Category: mathematics

Credit: Getty Images

Better living through set-builder notation

As a publicly mathematical person, one of the matters upon which I am called to adjudicate is what I think of as “viral order of operations questions” with a math problem along the lines of “48÷2(9+3) = ?”

In the past, I used to tell people who asked me about one of those questions something like, “I think the correct answer is __, but it’s better to write the expression unambiguously.” But recently I decided I need to put my foot down. I will no longer give an answer to those questions. The only way to win is not to play.

An ambiguous sequence of digits and mathematical operation symbols is not interesting to mathematicians. Most of us learn something about the order of operations fairly early on in our mathematical educations. We might learn PEMDAS or BIDMAS, or “Please excuse my dear aunt Sally.” All of these expressions tell us that we’re supposed to take care of expressions inside parentheses or brackets first, followed by exponential expressions, followed by multiplication and division, then addition and subtraction. It’s good that we have a system, but mathematicians and other people who use mathematical expressions regularly would never write down something like “48÷2(9+3) = ?” because its potential for causing confusion is too great.

In general, mathematicians strive to reduce ambiguity when possible. A mathematician would write (48÷2)(9+3) or 48÷(2(9+3)), depending on which one they meant. Viral order of operations problems are unappealing. Just toss in a few more parentheses to clarify your meaning and move on. There are cat pictures to scroll through, for goodness’ sake!

In fact, I think if mathematicians had their way, they would get rid of easily-fixed ambiguous order of operations problems altogether, and I don’t think they’d stop there. The English language often leaves room for ambiguity, and I think mathematical notation could help us make some improvements.

I remember chuckling to myself when I saw the phrase “I like to play board games and read a book while taking a bath.” It conjures up an image of a very exciting game of Monopoly where Baltic Avenue is replaced by the Baltic Sea. Set-builder notation could resolve that titillating ambiguity there while simultaneously freeing me of a tiny shred of joy in this world of woe. Mathematicians use curly brackets to indicate things that are in one set and can be treated as one object. A person who enjoys two separate activities, one of which is playing board games and one of which is reading a book while taking a bath, could write “I like to {play board games} and {read a book while taking a bath}.” A person with a much more interesting bathtime routine than mine could write “I like to {play board games and read a book} while taking a bath.”

Curly brackets take care of written English, but we communicate through speech as well. The late Danish pianist and comedian Victor Borge had a routine about phonetic punctuation. He assigned sounds to some common punctuation marks and inserted them into sentences.

He didn’t include a sound for curly brackets, and I’m not sure the best option. Perhaps a “zzp” sound would work, but I’m open to other suggestions. In the meantime, there is a precedent for air quotes, and perhaps we could add extend that to air parentheses.

When I started thinking about using brackets and air-parentheses in English writing and speech, I wondered if I was just reinventing sentence diagramming. I don’t know how many people learned to diagram a sentence in school, but a sentence diagram is a graphical representation of a sentence that shows how each word and phrase functions in the sentence. The diagram for the sentence “I like to {play board games} and {read a book while taking a bath}” is different from the diagram for “I like to {play board games and read a book} while taking a bath.” It’s been a while since I did any sentence diagramming, so I beg for lenience from any grammar teachers reading, but these were the diagrams I came up with. The placement of the “while” clause is the only difference between the two diagrams.

Two possible diagrams for the sentence “I like to play board games and read a book while taking a bath.” Credit: Evelyn Lamb

Diagramming the sentence does remove the ambiguity, and it gives even more information than the set-builder notation I’m suggesting, but it comes at a cost of both space and effort. It’s much more practical to throw a few brackets into English prose than to draw every sentence as a complex, multi-storied building.

Set-builder brackets are just the tip of the iceberg when it comes to resolving linguistic ambiguity through mathematical notation. After we’ve mastered those, we can consider incorporating union and disjoint union symbols and the distinction between or and xor into our speech and writing. But that will have to wait until after we’ve relaxed with an extra-realistic game of Battleship in the bathtub.

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

Credit of the article given to Evelyn Lamb

Credit: Getty Images/iStockphoto Thinkstock Images/photoncatcher

An embellished account of a border crossing

“How many days will you be staying?” The immigration officer’s question made my blood run cold. I could easily tell him my origin city, nationality, flight number, and eventual destination, but this question was different. It was a fencepost question.

