Author: IMC

Dialogue is a key part of all lively classrooms, but how do we ensure this dialogue is effective? How do we give students the tools they need to discuss mathematical ideas? What are the components of effective classroom dialogue? Let’s take a look at dialogic teaching and explore how it can encourage deeper mathematical learning.

What is dialogic teaching?

Dialogic teaching is grounded in active and meaningful dialogue between teacher and students. A dialogue is not a teacher standing in front of a class delivering a lesson – it is an active back and forth that promotes questioning and reasoning. The goal is to foster a collaborative and interactive learning environment where students actively engage in building their understanding of the subject.

Shyam Drury from Scitech explores the specifics of dialogic teaching in mathematics in this fantastic podcast  on the Maths in Schools Strategies for Explicit Teaching podcast series. Drury observes, ‘When teaching mathematics in a dialogical classroom, the authority in the room is not the teacher or the student, instead it is mathematical truth.’

Questioning

Questions are a part of every classroom, but what kind of questions encourage dialogue? Is it a matter of open-ended versus closed questions? Closed questions are important for checking comprehension, but they don’t promote dialogue. Open-ended questions promote dialogue, but discussions can easily get off track.

The key to constructing productive questions is to ensure that they promote focused dialogue. The mathematical idea you are teaching – and the desired learning outcome – should always direct the conversation and inform the questions you ask. This may include both open-ended and closed questions.

If you want to learn more about questioning, listen  to the fascinating conversation with Professor Helen Chick from the University of Tasmania as she explores the ‘how to’ of questions in teaching.

ow do you build a dialogic classroom?

Now we understand what dialogical teaching is, let’s explore how we put it into practice.

Mathematics teaching is largely based around an IRE dialogue pattern: initiate, respond, evaluate. For example:

• Initiate: What’s 6 x 7?
• Respond: 42
• Evaluate: That’s right!

How do we extend this dialogue pattern? Instead of the conversation ending with the ‘evaluate’ response, ask your students a question. How did you come to that answer? Why did you use that method?

Bring in how and why questions to provoke thinking. How questions unpack and make more explicit a student’s approach to a mathematical problem and why questions promote reasoning.

How does it feel as a learner?

Dialogic teaching is only effectively within the right classroom culture. A learner needs to feel safe to engage, enquire and take risks.

Here are some tools to help create a safe space for dialogic teaching:

• Provide opportunities for students to speak to each other in low-risk situations, such as peer-to-peer discussions or small-group discussions. This way, every student in the room is expressing their thinking, not just those who arrive at the answer first.
• Place whiteboards around the room displaying what you want your class to tackle. When everyone is looking at the same piece of mathematics, it encourages working together.
• Set up challenges that provide a framework for students to share ideas.

Dialogue leads to deeper understanding

When learners feel safe, dialogic teaching helps construct a shared understanding of the strategies and tools required for mathematical learning.

Encouraging dialogue helps students develop the language they need to unpack and explore mathematics.

If they can talk about, they can share it!

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

Credit: PK Flickr(CC BY 2.0)

The mathematics that is most important to society is the province of the exceptional few—and that’s always been true

Most people never become mathematicians, but everyone has a stake in mathematics. Almost since the dawn of human civilization, societies have vested special authority in mathematical experts. The question of how and why the public should support elite mathematics remains as pertinent as ever, and in the last five centuries (especially the last two) it has been joined by the related question of what mathematics most members of the public should know.

Why does mathematics matter to society at large? Listen to mathematicians, policymakers, and educators and the answer seems unanimous: mathematics is everywhere, therefore everyone should care about it. Books and articles abound with examples of the math that their authors claim is hidden in every facet of everyday life or unlocks powerful truths and technologies that shape the fates of individuals and nations. Take math professor Jordan Ellenberg, author of the bestselling book How Not to Be Wrong, who asserts “you can find math everywhere you look.”

To be sure, numbers and measurement figure regularly in most people’s lives, but this risks conflating basic numeracy with the kind of math that most affects your life. When we talk about math in public policy, especially the public’s investment in mathematical training and research, we are not talking about simple sums and measures. For most of its history, the mathematics that makes the most difference to society has been the province of the exceptional few. Societies have valued and cultivated math not because it is everywhere and for everyone but because it is difficult and exclusive. Recognizing that math has elitism built into its historical core, rather than pretending it is hidden all around us, furnishes a more realistic understanding of how math fits into society and can help the public demand a more responsible and inclusive discipline.

In the first agricultural societies in the cradle of civilization, math connected the heavens and the earth. Priests used astronomical calculations to mark the seasons and interpret divine will, and their special command of mathematics gave them power and privilege in their societies. As early economies grew larger and more complex, merchants and craftsmen incorporated more and more basic mathematics into their work, but for them mathematics was a trick of the trade rather than a public good. For millennia, advanced math remained the concern of the well-off, as either a philosophical pastime or a means to assert special authority.

The first relatively widespread suggestions that anything beyond simple practical math ought to have a wider reach date to what historians call the Early Modern period, beginning around five centuries ago, when many of our modern social structures and institutions started to take shape. Just as Martin Luther and other early Protestants began to insist that Scripture should be available to the masses in their own languages, scientific writers like Welsh polymath Robert Recorde used the relatively new technology of the printing press to promote math for the people. Recorde’s 1543 English arithmetic textbook began with an argument that “no man can do any thing alone, and much less talk or bargain with another, but he shall still have to do with number” and that numbers’ uses were “unnumerable” (pun intended).

Far more influential and representative of this period, however, was Recorde’s contemporary John Dee, who used his mathematical reputation to gain a powerful position advising Queen Elizabeth I. Dee hewed so closely to the idea of math as a secret and privileged kind of knowledge that his detractors accused him of conjuring and other occult practices. In the seventeenth century’s Scientific Revolution, the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty. During the eighteenth-century Enlightenment, by contrast, the savants of the French Academy of Sciences parlayed their mastery of difficult mathematics into a special place of authority in public life, weighing in on philosophical debates and civic affairs alike while closing their ranks to women, minorities, and the lower social classes.

Societies across the world were transformed in the nineteenth century by wave after wave of political and economic revolution, but the French model of privileged mathematical expertise in service to the state endured. The difference was in who got to be part of that mathematical elite. Being born into the right family continued to help, but in the wake of the French Revolution successive governments also took a greater interest in primary and secondary education, and strong performance in examinations could help some students rise despite their lower birth. Political and military leaders received a uniform education in advanced mathematics at a few distinguished academies which prepared them to tackle the specialized problems of modern states, and this French model of state involvement in mass education combined with special mathematical training for the very best found imitators across Europe and even across the Atlantic. Even while basic math reached more and more people through mass education, math remained something special that set the elite apart. More people could potentially become elites, but math was definitely not for everyone.

Entering the twentieth century, the system of channeling students through elite training continued to gain importance across the Western world, but mathematics itself became less central to that training. Partly this reflected the changing priorities of government, but partly it was a matter of advanced mathematics leaving the problems of government behind. Where once Enlightenment mathematicians counted practical and technological questions alongside their more philosophical inquiries, later modern mathematicians turned increasingly to forbiddingly abstract theories without the pretense of addressing worldly matters directly.

