Author: IMC

Bourbaki Congress of 1938.

By many measures, Nicolas Bourbaki ranks among the greatest mathematicians of the 20th century.

Largely unknown today, Bourbaki is likely the last mathematician to master nearly all aspects of the field. A consummate collaborator, he made fundamental contributions to important mathematical fields such as set theory and functional analysis. He also revolutionized mathematics by emphasizing rigor in place of conjecture.

There’s just one problem: Nicolas Bourbaki never existed.

Never existed?

The cover of the first volume in Bourbaki’s textbook. Maitrier/Wikimedia, CC BY-SA

While it is now widely accepted that there never was a Nicolas Bourbaki, there is evidence to the contrary.

For example, there are wedding announcements for his daughter Betty, a baptismal certificate in his name and an impressive family lineage extending back to an ancestor Napoleon raised as his own son.

Even the professional mathematics community was misled for a time. When Ralph Boas, an editor of the journal Mathematical Reviews, wrote that Bourbaki was a pseudonym, he was promptly refuted by none other than Bourbaki himself. Bourbaki countered with a letter stating that B.O.A.S. actually just was an acronym of the last names of the editors of the Reviews.

These cases of confused identity were not all fun and games. For example, it is alleged that, while visiting Finland at the outset of World War II, French mathematician André Weil was investigated for spying. The authorities found suspicious papers in his possession: a fake identity, a set of business cards and even invitations from the Russian Academy of Science – all in Bourbaki’s name. Supposedly, Weil was freed only after an officer recognized him as a preeminent mathematician.

Who was Bourbaki?

If Bourbaki never existed, who – or what – was he?

The name Nicolas Bourbaki first appeared in a place rocked by turmoil at a volatile time in history: Paris in 1934.

World War I had wiped out a generation of French intellectuals. As a result, the standard university-level calculus textbook had been written more than two and half decades before and was out of date.

Newly minted professors André Weil and Henri Cartan wanted a rigorous method to teach Stokes’ theorem, a key result of calculus. After realizing that others had similar concerns, Weil organized a meeting. It took place December 10, 1934 at a Parisian café called Capoulade.

The nine mathematicians in attendance agreed to write a textbook “to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis,” which they hoped to complete in just six months.

Cafe Capoulade in 1943. Langhaus, German Federal Archive/Wikimedia, CC BY-SA

As a joke, they named themselves after an old French general who had been duped in the Franco-Prussian war.

As they proceeded, their original goal of elucidating Stokes’ theorem expanded to laying out the foundations of all mathematics. Eventually, they began to hold regular Bourbaki “conferences” three times a year to discuss new chapters for the treatise.

Individual members were encouraged to engage with all aspects of the effort, to ensure that the treatise would be accessible to nonspecialists. According to one of the founders, spectators invariably came away with the impression that they were witnessing “a gathering of madmen.” They could not imagine how people, shouting – “sometimes three or four at the same time” – could ever come up with something “intelligent.”

Top mathematicians from across Europe, intrigued by the group’s work and style, joined to augment the group’s ranks. Over time, the name Bourbaki became a collective pseudonym for dozens of influential mathematicians spanning generations, including Weil, Dieudonne, Schwartz, Borel, Grothendieck and many others.

Since then, the group which has added new members over time, has proved to have a profound impact on mathematics, certainly rivaling any of its individual contributors.

Profound impact

Mathematicians have made a plethora of important contributions under Bourbaki’s name.

To name a few, the group introduced the null set symbol; the ubiquitous terms injective, surjective, bijective; and generalizations of many important theorems, including the Bourbaki-Witt theorem, the Jacobson-Bourbaki theorem and the Bourbaki-Banach-Alaoglu theorem.

Their text, “Elements of Mathematics,” has swelled to more than 6,000 pages. It provides a “solid foundation for the whole body of modern mathematics,” according to mathematician Barbara Pieronkiewicz.

Bourbaki’s influence is still alive and well. Now in “his” 80th year of research, in 2016 “he” published the 11th volume of the “Elements of Mathematics.” The Bourbaki group, with its ever-changing cast of members, still holds regular seminars at the University of Paris.

