Category: Skills and Learning

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Last digits of nearby primes have “anti-sameness” bias

Two mathematicians have found a strange pattern in prime numbers—showing that the numbers are not distributed as randomly as theorists often assume.

“Every single person we’ve told this ends up writing their own computer program to check it for themselves,” says Kannan Soundararajan, a mathematician at Stanford University in California, who reported the discovery with his colleague Robert Lemke Oliver in a paper submitted to the arXiv preprint server on March 11. “It is really a surprise,” he says.

Prime numbers near to each other tend to avoid repeating their last digits, the mathematicians say: that is, a prime that ends in 1 is less likely to be followed by another ending in 1 than one might expect from a random sequence. “As soon as I saw the numbers, I could see it was true,” says mathematician James Maynard of the University of Oxford, UK. “It’s a really nice result.”

Although prime numbers are used in a number of applications, such as cryptography, this ‘anti-sameness’ bias has no practical use or even any wider implication for number theory, as far as Soundararajan and Lemke Oliver know. But, for mathematicians, it’s both strange and fascinating.

Not so random

A clear rule determines exactly what makes a prime: it’s a whole number that can’t be exactly divided by anything except 1 and itself. But there’s no discernable pattern in the occurrence of the primes. Beyond the obvious—after the numbers 2 and 5, primes can’t be even or end in 5—there seems to be little structure that can help to predict where the next prime will occur.

As a result, number theorists find it useful to treat the primes as a ‘pseudorandom’ sequence, as if it were created by a random-number generator.

But if the sequence were truly random, then a prime with 1 as its last digit should be followed by another prime ending in 1 one-quarter of the time. That’s because after the number 5, there are only four possibilities—1, 3, 7 and 9—for prime last digits. And these are, on average, equally represented among all primes, according to a theorem proved around the end of the nineteenth century, one of the results that underpin much of our understanding of the distribution of prime numbers. (Another is the prime number theorem, which quantifies how much rarer the primes become as numbers get larger.)

Instead, Lemke Oliver and Soundararajan saw that in the first billion primes, a 1 is followed by a 1 about 18% of the time, by a 3 or a 7 each 30% of the time, and by a 9 22% of the time. They found similar results when they started with primes that ended in 3, 7 or 9: variation, but with repeated last digits the least common. The bias persists but slowly decreases as numbers get larger.

The k-tuple conjecture

The mathematicians were able to show that the pattern they saw holds true for all primes, if a widely accepted but unproven statement called the Hardy–Littlewood k-tuple conjecture is correct. This describes the distributions of pairs, triples and larger prime clusters more precisely than the basic assumption that the primes are evenly distributed.

The idea behind it is that there are some configurations of primes that can’t occur, and that this makes other clusters more likely. For example, consecutive numbers cannot both be prime—one of them is always an even number. So if the number n is prime, it is slightly more likely that n + 2 will be prime than random chance would suggest. The k-tuple conjecture quantifies this observation in a general statement that applies to all kinds of prime clusters. And by playing with the conjecture, the researchers show how it implies that repeated final digits are rarer than chance would suggest.

At first glance, it would seem that this is because gaps between primes of multiples of 10 (20, 30, 100 and so on) multiples of 10 are disfavoured. But the finding gets much more general—and even more peculiar. A prime’s last digit is its remainder when it is divided by 10. But the mathematicians found that the anti-sameness bias holds for any divisor. Take 6, for example. All primes have a remainder of 1 or 5 when divided by 6 (otherwise, they would be divisible by 2 or 3) and the two remainders are on average equally represented among all primes. But the researchers found that a prime that has a remainder of 1 when divided by 6 is more likely to be followed by one that has a remainder of 5 than by another that has a remainder of 1. From a 6-centric point of view, then, gaps of multiples of 6 seem to be disfavoured.

Paradoxically, checking every possible divisor makes it appear that almost all gaps are disfavoured, suggesting that a subtler explanation than a simple accounting of favoured and disfavoured gaps must be at work. “It’s a completely weird thing,” says Soundararajan.

Mystifying phenomenon

The researchers have checked primes up to a few trillion, but they think that they have to invoke the k-tuple conjecture to show that the pattern persists. “I have no idea how you would possibly formulate the right conjecture without assuming it,” says Lemke Oliver.