The next turning point, which continues in many ways to define the relations between math and society today, was World War II. Fighting a war on that scale, the major combatants encountered new problems in logistics, weapons design and use, and other areas that mathematicians proved especially capable of solving. It wasn’t that the most advanced mathematics suddenly got more practical, but that states found new uses for those with advanced mathematical training and mathematicians found new ways to appeal to states for support. After the war, mathematicians won substantial support from the United States and other governments on the premise that regardless of whether their peacetime research was useful, they now had proof that highly trained mathematicians would be needed in the next war.

Some of those wartime activities continue to occupy mathematical professionals, both in and beyond the state—from security scientists and code-breakers at technology companies and the NSA to operations researchers optimizing factories and supply chains across the global economy. Postwar electronic computing offered another area where mathematicians became essential. In all of these areas, it is the special mathematical advances of an elite few that motivate the public investments mathematicians continue to receive today. It would be great if everyone were confident with numbers, could write a computer program, and evaluate statistical evidence, and these are all important aims for primary and secondary education. But we should not confuse these with the main goals and rationales of public support for mathematics, which have always been about math at the top rather than math for everyone.

Imagining math to be everywhere makes it all too easy to ignore the very real politics of who gets to be part of the mathematical elite that really count—for technology, security, and economics, for the last war and the next one. Instead, if we see that this kind of mathematics has historically been built by and for the very few, we are called to ask who gets to be part of that few and what are the responsibilities that come with their expertise. We have to recognize that elite mathematics today, while much more inclusive than it was one or five or fifty centuries ago, remains a discipline that vests special authority in those who, by virtue of gender, race, and class, are often already among our society’s most powerful. If math were really everywhere, it would already belong to everyone equally. But when it comes to accessing and supporting math, there is much work to be done. Math isn’t everywhere.

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

Credit of the article given to New York University

Archaeologists recently uncovered a 600-year-old die that was probably used for cheating. The wooden die from medieval Norway has two fives, two fours, a three and a six, while the numbers one and two are missing. It is believed that the die was used to cheat in games, rather than being for a game that requires that specific configuration of numbers.

Today, dice like this with missing numbers are known as tops and bottoms. They can be a useful way to cheat if you’re that way inclined, although they don’t guarantee a win every time and they don’t stand up to scrutiny from suspicious opponents (they only have to ask to take a look and you’ll be found out). But there are several other options of cheating at dice too, and I’ll talk you through some of them here.

It should be noted that using these methods in a casino are illegal and I’m not suggesting you adopt them in such establishments – but it’s an interesting look at how probabilities work.

Probabilities of a fair die and a top and bottom die. Graham Kendall

For a fair die, each number has an equal one in six, or 16.67%, chance of appearing. In the case of the die found in Norway, the numbers four and five are twice as likely to appear (as there are two of them), so have a one in three, or 33.33%, chance. The table shows these probabilities.

It does not take too much imagination to see how tops and bottoms can be used to your advantage. Let’s assume that we are playing with two normal dice. There are 36 possible outcomes but only 11 possible total values the dice can produce. For example, six-four, four-six and five-five all add up to ten.

If we instead used two top and bottom dice with only the numbers one, four and five on them, we can never roll a total of 11 or 12 as we don’t have a six to make that total. Similarly, we can never get a total of three as we don’t have a two and a one. But we also cannot get any combination that would produce a total of seven, which would otherwise be the most likely total to appear with a probability of 16.67%. In a game of craps there are times when it can be really bad to throw a seven. So if you are playing with dice where a combination of seven is impossible, you have a distinct advantage.

As these kind of tops and bottoms dice will not pass even a cursory, closer inspection, they have to be brought into the game for a short time and then switched out again. This requires the cheat to be an expert at palming, meaning being able to conceal one set of dice in your hand and then bring them into play while simultaneously removing the other dice.

Using two dice, with the same three numbers repeated, might be too risky so a cheat would probably only want to switch in a single die into the game. In our example, this would mean no longer avoiding a total of seven, which would still have a probability of 16.67%. But now the totals of five and six would also have this probability.

In craps the odds are such that when you are required to avoid a seven, it is the number most likely to appear. Switching in a single dice can still reduce the house’s chances of winning, by making other totals equally likely to appear.

Loaded dice

Loaded dice can make cheating harder to spot. These can take a number of different forms. For example, some of the spots on one face could be drilled out and the holes filled with a heavy substance so the die is more likely to land with this face down. If you were to drill out the number one, this means that the number six is more likely to appear, as the six is always on the opposite face to the one. Another way of loading a die would be to slightly change its shape, so that it is more likely to keep rolling. This may only give a small advantage, but it could be enough to tip the game in the cheat’s favour.

With tops and bottoms it is easy to know the probabilities of various totals appearing. This is not the case with loaded dice. One way of gauging the probabilities is to toss the dice a number of times (possibly thousands) and work out what numbers appear and how often. If you know that seven is less likely to appear than it would with fair dice then, over the long run, it would be a cheat’s advantage.

Controlled throws

One other way to cheat doesn’t require an unfair die at all but involves learning how to throw in a very controlled way. This can involve effectively sliding or dropping the die so the desired number appears. If two dice are used, one can be used to trap the other and stop it bouncing. If this is done by a skilled operator, it is very difficult to see.

Dominic LoRiggio, the “Dice Dominator”, was able to throw dice in what appeared a normal way but so that they would land on certain numbers. This was done by understanding how dice travel thorough the air and controlling each part of the throw. It took many (many, many) hours of practice to perfect, but he was able to consistently win at the craps table.

Many would consider what LoRiggio did to be advantage play, meaning using the rules to your advantage. This is similar to card counting in blackjack. The casinos may not like it, but you are technically not cheating – though some casino may try to make you shoot the dice in a different way if they suspect you are doing controlled throws.

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

Credit: Count your pennies! Learn a fun puzzle to test your quick computation skills—and see if you can find new strategies for getting speedy solutions. George Retseck

A puzzling activity from Science Buddies

Introduction
Do you ever use math as a tool to solve interesting problems? In the 1970s math was often taught with simple worksheets. One teacher was looking for a way to help his students have more fun with math and logic. So he developed what is now known as the perimeter magic triangle puzzles. Try them out—and have some fun as you start thinking about counting in a whole new way!

Background
Counting is so common that we forget how it is connected to the broader area of mathematics that studies numbers, known as arithmetic. We can see counting as repeatedly adding one: when you add one object to another you have two objects. Add one more and you have three, and so on. Addition is the process of adding numbers. The result of the addition is called the sum. With smaller numbers you might use counting to find the sum. When you have three and want to add two, for example, you can count two numbers beyond three to get to five. With plenty of practice you can often memorize the sums of the numbers one through 10—at which point in can be fun to play with numbers to find all the ways you can make a particular sum.