Partly thanks to the breadth and significance of “his” mathematical contributions, and also because – ageless, unchanging and operating in multiple places at once – “he” seems to defy the very laws of physics, Bourbaki’s mathematical prowess will likely never be equaled.

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

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

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

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

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

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

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

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

Julia Robinson (1919-1985)

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

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

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

Emmy Noether (1882-1935)

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

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

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

Ada Lovelace (1815-1852)

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

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

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

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

Credit of the article given to Avery Carr

Grab a few balls and get calculating pi

Pi Day, which occurs every 14 March – or 3/14, in the US date format – celebrates the world’s favourite mathematical constant. This year, why not try an experiment to calculate its value? All you will need is a cardboard tube and a series of balls, each 100 times lighter than the next. You have those lying around the house, right?

This experiment was first formulated by mathematician Gregory Galperin in 2001. It works because of a mathematical trick involving the masses of a pair of balls and the law of conservation of energy.

First, take the tube and place one end up against a wall. Place two balls of equal mass in the tube. Let’s say that the ball closer to the wall is red, and the other is blue.

Next, bounce the blue ball off the red ball. If you have rolled the blue ball hard enough, there should be three collisions: the blue ball hits the red one, the red ball hits the wall, and the red ball bounces back to hit the blue ball once more. Not-so-coincidentally, three is also the first digit of pi.

To calculate pi a little bit more precisely, replace the red ball with one that is 100 times less massive than the blue ball – a ping pong ball might work, so we will call this the white ball.

When you perform the experiment again, you will find that the blue ball hits the white ball, the white ball hits the wall and then the white ball continues to bounce back and forth between the blue ball and the wall as it slows down. If you count the bounces, you’ll find that there are 31 collisions. That gives you the first two digits of pi: 3.1.

Galperin calculated that if you continue the same way, you will keep getting more digits of pi. If you replace the white ball with another one that is 10,000 times less massive than the blue ball, you will find that there are 314 collisions, and so on. If you have enough balls, you can count as many digits of pi as you like.

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

*Credit for article given to Leah Crane*

Machines are getting better at maths – artificial intelligence has learned to solve university-level calculus problems in seconds.

François Charton and Guillaume Lample at Facebook AI Research trained an AI on tens of millions of calculus problems randomly generated by a computer. The problems were mathematical expressions that involved integration, a common technique in calculus for finding the area under a curve.

To find solutions, the AI used natural language processing (NLP), a computational tool commonly used to analyse language. This works because the mathematics in each problem can be thought of as a sentence, with variables, normally denoted x, playing the role of nouns and operations, such as finding the square root, playing the role of verbs. The AI then “translates” the problem into a solution.

When the pair tested the AI on 500 calculus problems, it found a solution with an accuracy of 98 per cent. A comparable standard program for solving maths problems had only an accuracy of 85 per cent on the same problems.

The team also gave the AI differential equations to solve, which are other equations that require integration to solve as well as other techniques. For these equations, the AI wasn’t quite as good, solving them correctly 81 per cent for one type of differential equation and 40 per cent on a harder type.

Despite this, it could still correctly answer questions that confounded other maths programs.

Doing calculus on a computer isn’t especially useful in practice, but with further training AI might one day be able to tackle maths problems that are too hard for humans to crack, says Charton.

The efficiency of the AI could save humans time in other mathematical tasks, for example, when proving theorems, says Nikos Aletras at the University of Sheffield, UK.

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

*Credit for article given to Gege Li*

The whole is equivalent to the sum of its parts

Which is bigger, 10 or 7?

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

How do you know?

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

Perhaps you will indulge me and delve a little deeper:

In how many ways do you know?

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

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

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

Part-whole

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

What are there more of:

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

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

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

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

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

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

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

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

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

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

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

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

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

How do you know?

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

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

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

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

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

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

Credit for the article given to Tabitha Gould

AI can search through numbers and equations to find patterns

Artificial intelligence’s ability to sift through large amounts of data is helping us tackle one of the most difficult unsolved problems in mathematics.

Yang-Hui He at City, University of London, and his colleagues are using the help of machine learning to better understand the Birch and Swinnerton-Dyer conjecture. This is one of the seven fiendishly difficult Millennium Prize Problems, each of which has a $1 million reward on offer for the first correct solution.