Without assuming unproven statements such as the k-tuple conjecture and the much-studied Riemann hypothesis, mathematicians’ understanding of the distribution of primes dries up. “What we know is embarrassingly little,” says Lemke Oliver. For example, without assuming the k-tuple conjecture, mathematicians have proved that the last-digit pairs 1–1, 3–3, 7–7 and 9–9 occur infinitely often, but they cannot prove that the other pairs do. “Perversely, given our work, the other pairs should be more common,” says Lemke Oliver.

He and Soundararajan feel that they have a long way to go before they understand the phenomenon on a deep level. Each has a pet theory, but none of them is really satisfying. “It still mystifies us,” says Soundararajan.

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

Credit of the article given to Evelyn Lamb & Nature magazine

Generally, the occurrence of students asking this question increases with growing age. Primary students know inside out that exams are very important. Brilliant middle school students consider a connection between their test results and semester mark sheets. Ultimately, upon graduation from secondary school, students have comprehended that the totality of their learning has less value than their results in the final exams.

Performance-Oriented Learning

Exam enthusiasm is an indication of performance-oriented learning, and it is intrinsic to our recent education management that needs standards-based reporting of student results. This focuses on performance apart from the method of learning and requests comparison of procurement amongst peers.

The focus for performance-aligned students is showing their capabilities. Fascinatingly, this leads to an affection of fixed mindset characteristics such as the ignorance of challenging tasks because of fear of failure and being intimidated by the success of other students.

Mastery-Oriented Learning

Mastery learning putting down a focus on students developing their competence. Goals are pliably positioned far away from reach, pushing regular growth. The phrase “how can this be even better?” changes the concept of “good enough”. Not to be bewildered with perfectionism, a mastery approach to learning encourages development mindset qualities such as determination, hard work, and facing challenges.

Most forms of mastery learning nowadays can be discovered in the work of Benjamin Bloom in the late 1960s. Bloom saw the important elements of one-to-one teaching that take to effective benefits over group-based classrooms and inspects conveyable instructional plans. Eventually, formative assessment was defined in the circumstances of teaching and learning as a major component for tracking student performance.

So where does mastery learning position in today’s classroom? The idea of formative assessment is frequent, as are posters and discussions encouraging a growth mindset. One significant missing element is making sure that students have a deep knowledge of concepts before moving to the next.

Shifting the Needle

With the growing possibilities offered by Edtech organizations, many are beginning to look to a tech-based solution like International Maths Olympiad Challenge to provide individualized learning possibilities and prepare for the maths Olympiad. The appropriate platform can offer personalized formative assessment and maths learning opportunities.

But we should take a careful viewpoint to utilize technology as a key solution. History shows us that implementing the principles of mastery learning in part restricts potential gains. Despite assessment plans, teachers will also have to promote a mastery-orientated learning approach in their classrooms meticulously. Some strategies are:

• Giving chances for student agency
• Encouraging learning from flaws
• Supporting individual growth with an effective response
• Overlooking comparing students and track performance

We think teaching students how to learn is far more necessary than teaching them what to learn.

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

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

Credit: Annalisa Crannell analyzes art with the help of chopsticks and projective geometry. Evelyn Lamb

To fully appreciate perspective art, mathematician Annalisa Crannell says both the artist and the art viewer need to do some math

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

Annalisa Crannell goes to art museums with chopsticks. She is not unusually hungry or over-prepared; she uses them to figure out how to look at the art.

Crannell, a mathematician at Franklin and Marshal College in Lancaster, Pennsylvania, studies mathematical perspective and applies her work to the world of art. She writes not only about how artists use perspective but also about how viewers can use it to see art in different ways.

In a 2014 Math Horizons article (pdf, also available in The Best Writing on Mathematics 2015, edited by Mircea Pitici), she and coauthors Marc Frantz and Fumiko Futamura take on the case of the mysterious table in 15th- and 16th-century German artist Albrecht Dürer’s famous engraving St. Jerome in His Study.

Credit: St. Jerome in His Study, by Albrecht Dürer. Public domain, via Wikimedia Commons.

This work is an early example of mathematical perspective in art, but some critics have maligned Dürer’s technique. William Mills Ivins Jr., a former curator at the Metropolitan Museum of Art in New York, described it as “the oddest trapezoidal shape” and claimed it wasn’t even level with the floor. Crannell and her coauthors say it’s a matter of perspective. They write, “Surprisingly, the answers to these questions depend not only on what Dürer did 500 years ago, but also on what Ivins did in 1938. And, as we will show, it depends on what you, the reader, do when you look at St. Jerome today.”