Math puzzles and games can be a fun way to get practice working with numbers. Puzzles also provide entertaining ways to build strategic and logical thinking. With a little trial and error you can often start to find new strategies to complete a puzzle faster. These are the very same techniques mathematicians use: starting small and trying to find patterns in the sequence of answers. These patterns are then used to predict the answers to even bigger puzzles.

If this is all too abstract, try the puzzle presented in this activity! It might make the process of learning arithmetic clear.

Materials

• Two sheets of 9 by 12-inch paper, such as construction or craft paper (if possible, choose contrasting colors)
• Pencil or marker
• Ruler
• Scissors
• A quarter or other round object of similar size
• 21 pennies, small blocks or other small stackable objects
• More sheets of paper (optional)

Preparation

• Draw a large triangle on a sheet of paper (you can use a ruler to help make straight lines).
• Use a quarter to trace a circle on each corner of the triangle. Now trace a circle onto the middle of each side of the triangle. You should have six circles.
• On the bottom of the second sheet of paper draw six circles similar in size to the ones drawn on the triangle.
• Cut out these circles, and number them 1 through 6. These circles will be referred to as number disks.
• Keep the top part of the second sheet of paper. You will use it to write down your results.

Procedure

• On the paper with the triangle use the 21 pennies to build towers on each circle. Each circle must have at least 1 penny, but no two towers can be of the same height. Can you do it?
• Keep trying until you find a solution!
• Count the number of pennies in each tower. Write down each sum in order from the smallest to the largest number. What do you notice about this set of numbers?
• Shift the towers around or rebuild them until you can fulfill one more requirement: The total number of pennies used to build the three towers on each side of the triangle must be the same. If you build towers of 1, 5 and 3 pennies in the circles lining up on one side of the triangle, for example, you used 1 + 5 + 3 = 9 pennies on that side. Lining up towers of 1, 2 and 4 pennies on the adjacent side would not work because 1+ 2 + 4 = 7 —not 9 like the first side. (Notice the tower of 1 penny was placed on the corner of this triangle, so it contributes to two sides.) If you tried 1, 2 and 6 for the adjacent side instead, that works because 1 + 2 + 6 = 9. Now you can place the one tower that is left and check if 9 pennies are used in the three towers on the third side of this triangle. Try it out! Did you find a solution?
• If this is not a solution, think. Can you rearrange a few towers and get a solution?
• If working with abstract numbers is easier for you, replace the towers with the number disks. Each number disks then represents a tower of pennies. The number written on the number disks informs you of the number of pennies in that tower.
• Using 9 pennies per side is possible! Did you find the solution? Are there several ways you can arrange the towers so there are 9 pennies used per side?
• Can you arrange the pennies so you use 10, 11 or even 12 pennies per side?
• Extra: Show that there are no solutions that use 8 or fewer pennies per side—or show that there are no solutions with a total of 13 or more pennies per side.
• Extra: The puzzle presented in this activity is called a “perimeter magic triangle of order three.” To extend it to a higher-order perimeter magic triangle start by drawing a new triangle. Add circles on the corners like you did the first time, but this time add two more circles on each side in between the corners. For this puzzle you will need nine number disks. Number them 1 through 9. Just like in the previous puzzle you need to find ways to place the disks on the circles so the sums of the numbers on each side of this triangle are identical. Mathematicians call this triangle a triangle of order four as it has four numbers on each side. Once you have solved this puzzle continue with a triangle of order five (add three more circles between the corners and cut 12 number disks), then order six, and so on.
• Extra: Can you create a strategy to find solutions for this type of puzzle quickly?

Observations and Results
Did you find that you can only arrange the 21 pennies in towers of 1, 2, 3, 4, 5 and 6 pennies if you need to make six towers of different heights? Could you come up with ways to arrange the towers so the sum of pennies used on each side of the triangle is identical for all three sides? It is possible for a total of 9, 10, 11, and 12 pennies per side.

To use a total of 9 pennies on each side, you place the towers with 1, 2, and 3 pennies on the corners of the triangle. The tower of 6 pennies goes in between the towers of 1 and 2 pennies because 1 + 2 + 6 = 9. The tower of 5 pennies stands between the tower of 1 and the tower of 3 pennies, as 1 + 3 + 5 also equals 9. The towers with 2, 4 and 3 pennies fill up the third row. Notice how the smallest towers are placed on the corners for this solution.

To arrange the towers so that you use 12 pennies on each side start by arranging the tallest towers (those with 6, 5 and 4 pennies) on the corners of your triangle and fill in the circles in between. Place the smallest tower you have left (1 penny tall) in between the two tallest towers (5 and 6 pennies each). Do you see that the smallest one you are left with (2 pennies tall) goes in between the tallest ones that need a tower in between (the towers with 6 and 4 pennies each)?

A strategy you could use to find the solution that has 10 pennies on each side is listing all the ways you can make 10 by adding three different numbers. You will find 3 + 2 + 5 = 10, 5 + 4 + 1 = 10, and 1 + 6 + 3 = 10. Can you see that 3, 5 and 1 are part of two of these sums? This means these go on the corners of your triangle. You can use the same strategy to find out how to place the pennies so there are 11 or 12 pennies used on each side.

Are you wondering how you can know that using 8 pennies per side is not possible? With 8 pennies per side you use 3 X  8, or 24, pennies for the triangle. Because you reuse the pennies on the corner towers you at most use 1 + 2 + 3 (the sum of the three smallest towers) or 6 pennies fewer. In other words you can use at most 18 pennies. The puzzle asks you to use 21.

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

Credit of the article given to Science Buddies & Sabine De Brabandere

The central limit theorem – the idea that plotting statistics for a large enough number of samples from a single population will result in a normal distribution – forms the basis of the majority of the inferential statistics that students learn in advanced school-level maths courses. Because of this, it’s a concept not normally encountered until students are much older. In our work on the Framework, however, we always ask ourselves where the ideas that make up a particular concept begin. And are there things we could do earlier in school that will help support those more advanced concepts further down the educational road?

The central limit theorem is an excellent example of just how powerful this way of thinking can be, as the key ideas on which it is built are encountered by students much earlier, and with a little tweaking, they can support deeper conceptual understanding at all stages.

The key underlying concept is that of a sampling distribution, which is a theoretical distribution that arises from taking a very large number of samples from a single population and calculating a statistic – for example, the mean – for each one. There is an immediate problem encountered by students here which relates to the two possible ways in which a sample can be conceptualised. It is common for students to consider a sample as a “mini-population;” this is often known as an additive conception of samples and comes from the common language use of the word, where a free “sample” from a homogeneous block of cheese is effectively identical to the block from which it came. If students have this conception, then a sampling distribution makes no sense as every sample is functionally identical; furthermore, hypothesis tests are problematic as every random sample is equally valid and should give us a similar estimate of any population parameter.

A multiplicative conception of a sample is, therefore, necessary to understand inferential statistics; in this frame, a sample is viewed as one possible outcome from a set of possible but different outcomes. This conception is more closely related to ideas of probability and, in fact, can be built from some simple ideas of combinatorics. In a very real sense, the sampling distribution is actually the sample space of possible samples of size n from a given population. So, how can we establish a multiplicative view of samples early on so that students who do go on to advanced study do not need to reconceptualise what a sample is in order to avoid misconceptions and access the new mathematics?

