Author: IMC

Primes still have the power to surprise. Chris-LiveLoveClick/shutterstock.com

On March 20, American-Canadian mathematician Robert Langlands received the Abel Prize, celebrating lifetime achievement in mathematics. Langlands’ research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers.

When the King of Norway presents the award to Langlands in May, he will honor the latest in a 2,300-year effort to understand prime numbers, arguably the biggest and oldest data set in mathematics.

As a mathematician devoted to this “Langlands program,” I’m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?

How to find primes

To study primes, mathematicians strain whole numbers through one virtual mesh after another until only primes remain. This sieving process produced tables of millions of primes in the 1800s. It allows today’s computers to find billions of primes in less than a second. But the core idea of the sieve has not changed in over 2,000 years.

“A prime number is that which is measured by the unit alone,” mathematician Euclid wrote in 300 B.C. This means that prime numbers can’t be evenly divided by any smaller number except 1. By convention, mathematicians don’t count 1 itself as a prime number.

Euclid proved the infinitude of primes – they go on forever – but history suggests it was Eratosthenes who gave us the sieve to quickly list the primes.

 

Here’s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7 – the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.

With eight filtering steps, one can isolate the primes up to 400. With 168 filtering steps, one can isolate the primes up to 1 million. That’s the power of the sieve of Eratosthenes.

Tables and tables

An early figure in tabulating primes is John Pell, an English mathematician who dedicated himself to creating tables of useful numbers. He was motivated to solve ancient arithmetic problems of Diophantos, but also by a personal quest to organize mathematical truths. Thanks to his efforts, the primes up to 100,000 were widely circulated by the early 1700s. By 1800, independent projects had tabulated the primes up to 1 million.

To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician Jakob Kulik had embarked on an ambitious project to find all the primes up to 100 million.

This “big data” of the 1800s might have only served as reference table, if Carl Friedrich Gauss hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “chiliad,” or group of 1000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on.

Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an “inverse-log” law. Gauss’s law doesn’t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The correct count is 75 primes, about a 4 percent error.

A century after Gauss’ first explorations, his law was proved in the “prime number theorem.” The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, a million-dollar prize problem today, also describes how accurate Gauss’ estimate really is.

The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.

Modern prime mysteries

Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.

Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.

A few years ago, Stanford number theorists Robert Lemke Oliver and Kannan Soundararajan were caught off guard by quirks in the final digits of primes. An experiment looked at the last digit of a prime, as well as the last digit of the very next prime. For example, the next prime after 23 is 29: One sees a 3 and then a 9 in their last digits. Does one see 3 then 9 more often than 3 then 7, among the last digits of primes?

Frequency of last-digit pairs, among successive prime numbers up to 100 million. Matching colors correspond to matching gaps. M.H. Weissman, CC BY

Number theorists expected some variation, but what they found far exceeded expectations. Primes are separated by different gaps; for example, 23 is six numbers away from 29. But 3-then-9 primes like 23 and 29 are far more common than 7-then-3 primes, even though both come from a gap of six.

Mathematicians soon found a plausible explanation. But, when it comes to the study of successive primes, mathematicians are (mostly) limited to data analysis and persuasion. Proofs – mathematicians’ gold standard for explaining why things are true – seem decades away.

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

Credit: The earth is massive, but you don’t need a massive ruler to measure its size. All you need are a few household items–and little bit of geometry! George Retseck

A geometry science project from Science Buddies

Introduction
If you wanted to measure the circumference of Earth, how long would your tape measure have to be? Would you need to walk the whole way around the world to find the answer? Do you think you can do it with just a meterstick in one location? Try this project to find out!

Before you begin, however, it is important to note this project will only work within about two weeks of either the spring or fall equinoxes (usually around March 20 and September 23, respectively).

Background
What is Earth’s circumference? In the age of modern technology this may seem like an easy question for scientists to answer with tools such as satellites and GPS—and it would be even easier for you to look up the answer online. It might seem like it would be impossible for you to measure the circumference of our planet using only a meterstick. The Greek mathematician Eratosthenes, however, was able to estimate Earth’s circumference more than 2,000 years ago, without the aid of any modern technology. How? He used a little knowledge about geometry!