Crannell and her coauthors describe how to use straight lines in St. Jerome in His Study to determine exactly where the viewer should stand to see the painting from the perspective Dürer probably intended, and therefore to see the table as a square instead of a trapezoid. The proper viewing location for that particular engraving turns out to be closer to the picture and farther to the right than most people would naturally stand. They write, “The oddness that Ivins saw in the table wasn’t because Dürer was in the wrong, but because Ivins was in the wrong, literally: he was looking from the wrong place!”

In February, I had the pleasure of going to the Brigham Young University Museum of Art with Crannell, and she shared some of her secrets with me. (The BYU Museum of Art does not normally allow photography, so I thank them for graciously making an exception for us.)

In perspective art, lines in the painting that represent parallel lines in the real world—say, train tracks or the opposite sides of a table—intersect on the canvas at so-called vanishing points. These vanishing points are the key to determining the optimal location from which to view a painting.

The most obvious way to find the vanishing points of a painting and thus to determine the optimal viewing location is to place rulers directly on “parallel” lines in the drawing, but shockingly, most museums frown on that practice. That’s where the chopsticks come in.

Standing in front of a piece of art, Crannell closes one eye and holds the chopsticks in front of her so they line up with lines in the artwork that represent parallel lines in the real world. The place the chopsticks appear to intersect is in front of the vanishing point of those lines. For art that has one vanishing point, the viewer should stand directly in front of that point. The viewing distance can be determined by trial and error or by some sneaky geometry with squares.

Credit: A diagram illustrating the orthocenter of a triangle. Each red line is one altitude of the triangle. Image: Public domain-self, via Wikimedia Commons.

For art with two vanishing points, the optimal viewing point is somewhere on the semicircle that connects the two vanishing points. For three vanishing points, determining the optimal vantage point is a bit more involved. It is at the intersection of three hemispheres, each one of which has two of the vanishing points as a diameter. Equivalently, it is somewhere in front of the orthocenter of the triangle whose vertices are the vanishing points. (The orthocenter of a triangle is the intersection point of the three altitidues of the triangle, as illustrated in the diagram on the right.) For a more complete description of how to find viewing points in art, check out Viewpoints: Mathematical Perspective and Fractal Geometry in Art by Crannell and Frantz.

In addition to giving me a way to look eccentric at the museum, Crannell’s technique helps me understand why some paintings seem to leap off the page, and some, even though they basically look realistic, don’t quite pop. In some pieces we looked at, lines that should have represented parallel lines in the real world didn’t end up determining a consistent vanishing point. Looking at any one part of the painting, nothing was clearly wrong, but the overall effect was slightly imperfect. When artists do manage to deploy perfect perspective—and viewers manage to find the correct vantage point—the effect can be startlingly realistic.

As Crannell and her coauthors describe it,  we can see the effect that the master geometer Albrecht Dürer intended. If you view St. Jerome in His Study [from the mathematically determined vantage point], you’ll see that the engraving takes on an amazing realism and depth. The gourd in the picture seems to hover over your head; you feel you could stick your hand in the space under the table; the bench off to the left invites you to come sit down and fluff up the pillows.

If you would like to know how to find those fluffy pillows for yourself, Crannell has information about projective geometry and perspective art on her website.

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

Credit of the article given to Evelyn Lamb |

Credit: Getty Images

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

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

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

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

A puzzle is born

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

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

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

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

Red and blue lines

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

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

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

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

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

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

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

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

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

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

Ramsey numbers

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

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

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

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

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

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

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

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

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

Credit: Jasmina81 Getty Images

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

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

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

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

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

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

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

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

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

Credit of the article given to Robert S. Siegler

Can three cubed numbers be added up to give 42? Until now, we didn’t know

It might not tell us the meaning of life, the universe, and everything, but mathematicians have cracked an elusive problem involving the number 42.

Since the 1950s, mathematicians have been puzzling over whether any integer – or whole number – can be represented as the sum of three cubed numbers.

Put another way: are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

Andrew Booker at Bristol University, UK, and Andrew Sutherland at the Massachusetts Institute of Technology, US, have solved the problem for the number 42, the only number less than 100 for which a solution had not been found.

