Category: mathematics

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A rare appearance by enigmatic Shinichi Mochizuki brings faint optimism about his famously impenetrable work

Nearly four years after Shinichi Mochizuki unveiled an imposing set of papers that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity — although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University’s Research Institute for Mathematical Sciences (RIMS).

Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego. Although at first Mochizuki’s papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae, experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.

And Jeffrey Lagarias, a number theorist at the University of Michigan in Ann Arbor, says that he got far enough to see that Mochizuki’s work is worth the effort. “It has some revolutionary new ideas,” he says.

Still, Kedlaya says that the more he delves into the proof, the longer he thinks it will take to reach a consensus on whether it is correct. He used to think that the issue would be resolved perhaps by 2017. “Now I’m thinking at least three years from now.”

Others are even less optimistic. “The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me,” says mathematician Vesselin Dimitrov of Yale University in New Haven, Connecticut. “Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature.”

The abc proof

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers — whole numbers that cannot be evenly divided by any smaller number except by 1. The conjecture comes in a number of different forms, but explains how the primes that divide two numbers, a and b, are related to those that divide their sum, c.

If Mochizuki’s proof is correct, it would have repercussions across the entire field, says Dimitrov. “When you work in number theory, you cannot ignore the abc conjecture,” he says. “This is why all number theorists eagerly wanted to know about Mochizuki’s approach.” For example, Dimitrov showed in January how, assuming the correctness of Mochizuki’s proof, one might be able to derive many other important results, including a completely independent proof of the celebrated Fermat’s last theorem.

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

Mochizuki in the room

The Kyoto workshop followed on the heels of one held last December in Oxford, UK. Mochizuki did not attend that first meeting, although he answered the audience’s questions over a Skype video link. This time, having him in the room — and hearing him present some of the materials himself — was helpful, says Taylor Dupuy, a mathematician at the Hebrew University of Jerusalem who participated in both workshops.

There are now around ten mathematicians who are putting substantial effort into digesting the material — up from just three before the Oxford workshop, says Ivan Fesenko, a mathematician at the University of Nottingham, UK, who co-organized both workshops. The group includes younger researchers, such as Dupuy.

In keeping with his reputation for being a very private person, Mochizuki — who is said to never eat meals in the presence of colleagues — did not take part in the customary mingling and social activities at the Kyoto meeting, according to several sources. And although he was unfailingly forthcoming in answering questions, it was unclear what he thought of the proceedings. “Mochizuki does not give a lot away,” Kedlaya says. “He’s an excellent poker player.”

Fellow mathematicians have criticized Mochizuki for his refusal to travel. After he posted his papers, he turned down multiple offers to spend time abroad and lecture on his ideas. Although he spent much of his youth in the United States, he is now said to rarely leave the Kyoto area. (Mochizuki does not respond to requests for interviews, and the workshop’s website contained the notice: “Activities aimed at interviewing or media coverage of any sort within the facilities of RIMS, Kyoto University, will not be accepted.”)

“He is very level-headed,” says another workshop participant, who did not want to be named. “The only thing that frustrates him is people making rash judgemental comments without understanding any details.”

Still, Dupuy says, “I think he does take a lot of the criticism about him really personally. I’m sure he’s sick of this whole thing, too.”

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

Credit of the article given to Davide Castelvecchi & Nature magazine

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

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

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

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

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

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

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

Julia Robinson (1919-1985)

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

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

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

Emmy Noether (1882-1935)

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

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

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

Ada Lovelace (1815-1852)

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

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

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

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

Credit of the article given to Avery Carr

The whole is equivalent to the sum of its parts

Which is bigger, 10 or 7?

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

How do you know?

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

Perhaps you will indulge me and delve a little deeper:

In how many ways do you know?

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

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

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

Part-whole

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

What are there more of:

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

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

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

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

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

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

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

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

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

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

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

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

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

How do you know?

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

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

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

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

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

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

Credit for the article given to Tabitha Gould

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*

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

Knot theory is linked to many other branches of science, including those that tell us about the cosmos.

The mathematical study of knots started with a mistake. In the 1800s, mathematician and physicist William Thomson, also known as Lord Kelvin, suggested that the elemental building blocks of matter were knotted vortices in the ether: invisible microscopic currents in the background material of the universe. His theory dropped by the wayside fairly quickly, but this first attempt to classify how curves could be knotted grew into the modern mathematical field of knot theory. Today, knot theory is not only connected to many branches of theoretical mathematics but also to other parts of science, like physics and molecular biology. It’s not obvious what your shoelace has to do with the shape of the universe, but the two may be more closely related than you think.