At the time Eratosthenes was in the city of Alexandria in Egypt. He read that in a city named Syene south of Alexandria, on a particular day of the year at noon, the sun’s reflection was visible at the bottom of a deep well. This meant the sun had to be directly overhead. (Another way to think about this is that perfectly vertical objects would cast no shadow.) On that same day in Alexandria a vertical object did cast a shadow. Using geometry, he calculated the circumference of Earth based on a few things that he knew (and one he didn’t):

• He knew there are 360 degrees in a circle.
• He could measure the angle of the shadow cast by a tall object in Alexandria.
• He knew the overland distance between Alexandria and Syene. (The two cities were close enough that the distance could be measured on foot.)
• The only unknown in the equation is the circumference of Earth!

The resulting equation was:

Angle of shadow in Alexandria / 360 degrees = Distance between Alexandria and Syene / Circumference of Earth

In this project you will do this calculation yourself by measuring the angle formed by a meterstick’s shadow at your location. You will need to do the test near the fall or spring equinoxes, when the sun is directly overhead at Earth’s equator. Then you can look up the distance between your city and the equator and use the same equation Eratosthenes used to calculate Earth’s circumference. How close do you think your result will be to the “real” value?

There is a geometric rule about the angles formed by a line that intersects two parallel lines. Eratosthenes assumed the sun was far enough away from our planet that its rays were effectively parallel when they arrived at Earth. This told him the angle of the shadow he measured in Alexandria was equal to the angle between Alexandria and Syene, measured at Earth’s center. If this sounds confusing, don’t worry! It is much easier to visualize with a picture. See the references in the “More to explore” section for some helpful diagrams and a more detailed explanation of the geometry involved.

Materials

• Sunny day on or near the spring or fall equinoxes (about March 20 or September 23, respectively)
• Flat, level ground that will be in direct sunlight around noon
• Meterstick
• Volunteer to help hold the meterstick while you take measurements (Or, if you are doing the test alone, you can use a bucket of sand or dirt to insert one end of the meter stick to hold it upright.)
• Stick or rock to mark the location of the shadow
• Calculator
• Protractor
• Long piece of string
• Optional: plumb bob (you can make one by tying a small weight to the end of a string) or post level to make sure the meter stick is vertical

Preparation

• Look at your local weather forecast a few days in advance and pick a day where it looks like it will be mostly sunny around noon. (You have a window of several weeks to do this project, so don’t get discouraged if it turns out to be cloudy! You can try again.)
• Look up the sunrise and sunset times for that day in your local newspaper or on a calendar, weather or astronomy Web site. You will need to calculate “solar noon,” the time exactly halfway between sunrise and sunset, which is when the sun will be directly overhead. This will probably not be exactly 12 o’clock noon.
• Go outside and set up for your materials about 10 minutes before solar noon so you have everything ready.

Procedure

• Set up your meter stick vertically, outside in a sunny spot just before solar noon.
• If you have a volunteer to help, have them hold the meterstick. Otherwise, bury one end of the meterstick in a bucket of sand or dirt so it stays upright.
• If you have a post level or plumb bob, use it to make sure the meterstick is perfectly vertical. Otherwise, do your best to eyeball it.
• At solar noon, mark the end of the meterstick’s shadow on the ground with a stick or a rock.
• Draw an imaginary line between the top of the meterstick and the tip of its shadow. Your goal is to measure the angle between this line and the meterstick. Have your volunteer stretch a piece of string between the top of the meterstick and the end of its shadow.
• Use a protractor to measure the angle between the string and the meterstick in degrees. Write this angle down.
• Look up the distance between your city and the equator.
• Calculate the circumference of the Earth using this equation:

Circumference = 360 x distance between your city and the equator / angle of shadow that you measured

• What value do you get? How close is your answer to the true circumference of Earth (see “Observations and results” section)?
• Extra: Try repeating your test on different days before, on and after the equinox; or at different times before, at and after solar noon. How much does the accuracy of your answer change?
• Extra: Ask a friend or family member in a different city to try the test on the same day and compare your results. Do you get the same answer?

