nonmathematical thinking.

The team used functional magnetic resonance imaging (fMRI) to scan the brains of 15 professional mathematicians and 15 nonmathematicians of the same academic standing. While in the scanner the subjects listened to a series of 72 high-level mathematical statements, divided evenly among algebra, analysis, geometry and topology, as well as 18 high-level nonmathematical (mostly historical) statements. They had four seconds to reflect on each proposition and determine whether it was true, false or meaningless.

The researchers found that in the mathematicians only, listening to math-related statements activated a network involving bilateral intraparietal, dorsal prefrontal, and inferior temporal regions of the brain. This circuitry is usually not associated with areas involved in language processing and semantics, which were activated in both mathematicians and nonmathematicians when they were presented with the nonmathematical statements. “On the contrary,” says study co-author and graduate student Marie Amalric, “our results show that high-level mathematical reflection recycles brain regions associated with an evolutionarily ancient knowledge of number and space.”

Previous research has found that these nonlinguistic areas are active when performing rudimentary arithmetic calculations and even simply seeing numbers on a page, suggesting a link between advanced and basic mathematical thinking. In fact, co-author Stanislas Dehaene, director of the Cognitive Neuroimaging Unit and experimental psychologist, has studied how humans (and even some animal species) are born with an intuitive sense of numbers—of quantity and arithmetic manipulation—closely related to spatial representation. How the connection between a hardwired “number sense” and higher-level math is formed, however, remains unknown. This work raises the intriguing question of whether an innate capability to recognize different quantities—that two pieces of fruit are greater than one—is the biological foundation on which can be built the capacity to master group theory. “It would be interesting to investigate the causal chain between lower-level and higher-level mathematical competency,” says Daniel Ansari, a cognitive neuroscientist at the University of Western Ontario who did not participate in the study. “Most of us master basic arithmetic, so we’re already recruiting these brain regions, but only a fraction of us go on to do high-level math. We don’t yet know whether becoming a mathematical expert changes the way you do arithmetic or whether learning arithmetic lays out the foundation for acquiring higher-level mathematical concepts.”

Ansari suggests that a training study, in which nonmathematicians are taught advanced mathematical concepts, could provide a better understanding of these connections and how they form. Moreover, achieving expertise in mathematics may affect neuronal circuitry in other ways. Amalric’s study found that mathematicians had reduced activity in the visual areas of the brain involved in facial processing. This could mean that the neural resources required to grasp and work with certain math concepts may undercut—or “use up”—some of the brain’s other capacities. Although additional studies are needed to determine whether mathematicians actually do process faces differently, the researchers hope to gain further insight into the effects that expertise has on how the brain is organized.

“We can start to investigate where exceptional abilities come from, and the neurobiological correlates of such high-level expertise,” Ansari says. “I just think it’s great that we now have the capability to use brain imaging to answer these deep questions about the complexity of human abilities.”

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to  Jordana Cepelewicz

The first modern electronic digital computer was called the Atanasoff–Berry computer, or ABC.

It was built by physics Professor John Vincent Atanasoff and his graduate student, Clifford Berry, in 1942 at Iowa State College, now known as Iowa State University.

That’s where I have been teaching computer engineering for over 30 years, and I’m also a collector of old computers. I got to meet Atanasoff when he visited Iowa State and got a signed copy of his book.

Before ABC, there were mechanical computing devices that could perform simple calculations. The first mechanical computer, The Babbage Difference Engine, was designed by Charles Babbage in 1822. The ABC was the basis for the modern computer we all use today.

The ABC’s drums. Courtesy of Iowa State University Library Special Collections and University Archives, CC BY-ND

The ABC weighed over 700 pounds and used vacuum tubes. It had a rotating drum, a little bigger than a paint can, that had small capacitors on it. A capacitor is device that can store an electric charge, like a battery.

The ABC was designed to solve problems with up to 29 different variables. You might be familiar with equations with one variable, like 2y = 14. Now imagine 29 different variables. These are common problems in physics and other sciences, but were difficult and time-consuming to solve by hand.

Atanasoff was credited with several breakthrough ideas that are still present in modern computers. The most important idea was using binary digits, just ones and zeroes, to represent all numbers and data. This allowed the calculations to be performed using electronics.

Another idea was the separation of the program (the computer instructions) and memory (places to store numbers).

The ABC completed one operation about every 15 seconds. Compared to the millions of operations per second of today’s computer, that probably seems very slow.