As it turns out, a tangled necklace offers a better model of a knot than a shoelace: to a mathematician, a knot is a loop in three-dimensional space rather than a string with loose ends. Just as a physical loop of string can stretch and twist and rotate, so can a mathematical knot – these loops are floppy rather than fixed. If we studied strings with free ends, they could wiggle around and untie themselves, but a loop stays knotted unless it’s cut.

Most questions in knot theory come in two varieties: sorting knots into classes and using knots to study other mathematical objects. I’ll try to give a flavour of both, starting with the simplest possible example: the unknot.

Draw a circle on a piece of paper. Congratulations, you’ve just constructed an unknot! This is the name for any loop in three-dimensional space that is the boundary of a disc. When you draw a circle on a piece of paper, you can see this disc as the space inside the circle, and your curve continues to be an unknot if you crumple the paper up, toss it through the air, flatten it out and then do some origami. As long as the disc is intact, no matter how distorted, the boundary is always an unknot.

Things get more interesting when you start with just the curve. How can you tell if it’s an unknot? There may secretly be a disc that can fill in the loop, but with no limits on how deformed the disc could be, it’s not clear how you can figure this out.

Two unknots

It turns out that this question is both hard and important: the first step in studying complicated objects is distinguishing them from simple ones. It’s also a question that gets answered inside certain bacterial cells each time they replicate. In the nuclei of these cells, the DNA forms a loop, rather than a strand with loose ends, and sometimes these loops end up knotted. However, the DNA can replicate only when the loop is an unknot, so the basic life processes of the cell require a process for turning a potentially complicated loop into an unknotted one.

A class of proteins called topoisomerases unknot tangled loops of DNA by cutting a strand, moving the free ends and then reattaching them. In a mathematical context, this operation is called a “crossing change”, and it’s known that any loop can be turned into the unknot by some number of crossing changes. However, there’s a puzzle in this process, since random crossing changes are unlikely to simplify a knot. Each topoisomerase operates locally, but collectively they’re able to reliably unknot the DNA for replication. Topoisomerases were discovered more than 50 years ago, but biologists are still studying how they unknot DNA so effectively.

When mathematicians want to identify a knot, they don’t turn to a protein to unknot it for them.  Instead, they rely on invariants, mathematical objects associated with knots. Some invariants are familiar things like numbers, while others are elaborate algebraic structures. The best invariants have two properties: they’re practical to compute, given the input of a specific knot, and they distinguish many different classes of knots from each other. It’s easy to define an invariant with only one of these properties, but a computable and effective knot invariant is a rare find.

The modern era of knot theory began with the introduction of an invariant called the Jones Polynomial in the 1980s. Vaughan Jones was studying statistical mechanics when he discovered a process that assigns a polynomial – a type of simple algebraic expression – to any knot. The method he used was technical, but the essential feature is that no amount of wiggling, stretching or twisting changes the output. The Jones Polynomial of an unknot is 1, no matter how complicated the associated disc might be.

Jones’s discovery caught the attention of other researchers, who found simpler techniques for computing the same polynomial. The result was an invariant that satisfies both the conditions listed above: the Jones Polynomial can be computed from a drawing of a knot on paper, and many thousands of knots can be distinguished by the fact that they have different Jones Polynomials.

However, there are still many things we don’t know about the Jones Polynomial, and one of the most tantalising questions is which knots it can detect. Most invariants distinguish some knots while lumping others together, and we say an invariant detects a knot if all the examples sharing a certain value are actually deformations of each other. There are certainly pairs of distinct knots with the same Jones Polynomial, but after decades of study, we still don’t know whether any knot besides the unknot has the polynomial 1. With computer assistance, experts have examined nearly 60 trillion examples of distinct knots without finding any new knots whose Jones Polynomials equal 1.

The Jones Polynomial has applications beyond knot detection. To see this, let’s return to the definition of an unknot as a loop that bounds a disc. In fact, every knot is the boundary of some surface – what distinguishes an unknot is that this surface is particularly simple. There’s a precise way to rank the complexity of surfaces, and we can use this to rank the complexity of knots. In this classification, the simplest knot is the unknot, and the second simplest is the trefoil, which is shown below.