Observations and results
In 200 B.C. Eratosthenes estimated Earth’s circumference at about 46,250 kilometers (28,735 miles). Today we know our planet’s circumference is roughly 40,000 kilometers (24,850 miles). Not bad for a more than 2,000-year-old estimate made with no modern technology! Depending on the error in your measurements—such as the exact day and time you did the test, how accurately you were able to measure the angle or length of the shadow and how accurately you measured the distance between your city and the equator—you should be able to calculate a value fairly close to 40,000 kilometers (within a few hundred or maybe a few thousand). All without leaving your own backyard!

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

Credit of the article given to Science Buddies & Ben Finio

They come from all over the world to see, touch and read the originals of tens of thousands of letters, to study boxes of drafts and revisions of his ideas and mathematical equations, to understand his complex personal relationships and to explore the commitment to peace and opposition to nuclear weapons that landed him in jail more than once.

Visitors love to look at the wiry thinker’s easy chair and imagine what he must have been pondering as he sat there.

These, together with a Nobel Prize for Literature, a desk, a tweed suit and a trademark pipe, were the belongings of Bertrand Russell, modern philosopher, social critic, mathematician and anti-war crusader who died in 1970 just a couple of years short of his 100th birthday on May 18.

Canada’s McMaster University obtained Russell’s vast archives 50 years ago this year, and the parade of scholars who continue to use them affirms that his ideas are at least as relevant as ever — perhaps more so today, when the threat to world peace seems so grave.

How can the ideas of a man who started teaching at the London School of Economics in 1896 — and who corresponded with Jean-Paul Sartre, Ho Chi Minh, T.S. Eliot and so many others, and lived long enough to protest both the First World War and the Vietnam War — still be so meaningful?

Russell remains pertinent

A few years ago, I was home with my teenaged son, Michael. He was supposed to be working on an essay for a high school philosophy course, but I could hear the distinct sound of laughter coming from his room.

I asked him what was so funny, and was happily surprised by his answer: He was reading Bertrand Russell’s History of Western Philosophy and found Russell’s wry commentary very funny.

If I had ever needed affirmation that Russell remains pertinent, I certainly had it, though few would doubt the value of a life’s work that generated more than 4,000 publications on such disparate topics as truth, geometry, morality, politics and the future of humanity.

Not even imprisonment could stem Russell’s spirit or the flow of his ideas. He managed to write his Introduction to Mathematical Philosophy while incarcerated for pacifism during the First World War, and even sent the warden a copy to thank him for the opportunity.

Russell is seen in this photo taken in 1939 at UCLA, where he was working as a philosophy professor at the time. (Creative Commons)

Russell’s papers and ephemera are gathered in the McMaster library’s William Ready Division of Archives and Research Collections, where his desk and chair and more than 3,000 books from his personal library are shelved in the same order in which he kept them.

The fact that these archives ever reached McMaster is testament to the vision and audacity of the chief librarian of the time, William Ready, who, with the firm backing of McMaster’s president Harry Thode, brought the load of trunks, boxes and cabinets across the Atlantic from Wales after outbidding serious international competitors.

Opposed the Vietnam War

Ready’s chances at securing the archives had been boosted by Russell’s opposition to the U.S. involvement in the Vietnam War, which dampened the enthusiasm of potential American buyers.

Ready was a literary adventurer and his coup with the Russell archives came as he also managed to land Anthony Burgess’s typescript of A Clockwork Orange and extensive collections of rare books that today comprise an internationally renowned collection.

Thode, a world-renowned nuclear scientist, had recently secured a nuclear research reactor for McMaster’s campus and, as university president, was eager to balance that scientific triumph by securing a research asset of similar importance for the humanities.

Half a century later, the reactor continues to facilitate scientific discovery and to provide valuable medical isotopes while, across campus, the Russell archives remain a magnet for scholars the world over.

Russell student still tends to collection

Amazingly, the Russell archives and its supporting collections continue to be under the able and conscientious care of a man who had worked for Russell himself.

Ken Blackwell, a Canadian, was a young man when he went to work for Russell — a philosopher in his own right and a devoted student of Russell who landed a job organizing the Russell collection for eventual sale.

When Ready imported the collection, Blackwell came with it, and he stayed. Today, he likes to joke about emerging from one of the packing boxes. He spent the rest of his career on those papers and, in his retirement, continues to do so as a volunteer.