Fencepost questions, dealing as they do with the difference between count and duration, discrete and continuous, have always been difficult for me. May 1st and 8th were both Sundays. There are 7 days in a week, and 8-1 is 7. 9 am on May 8th is 7 days after 9 am on May 1st, but if I do something every day from the 1st to the 8th, I do it 8 times. If someone asks me about something that takes place from the 1st to the 8th, mere subtraction is not enough. I need to know the context of the question and what type of answer is expected.

For this trip, I left on the 14th and would return on the 24th. 24-14 is 10, but I would be there for portions of 11 calendar days and the entirety of only 9 of them. I arrived at 5 pm and would leave at 10 am, so I would be there for less than 240 hours, the length of 10 days. On the other hand, I would be away from my home for well over 240 hours. Would the officer just look at my arrival and departure dates, subtract, and be done with it? Would it confuse him if I said 11? Or did he know about fenceposts too? Was he trying to trap me? Luring me into the false certainty of subtraction, ready to pounce when I gave him a number that failed to reflect the nuances of my visit?

I felt like a cornered animal who doesn’t know whether, if I do something every other day, I do it 3 or 4 times in a week, and if I do it 4 times, whether that means a week has 8 days in it.

My mind was racing, my palms sweaty. The officer looked at me expectantly. Would he think I was flustered because I had something to hide? Would I be dragged away for more questioning, as another mathematician was last week?

Trying to keep my voice steady, I replied: “Ten.” I resisted the urge to explain the situation further.

I held my breath. As if in slow motion, he looked down, reached for the stamp, and flipped to a blank page in my passport. “Have a nice visit.”

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

Credit of the article given to Evelyn Lamb

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

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

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

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.

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

Credit of the article given to Robert S. Siegler

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.”

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

Credit of the article given to Davide Castelvecchi & Nature magazine

Credit: Ada Lovelace, Julia Robinson, and Emmy Noether. IanDagnall Computing/Alamy Stock Photo (Lovelace and Noether); George M. Bergman/Wikimedia Commons (CC BY-SA 4.0) (Robinson)

Everyone knows that history’s great mathematicians were all men—but everybody is wrong

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

From the profound revelations of the shape of space to the furthest explorations reachable by imagination and logic, the history of mathematics has always been seen as a masculine endeavor. Names like Gauss, Euler, Riemann, Poincare, Erdős, and the more modern Wiles, Tao, Perelman, and Zhang, all of them associated with the most beautiful mathematics discovered since the dawn of humanity, are all men. The book Men of Mathematics, written by E.T. Bell in 1937, is just one example of how this “fact” has been reinforced in in the public consciousness.

Even today, it is no secret that male mathematicians still dominate the field. But this should not distract us from the revolutionary contributions women have made. We have notable women to thank for modern computation, revelations on the geometry of space, cornerstones of abstract algebra, and major advances in decision theory, number theory, and celestial mechanics that continue to provide crucial breakthroughs in applied areas like cryptography, computer science, and physics.

The works of geniuses like Julia Robinson on Hilbert’s Tenth Problem in number theory, Emmy Noether in abstract algebra and physics, and Ada Lovelace in computer science, are just three examples of women whose contributions have been absolutely essential.

Julia Robinson at Berkeley, California, 1975. Credit: George M. Bergman/Wikimedia Commons (CC BY-SA 4.0)

Julia Robinson (1919-1985)

At the turn of the twentieth century the famed German mathematician David Hilbert published a set of twenty-three tantalizing problems that had evaded the most brilliant of mathematical minds. Among them was his tenth problem, which asked if a general algorithm could be constructed to determine the solvability of any Diophantine equation (those polynomial equations with only integer coefficients and integer solutions). Imagine, for any Diophantine equation of the infinite set of such equations a machine that can tell whether it can be solved. Mathematicians often deal with infinite questions of this nature that exist far beyond resolution by simple extensive observations. This particular problem drew the attention of a Berkeley mathematician named Julia Robinson. Over several decades, Robinson collaborated with colleagues including Martin Davis and Hillary Putnam that resulted in formulating a condition that would answer Hilbert’s question in the negative.