Trefoil knot

To build a surface with a trefoil boundary, start with a strip of paper. Twist it three times and then glue the ends together. This is more complicated than a disc, but still pretty simple. It also gives us a new question to investigate: given an arbitrary knot, where does it fit in the ranking of knot complexity? What’s the simplest surface it can bound? Starting with a curve and then hunting for a surface may seem backwards, but in some settings, the Jones Polynomial answers this question: the coefficients of the knot polynomial can be used to estimate the complexity of the surfaces it bounds.

Joan Licata

Knots also help us classify other mathematical objects. We can visually distinguish the two-dimensional surface of sphere from the surface a torus (the shape of a ring donut), but an ant walking on one of these surfaces might need knot theory to tell them apart. On the surface of a torus, there are loops that can’t be pulled any tighter, while any loop lying on a sphere can contract to a point.

We live inside a universe of three physical dimensions, so like the ant on a surface, we lack a bird’s eye view that could help us identify its global shape. However, we can ask the analogous question: can each loop we encounter shrink without breaking, or is there a shortest representative? Mathematicians can classify three-dimensional spaces by the existence of the shortest knots they contain. Presently, we don’t know if some knots twisting through the universe are unfathomably long or if every knot can be made as small as one of Lord Kelvin’s knotted vortices.

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*Credit for article given to Joan Licata*

Artificial intelligence firm DeepMind has tackled games like Go and Starcraft, but now it is turning its attention to more sober affairs: how to solve school-level maths problems.

Researchers at the company tasked an AI with teaching itself to solve arithmetic, algebra and probability problems, among others. It didn’t do a very good job: when the neural network was tested on a maths exam taken by 16-year-olds in the UK, it got just 14 out of 40 questions correct, or the equivalent of an E grade.

There were also strange quirks in the AI’s ability. For example, it could successfully add up 1+1+1+1+1+1 to make 6, but failed when an extra 1 was added. On the other hand, it gave the correct answer for longer sequences and much bigger numbers.

Other oddities included the ability to correctly answer 68 to the question “calculate 17×4.”, but when the full stop was removed, the answer came out at 69.

Puzzling behaviour

The DeepMind researchers concede they don’t have a good explanation for this behaviour. “At the moment, learning systems like neural networks are quite bad at doing ‘algebraic reasoning’,” says David Saxton, one of the team behind the work.

Despite this, it is still worth trying to teach a machine to solve maths problems, says Marcus du Sautoy, a mathematician at the University of Oxford.

“There are already algorithms out there to do these problems much faster, much better than machine-learning algorithms, but that’s not the point,” says du Sautoy. “They are setting themselves a different target – we want to start from nothing, by being told whether you got that one wrong, that one right, whether it can build up how to do this itself. Which is fascinating.”

An AI capable of solving advanced mathematics problems could put him out of a job, says du Sautoy. “That’s my fear. It may not take too much for an AI to get maturity in this world, whereas a maturity in the musical or visual or language world might be much harder for it. So I do think my subject is vulnerable.”

However, he takes some comfort that machine learning’s general weakness in remaining coherent over a long form – such as a novel, rather than a poem – will keep mathematicians safe for now. Creating mathematical proofs, rather than solving maths problems for 16-year-olds, will be difficult for machines, he says.

Noel Sharkey at the University of Sheffield, UK, says the research is more about finding the limits of machine-learning techniques, rather than promoting advancements in mathematics.

The interesting thing, he says, will be to see how the neural networks can adapt to challenges outside of those they were trained on. “The big question is to ask how well they can generalise to novel examples that were not in the training set. This has the potential to demonstrate formal limits to what this type of learning is capable of.”

Saxton says training a neural network on maths problems could help provide AI with reasoning skills for other applications.

“Humans are good at maths, but they are using general reasoning skills that current artificial learning systems don’t possess,” he says. “If we can develop models that are good at solving these problems, then these models would likely be using general skills that would be good at solving other hard problems in AI as well.”

He hopes the work could make a small contribution towards more general mathematical AIs that could tackle things such as proving theorems.

The DeepMind team has published its data set of maths questions, and encouraged people to train their own AI.

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*Credit for article given to Adam Vaughan*

It’s a bit more complicated than this

Forget your times tables – mathematicians have found a new, faster way to multiply two numbers together. The method, which works only for whole numbers, is a landmark result in computer science. “This is big news,” says Joshua Cooper at the University of South Carolina.