Unlike today’s computers, the ABC did not have a changeable stored program. This meant the program was fixed and designed to do a single task. This also meant that, to solve these problems, an operator had to write down the intermediate answer and then feed that back into the ABC. Atanasoff left Iowa State before he perfected a storage method that would have eliminated the need for the operator to reenter the intermediate results.

Part of the ABC. Courtesy of Iowa State University Library Special Collections and University Archives, CC BY-ND

Shortly after Atanasoff left Iowa State, the ABC was dismantled. Atanasoff never filed a patent for his invention. That means that, for a long time, many people weren’t aware of the ABC.

In 1947, the creators of the Electronic Numerical Integrator And Computer, or ENIAC, filed a patent. This allowed them to claim they were the inventors of the digital computer. For several decades, most people thought that the ENIAC was the first modern computer.

But one of the inventors of the ENIAC had visited Atanasoff in 1941. The courts later ruled that this visit influenced the design of the ENIAC. The ENIAC patent was thrown out by a judge in 1973.

The holders of the ENIAC patent argued that the ABC never really worked. Since all that remained was one of the drum memory units, it was hard to prove otherwise.

In 1997 a team of faculty, researchers and students at Iowa State University finished building a replica of the ABC. They were able to demonstrate that the ABC did function. You can see the replica today at the Computer History Museum in Mountain View, California.

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Credit of the article given to Doug Jacobson

Credit: Jasmina81 Getty Images

Why do Americans have such trouble with fractions—and what can be done?

Many children never master fractions. When asked whether 12/13 + 7/8 was closest to 1, 2, 19, or 21, only 24% of a nationally representative sample of more than 20,000 US 8th graders answered correctly. This test was given almost 40 years ago, which gave Hugo Lortie-Forgues and me hope that the work of innumerable teachers, mathematics coaches, researchers, and government commissions had made a positive difference. Our hopes were dashed by the data, though; we found that in all of those years, accuracy on the same problem improved only from 24% to 27% correct.

Such difficulties are not limited to fraction estimation problems nor do they end in 8th grade. On standard fraction addition, subtraction, multiplication, and division problems with equal denominators (e.g., 3/5+4/5) and unequal denominators (e.g., 3/5+2/3), 6th and 8th graders tend to answer correctly only about 50% of items. Studies of community college students have revealed similarly poor fraction arithmetic performance. Children in the US do much worse on such problems than their peers in European countries, such as Belgium and Germany, and in Asian countries such as China and Korea.

This weak knowledge is especially unfortunate because fractions are foundational to many more advanced areas of mathematics and science. Fifth graders’ fraction knowledge predicts high school students’ algebra learning and overall math achievement, even after controlling for whole number knowledge, the students’ IQ, and their families’ education and income. On the reference sheets for recent high school AP tests in chemistry and physics, fractions were part of more than half of the formulas. In a recent survey of 2300 white collar, blue collar, and service workers, more than two-thirds indicated that they used fractions in their work. Moreover, in a nationally representative sample of 1,000 Algebra 1 teachers in the US, most rated as “poor” their students’ knowledge of fractions and rated fractions as the second greatest impediment to their students mastering algebra (second only to “word problems”).

Why are fractions so difficult to understand? A major reason is that learning fractions requires overcoming two types of difficulty: inherent and culturally contingent. Inherent sources of difficulty are those that derive from the nature of fractions, ones that confront all learners in all places. One inherent difficulty is the notation used to express fractions. Understanding the relation a/b is more difficult than understanding the simple quantity a, regardless of the culture or time period in which a child lives. Another inherent difficulty involves the complex relations between fraction arithmetic and whole number arithmetic. For example, multiplying fractions involves applying the whole number operation independently to the numerator and the denominator (e.g., 3/7 * 2/7 = (3*2)/(7*7) = 6/49), but doing the same leads to wrong answers on fraction addition (e.g., 3/7 + 2/7 ≠ 5/14). A third inherent source of difficulty is complex conceptual relations among different fraction arithmetic operations, at least using standard algorithms. Why do we need equal denominators to add and subtract fractions but not to multiply and divide them? Why do we invert and multiply to solve fraction division problems, and why do we invert the fraction in the denominator rather than the one in the numerator? These inherent sources of difficulty make understanding fraction arithmetic challenging for all students.