The conjecture describes solutions to equations known as elliptic curves, equations in the form of y² = x³ + ax + b, where x and y are variables and a and b are fixed constants.

Elliptic curves were essential in cracking the long-standing Fermat’s last theorem, which was solved by mathematician Andrew Wiles in the 1990s, and are also used in cryptography.

To study the behaviour of elliptic curves, mathematicians also use an equation called the L-series. The conjecture, first stated by mathematicians Bryan Birch and Peter Swinnerton-Dyer in the 1960s, says that if an elliptic curve has an infinite number of solutions, its L-series should equal 0 at certain points.

“It turns out to be a very, very difficult problem to find a set of integer solutions on such equations,” says He, meaning solutions only involving whole numbers. “This is part of the biggest problem in number theory: how do you find integer solutions to polynomials?”

Finding integer solutions or showing that they cannot exist has been crucial. “For example, Fermat’s last theorem is reduced completely to the statement of whether you can find certain properties of elliptic curves,” says He.

He and his colleagues used an AI to analyse close to 2.5 million elliptic curves that had been compiled in a database by John Cremona at the University of Warwick, UK. The rationale was to search the equations to see if any statistical patterns emerged.

Plugging different values into the elliptic curve equation and plotting the results on a graph, the team found that the distribution takes the shape of a cross, which mathematicians hadn’t previously observed. “The distribution of elliptic curves seems to be symmetric from left to right, and up and down,” says He.

“If you spot any interesting patterns, then you can raise a conjecture which may later lead to an important result,” says He.  “We now have a new, really powerful thing, which is machine learning and AI, to do this.”

To see whether a theoretical explanation exists for the cross-shaped distribution, He consulted number theorists. “Apparently, nobody knows,” says He.

“Machine learning hasn’t yet been applied very much to problems in pure maths,” says Andrew Booker at the University of Bristol, UK. “Elliptic curves are a natural place to start.”

“Birch and Swinnerton-Dyer made their conjecture based on patterns that they observed in numerical data in the 1960s, and I could imagine applications of machine learning that tried to detect those patterns efficiently,” says Booker, but the approach so far is too simple to turn up any deep patterns, he says.

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

*Credit for article given to Donna Lu*

Third time’s a charm: just weeks after cracking an elusive problem involving the number 42, mathematicians have found a solution to an even harder problem for the number 3.

Andrew Booker at Bristol University, UK, and Andrew Sutherland at the Massachusetts Institute of Technology have found a big solution to a maths problem known as the sum of three cubes.

The problem asks whether any integer, or whole number, can be represented as the sum of three cubed numbers.

There were already two known solutions for the number 3, both of which involve small numbers: 1³ + 1³ + 1³ and 4³ + 4³ + (-5)³.

But mathematicians have been searching for a third for decades. The solution that Booker and Sutherland found is:

569936821221962380720³ + (-569936821113563493509) ³ + (-472715493453327032) ³ = 3

Earlier this month, the pair also found a solution to the same problem for 42, which was the last remaining unsolved number less than 100.

To find these solutions, Booker and Sutherland worked with software firm Charity Engine to run an algorithm across the idle computers of half a million volunteers.

For the number 3, the amount of processing time was equivalent to a single computer processor running continuously for 4 million hours, or more than 456 years.

When a number can be expressed as the sum of three cubes, there are infinitely many possible solutions, says Booker. “So there should be infinitely many solutions for three, and we’ve just found the third one,” he says.

There’s a reason the third solution for 3 was so hard to find. “If you look at just the solutions for any one number, they look random,” he says. “We think that if you could get your hands on loads and loads of solutions – of course, that’s not possible, just because the numbers get so huge so quickly – but if you could, there’s kind of a general trend to them: that the digit sizes are growing roughly linearly with the number of solutions you find.”

It turns out that this rate of growth is extremely small for the number 3 – only 114, now the smallest unsolved number, has a smaller rate of growth. In other words, numbers with a slow rate of growth have fewer solutions with a lower number of digits.