In addition to their own research value, Russell’s archives have helped McMaster draw other collections, including that of McMaster’s own peace advocate and cultural critic Henry Giroux, who carries Russell’s torch into new battles against ignorance, violence and unchecked corporate and political power.

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

Credit: Saul Gravy Getty Images

Proof rests on a surprising link between infinity size and the complexity of mathematical theories

In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.

The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University.

In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians.

“It was certainly my opinion, and the general opinion, that p should be less than t,” Shelah said.

Malliaris and Shelah published their proof last year in the Journal of the American Mathematical Society and were honored this past Julywith one of the top prizes in the field of set theory. But their work has ramifications far beyond the specific question of how those two infinities are related. It opens an unexpected link between the sizes of infinite sets and a parallel effort to map the complexity of mathematical theories.

Many Infinities

The notion of infinity is mind-bending. But the idea that there can be different sizes of infinity? That’s perhaps the most counterintuitive mathematical discovery ever made. It emerges, however, from a matching game even kids could understand.

Suppose you have two groups of objects, or two “sets,” as mathematicians would call them: a set of cars and a set of drivers. If there is exactly one driver for each car, with no empty cars and no drivers left behind, then you know that the number of cars equals the number of drivers (even if you don’t know what that number is).

In the late 19th century, the German mathematician Georg Cantor captured the spirit of this matching strategy in the formal language of mathematics. He proved that two sets have the same size, or “cardinality,” when they can be put into one-to-one correspondence with each other—when there is exactly one driver for every car. Perhaps more surprisingly, he showed that this approach works for infinitely large sets as well.

Consider the natural numbers: 1, 2, 3 and so on. The set of natural numbers is infinite. But what about the set of just the even numbers, or just the prime numbers? Each of these sets would at first seem to be a smaller subset of the natural numbers. And indeed, over any finite stretch of the number line, there are about half as many even numbers as natural numbers, and still fewer primes.

Yet infinite sets behave differently. Cantor showed that there’s a one-to-one correspondence between the elements of each of these infinite sets.

1	2	3	4	5	…	(natural numbers)
2	4	6	8	10	…	(evens)
2	3	5	7	11	…	(primes)

Because of this, Cantor concluded that all three sets are the same size. Mathematicians call sets of this size “countable,” because you can assign one counting number to each element in each set.

After he established that the sizes of infinite sets can be compared by putting them into one-to-one correspondence with each other, Cantor made an even bigger leap: He proved that some infinite sets are even larger than the set of natural numbers.

Consider the real numbers, which are all the points on the number line. The real numbers are sometimes referred to as the “continuum,” reflecting their continuous nature: There’s no space between one real number and the next. Cantor was able to show that the real numbers can’t be put into a one-to-one correspondence with the natural numbers: Even after you create an infinite list pairing natural numbers with real numbers, it’s always possible to come up with another real number that’s not on your list. Because of this, he concluded that the set of real numbers is larger than the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite.

What Cantor couldn’t figure out was whether there exists an intermediate size of infinity—something between the size of the countable natural numbers and the uncountable real numbers. He guessed not, a conjecture now known as the continuum hypothesis.

In 1900, the German mathematician David Hilbert made a list of 23 of the most important problems in mathematics. He put the continuum hypothesis at the top. “It seemed like an obviously urgent question to answer,” Malliaris said.

In the century since, the question has proved itself to be almost uniquely resistant to mathematicians’ best efforts. Do in-between infinities exist? We may never know.

Forced Out

Throughout the first half of the 20th century, mathematicians tried to resolve the continuum hypothesis by studying various infinite sets that appeared in many areas of mathematics. They hoped that by comparing these infinities, they might start to understand the possibly non-empty space between the size of the natural numbers and the size of the real numbers.

Many of the comparisons proved to be hard to draw. In the 1960s, the mathematician Paul Cohen explained why. Cohen developed a method called “forcing” that demonstrated that the continuum hypothesis is independent of the axioms of mathematics—that is, it couldn’t be proved within the framework of set theory. (Cohen’s work complemented work by Kurt Gödel in 1940 that showed that the continuum hypothesis couldn’t be disproved within the usual axioms of mathematics.)