One possible approach is to begin by exploring a small data set by considering the following:

“Imagine you want to know something about six people, but you only have time to actually ask four of them. How many different combinations of four people are there?”

There are lots of ways to explore this question that make it more concrete – perhaps by giving a list of names of the people along with some characteristics, such as number of siblings, hair colour, method of travel to school, and so on. Early explorations could focus on simply establishing that there are in fact 15 possible samples of size four through a systematic listing and other potentially more creative representations, but then more detailed questions could be asked that focus on the characteristics of the samples; for example, is it common that three of the people in the sample have blonde hair? Is an even split between blue and brown eyes more or less common? How might these things change if a different population of six people was used?

Additionally, there are opportunities to practise procedures within a more interesting framework; for example, if one of the characteristics was height then students could calculate the mean height for each of their samples – a chance to practise the calculation as part of a meaningful activity – and then examine this set of averages. Are they close to a particular value? What range of values are covered? How are these values clustered? Hey presto – we have our first sampling distribution without having to worry about the messy terminology and formal definitions.

In the Cambridge Mathematics Framework, this approach is structured as exploratory work in which students play with the idea of a small sample as a combinatorics problem in order to motivate further exploration. Following this early work, they eventually created their first sampling distribution for a more realistic population and explored its properties such as shape, spread, proportions, etc. This early work lays the ground to look at sampling from some specific population distributions – uniform, normal, and triangular – to get a sense of how the underlying distribution impacts the sampling distribution. Finally, this is brought together by using technology to simulate the sampling distribution for different empirical data sets using varying sizes of samples in order to establish the concept of the central limit theorem.

While sampling distributions and the central limit theorem may well remain the preserve of more advanced mathematics courses, considering how to establish the multiplicative concept of a sample at the very beginning of students’ work on sampling may well help lay more secure foundations for much of the inferential statistics that comes later, and may even support statistical literacy for those who don’t go on to learn more formal statistical techniques.

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

Credit for the article given to Darren Macey

DNA knot as seen under the electron microscope. Javier Arsuaga, CC BY-ND

If you’ve ever seen a picture of a DNA molecule, you probably saw it in its famous B-form: two strands coiling around each other in a right-handed fashion to form a double helix. But did you know that DNA can change its shape?

DNA molecules, which carry the genetic code of an organism, have to be tightly packed to fit inside a cell. However, every few hours, the cell produces a faithful copy of its genome in preparation for cell division. This replication process puts tremendous stress on the DNA and can change its shape in lethal ways.

As a mathematician and a biologist, I am interested in how mathematics can describe the many shapes of DNA, as well as cellular processes like DNA replication. The answers to these questions inspire new mathematics and possibly a better understanding of the molecule of life.

The shape of DNA

To understand the mathematics of the shape of DNA, you need to consider both its geometry and its topology. These are related but distinct concepts.

Geometry describes an object at a particular moment in time – frozen rigid in space, like a sculpture. In the cell, the DNA helix coils upon itself, or “supercoils.” The way DNA folds and coils encodes valuable geometric information that can be crucial to control the way genes are expressed.

Topology describes how an object deforms smoothly, as if made out of clay without making new holes or breaks. For example, imagine a rubber band tumbling around in a whirlpool. As the water swirls, the rubber band twists, stretches and shrinks. All of the shapes adopted by the band as it moves are topologically identical, but geometrically different.

These three objects have very different geometries, but are topologically the same – meaning that the objects can be bent or twisted from one shape into another. Mariel Vazquez, CC BY

Merely copying DNA creates a large number of shape-related problems, but textbook images rarely illustrate this topological conundrum.

During the cell cycle, each chromosome is replicated into two identical copies. In order for that to happen, the DNA helix must unwind, causing stress on the DNA. DNA responds to this stress by supercoiling, just like an old telephone cord. But the cell cannot tolerate too much supercoiling. If DNA contorts too much, the cell will suffer.

Sketch of a right handed DNA double helix (left). The opening of the helix, indicated by a triangle, causes the DNA to supercoil (right). A supercoil occurs when the axis of the helix, indicated in purple, coils upon itself. Mariel Vazquez, CC BY

A DNA molecule can be linear – as in the case of human chromosomes – or circular. Examples of circular DNA molecules include bacterial chromosomes and human mitochondrial DNA. If the DNA molecule is circular, then cellular processes such as replication may tie DNA into knots or links, like rings in a keychain. DNA knots and links can cause cells to malfunction or even die.

Stabilizing DNA

Consider the bacterium E. coli. Its genetic code is found in one single DNA chromosome. In E. coli and other bacteria, the DNA double helix closes into a circle, like a twisted rubber band.

Replication of the E. coli chromosome can happen in as short as 20 minutes in a test tube. But when a circular chromosome is replicated, the process yields two interlinked chromosomes. That is, the new chromosomes form two rings linked through each other. The new chromosomes must unlink before the cell divides into two cells. Otherwise they would either break on the way to their target cell, or one cell would inherit two interlinked copies of one chromosome and the other one would be missing the chromosome altogether.

The cell recruits enzymes to unlink the DNA. Enzymes called topoisomerases and recombinases act as scissors and glue for DNA. They can change the geometry and topology of DNA, thus maintaining a stable genome. In E. coli, topoisomerases work tirelessly during and after replication to maintain healthy levels of supercoiling and to safely unlink the chromosomes.

Replication of a circular DNA molecule. The arrows show the direction of replication (left). The new molecules interlink in this process (right). Mariel Vazquez, CC BY

When topoisomerases don’t work

When topoisomerases don’t work, the cell eventually dies. This makes them good targets for drug design. But cells have different types of topoisomerases and other enzymes such as recombinases that may be able to come to the rescue. For example, we showed that, in E. coli cells where the topoisomerases in charge of unlinking have been disabled, other enzymes called site-specific recombinases can untie replication links.

Both topoisomerases and site-specific recombinases bind double stranded DNA and can change its shape by introducing breaks. Type II topoisomerases introduce a break along the DNA molecule and transport another piece of DNA through the break before resealing it. Site-specific recombinases attach to two sites along the DNA, introduce one cut in each, then reconnect the ends.

My lab uses mathematics and computer simulations to understand how these enzymes unlink DNA molecules. While the local action is well understood on a biochemical level, how exactly enzymes simplify the topology of DNA is still a mystery.

In one of our studies, we focused on E. coli cells where the topoisomerases don’t work. We showed how to untie a replication link in the minimum number of steps.

The unlinking pathway of a 6-crossing replication link. This is the only pathway that simplifies the link at each step. Replication links can have any even number of crossings and similar unlinking pathways. Arsuaga-Vazquez lab, Author provided (no reuse)

In general, there can be many unlinking pathways. We use computer simulations to assign probabilities to each pathway. Our work indicates that, in the case of replication links, the simplest pathway is the one that enzymes most likely take.

Sophisticated mathematical methods can help explain how enzymes unlink DNA. Without mathematical modeling, researchers would be restricted to simplified models suggested by biological experiments.