To understand the new technique, which was devised by David Harvey at the University of New South Wales, Australia, and Joris van der Hoeven at the Ecole Polytechnique near Paris, France, it helps to think back to the longhand multiplication you learned at school.

We write down two numbers, one on top of the other, and then painstakingly multiply each digit of one by each digit of the other, before adding all the results together. “This is an ancient algorithm,” says Cooper.

If your two numbers each have n digits, this way of multiplying will require roughly n² individual calculations. “The question is, can you do better?” says Cooper.

Lots of logs

Starting in the 1960s, mathematicians began to prove that they could. First Anatoly Karatsuba found an algorithm that could turn out an answer in no more than n^1.58 steps, and in 1971, Arnold Schönhage and Volker Strassen found a way to peg the number of steps to the complicated expression n*(log(n))*log(log(n)) – here “log” is short for logarithm.

These advances had a major impact on computing. Whereas a computer using the longhand multiplication method would take about six months to multiply two billion-digit numbers together, says Harvey, the Schönhage-Strassen algorithm can do it in 26 seconds.

The landmark 1971 paper also suggested a possible improvement, a tantalising prediction that multiplication might one day be possible in no more than n*log(n) steps. Now Harvey and van der Hoeven appear to have proved this is the case. “It finally appears to be possible,” says Cooper. “It passes the smell test.”

“If the result is correct, it’s a major achievement in computational complexity theory,” says Fredrik Johansson at INRIA, the French research institute for digital sciences, in Bordeaux. “The new ideas in this work are likely to inspire further research and could lead to practical improvements down the road.”

Cooper also praises the originality of the research, although stresses the complexity of the mathematics involved. “You think, jeez, I’m just multiplying two integers, how complicated can it get?” says Cooper. “But boy, it gets complicated.”

So, will this make calculating your tax returns any easier? “For human beings working with pencil and paper, absolutely not,” says Harvey. Indeed, their version of the proof only works for numbers with more than 10 to the power of 200 trillion trillion trillion digits. “The word ‘astronomical’ falls comically short in trying to describe this number,” says Harvey.

While future improvements to the algorithm may extend the proof to more humdrum numbers only a few trillion digits long, Cooper thinks its real value lies elsewhere. From a theoretical perspective, he says, this work allows programmers to provide a definitive guarantee of how long a certain algorithm will take. “We are optimistic that our new paper will allow us to achieve further practical speed-ups,” says van der Hoeven.

Harvey thinks this may well be the end of the story, with no future algorithm capable of beating n*log(n). “I would be extremely surprised if this turned out to be wrong,” he says, “but stranger things have happened.”

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*Credit for article given to Gilead Amit*

The number 33 has surprising depth

Add three cubed numbers, and what do you get? It is a question that has puzzled mathematicians for centuries.

In 1825, a mathematician known as S. Ryley proved that any fraction could be represented as the sum of three cubes of fractions. In the 1950s, mathematician Louis Mordell asked whether the same could be done for integers, or whole numbers. In other words, are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

We still don’t know. “It’s long been clear that there are maths problems that are easy to state, but fiendishly hard to solve,” says Andrew Booker at the University of Bristol, UK – Fermat’s last theorem is a famous example.

Booker has now made another dent in the cube problem by finding a sum for the number 33, previously the lowest unsolved example. He used a computer algorithm to search for a solution:

33 = 8,866,128,975,287,528³ + (-8,778,405,442,862,239)³ + (-2,736,111,468,807,040)³

To cut down calculation time, the program eliminated certain combinations of numbers. “For instance, if x, y and z are all positive and large, then there’s no way that x³ + y³ + z³ is going to be a small number,” says Booker. Even so, it took 15 years of computer-processing time and three weeks of real time to come up with the result.

For some numbers, finding a solution to the equation k = x³ + y³ + z³ is simple, but others involve huge strings of digits. “It’s really easy to find solutions for 29, and we know a solution for 30, but that wasn’t found until 1999, and the numbers were in the millions,” says Booker.

Another example is for the number 3, which has two simple solutions: 1³ + 1³ + 1³  and 4³ + 4³ + (-5) ³ . “But to this day, we still don’t know whether there are more,” he says.

There are certain numbers that we know definitely can’t be the sum of three cubes, including 4, 5, 13, 14 and infinitely many more.

The solution to 74 was only found in 2016, which leaves 42 as the only number less than 100 without a possible solution. There are still 12 unsolved numbers less than 1000.

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*Credit for article given to Donna Lu*