Author: IMC

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

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

Most middle and high school math curricula follow a well-defined path:

Pre-algebra → Algebra 1 → Geometry → Algebra 2 → Trig / Precalculus → Calculus

Other middle and high schools prefer an “integrated” curriculum, wherein elements of algebra,

geometry and trigonometry are mixed together over a 3-year or 4-year sequence. However, both of these approaches generally lack a great deal of emphasis on discrete math: topics such as combinatorics, probability, number theory, set theory, logic, algorithms, and graph theory. Because discrete math does not figure prominently in most states’ middle or high school “high-stakes” progress exams, and because it also does not figure prominently on college admissions exams such as the SAT, it is often overlooked.

However, discrete math has become increasingly important in recent years, for a number of reasons:

Discrete math is essential to college-level mathematics and beyond.

Discrete math—together with calculus and abstract algebra—is one of the core components of mathematics at the undergraduate level. Students who learn a significant quantity of discrete math before entering college will be at a significant advantage when taking undergraduate-level math courses.

Discrete math is the mathematics of computing.

The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory. This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects. Indeed, at most universities, an undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree.

Discrete math is very much “real-world” mathematics.

Many students’ complaints about traditional high school math—algebra, geometry, trigonometry, and the like—are, “What good is this for?” The somewhat abstract nature of these subjects often turns off students. By contrast, discrete math, in particular counting and probability, allows students—even at the middle school level—to very quickly explore non-trivial “real world” problems that are challenging and interesting.

Discrete math shows up in most middle and high school math contests.

Prominent math competitions such as the International Maths Olympiad feature discrete math questions as a significant portion of their contests. On harder high school contests, such as the AIME, the quantity of discrete math is even larger. Students who do not have a discrete math background will be at a significant disadvantage in these contests. In fact, one prominent Math coach tells us that he spends nearly 50% of his preparation time with his students covering counting and probability topics, because of their importance in global contests.

Discrete math teaches mathematical reasoning and proof techniques.

Algebra is often taught as a series of formulas and algorithms for students to memorize (for example, the quadratic formula, solving systems of linear equations by substitution, etc.), and geometry is often taught as a series of “definition-theorem-proof” exercises that are often done by rote (for example, the infamous “two-column proof”). While undoubtedly the subject matter being taught is important, the material (at least at the introductory level) does not lend itself to a great deal of creative mathematical thinking. By contrast, with discrete mathematics, students will be thinking flexibly and creatively right out of the box. There are relatively few formulas to memorize; rather, there are a number of fundamental concepts to be mastered and applied in many different ways.

Discrete math is fun.

Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring. Rarely is this the case with most discrete math topics. When we ask students what their favourite topic is, most respond either “combinatorics” or “number theory.” (When we ask them what their least favourite topic is, the overwhelming response is “geometry.”) Simply put, most students find discrete math more fun than algebra or geometry.

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

Credit for the article given to David Patrick

After losing an eye at the age of 5, the 2024 Abel prize winner Michel Talagrand found comfort in mathematics.

French mathematician Michel Talagrand has won the 2024 Abel prize for his work on probability theory and describing randomness. Shortly after he had heard the news, New Scientist spoke with Talagrand to learn more about his mathematical journey.

Alex Wilkins: What does it mean to win the Abel prize?

Michel Talagrand: I think everybody would agree that the Abel prize is really considered like the equivalent of the Nobel prize in mathematics. So it’s something for me totally unexpected, I never, ever dreamed I would receive this prize. And actually, it’s not such an easy thing to do, because there is this list of people who already received it. And on that list, they are true giants of mathematics. And it’s not such a comfortable feeling to sit with them, let me tell you, because it’s clear that their achievements are on an entirely other scale than I am.

What are your attributes as a mathematician?

I’m not able to learn mathematics easily. I have to work. It takes a very long time and I have a terrible memory. I forget things. So I try to work, despite handicaps, and the way I worked was trying to understand really well the simple things. Really, really well, in complete detail. And that turned out to be a successful approach.