Some numbers have simple solutions. The number 3, for example, can be expressed as 1³ + 1³ + 1³ and 4³ + 4³ + (-5) ³ . But solving the problem for other numbers requires vast strings of digits and, correspondingly, computing power.

The solution for 42, which Booker and Sutherland found using an algorithm, is:

42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³

They worked with software firm Charity Engine to run the program across more than 400,000 volunteers’ idle computers, using computing power that would otherwise be wasted. The amount of processing time is equivalent to a single computer processor running continuously for more than 50 years, says Sutherland.

Earlier this year, Booker found a sum of cubes for the number 33, which was previously the lowest unsolved example.

We know for certain that some numbers, such as 4, 5 and 13, can’t be expressed as the sum of three cubes.

The problem is still unsolved for 10 numbers less than 1000, the smallest of which is 114.

The team will next search for another solution to the number 3.

“It’s possible we’ll find it in the next few months; it’s possible it won’t be for another hundred years,” says Booker.

People interested in aiding the search can can volunteer computing power through Charity Engine, says Sutherland.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Donna Lu*

The importance of mathematics in daily and professional life has been increasing with the contribution of developing technology. The level of mathematical knowledge and skills directly influence the quality standards of our individual and social life. However, mathematics the importance of which we feel in every aspect of our life is unfortunately not learned enough by many individuals for many reasons. The leading reasons regarding this issue are as follows: the abstract and hierarchical structure of mathematics, methods and strategies in learning mathematics, and the learning difficulties in mathematics. Developmental Dyscalculia (DD)/Mathematics Learning Difficulty (MLD) is a brain-based condition that negatively affects mathematics acquisition.

The mathematical performance of a student with MLD is much lower than expected for age, intelligence, and education, although there are no conditions such as intellectual disability, emotional disturbances, cultural deprivation, or lack of education. Difficulties in mathematics result from a number of cognitive and emotional factors. Math anxiety is one of the emotional factors that may severely disrupt a significant number of children and adults in learning and achievement in math.

Math anxiety is defined as “the feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations”. Sherard describes math anxiety as the fear of math or an intense and negative emotional response to mathematics. There are many reasons for the cause of the math anxiety. These include lack of the appropriate mathematical background of the students, study habits of memorizing formulas, problems and applications that are not related to real life, challenging and time-limited exams, lack of concrete materials, the difficulty of some subjects in mathematics, type of personality, negative approach on mathematics, lack of confidence, the approaches, feelings, and thoughts of teachers and parents on mathematics.

The negative relationship between math anxiety and math performance is an international issue. The PISA (Programme for International Student Assessment) statistics measuring a wide variety of countries and cultures depict that the high level of negative correlation between math anxiety and mathematical performance is remarkable. Some studies showed that highly math-anxious individuals are worse than those with low mathematics anxiety in terms of solving mathematical problems. These differences are not typically observed in simple arithmetic operations such as 7 + 9 and 6 × 8, but it is more evident when more difficult arithmetic problems are tested.

Math anxiety is associated with cognitive information processing resources during arithmetic task performance in a developing brain. It is generally accepted that math anxiety negatively affects mathematical performance by distorting sources of working memory. The working memory is conceptualized as a limited source of cognitive systems responsible for the temporary storage and processing of information in momentary awareness.

The learning difficulties in mathematics relate to deficiencies in the central executive component of the working memory. Many studies suggest that individuals with learning difficulties in mathematics have a lack of working memory. It is stated that students with learning difficulties in mathematics use more inferior strategies than their peers for solving basic (4 + 3) and complex (16 + 8) addition and fall two years behind their peers while they fall a year behind in their peers’ working memory capacities.

Highly math-anxious individuals showed smaller working memory spans, especially when evaluated with a computationally based task. This reduced working memory capacity, when implemented simultaneously with a memory load task, resulting in a significant increase in the reaction time and errors. A number of studies showed that working memory capacity is a robust predictor of arithmetic problem-solving and solution strategies.

Although it is not clear to what extent math anxiety affects mathematical difficulties and how much of the experience of mathematical difficulties causes mathematical anxiety, there is considerable evidence that math anxiety affects mathematical performance that requires working memory. Figure below depicts these reciprocal relationships among math anxiety, poor math performance, and lack of working memory. The findings of the studies mentioned above, make it possible to draw this figure.