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

Philosophers cannot agree on whether mathematical objects exist or are pure fictions

Credit: Brook VanDevelder

When I tell someone I am a mathematician, one of the most curious common reactions is: “I really liked math class because everything was either right or wrong. There is no ambiguity or doubt.” I always stutter in response. Math does not have a reputation for being everyone’s favorite subject, and I hesitate to temper anyone’s enthusiasm. But math is full of uncertainties—it just hides them well.

Of course, I understand the point. If your teacher asks whether 7 is a prime number, the answer is definitively “yes.” By definition, a prime number is a whole number greater than 1 that is only divisible by itself and 1, such as 2, 3, 5, 7, 11, 13, and so on. Any math teacher, anywhere in the world, anytime in the past several thousand years, will mark you correct for stating that 7 is prime and incorrect for stating that 7 is not prime. Few other disciplines can achieve such incredible consensus. But if you ask 100 mathematicians what explains the truth of a mathematical statement, you will get 100 different answers. The number 7 might really exist as an abstract object, with primality being a feature of that object. Or it could be part of an elaborate game that mathematicians devised. In other words, mathematicians agree to a remarkable degree on whether a statement is true or false, but they cannot agree on what exactly the statement is about.

One aspect of the controversy is the simple philosophical question: Was mathematics discovered by humans, or did we invent it? Perhaps 7 is an actual object, existing independently of us, and mathematicians are discovering facts about it. Or it might be a figment of our imaginations whose definition and properties are flexible. The act of doing mathematics actually encourages a kind of dual philosophical perspective, where math is treated as both invented and discovered.

This all seems to me a bit like improv theater. Mathematicians invent a setting with a handful of characters, or objects, as well as a few rules of interaction, and watch how the plot unfolds. The actors rapidly develop surprising personalities and relationships, entirely independent of the ones mathematicians intended. Regardless of who directs the play, however, the denouement is always the same. Even in a chaotic system, where the endings can vary wildly, the same initial conditions will always lead to the same end point. It is this inevitability that gives the discipline of math such notable cohesion. Hidden in the wings are difficult questions about the fundamental nature of mathematical objects and the acquisition of mathematical knowledge.

Invention

How do we know whether a mathematical statement is correct or not? In contrast to scientists, who usually try to infer the basic principles of nature from observations, mathematicians start with a collection of objects and rules and then rigorously demonstrate their consequences. The result of this deductive process is called a proof, which often builds from simpler facts to a more complex fact. At first glance, proofs seem to be key to the incredible consensus among mathematicians.

But proofs confer only conditional truth, with the truth of the conclusion depending on the truth of the assumptions. This is the problem with the common idea that consensus among mathematicians results from the proof-based structure of arguments. Proofs have core assumptions on which everything else hinges—and many of the philosophically fraught questions about mathematical truth and reality are actually about this starting point. Which raises the question: Where do these foundational objects and ideas come from?

Often the imperative is usefulness. We need numbers, for example, so that we can count (heads of cattle, say) and geometric objects such as rectangles to measure, for example, the areas of fields. Sometimes the reason is aesthetic—how interesting or appealing is the story that results? Altering the initial assumptions will sometimes unlock expansive structures and theories, while precluding others. For example, we could invent a new system of arithmetic where, by fiat, a negative number times a negative number is negative (easing the frustrated explanations of math teachers), but then many of the other, intuitive and desirable properties of the number line would disappear. Mathematicians judge foundational objects (such as negative numbers) and their properties (such as the result of multiplying them together) within the context of a larger, consistent mathematical landscape. Before proving a new theorem, therefore, a mathematician needs to watch the play unfold. Only then can the theorist know what to prove: the inevitable, unvarying conclusion. This gives the process of doing mathematics three stages: invention, discovery and proof.

The characters in the play are almost always constructed out of simpler objects. For example, a circle is defined as all points equidistant from a central point. So its definition relies on the definition of a point, which is a simpler type of object, and the distance between two points, which is a property of those simpler objects. Similarly, multiplication is repeated addition, and exponentiation is repeated multiplication of a number by itself. In consequence, the properties of exponentiation are inherited from the properties of multiplication. Conversely, we can learn about complicated mathematical objects by studying the simpler objects they are defined in terms of. This has led some mathematicians and philosophers to envision math as an inverted pyramid, with many complicated objects and ideas deduced from a narrow base of simple concepts.

In the late 19th and early 20th centuries a group of mathematicians and philosophers began to wonder what holds up this heavy pyramid of mathematics. They worried feverishly that math has no foundations—that nothing was grounding the truth of facts like 1 + 1 = 2. (An obsessive set of characters, several of them struggled with mental illness.) After 50 years of turmoil, the expansive project failed to produce a single, unifying answer that satisfied all the original goals, but it spawned various new branches of mathematics and philosophy.

Some mathematicians hoped to solve the foundational crisis by producing a relatively simple collection of axioms from which all mathematical truths can be derived. The 1930s work of mathematician Kurt Gödel, however, is often interpreted as demonstrating that such a reduction to axioms is impossible. First, Gödel showed that any reasonable candidate system of axioms will be incomplete: mathematical statements exist that the system can neither prove nor disprove. But the most devastating blow came in Gödel’s second theorem about the incompleteness of mathematics. Any foundational system of axioms should be consistent—meaning, free of statements that can be both proved and disproved. (Math would be much less satisfying if we could prove that 7 is prime and 7 is not prime.) Moreover, the system should be able to prove—to mathematically guarantee—its own consistency. Gödel’s second theorem states that this is impossible.

The quest to find the foundations of mathematics did lead to the incredible discovery of a system of basic axioms, known as Zermelo-Fraenkel set theory, from which one can derive most of the interesting and relevant mathematics. Based on sets, or collections of objects, these axioms are not the idealized foundation that some historical mathematicians and philosophers had hoped for, but they are remarkably simple and do undergird the bulk of mathematics.

Throughout the 20th century mathematicians debated whether Zermelo-Fraenkel set theory should be augmented with an additional rule, known as the axiom of choice: If you have infinitely many sets of objects, then you can form a new set by choosing one object from each set. Think of a row of buckets, each containing a collection of balls, and one empty bucket. From each bucket in the row, you can choose one ball and place it in the empty bucket. The axiom of choice would allow you to do this with an infinite row of buckets. Not only does it have intuitive appeal, it is necessary to prove several useful and desirable mathematical statements. But it also implies some strange things, such as the Banach-Tarski paradox, which states that you can break a solid ball into five pieces and reassemble those pieces into two new solid balls, each equal in size to the first. In other words, you can double the ball. Foundational assumptions are judged by the structures they produce, and the axiom of choice implies many important statements but also brings extra baggage. Without the axiom of choice, math seems to be missing crucial facts, though with it, math includes some strange and potentially undesirable statements.