Why does maths appeal to you?

Once you are in mathematics, and you start to understand how it works, it’s completely fascinating and it’s very attractive. There are all kinds of levels, you are an explorer. First, you have to understand what people before you did, and that’s pretty challenging, and then you are on your own to explore, and soon you love it. Of course, it is extremely frustrating at the same time. So you have to have the personality that you will accept to be frustrated.

But my solution is when I’m frustrated with something, I put it aside, when it’s obvious that I’m not going to make any more progress, I put it aside and do something else, and I come back to it at a later date, and I have used that strategy with great efficiency. And the reason why it succeeds is the function of the human brain, things mature when you don’t look at them. There are questions which I’ve literally worked on for a period of 30 years, you know, coming back to them. And actually at the end of the 30 years, I still made progress. That’s what is incredible.

How did you get your start?

Now, that’s a very personal story. First, it helps that my father was a maths teacher, and of course that helped. But really, the determining factor is I was unlucky to have been born with a deficiency in my retinas. And I lost my right eye when I was 5 years old. I had multiple retinal detachments when I was 15. I stayed in the hospital a long time, I missed school for six months. And that was extremely traumatic, I lived in constant terror that there will be a next retinal detachment.

To escape that, I started to study. And my father really immensely helped me, you know, when he knew how hard it was, and when I was in hospital, he came to see me every day and he started talking about some simple mathematics, just to keep my brain functioning. I started studying hard mathematics and physics to really, as I say, to fight the terror and, of course, when you start studying, then you become good at it and once you become good, it’s very appealing.

What is it like to be a professional mathematician?

Nobody tells me what I have to do and I’m completely free to use my time and do what I like. That fitted my personality well, of course, and it’s helped me to devote myself totally to my work.

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

*Credit for article given to Alex Wilkins*

With its bright colors, easy-to-learn rules and familiar music, the video game Tetris has endured as a pop culture icon over the last 40 years. Many people, like me, have been playing the game for decades, and it has evolved to adapt to new technologies like game systems, phones and tablets. But until January 2024, nobody had ever been able to beat it.

A teen from Oklahoma holds the Tetris title after he crashed the game on Level 157 and beat the game. Beating it means the player moved the tiles too fast for the game to keep up with the score, causing the game to crash. Artificial intelligence can suggest strategies that allow players to more effectively control the game tiles and slot them into place faster—these strategies helped crown the game’s first winner.

But there’s far more to Tetris than the elusive promise of winning. As a mathematician and mathematics educator, I recognize that the game is based on a fundamental element of geometry, called dynamic spatial reasoning. The player uses these geometric skills to manipulate the game pieces, and playing can both test and improve a player’s dynamic spatial reasoning.

Playing the game

A Russian computer scientist named Alexey Pajitnov invented Tetris in 1984. The game itself is very simple: The Tetris screen is composed of a rectangular game board with dropping geometric figures. These figures are called tetrominoes, made up of four squares connected on their sides in seven different configurations.

The game pieces drop from the top, one at a time, stacking up from the bottom. The player can manipulate each one as it falls by turning or sliding it and then dropping it to the bottom. When a row completely fills up, it disappears and the player earns points.

As the game progresses, the pieces appear at the top more quickly, and the game ends when the stack reaches the top of the board.

Dynamic spatial reasoning

Manipulating the game pieces gives the player an exercise in dynamic spatial reasoning. Spatial reasoning is the ability to visualize geometric figures and how they will move in space. So, dynamic spatial reasoning is the ability to visualize actively moving figures.

The Tetris player must quickly decide where the currently dropping game piece will best fit and then move it there. This movement involves both translation, or moving a shape right and left, and rotation, or twirling the shape in increments of 90 degrees on its axis.

Spatial visualization is partly inherent ability, but partly learned expertise. Some researchers identify spatial skill as necessary for successful problem solving, and it’s often used alongside mathematics skills and verbal skills.