Culturally contingent sources of difficulty, in contrast, can mitigate or exacerbate the inherent challenges of learning fractions. Teacher understanding is one culturally-contingent variable: When asked to explain the meaning of fraction division problems, few US teachers can provide any explanation, whereas the large majority of Chinese teachers provide at least one good explanation. Language is another culturally-contingent factor; East Asian languages express fractions such as 3/4 as “out of four, three,” which makes it easier to understand their meaning than relatively opaque terms such as “three fourth.” A third such variable is textbooks. Despite division being the most difficult operation to understand, US textbooks present far fewer problems with fraction division than fraction multiplication; the opposite is true in Chinese and Korean textbooks. Probably most fundamental are cultural attitudes: Math learning is viewed as crucial throughout East Asia, but US attitudes about its importance are far more variable.

Given the importance of fractions in and out of school, the extensive evidence that many children and adults do not understand them, and the inherent difficulty of the topic, what is to be done? Considering culturally contingent factors points to several potentially useful steps. Inculcating a deeper understanding of fractions among teachers will likely help them to teach more effectively. Explaining the meaning of fractions to students using clear language (for example, explaining that 3/4 means 3 of the 1/4 units), and requesting textbook writers to include more challenging problems are other promising strategies. Addressing inherent sources of difficulty in fraction arithmetic, in particular understanding of fraction magnitudes, can also make a large difference.

Fraction Face-off!, a 12-week program designed by Lynn Fuchs to help children from low-income backgrounds improve their fraction knowledge, seems especially promising. The program teaches children about fraction magnitudes through tasks such as comparing and ordering fraction magnitudes and locating fractions on number lines. After participating in Fraction Face-off!, fourth graders’ fraction addition and subtraction accuracy consistently exceeds of children receiving the standard classroom curriculum. This finding was especially striking because Fraction Face-off! devoted less time to explicit instruction in fraction arithmetic procedures than did the standard curriculum. Similarly encouraging findings have been found for other interventions that emphasize the importance of fraction magnitudes. Such programs may help children learn fraction arithmetic by encouraging them to note that answers such as 1/3+1/2 = 2/5 cannot be right, because the sum is less than one of the numbers being added, and therefore to try procedures that generate more plausible answers. These innovative curricula seem well worth testing on a wider basis.

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Credit of the article given to Robert S. Siegler

Bees are pretty good at maths – as far as insects go, at least. We already know, for example, that they can count up to four and even understand the concept of zero.

But in a new study, published today in the Journal of Experimental Biology, we show honeybees can also understand numbers higher than four – as long as we provide feedback for both correct and incorrect responses as they learn.

Even our own brains are less adept at dealing with numbers greater than four. While we can effortlessly estimate up to four items, processing larger numbers requires more mental effort. Hence why when asked to count, a young child will sometimes answer with “1, 2, 3, 4, more”!

If you don’t believe me, try the test below. The various colour groupings representing 1-4 stars are easy to count quickly and accurately. However, if we try estimating the number of all stars at once by ignoring colours, it requires more concentration, and even then our accuracy tends to be poorer.

For numbers of elements ranging from 1-4, as represented here in different colours, we very efficiently process the exact number. However, if we try estimating the number of all stars at once by ignoring colour, it requires a lot more cognitive effort.

This effect isn’t unique to humans. Fish, for example, also show a threshold for accurate quantity discrimination at four.

One theory to explain this is that counting up to four isn’t really counting at all. It may be that many animals’ brains can innately recognise groups of up to four items, whereas proper counting (the process of sequentially counting the number of objects present) is needed for numbers beyond that.

By comparing the performance of different animal species in various number processing tasks we can better understand how differences in brain size and structure enable number processing. For example, honeybees have previously been shown to be able to count and discriminate numbers up to four, but not beyond. We wanted to know why there was a limit at four – and whether they can go further.

Best bee-haviour

Bees are surprisingly good at maths. We recently discovered that bees can learn to associate particular symbols with particular quantities, much like the way we use numerals to represent numbers.

Bees learn to do this type of difficult task if given a sugary reward for choosing the correct association, and a bitter liquid for choosing incorrectly. So if we were to push bees beyond the four threshold, we knew success would depend on us asking the right question, in the right way, and providing useful feedback to the bees.

We trained two different groups of bees to perform a task in which they were presented with a choice of two different patterns, each containing a different number of shapes. They could earn a reward for choosing the group of four shapes, as opposed to other numbers up to ten.

We used two different training strategies. One group of ten bees received only a reward for a correct choice (choosing a quantity of four), and nothing for an incorrect choice. A second group of 12 bees received a sugary reward for picking four, or a bitter-tasting substance if they made a mistake.

In the test, bees flew into a Y-shaped maze to make a choice, before returning to their hive to share their collected sweet rewards.