The bulk of modern mathematics uses a standard set of definitions and conventions that have taken shape over time. For example, mathematicians used to regard 1 as a prime number but no longer do. They still argue, however, whether 0 should be considered a natural number (sometimes called the counting numbers, natural numbers are defined as 0,1,2,3… or 1,2,3…, depending on who you ask). Which characters, or inventions, become part of the mathematical canon usually depends on how intriguing the resulting play is—observing which can take years. In this sense, mathematical knowledge is cumulative. Old theories can be neglected, but they are rarely invalidated, as they often are in the natural sciences. Instead mathematicians simply choose to turn their attention to a new set of starting assumptions and explore the theory that unfolds.

Discovery

As noted earlier, mathematicians often define objects and axioms with a particular application in mind. Over and over again, however, these objects surprise them during the second stage of the mathematical process: discovery. Prime numbers, for example, are the building blocks of multiplication, the smallest multiplicative units. A number is prime if it cannot be written as the product of two smaller numbers, and all the nonprime (composite) numbers can be constructed by multiplying a unique set of primes together.

In 1742 mathematician Christian Goldbach hypothesized that every even number greater than 2 is the sum of two primes. If you pick any even number, the so-called Goldbach conjecture predicts that you can find two prime numbers that add up to that even number. If you pick 8, those two primes are 3 and 5; pick 42, and that is 13 + 29. The Goldbach conjecture is surprising because although primes were designed to be multiplied together, it suggests amazing, accidental relations between even numbers and the sums of primes.

An abundance of evidence supports Goldbach’s conjecture. In the 300 years since his original observation, computers have confirmed that it holds for all even numbers smaller than 4 × 10¹⁸. But this evidence is not enough for mathematicians to declare Goldbach’s conjecture correct. No matter how many even numbers a computer checks, there could be a counterexample—an even number that is not the sum of two primes—lurking around the corner.

Imagine that the computer is printing its results. Each time it finds two primes that add up to a specific even number, the computer prints that even number. By now it is a very long list of numbers, which you can present to a friend as a compelling reason to believe the Goldbach conjecture. But your clever friend is always able to think of an even number that is not on the list and asks how you know that the Goldbach conjecture is true for that number. It is impossible for all (infinitely many) even numbers to show up on the list. Only a mathematical proof—a logical argument from basic principles demonstrating that Goldbach’s conjecture is true for every even number—is enough to elevate the conjecture to a theorem or fact. To this day, no one has been able to provide such a proof.

The Goldbach conjecture illustrates a crucial distinction between the discovery stage of mathematics and the proof stage. During the discovery phase, one seeks overwhelming evidence of a mathematical fact—and in empirical science, that is often the end goal. But mathematical facts require a proof.

Patterns and evidence help mathematicians sort through mathematical findings and decide what to prove, but they can also be deceptive. For example, let us build a sequence of numbers: 121, 1211, 12111, 121111, 1211111, and so on. And let us make a conjecture: all the numbers in the sequence are not prime. It is easy to gather evidence for this conjecture. You can see that 121 is not prime, because 121 = 11 × 11. Similarly, 1211, 12111 and 121111 are all not prime. The pattern holds for a while—long enough that you would likely get bored checking—but then it suddenly fails. The 136th element in this sequence (that is, the number 12111…111, where 136 “1”s follow the “2”) is prime.

It is tempting to think that modern computers can help with this problem by allowing you to test the conjecture on more numbers in the sequence. But there are examples of mathematical patterns that hold true for the first 10⁴² elements of a sequence and then fail. Even with all the computational power in the world, you would never be able to test that many numbers.

Even so, the discovery stage of the mathematical process is extremely important. It reveals hidden connections such as the Goldbach conjecture. Often two entirely distinct branches of math are intensively studied in isolation before a profound relation between them is uncovered. A relatively simple example is Euler’s identity, e^iπ + 1 = 0, which connects the geometric constant π with the number i, defined algebraically as the square root of –1, via the number e, the base of natural logarithms. These surprising discoveries are part of the beauty and curiosity of math. They seem to point at a deep underlying structure that mathematicians are only beginning to understand.

In this sense, math feels both invented and discovered. The objects of study are precisely defined, but they take on a life of their own, revealing unexpected complexity. The process of mathematics therefore seems to require that mathematical objects be simultaneously viewed as real and invented—as objects with concrete, discoverable properties and as easily manipulable inventions of mind. As philosopher Penelope Maddy writes, however, the duality makes no difference to how mathematicians work, “as long as double-think is acceptable.”

Real or unreal?

Mathematical realism is the philosophical position that seems to hold during the discovery stage: the objects of mathematical study—from circles and prime numbers to matrices and manifolds—are real and exist independently of human minds. Like an astronomer exploring a far-off planet or a paleontologist studying dinosaurs, mathematicians are gathering insights into real entities. To prove that Goldbach’s conjecture is true, for example, is to show that the even numbers and the prime numbers are related in a particular way through addition, just like a paleontologist might show that one type of dinosaur descended from another by showing that their anatomical structures are related.

Realism in its various manifestations, such as Platonism (inspired by the Greek philosopher’s theory of Platonic forms), makes easy sense of mathematics’ universalism and usefulness. A mathematical object has a property, such as 7 being a prime number, in the same way that a dinosaur might have had the property of being able to fly. And a mathematical theorem, such as the fact that the sum of two even numbers is even, is true because even numbers really exist and stand in a particular relation to each other. This explains why people across temporal, geographical and cultural differences generally agree about mathematical facts—they are all referencing the same fixed objects.

But there are some important objections to realism. If mathematical objects really exist, their properties are certainly very peculiar. For one, they are causally inert, meaning they cannot be the cause of anything, so you cannot literally interact with them. This is a problem because we seem to gain knowledge of an object through its impact. Dinosaurs decomposed into bones that paleontologists can see and touch, and a planet can pass in front of a star, blocking its light from our view. But a circle is an abstract object, independent of space and time. The fact that π is the ratio of the circumference to the diameter of a circle is not about a soda can or a doughnut; it refers to an abstract mathematical circle, where distances are exact and the points on the circle are infinitesimally small. Such a perfect circle is causally inert and seemingly inaccessible. So how can we learn facts about it without some type of special sixth sense?

That is the difficulty with realism—it fails to explain how we know facts about abstract mathematical objects. All of which might cause a mathematician to recoil from his or her typically realist stance and latch onto the first step of the mathematical process: invention. By framing mathematics as a purely formal mental exercise or a complete fiction, antirealism easily skirts problems of epistemology.

Formalism, a type of antirealism, is a philosophical position that asserts that mathematics is like a game, and mathematicians are just playing out the rules of the game. Stating that 7 is a prime number is like stating that a knight is the only chess piece that can move in an L shape. Another philosophical position, fictionalism, claims that mathematical objects are fictions. Stating that 7 is a prime number is then like stating that unicorns are white. Mathematics makes sense within its fictional universe but has no real meaning outside of it.

There is an inevitable trade-off. If math is simply made up, how can it be such a necessary part of science? From quantum mechanics to models of ecology, mathematics is an expansive and precise scientific tool. Scientists do not expect particles to move according to chess rules or the crack in a dinner plate to mimic Hansel and Gretel’s path—the burden of scientific description is placed exclusively on mathematics, which distinguishes it from other games or fictions.