Spatial visualization is a key component of a mathematics discipline called transformational geometry, which is usually first taught in middle school. In a typical transformational geometry exercise, students might be asked to represent a figure by its x and y coordinates on a coordinate graph and then identify the transformations, like translation and rotation, necessary to move it from one position to another while keeping the piece the same shape and size.

Reflection and dilation are the two other basic mathematical transformations, though they’re not used in Tetris. Reflection flips the image across any line while maintaining the same size and shape, and dilation changes the size of the shape, producing a similar figure.

For many students, these exercises are tedious, as they involve plotting many points on graphs to move a figure’s position. But games like Tetris can help students grasp these concepts in a dynamic and engaging way.

Transformational geometry beyond Tetris

While it may seem simple, transformational geometry is the foundation for several advanced topics in mathematics. Architects and engineers both use transformations to draw up blueprints, which represent the real world in scale drawings.

Animators and computer graphic designers use concepts of transformations as well. Animation involves representing a figure’s coordinates in a matrix array and then creating a sequence to change its position, which moves it across the screen. While animators today use computer programs that automatically move figures around, they are all based on translation.

Calculus and differential geometry also use transformation. The concept of optimization involves representing a situation as a function and then finding the maximum or minimum value of that function. Optimization problems often involve graphic representations where the student uses transformations to manipulate one or more of the variables.

Lots of real-world applications use optimization—for example, businesses might want to find out the minimum cost of distributing a product. Another example is figuring out the size of a theoretical box with the largest possible volume.

All of these advanced topics use the same concepts as the simple moves of Tetris.

Tetris is an engaging and entertaining video game, and players with transformational geometry skills might find success playing it. Research has found that manipulating rotations and translations within the game can provide a solid conceptual foundation for advanced mathematics in numerous science fields.

Playing Tetris may lead students to a future aptitude in business analytics, engineering or computer science—and it’s fun. As a mathematics educator, I encourage students and friends to play on.

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

Credit of the article to be given Leah McCoy, The Conversation

Is it possible to deduce the shape of a drum from the sounds it makes? This is the kind of question that Iosif Polterovich, a professor in the Department of Mathematics and Statistics at Université de Montréal, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena involving wave propagation.

Last summer, Polterovich and his international collaborators—Nikolay Filonov, Michael Levitin and David Sher—proved a special case of a famous conjecture in spectral geometry formulated in 1954 by the eminent Hungarian-American mathematician George Pólya.

The conjecture bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.

Pólya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. Until last year, the conjecture was known only for these cases. The disk, despite its apparent simplicity, remained elusive.

“Imagine an infinite floor covered with tiles of the same shape that fit together to fill the space,” Polterovich said. “It can be tiled with squares or triangles, but not with disks. A disk is actually not a good shape for tiling.”

The universality of mathematics

In an article published in the mathematical journal Inventiones Mathematicae, the researchers show that Pólya’s conjecture is true for the disk, a case considered particularly challenging.

Though their result is essentially of theoretical value, their proof method has applications in computational mathematics and numerical computation. The authors are now investigating this avenue.

“While mathematics is a fundamental science, it is similar to sports and the arts in some ways,” Polterovich said.

“Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. And in many cases, beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.”

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

Credit of the article to be given Béatrice St-Cyr-Leroux, University of Montreal

In times of crisis, effective humanitarian aid depends largely on the fast and efficient allocation of resources and personnel. Accurate data about the locations and movements of affected people in these situations is essential for this.

Researchers from the University of Tokyo, working with the World Bank, have produced a framework to analyse and visualize population mobility data, which could help in such cases. The research is publishedin the journal Scientific Reports.

Wars, famines, outbreaks, natural disasters—there are unfortunately many reasons why populations might be forced or feel compelled to leave their homes in search of refuge elsewhere, and these cases continue to grow.

The United Nations estimated in 2023 that there were more than 100 million forcibly displaced people in the world. More than 62 million of these individuals are considered internally displaced people (IDPs), those in particularly vulnerable situations due to being stuck within the borders of their countries, from which they might be trying to flee.