Basic numerical and mathematical skills have been crucial predictors of an individual’s vital success. When anxiety is controlled, it is seen that the mathematical performance of the students increases significantly. Hence, early identification and treatment of math anxiety is of importance. Otherwise, early anxieties can have a snowball effect and eventually lead students to avoid mathematics courses and career options for math majors. Although many studies confirm that math anxiety is present at high levels in primary school children, it is seen that the studies conducted at this level are relatively less when the literature on math anxiety is examined. In this context, this study aims to determine the dimensions of the relationship between math anxiety and mathematics achievement of third graders by their mathematics achievement levels.

Methods

The study was conducted by descriptive method. The purpose of the descriptive method is to reveal an existing situation as it is. This study aims to examine the relationship between math anxiety and mathematics achievement of third graders in primary school in terms of student achievement levels.

Participants

Researchers of mathematics learning difficulties (MLD) commonly use cutoff scores to determine which participants have MLD. These cutoff scores vary between -2 ss and -0.68 ss. Some researchers apply more restrictive cutoffs than others (e.g., performance below the 10th percentile or below the 35th percentile). The present study adopted the math achievement test to determine children with MLD based below the 10th percentile. The unit of analysis was third graders of an elementary school located in a low socioeconomic area. The study reached 288 students using math anxiety scale and math achievement test tools. The students were classified into four groups by their mathematics achievement test scores: math learning difficulties (0-10%), low achievers (11-25%), normal achievers (26-95%), and high achievers (96-100%).

Table 1. Distribution of participants by gender and groups

Data Collection Tools

Two copyrighted survey scales, consisting of 29 items were used to construct a survey questionnaire. The first scale is the Math Anxiety Scale developed by Mutlu & Söylemez for 3rd and 4th graders with a 3-factor structure of 13 items. The Cronbach’s Alpha coefficient is adopted by the study to evaluate the extent to which a measurement produces reliable results at different times. The Cronbach Alpha coefficient of the scale is .75 which confirms the reliability of and internal consistency of the study. The response set was designed in accordance with the three- point Likert scale with agree, neutral, and disagree. Of the 13 items in the scale, 5 were positive and 8 were negative. Positive items were rated as 3-2-1, while negative items were rated as 1-2-3. The highest score on the scale was 39 and the lowest on the scale was 13.

The second data collection tool adopted by this study is the math achievement test for third graders developed by Fidan (2013). It has 16 items designed in accordance with the national math curriculum. Correct responses were scored one point while wrong responses were scored zero point.

Data Analysis

The study mainly utilized five statistical analyses which are descriptive analysis, independent samples t-test, Pearson product-moment correlation analysis, linear regression and ANOVA. First, an independent samples t-test was performed to determine whether there was a significant difference between the levels of math anxiety by gender. Then, a Pearson product-moment correlation analysis was performed to determine the relationship between the math anxiety and mathematics achievement of the students. After that, a linear regression analysis was performed to predict the mathematics achievement of the participants based on their math anxiety. Finally, an ANOVA was performed to determine if there was a significant difference between the math anxiety of the groups determined in terms of mathematics achievement.

Results

The findings of the math anxiety scores by gender of the study found no significant difference between the averages [t(286)= 1.790, p< .05]. This result shows that the math anxiety levels of girls and boys are close to each other. Since there is no difference between math anxiety scores by gender, the data in the study were combined.

Table 2. Comparison of anxiety scores by gender

There was a strong and negative correlation between math anxiety and mathematics achievement with the values of r= -0.597, n= 288, and p= .00. This result indicates that the highly math-anxious students and decreases in math anxiety were correlated with increases in rating of math achievement.

A simple linear regression was calculated to predict math achievement level based on the math anxiety. A significant regression equation was found (F(1,286)= 158.691, p< .000) with an R2 of .357. Participants’ predicted math achievement is equal to 20.153 + -6.611 when math anxiety is measured in unit. Math achievement decreased -6.611 for each unit of the math anxiety.

Figure below shows the relationship between the math anxiety of the children and their mathematics achievement on a group basis. Figure 1 provides us that there is a negative correlation between mathematical performance and math anxiety. The results depict that the HA group has the lowest math anxiety score, while the MLD group has the highest math anxiety.