In the end, these questions do not affect the practice of mathematics. Mathematicians are free to choose their own interpretations of their profession. In The Mathematical Experience, Philip Davis and Reuben Hersh famously wrote that “the typical working mathematician is a Platonist on weekdays and a formalist on Sundays.” By funneling all disagreements through a precise process—which embraces both invention and discovery—mathematicians are incredibly effective at producing disciplinary consensus.

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

Credit of the article given to Kelsey Houston-Edwards |

“When will I ever use this?” It’s a question math and science teachers hear all the time from their high school students.

Teaching science, technology, engineering and math (STEM) skills is more important than ever, but it’s often difficult for students to understand the practical applications of such fundamental learning and how it will help them down the road.

Classroom activities should be relevant, meaningful and connected to students’ prior knowledge and experiences. Learning must be based on lived experiences within both formal and informal educational settings.

Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling. Educators must lead by example and provide students with opportunities to explore interdisciplinary approaches to learning.

Creative thinking

The new British Columbia curriculum embraces these principles of learning. In the same spirit, I’m part of a new and unique Bachelor of Education program at Thompson Rivers University where teacher candidates are learning to teach STEM by actively engaging students. The program promotes cross-curricular and interdisciplinary approaches to learning and is tied to the provincial curriculum core competencies of communication, critical and creative thinking.

So how do you teach a subject like math differently in a way that can help students learn through creative thinking and experience, rather than rote memorization?

Let’s take, for example, Pi.

I often ask my teacher candidates: What is π? Many respond “3.14” and, if probed further, explain the meaning by merely stating an equation like A=πr² (where A is the area of a circle and r is the radius of a circle). Or they may tell me C=2πr (where C is the circumference of a circle).

A door handle in the shape of Pi at the National Museum of Mathematics in New York, (AP Photo/Seth Wenig)

Teaching through discovery

I encourage these teacher candidates to think differently and to help students discover mathematical concepts for themselves. What better way to teach students that π is the ratio of a circle’s circumference to its diameter than to have them trace any circle and then measure it with a piece of string?

They will soon learn that regardless of the size of the circle, the ratio of circumference to diameter will always be 22/7, an approximation of π.

Innovative educators can integrate history, geography, math and science lessons by teaching a thematic unit on ancient civilizations.

For example, the Egyptians succeeded in building great pyramids with incredible precision and accuracy. These magnificent architectural accomplishments have stood the test of time, remaining largely intact after centuries — a tribute to their construction.

The ancient Egyptians understood the significance of mathematics through the very beauty and symmetry of nature. They used geometry to solve everyday problems.

Tearing down silos

Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling — exactly the opposite of the “back to basics” approach suggested by populist politicians like new Ontario Premier Doug Ford.

Students benefit from learning experiences that are meaningful, relevant and well-connected to their own experiences. For that to happen, the people teaching those students must be prepared to take on new attitudes of reflectiveness and inquisitiveness.

What is necessary is to follow in the footsteps of the great thinkers like Galileo and Newton, who questioned our perceptions of reality and sought answers from tactile experiences rather than textbooks or teachers.

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

Credit: The mathematician Ken Ono in his office at Emory University in Atlanta. Raymond McCrea Jones for Quanta Magazine

The mathematician Ken Ono believes that the story of Srinivasa Ramanujan—mathematical savant and two-time college dropout—holds valuable lessons for how we find and reward hidden genius

For the first 27 years of his life, the mathematician Ken Ono was a screw-up, a disappointment and a failure. At least, that’s how he saw himself. The youngest son of first-generation Japanese immigrants to the United States, Ono grew up under relentless pressure to achieve academically. His parents set an unusually high bar. Ono’s father, an eminent mathematician who accepted an invitation from J. Robert Oppenheimer to join the Institute for Advanced Study in Princeton, N.J., expected his son to follow in his footsteps. Ono’s mother, meanwhile, was a quintessential “tiger parent,” discouraging any interests unrelated to the steady accumulation of scholarly credentials.

This intellectual crucible produced the desired results—Ono studied mathematics and launched a promising academic career—but at great emotional cost. As a teenager, Ono became so desperate to escape his parents’ expectations that he dropped out of high school. He later earned admission to the University of Chicago but had an apathetic attitude toward his studies, preferring to party with his fraternity brothers. He eventually discovered a genuine enthusiasm for mathematics, became a professor, and started a family, but fear of failure still weighed so heavily on Ono that he attempted suicide while attending an academic conference. Only after he joined the Institute for Advanced Study himself did Ono begin to make peace with his upbringing.

Through it all, Ono found inspiration in the story of Srinivasa Ramanujan, a mathematical prodigy born into poverty in late-19th-century colonial India. Ramanujan received very little formal schooling, yet he still produced thousands of independent mathematical results, some of which—like the Ramanujan theta function, which has found applications in string theory—are still intensely studied. But despite his genius, Ramanujan’s achievements didn’t come easily. He struggled to gain acceptance from Western mathematicians and dropped out of university twice before dying of illness at the age of 32.

While Ono, now 48, doesn’t compare himself to Ramanujan in terms of ability, he has built his career in part from Ramanujan’s insights. In 2014, Ono and his collaborators Michael Griffin and Ole Warnaar published a breakthrough result in algebraic number theory that generalized one of Ramanujan’s own results. Ono’s work, which is based on a pair of equations called the Rogers-Ramanujan identities, can be used to easily produce algebraic numbers (such as phi, better known as the “golden ratio”).

More recently, Ono served as an associate producer and mathematical consultant for The Man Who Knew Infinity, a recently released feature film about Ramanujan’s life. And his new memoir, My Search for Ramanujan: How I Learned to Count (co-authored with Amir D. Aczel), draws connections between Ramanujan’s life and Ono’s own circuitous path to mathematical and emotional fulfillment. “I wrote this book to show off my weaknesses, to show off my struggles,” Ono said. “People who are successful in their careers were not always successful from day one.”

Like Ramanujan, who benefited from years of mentorship by the British mathematician G.H. Hardy, Ono credits his own success to serendipitous encounters with teachers who helped his talents flourish. He now spends a great deal of time mentoring his own students at Emory University. Ono has also helped launch the Spirit of Ramanujan Math Talent Initiative, a venture that “strives to find undiscovered mathematicians around the world and match them with advancement opportunities in the field.”

Quanta Magazine spoke with Ono about finding his way as a mathematician and a mentor, and about Ramanujan’s inspiring brand of creativity. An edited and condensed version of the interview follows.

QUANTA MAGAZINE: What was so special about Ramanujan’s approach to doing mathematics?
KEN ONO: First, he was really a poet, not a problem solver. Most professional mathematicians, whether they’re in academia or industry, have problems that they’re aiming to solve. Somebody wants to prove the Riemann hypothesis, and sets out to do it. That’s how we think science should proceed, and in fact almost every scientist should work that way, because in reality science develops through the work of thousands of individuals slowly adding to a body of knowledge. But what you find in Ramanujan’s original notebooks is just formula after formula, and it’s not apparent where he’s going with his ideas. He was someone who could set down the paths of beginnings of important theories without knowing for sure why we would care about them as mathematicians of the future.