The circumstances that displace populations are inevitably chaotic and certainly, but not exclusively, in cases of conflict, information infrastructure can be impeded. So, authorities and agencies trying to get a handle on crises are often operating with limited data on the people they are trying to help. But the lack of data alone is not the only problem; being able to easily interpret data, so that nonexperts can make effective decisions based on it, is also an issue, especially in rapidly evolving situations where the stakes, and tensions, are high.

“It’s practically impossible to provide aid agencies and others with accurate real time data on affected populations. The available data will often be too fragmented to be useful directly,” said Associate Professor Yuya Shibuya from the Interfaculty Initiative in Information Studies.

“There have been many efforts to use GPS data for such things, and in normal situations, it has been shown to be useful to model population behaviour. But in times of crisis, patterns of predictability break down and the quality of data decreases.

“As data scientists, we explore ways to mitigate these problems and have developed a tracking framework for monitoring population movements by studying IDPs displaced in Russia’s invasion of Ukraine in 2022.”

Even though Ukraine has good enough network coverage throughout to acquire GPS data, the data generated is not representative of the entire population. There are also privacy concerns, and likely other significant gaps in data due to the nature of conflict itself. As such, it’s no trivial task to model the way populations move.

Shibuya and her team had access to a limited dataset which covered the period a few weeks before and a few weeks after the initial invasion on Feb. 24, 2022. This data contained more than 9 million location records from more than 100,000 anonymous IDPs who opted in to share their location data.

“From these records, we could estimate people’s home locations at the regional level based on regular patterns in advance of the invasion. To make sure this limited data could be used to represent the entire population, we compared our estimates to survey data from the International Organization for Migration of the U.N.,” said Shibuya.

“From there, we looked at when and where people moved just prior to and for some time after the invasion began. The majority of IDPs were from the capital, Kyiv, and some people left as early as five weeks before Feb. 24, perhaps in anticipation, though it was two weeks after that day that four times as many people left. However, a week later still, there was evidence some people started to return.”

That some people return to afflicted areas is just one factor that confounds population mobility models—in actual fact, people may move between locations, sometimes multiple times. Trying to represent this with a simple map with arrows to show populations could get cluttered fast. Shibuya’s team used color-coded charts to visualize its data, which allow you to see population movements in and out of regions at different times, or dynamic data, in a single image.

“WE want visualizations like these to help humanitarian agencies gauge how to allocate human resources and physical resources like food and medicine. As they tell you about dynamic changes in populations, not just A to B movements, WEthink it could mean aid gets to where it’s needed and when it’s needed more efficiently, reducing waste and overheads,” said Shibuya.

“Another thing we found that could be useful is that people’s migration patterns vary, and socioeconomic status seems to be a factor in this. People from more affluent areas tended to move farther from their homes than others. There is demographic diversity and good simulations ought to reflect this diversity and not make too many assumptions.”

The team worked with the World Bank on this study, as the international organization could provide the data necessary for the analyses. They hope to look into other kinds of situations too, such as natural disasters, political conflicts, environmental issues and more. Ultimately, by performing research like this, Shibuya hopes to produce better general models of human behaviour in crisis situations in order to alleviate some of the impacts those situations can create.

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

Credit of the article given to University of Tokyo

There’s a strange trend in mathematics education in England. Maths is the most popular subject at A-level since overtaking English in 2014. It’s taken by around 85,000 and 90,000 students a year.

But many universities – particularly lower-tariff institutions, which accept students with lower A-level grades – are recruiting far fewer students for maths degrees. There’s been a 50% drop in numbers of maths students at the lowest tariff universities over the five years between 2017 and 2021. As a result, some universities are struggling to keep their mathematics departments open.

The total number of students studying maths has remained largely static over the last decade. Prestigious Russell Group universities which require top A-level grades have increased their numbers of maths students.

This trend in degree-level mathematics education is worrying. It restricts the accessibility of maths degrees, especially to students from poorer backgrounds who are most likely to study at universities close to where they live. It perpetuates the myth that only those people who are unusually gifted at mathematics should study it – and that high-level maths skills are not necessary for everyone else.