Table 3. Comparison of the mathematical anxiety scores of the groups

The table indicates that there is a statistically significant difference between groups as determined at the p<.05 level by one-way ANOVA (F(3,284)= 36.584, p= .000). Post hoc comparisons using the Tukey test indicated that the mean score for MLD group (M= 1.96, sd= 0.30) was significantly different than the NA group (M= 1.41, sd= 0.84) and HA group (M= 1.24, sd= 0.28). However, the MLD group (M= 1.96, sd= 0.30) did not significantly differ from the LA group (M= 1.76, sd= 0.27).

Discussion and Conclusion

Math anxiety is a problem that can adversely affect the academic success and employment prospects of children. Although the literature on math anxiety is largely focused on adults, recent studies have reported that some children begin to encounter math anxiety at the elementary school level. The findings of the study depict that the correlation level of math anxiety and math achievement is -.597 among students. In a meta-analysis study of Hembre and Ma, found that the level of relationship between mathematical success and math anxiety is -.34 and -.27, respectively. In a similar meta-analysis study performed in Turkey, the correlation coefficient was found to be -.44. The different occurrence of the coefficients is probably dependent on the scales used and the sample variety.

The participants of the study were classified into four groups: math learning difficulties (0-10%), low success (11-25%), normal (26-95%), and high success (96-100%) by the mathematics achievement test scores. The study compared the math anxiety scores of the groups and found no significant difference between the mean scores of the math anxiety of the lower two groups (mean of MLD math anxiety, .196; mean of LA math anxiety .177) as it was between the upper two groups (mean of NA math anxiety, .142; mean of HA math anxiety .125). This indicates that the math anxiety level of the students with learning difficulties in math does not differ from the low math students. However, a significant difference was found between the mean scores of math anxiety of the low successful and the normal group.

It may be better for some students to maintain moderate levels of math anxiety to make their learning and testing materials moderately challenging, but it can be clearly said that high math anxiety has detrimental effects on the mathematical performance of the individuals. Especially for students with learning difficulties in math, the high level of math anxiety will lead to destructive effects in many dimensions, primarily a lack of working memory.

Many of the techniques employed to reduce or eliminate the link between math anxiety and poor math performance involve addressing the anxiety rather than training math itself. Some methods for reducing math anxiety can be used in teaching mathematics. For instance, effective instruction for struggling mathematics learners includes instructional explicitness, a strong conceptual basis, cumulative review and practice, and motivators to help maintain student interest and engagement.

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

Credit for the article given to Yılmaz Mutlu

Today I’d like to share an idea. It’s a very simple idea. It’s not fancy and it’s certainly not new. In fact, I’m sure many of you have thought about it already. But if you haven’t—and even if you have!—I hope you’ll take a few minutes to enjoy it with me. Here’s the idea:

So simple! But we can get a lot of mileage out of it.

To start, I’ll be a little more precise: every matrix corresponds to a weighted bipartite graph. By “graph” I mean a collection of vertices (dots) and edges; by “bipartite” I mean that the dots come in two different types/colors; by “weighted” I mean each edge is labeled with a number.

The graph above corresponds to a 3×23×2 matrix MM. You’ll notice I’ve drawn three greengreen dots—one for each row of MM—and two pinkpink dots—one for each column of MM. I’ve also drawn an edge between a green dot and a pink dot if the corresponding entry in MM is non-zero.

For example, there’s an edge between the second green dot and the first pink dot because M21=4M21=4, the entry in the second row, first column of MM, is not zero. Moreover, I’ve labeled that edge by that non-zero number. On the other hand, there is no edge between the first green dot and the second pink dot because M12M12, the entry in the first row, second column of the matrix, is zero.

Allow me to describe the general set-up a little more explicitly.

Any matrix MM is an array of n×mn×m numbers. That’s old news, of course. But such an array can also be viewed as a function M:X×Y→RM:X×Y→R where X={x1,…,xn}X={x1,…,xn} is a set of nn elements and Y={y1,…,ym}Y={y1,…,ym} is a set of mm elements. Indeed, if I want to describe the matrix MM to you, then I need to tell you what each of its ijijth entries are. In other words, for each pair of indices (i,j)(i,j), I need to give you a real number MijMij. But that’s precisely what a function does! A function M:X×Y→RM:X×Y→R associates for every pair (xi,yj)(xi,yj) (if you like, just drop the letters and think of this as (i,j)(i,j)) a real number M(xi,yj)M(xi,yj). So simply write MijMij for M(xi,yj)M(xi,yj).

Et voila. A matrix is a function.

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*Credit for article given to Tai-Danae Bradley*