He’s credited with compiling thousands of identities—that is, equations that are true regardless of what values the variables take. Why is that important?
It is true that the vast majority of the contents of his notebooks are what you would call identities. Identities that relate continued fractions to other functions, expressions for integrals, expressions for hypergeometric functions, and expressions for objects that we call q-series.

But that would be a literal interpretation of his notebooks. In my opinion, that would be like taking a cookbook by Julia Child, reading the recipes and saying that it’s about assembling chemical compounds into something more complicated. Strictly speaking that would be true, but you would be missing out on what makes delicious recipes so important to us.

Ramanujan’s work came through flights of fancy. If he had been asked to explain why he did his work, he would probably say that he recorded formulas that he found beautiful, and they were beautiful because they revealed some unexpected phenomenon. And they’re important to us today because these special phenomena that Ramanujan identified, over and over again, have ended up becoming prototypes for big mathematical theories in the 20th and 21st centuries.

Here’s an example. In one of his published manuscripts, Ramanujan recorded a lot of elementary-looking results called congruences. In the 1960s, Jean-Pierre Serre, himself a Fields medalist, revisited some of these results, and in them he found glimpses of a theory that he named the theory of Galois representations. This theory of Galois representations is the language that Andrew Wiles used in the 1990s to prove Fermat’s last theorem.

There’s no “theory of Ramanujan,” but he anticipated mathematical structures that would be important to all of these other more contemporary works. He lived 80 years before his time.

How do you approach your own mathematical work—more as an artist, like Ramanujan, or with the aim of solving specific problems, like a scientist?
I’m definitely much more of a scientist. Science proceeds at a much faster rate than when I started in my career in the early 1990s, and I have to stop often to recognize the beauty in it and try not to be so caught up in the more professionalized side of how one does science. The grant getting, the publications, and all of that—I have to admit, I don’t like it.

What compelled you to juxtapose your own story with his?
Well, I almost didn’t write it. There are a lot of very personal things that I’ve never told anyone before. It wasn’t until I started writing this book that I was mature enough as a parent myself to try to understand the circumstances that led my parents to raise us the way they did. And as a professor at Emory, I see all these kids under tremendous pressure—rarely pressure that they understand the origin of. So many of these super-talented kids are just going through the motions, and aren’t passionate about their studies at all, and that’s terrible. I was like that too. I’d given up on ever trying to live up to my parents’ expectations, but somehow because I’ve had Ramanujan as a guardian angel, things have worked out well for me. It makes you a better teacher when you just tell people how hard it was for you.

This book and your story don’t fit the typical “great man of science” narrative.
I think you’ll find that’s much more common than people are willing to admit. I didn’t discover my passion for mathematics until my early 20s—that’s when [my doctoral adviser Basil] Gordon turned me on to mathematics at a time when I didn’t think anything was beautiful. I thought it was all about test scores, grades and trying to do as well as possible without putting in effort. Colleges are full of kids who think that way. How do you beat the system, right? I wasn’t beating the system. The system was beating me, and Gordon turned me around. When I’ve told people the story I’ve discovered that I’m really not alone.

That’s what I see in Ramanujan. He was a two-time college dropout whom my father looked up to as a hero—which made no sense to me when I was 16, because I was told I had to be a child prodigy. I was supposed to do my geometry problems during the summer sitting next to my dad while he did his research. I wasn’t even really allowed to go out and play, and then to just have my father tell me about Ramanujan out of the blue—it was beyond earth-shattering.

If you’d been interested in something conventionally “artistic,” like music, this kind of painful journey toward success would not seem so surprising. Why does it surprise us to hear about a mathematician having the same struggles?
For whatever reason, we live in a culture where we think that the abilities of our best scientists and our best mathematicians are somehow just God-given. That either you have this gift or you don’t, and it’s not related to help, to hard work, to luck. I think that’s part of the reason why, when we try to talk about mathematics to the public, so many people just immediately respond by saying, “Well, I was never very good at math. So I’m not really supposed to understand it or identify with it.” I might have had some mathematical talent passed through my father genetically, but that was by no means enough. You have to be passionate about a subject.

At the same time, I want it to be known that it’s totally OK to fail. In fact, you learn from your mistakes. We learn early on if that you want to be good at playing the violin, you’ve got to practice. If you want to be good at sports, you practice. But for some crazy reason, our culture assumes that if you’re good at math, you were just born with it, and that’s it. But you can be so good at math in so many different ways. I failed my [graduate-school] algebra qualifications! That doesn’t mean I can’t end up being a successful mathematician. But when I tell people I failed at this, nobody believes me.

But Ramanujan seems to be just that: a unique genius who appeared out of nowhere. What does that have to do with a regular person’s life?
You think no one can be like Ramanujan? Well, I disagree. I think we can search the world looking for a mathematical talent, just not by the usual metrics. I want teachers and parents to recognize that when you do see unusual talent, instead of demanding that these people have certain test scores, let’s find a way to help nurture them. Because I think humanity needs it. I think these are the lessons we learn from Ramanujan.

You’re leading the Spirit of Ramanujan Math Talent Initiative. What is this spirit? How do we recognize it?
First of all, it’s the idea that talent is often found in the most unforgiving and unpromising of circumstances. It’s the responsibility of mentors, teachers and parents first to recognize that talent, which is not always easy to do, and then to offer opportunities that nurture that talent.

There are no age limits, and I don’t want this to be a competition where you’re recognized for high test scores. I have no trouble finding people who can get an 800 on the math SAT. That’s easy. Those people don’t need to be identified. They’ve already self-identified. I’m searching for creativity.

That said, the spirit of Ramanujan does not require finding the next Ramanujan. We would be super lucky to do that, but if we make opportunities for 30 talented people around the world who are presently working in an intellectual desert, or are subjected to inelastic educational systems where they’re not allowed to flourish—or if we can provide an opportunity for someone to work with a scientist who could be their G.H. Hardy—then this initiative will be successful.

Do you wish you had been nurtured differently? Do you resent your parents?
I love my parents. We discussed the draft of the book for months last summer. They were very upset with me at first, because it was difficult for them to get past the first 30 pages, but now they embrace it. One reviewer actually saw the book as a love letter to my parents and to my mentors, because they taught me skills I needed.

If you had never joined the Institute for Advanced Study, would you still be struggling to reconcile your own path with your parents’ expectations?
I think I would still be searching for that recognition today if I hadn’t gotten there.

Both my parents will tell you that you only get to live once, so you might as well be the very, very best that you can be at whatever you choose. Which I don’t necessarily agree with, because if everyone lived that way, there would be nothing but a whole bunch of unhappy people in the world. But that’s how they brought us up. They taught me to be competitive. They taught me not to falsely believe I had done well when I hadn’t. They taught me standards, and those are important. But it’s true that if I hadn’t had the opportunity to work at the Institute, I’m not sure I would have been able to write this book. I might still be struggling with these things.

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

Credit of the article given to John Pavlus & Quanta Magazine

Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis.

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002.

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

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Credit of the article given to William Ross