Each experiment conducted with a single bee lasted about four hours, by which time each bee had made 50 choices.

Bees were individually trained and tested in a Y-shaped maze where a sugar reward was presented on the pole directly in front of the correct stimulus. Author provided

The group that only received sweet rewards could not successfully learn to discriminate between four and higher numbers. But the second group reliably discriminated the group of four items from other groups containing higher numbers.

Thus, bees’ ability to learn higher number discrimination depends not just on their innate abilities, but also on the risks and rewards on offer for doing so.

Bee’s-eye view of either four or five element displays that could be discriminated. Inserts show how we normally see these images.

Our results have important implications for understanding how animals’ brains may have evolved to process numbers. Despite being separated by 600 million years of evolution, invertebrates such as bees and vertebrates such as humans and fish all seem to share a common threshold for accurately and quickly processing small numbers. This suggests there may be common principles behind how our brains tackle the question of quantity.

The evidence from our new study shows bees can learn to process higher numbers if the question and training are presented in the right way. These results suggest an incredible flexibility in animal brains, of all sizes, for learning to become maths stars.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Adrian Dyer, Jair Garcia, Scarlett Howard

Credit: Pavel Bolotov/Thinkstock (MARS)

A rare appearance by enigmatic Shinichi Mochizuki brings faint optimism about his famously impenetrable work

Nearly four years after Shinichi Mochizuki unveiled an imposing set of papers that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity — although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University’s Research Institute for Mathematical Sciences (RIMS).

Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego. Although at first Mochizuki’s papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae, experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.

And Jeffrey Lagarias, a number theorist at the University of Michigan in Ann Arbor, says that he got far enough to see that Mochizuki’s work is worth the effort. “It has some revolutionary new ideas,” he says.

Still, Kedlaya says that the more he delves into the proof, the longer he thinks it will take to reach a consensus on whether it is correct. He used to think that the issue would be resolved perhaps by 2017. “Now I’m thinking at least three years from now.”

Others are even less optimistic. “The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me,” says mathematician Vesselin Dimitrov of Yale University in New Haven, Connecticut. “Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature.”

The abc proof

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers — whole numbers that cannot be evenly divided by any smaller number except by 1. The conjecture comes in a number of different forms, but explains how the primes that divide two numbers, a and b, are related to those that divide their sum, c.

If Mochizuki’s proof is correct, it would have repercussions across the entire field, says Dimitrov. “When you work in number theory, you cannot ignore the abc conjecture,” he says. “This is why all number theorists eagerly wanted to know about Mochizuki’s approach.” For example, Dimitrov showed in January how, assuming the correctness of Mochizuki’s proof, one might be able to derive many other important results, including a completely independent proof of the celebrated Fermat’s last theorem.

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

Mochizuki in the room

The Kyoto workshop followed on the heels of one held last December in Oxford, UK. Mochizuki did not attend that first meeting, although he answered the audience’s questions over a Skype video link. This time, having him in the room — and hearing him present some of the materials himself — was helpful, says Taylor Dupuy, a mathematician at the Hebrew University of Jerusalem who participated in both workshops.

There are now around ten mathematicians who are putting substantial effort into digesting the material — up from just three before the Oxford workshop, says Ivan Fesenko, a mathematician at the University of Nottingham, UK, who co-organized both workshops. The group includes younger researchers, such as Dupuy.

In keeping with his reputation for being a very private person, Mochizuki — who is said to never eat meals in the presence of colleagues — did not take part in the customary mingling and social activities at the Kyoto meeting, according to several sources. And although he was unfailingly forthcoming in answering questions, it was unclear what he thought of the proceedings. “Mochizuki does not give a lot away,” Kedlaya says. “He’s an excellent poker player.”

Fellow mathematicians have criticized Mochizuki for his refusal to travel. After he posted his papers, he turned down multiple offers to spend time abroad and lecture on his ideas. Although he spent much of his youth in the United States, he is now said to rarely leave the Kyoto area. (Mochizuki does not respond to requests for interviews, and the workshop’s website contained the notice: “Activities aimed at interviewing or media coverage of any sort within the facilities of RIMS, Kyoto University, will not be accepted.”)

“He is very level-headed,” says another workshop participant, who did not want to be named. “The only thing that frustrates him is people making rash judgemental comments without understanding any details.”

Still, Dupuy says, “I think he does take a lot of the criticism about him really personally. I’m sure he’s sick of this whole thing, too.”

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Credit of the article given to Davide Castelvecchi & Nature magazine