The duo also found a solution to the problem for 906. We know for sure that certain numbers, such as 4, 5 and 13, can’t be expressed as the sum of three cubes. There now remain nine unsolved numbers under 1000. Mathematicians think these can be written as the sum of three cubes, but we don’t yet know how.

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

*Credit for article given to Donna Lu*

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*

Planning is a key ingredient for effective teaching. But why is it so important? What is ‘good’ planning? How do we make our planning purposeful and focused? Let’s delve further into planning, and what it encompasses.

Curriculum is central to planning. It guides what we teach. The other crucial factor is to know the needs of your students. Effective planning begins with finding where these two elements meet.

The Purposeful planning podcast has been developed to complement the Explicit Teaching in Maths professional learning modules. In this podcast, Dr Emily Ross from The University of Queensland uses the analogy of planning being like a road trip. Dr Ross explains that the curriculum (the knowledge and skills taught) is the destination of this road trip. It’s where you want your students to get to by the end of the lesson or topic.

The road trip itself includes the places you see and the stops on the way to your destination, and this is likened to the teaching and learning. It’s the steps in the lesson or unit plan that enable your students to reach their destination.

Using this road trip analogy, we can ask ourselves two big questions in terms of planning: where are we going? and how will we get there?

Let’s breakdown this analogy further.

• Some people like to be very well planned and outline the detailed steps required to reach the destination.
• Some people like to make a more general plan and they outline the main signposts required to travel past to get to the destination.
• Sometimes you may need to take some detours along the way depending on the needs of your learners. Listen to your learners: Are you moving too slowly? Are you going too fast? Or are you on the wrong road?There are different ways of getting to your end destination, and understanding your students and their needs will determine the path you take.

The process of planning

Planning involves interpreting the curriculum and working out how we can support our students to learn knowledge and develop skills. So, what might ‘good’ planning look like?

Start with the learner

What do your students know about the topic? What prior knowledge do they have? This isn’t always straightforward as students in your class will bring a range of skills and knowledge to each topic, and your planning needs to reflect this.

The curriculum

Use your curriculum knowledge and understandings and know exactly what you want your students to achieve.

The steps

Break the learning into small steps. Think about the chunks of knowledge and skills the students need to learn and build upon this throughout the lesson and topic.

Learning sequences

Build authentic teaching and learning sequences to support students to learn knowledge, develop skills, and understand and apply concepts.

Learning intentions and success criteria

Planning is enhanced by including purposeful learning intentions and success criteria.

A learning intention states the goal of the lesson. What will you learn?  The success criteria outlines how the students will know they have succeeded. How do you know you have learnt it? How do you know you can now do it?

Learning intention (LI) and success criteria (SC) checklist

• Sharethe LI and SC with your students.
• Make the LI and SC explicitso that students know exactly what is expected.
• Make the LI challenging,but not too difficult. Students need to be learning new knowledge and skills, and experience success in doing so.
• Make the SC measurableso students can easily see if they have been successful or not.
• Provide feedbackto students throughout the lesson, so they know what to do next to achieve the learning intention.

Good planning is essential for quality maths teaching and learning. If you’d like to know more about planning:

• sign-up to The Maths in schools: Explicit teaching in Maths learning modules. This self-paced, professional leaning course offers five modules that are designed around the seven components of explicit teaching. The modules are aligned to the Australian Institute and School Leadership (ASITSL) professional standards, and they include lessons and activities you can use to teach maths concepts from the Australian Curriculum.
• listen to the Maths hub podcast, Episode 1: Purposeful planning. This engaging and informative podcast is hosted by Allan Dougan, the CEO of the Australian Association of Mathematics Teachers (AAMT). Allan chats to expert Dr Emily Ross about planning. The podcast provides practical ideas you can readily use as you plan your maths lessons. A highlight is the discussion on how to incorporate learning intentions and success criteria in creative and flexible ways.
• watch the Explicit teaching in mathematics: purposeful planning webinar. Associate Professor Helen Chick, University of Tasmania, discusses purposeful planning when thinking through and constructing maths lessons using the explicit teaching model and how careful planning of lessons is just as important as teaching the lesson and can enable explicit teaching to be successful.

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

Credit of the article given to The Mathematics Hub

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