In 1970 a young Russian mathematician named Yuri Matiyasevich solved the problem using the insight provided by Robinson, Davis, and Putnam. With her brilliant contributions in number theory, Robinson was a remarkable mathematician who paved the way to answering one of the greatest pure math questions ever proposed. In a Mathematical Association of America article, “The Autobiography of Julia Robinson”, her sister and biographer Constance Read wrote, “She herself, in the normal course of events, would never have considered recounting the story of her own life. As far as she was concerned, what she had done mathematically was all that was significant.”

Portrait of the German mathematician Amalie Emmy Noether, c.1910. Credit: IanDagnall Computing/Alamy Stock Photo

Emmy Noether (1882-1935)

Sitting in an abstract math course for any length of time, one is bound to hear the name Emmy Noether. Her notable work spans subjects from physics to modern algebra, making Noether one of the most important figures in mathematical history. Her 1913 result on the calculus of variations, leading to Noether’s Theorem is considered one of the most important theorems in mathematics—and one that shaped modern physics. Noether’s theory of ideals and commutative rings forms a foundation for any researcher in the field of higher algebra.

The influence of her work continues to shine as a beacon of intuition for those who grapple with understanding physical reality more abstractly. Mathematicians and physicists alike admire her epoch contributions that provide deep insights within their respective disciplines. In 1935, Albert Einstein wrote in a letter to the New York Times, “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

Ada Lovelace, Portrait by Margaret Sarah Carpenter, oil on canvas, 1836. Credit: IanDagnall Computing/Alamy Stock Photo

Ada Lovelace (1815-1852)

In 1842, Cambridge mathematics professor Charles Babbage gave a lecture at the University of Turin on the design of his Analytical Engine (the first computer). Mathematician Luigi Menabrea later transcribed the notes of that lecture to French. The young Countess Ada Lovelace was commissioned by Charles Wheatstone (a friend of Babbage) to translate the notes of Menabrea into English. She is known as the “world’s first programmer” due to her insightful augmentation of that transcript. Published in 1843, Lovelace added her own notes including Section G, which outlined an algorithm to calculate Bernoulli numbers. In essence, she took Babbage’s theoretical engine and made it a computational reality. Lovelace provided a path for others to shed light on the mysteries of computation that continues to impact technology.

Despite their profound contributions, the discoveries made by these three women are often overshadowed by the contributions of their male counterparts. According to a 2015 United Nations estimate, the number of men and women in the world is almost equal (101.8 men for every 100 women). One could heuristically argue, therefore that we should see roughly the same number of women as men working in the field of mathematics.

One large reason that we don’t is due to our failure to recognize the historical accomplishments of female mathematicians. Given the crucial role of science and technology in the modern world, however, it is imperative as a civilization to promote and encourage more women to pursue careers in mathematics.

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

Credit of the article given to Avery Carr

The whole is equivalent to the sum of its parts

Which is bigger, 10 or 7?

I suspect that for most, the response to this question is instinctive, unconscious, and immediate. So how about I pose a follow-up question:

How do you know?

If you can refrain from dismissing this question as trivial, I invite you to pause and try to reflect on what happened in your mind in that instant – is this factual recall, was there something visual, was it something contextual, or was it something else?

Perhaps you will indulge me and delve a little deeper:

In how many ways do you know?

Here again, I invite you to pause and consider your response before continuing. Maybe you would like to imagine that you are trying to convince someone or different people. Pick up a piece of paper and draw pictures, write things down, and try to form another approach that is different in some way from the others.

When we compare numerical values, there are many helpful approaches that we can take. These might be based on processes such as: counting, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I said 7 first so it must be smaller (that’s how numbers work!)”; motion/movement, “If we start together at the bottom, then I climb 7 stairs and you climb 10 stairs, you will be higher than me (and more tired!)”; measurement/length, “This length (7cm) is shorter than this length (10cm)”; matching/creating correspondences, “There are 10 people and 7 cupcakes if I hand out a cupcake to each person I will run out – not everyone will get one!”

Each of these approaches (and the many more you might imagine) might be grounded in two prominent types of reasoning: part-whole and/or correspondence. These two ideas are used pervasively, interchangeably, and often simultaneously when reasoning with numbers in most of school mathematics and in our daily experiences.