Cohen’s work won him the Fields Medal (one of math’s highest honors) in 1966. Mathematicians subsequently used forcing to resolve many of the comparisons between infinities that had been posed over the previous half-century, showing that these too could not be answered within the framework of set theory. (Specifically, Zermelo-Fraenkel set theory plus the axiom of choice.)

Some problems remained, though, including a question from the 1940s about whether p is equal to t. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of the natural numbers in precise (and seemingly unique) ways.

The details of the two sizes don’t much matter. What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity—something between the size of the natural numbers and the size of the real numbers. The continuum hypothesis would be false.

Mathematicians tended to assume that the relationship between p and t couldn’t be proved within the framework of set theory, but they couldn’t establish the independence of the problem either. The relationship between p and t remained in this undetermined state for decades. When Malliaris and Shelah found a way to solve it, it was only because they were looking for something else.

An Order of Complexity

Around the same time that Paul Cohen was forcing the continuum hypothesis beyond the reach of mathematics, a very different line of work was getting under way in the field of model theory.

For a model theorist, a “theory” is the set of axioms, or rules, that define an area of mathematics. You can think of model theory as a way to classify mathematical theories—an exploration of the source code of mathematics. “I think the reason people are interested in classifying theories is they want to understand what is really causing certain things to happen in very different areas of mathematics,” said H. Jerome Keisler, emeritus professor of mathematics at the University of Wisconsin, Madison.

In 1967, Keisler introduced what’s now called Keisler’s order, which seeks to classify mathematical theories on the basis of their complexity. He proposed a technique for measuring complexity and managed to prove that mathematical theories can be sorted into at least two classes: those that are minimally complex and those that are maximally complex. “It was a small starting point, but my feeling at that point was there would be infinitely many classes,” Keisler said.

It isn’t always obvious what it means for a theory to be complex. Much work in the field is motivated in part by a desire to understand that question. Keisler describes complexity as the range of things that can happen in a theory—and theories where more things can happen are more complex than theories where fewer things can happen.

A little more than a decade after Keisler introduced his order, Shelah published an influential book, which included an important chapter showing that there are naturally occurring jumps in complexity—dividing lines that distinguish more complex theories from less complex ones. After that, little progress was made on Keisler’s order for 30 years.

Then, in her 2009 doctoral thesis and other early papers, Malliaris reopened the work on Keisler’s order and provided new evidence for its power as a classification program. In 2011, she and Shelah started working together to better understand the structure of the order. One of their goals was to identify more of the properties that make a theory maximally complex according to Keisler’s criterion.

Malliaris and Shelah eyed two properties in particular. They already knew that the first one causes maximal complexity. They wanted to know whether the second one did as well. As their work progressed, they realized that this question was parallel to the question of whether p and t are equal. In 2016, Malliaris and Shelah published a 60-page paper that solved both problems: They proved that the two properties are equally complex (they both cause maximal complexity), and they proved that p equals t.

“Somehow everything lined up,” Malliaris said. “It’s a constellation of things that got solved.”

This past July, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in set theory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Most mathematicians had expected that p was less than t, and that a proof of that inequality would be impossible within the framework of set theory. Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between p and t has much more depth to it than mathematicians had realized.

“I think people thought that if by chance the two cardinals were provably equal, the proof would maybe be surprising, but it would be some short, clever argument that doesn’t involve building any real machinery,” said Justin Moore, a mathematician at Cornell University who has published a brief overview of Malliaris and Shelah’s proof.

Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory that is already opening new frontiers of research in both fields. Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis. Still, the overwhelming feeling among experts is that this apparently unresolvable proposition is false: While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found.

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

Credit of the article given to Kevin Hartnett & Quanta Magazine

 

The outmoded gender stereotype that women lack mathematical ability suffered a major blow in 2014, when Maryam Mirzakhani became the first woman to receive the Fields Medal, math’s most prestigious award.

An equally important blow was struck by an Italian mathematician Maria Gaetana Agnesi in the 18th century. Agnesi was the first woman to write a mathematics textbook and to be appointed to a university chair in math, yet her life was marked by paradox.

Though brilliant, rich and famous, she eventually opted for a life of poverty and service to the poor. Her remarkable story serves as a source for mathematical inspiration even today.