Part-whole

Let’s take another comparison problem, this time inspired by questions posed to children in a study by Falk (2010):

What are there more of:

1. Hairs on your head OR fingers on two hands?
2. Fingers on two hands OR days in a month?
3. Grains of sand on Earth OR hairs on your head?
4. All numbers OR grains of sand on Earth?

This time, I suspect, your responses were not always instantaneous and more conscious thought was required. How convinced are you of your responses? Did you feel as though more information was required?

When you reflect on the reasoning you employed in making these comparisons, I wonder whether you assigned numerical values to the quantities – did you feel an urge to do so, as a first step, before applying similar techniques to those used before?

When the children in this study were faced with such comparisons, an interesting misconception revealed itself: many of them considered a very large number, for instance, the number of grains of sand on Earth, to be synonymous with infinity. This, of course, presents a potential difficulty with question 4; I suspect you won’t be alone if you encounter this, too.

When we encounter numbers or quantities that are so large/vast that they are beyond our comprehension, it is perhaps unsurprising that we equate these with infinity – that magical word that creeps into our consciousness from a very young age as the default answer to any questions about “biggest number.” So, is this a problematic concept to hold? In practical terms, for most people, probably not. But mathematically it is, and actually confronting it offers some wonderful opportunities to explore, discuss and better understand the numbers that we work with, the structure of mathematical systems, and the nature of the mathematics that we study.

So how could we confront this misconception? How can we take advantage of the opportunities alluded to above? Well, one possibility is purposefully to create situations where the misconception might arise.

Position the quantities representing the grains of sand on Earth and all numbers on a number line.

• Would they be in the same place, or would one be closer to zero than the other?
• If they are not in the same place, are they very close together or very far apart?
• Is it possible to measure the gap between these two quantities?

Talking around this task is likely to draw attention to the fact that some quantities may be large and unknown, but we can be certain they are finite – a single number exists to represent them, we just don’t know what it is. Others, however, are large, unknown, and also not finite – they are not represented by a single large number but are unbounded, often the result of an infinite process such as counting. These infinite quantities cannot be positioned on a number line, and the gap (the difference) between any finite quantity and an infinite one is immeasurable – it is infinitely large in itself!

So, is it possible to make comparisons with infinite quantities? Or is this “not allowed?!” Well, we can certainly say that any finite quantity is smaller than any infinite quantity. But how about this:

What are there more of: natural numbers or even numbers?

I would encourage you, once again, to establish and hold your own response to this question in your mind before reading on.

As at the beginning of this blog, the follow-up question is:

How do you know?

Intuition tends to be strong here, grounded in our experiences with finite quantities and part-whole reasoning: the even numbers are a part of the natural numbers so there must be more natural numbers (twice as many, we might argue). We can confirm this with examples; for instance, by comparing the number of natural numbers and even numbers there are up to a fixed point, say 100:

Now, what if I asked you to find an alternative approach, another way of explaining how you know that there are more natural numbers than even numbers? When we compared 7 and 10, we discussed two main approaches, those based on part-whole reasoning and those based on matching / correspondences. What would a correspondence approach look like here?

It looks as though I can pair up the two sets of numbers, I can match every natural number, one-to-one, with an even number, so the two sets are equal… Uh oh! And, more than that, our two methods of comparison, which are usually used interchangeably, lead to different results!

How do you feel about this seemingly contradictory situation? Maybe this example is something you are comfortable with, but most likely not! For many students, and indeed teachers, this is a troubling situation, causing us to throw up our hands in despair and confusion! However, if we can overcome this sensation and recognise that the conflict is real (it’s not that we’ve made an error), then the stage is set for thinking more carefully about assumptions that might have been made and when and where our mathematical rules and procedures are used and valid. Giving students similar opportunities to encounter situations where their intuition is called into question, inviting them to discuss (and argue!), expose their own lines of reasoning, and compare contexts and situations in the search for an explanation, is surely a good thing! Perhaps, when prompted in this way they might also be more receptive to the introduction of standard, accepted approaches within mathematics.

As a closing comment, let’s notice that our discussions are touching on the most fundamental property of any infinite set: that it can be matched, one-to-one, with a proper subset of itself. In other words, in the case of infinite sets, the whole is equivalent to some of its parts!

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

Credit for the article given to Tabitha Gould