Early years

Born May 16, 1718 in Milan, Agnesi was the eldest of her wealthy silk merchant father’s 21 children. By age 5 she could speak French, and by 11 she was known to Milanese society as the “seven-tongued orator” for her mastery of modern and classical languages. In part to give Agensi the best education possible, her father invited leading intellectuals of the day to the family’s home, where his daughter’s gifts shone.

When Agnesi was 9, she recited from memory a Latin oration, likely composed by one of her tutors. The oration decried the widespread prejudice against educating women in the arts and sciences, which had been grounded in the view that a life of managing a household would require no such learning. Agnesi presented a clear and convincing argument that women should be free to pursue any kind of knowledge available to men.

Agnesi eventually became tired of displaying her intellect and expressed a desire to enter a convent. When her father’s second wife died, however, she assumed responsibility for his household and the education of her many younger siblings.

Through this role, she recognized that teachers and students needed a comprehensive mathematics textbook to introduce Italian students to the many recent Enlightenment-era mathematical discoveries.

Agnesi’s textbook

Portrait of Maria Agnesi by an unknown artist.

Agnesi found a special appeal in mathematics. Most knowledge derived from experience, she believed, is fallible and open to dispute. From mathematics, however, come truths that are wholly certain, the contemplation of which brings particularly great joy. In writing her textbook, she was not only teaching a useful skill, but opening to her students the door to such contemplation.

Published in two volumes in 1748, Agnesi’s work was entitled the “Basic Principles of Analysis.” It was composed not in Latin, as was the custom for great mathematicians such as Newton and Euler, but Italian vernacular, to make it more accessible to students.

Hers represented one of the first textbooks in the relatively new field of calculus. It helped to shape the education of mathematics students for several generations that followed. Beyond Italy, contemporary scholars in Paris and Cambridge translated the textbook for use in their university classrooms.

Agnesi’s textbook was praised in 1749 by the French Academy: “It took much skill and sagacity to reduce to almost uniform methods discoveries scattered among the works of many mathematicians very different from each other. Order, clarity, and precision reign in all parts of this work. … We regard it as the most complete and best made treatise.”

In offering similarly fine words of praise, another contemporary mathematician, Jean-Etienne Montucla, also revealed some of the mathematical sexism that persists down to the present day. He wrote: “We cannot but behold with the greatest astonishment how a person of a sex that seems so little fitted to tread the thorny paths of these abstract sciences penetrates so deeply as she has done into all the branches of algebra.”

Agnesi dedicated the “Basic Principles” to Empress Maria Theresa of Austria, who acknowledged the favor with a letter of thanks and a diamond-bearing box and ring. Pope Benedict XIV praised the work and predicted that it would enhance the reputation of the Italians. He also appointed her to the chair of mathematics at the University of Bologna, though she never traveled there to accept it.

A life of service

A passionate advocate for the education of women and the poor, Agnesi believed that the natural sciences and math should play an important role in an educational curriculum. As a person of deep religious faith, however, she also believed that scientific and mathematical studies must be viewed in the larger context of God’s plan for creation.

When Maria’s father died in 1752, she was free to answer a religious calling and devote herself to her other great passion: service to the poor, sick and homeless. She began by founding a small hospital in her home. She eventually gave away her wealth, including the gifts she had received from the empress. When she died at age 80, she was buried in a pauper’s grave.

To this day, some mathematicians express surprise at Maria’s apparent turn from learning and mathematics to a religious vocation. To her, however, it made perfect sense. In her view, human beings are capable of both knowing and loving, and while it is important for the mind to marvel at many truths, it’s ultimately even more important for the heart to be moved by love.

“Man always acts to achieve goals; the goal of the Christian is the glory of God,” she wrote. “I hope my studies have brought glory to God, as there were useful to others, and derived from obedience, because that was my father’s will. Now I have found better ways and means to serve God, and to be useful to others.”

Though few remember Agnesi today, her pioneering role in the history of mathematics serves as an inspiring story of triumph over gender stereotypes. She helped to blaze a trail for women in math and science for generations to follow. Agnesi excelled at math, but she also loved it, perceiving in its mastery an opportunity to serve both her fellow human beings and a higher order.

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

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