Research carried out in 2019 by King’s College London and Ipsos found that half of the working age population had the numeracy skills expected of a child at primary school. Just as worrying was that despite this, 43% of those polled said “they would not like to improve their numeracy skills”. Nearly a quarter (23%) stated that “they couldn’t see how it would benefit them”.

Mathematics has been fundamental in recent technological developments such as quantum computing, information security and artificial intelligence. A pipeline of more mathematics graduates from more diverse backgrounds will be essential if the UK is to remain a science and technology powerhouse into the future.

But maths is also vital to a huge range of careers, including in business and government. In March 2024, campaign group Protect Pure Maths held a summit to bring together experts from industry, academia and government to discuss concerns about poor maths skills and the continuing importance of high-quality mathematics education.

Prior to the summit, the London Mathematical Society commissioned a survey of over 500 businesses to gauge their concerns about the potential lack of future graduates with strong mathematical skills.

They found that 72% of businesses agree they would benefit from more maths graduates entering the workforce. And 75% would worry if UK universities shrunk or closed their maths departments.

A 2023 report on MPs’ staff found that skills in Stem subjects (science, technology, engineering and mathematics) were particularly hard to find among those who worked in Westminster. As many as 90% of those who had taken an undergraduate degree had studied humanities or social sciences. While these subject backgrounds are valuable, the lack of specialised maths skills is stark.

Limited options

The mathematics department at Oxford Brookes has closed and other universities have seen recruitment reductions or other cuts. The resulting maths deserts will remove the opportunity for students to gain a high-quality mathematics education in their local area. Universities should do their best to keep these departments open.

This might be possible if the way that degrees are set up changes. For many degree courses in countries such as the US and Australia, students are able to take a broad selection of subjects, from science and maths subjects through to the humanities. Each are taught in their respective academic departments. This allows students to gain advanced knowledge and see how each field feeds into others.

This is scarcely possible in the UK, where students must choose a specialist and narrow degree programme at age 18.

Another possible solution would be to put core mathematics modules in degree disciplines that rely so heavily on it – such as engineering, economics, chemistry, physics, biology and computer science – and have them taught by specialist mathematicians. This would help keep mathematics departments open, while also ensuring that general mathematical literacy improves in the UK.

The relevance of mathematics and its vast range of applications would be abundantly clear, better equipping every student with the necessary mathematical skills the workforce needs.

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

Credit of the article given to Neil Saunders, The Conversation

Bob Moses, who helped register Black residents to vote in Mississippi during the Civil Rights Movement, believed civil rights went beyond the ballot box. To Moses, who was a teacher as well as an activist, math literacy is a civil right: a requirement to earning a living wage in modern society. In 1982, he founded the Algebra Project to ensure that “students at the bottom get the math literacy they need.”

As a researcher who studies ways to improve the math experiences of students, we believe a new approach that expands access to algebra may help more students get the math literacy Moses, who died in 2021, viewed as so important. It’s a goal districts have long been struggling to meet.

Efforts to improve student achievement in algebra have been taking place for decades. Unfortunately, the math pipeline in the United States is fraught with persistent opportunity gaps. According to the Nation’s Report Card—a congressionally mandated project administered by the Department of Education—in 2022 only 29% of U.S. fourth graders and 20% of U.S. eighth graders were proficient in math. Low-income students, students of colour and multilingual learners, who tend to have lower scores on math assessments, often do not have the same access as others to qualified teachers, high-quality curriculum and well-resourced classrooms.

A new approach

The Dallas Independent School District—or Dallas ISD—is gaining national attention for increasing opportunities to learn by raising expectations for all students. Following in the footsteps of more than 60 districts in the state of Washington, in 2019 the Dallas ISD implemented an innovative approach of having students be automatically enrolled rather than opt in to honours math in middle school.

Under an opt-in policy, students need a parent or teacher recommendation to take honours math in middle school and Algebra 1 in eighth grade. That policy led both to low enrolment and very little diversity in honours math. Some parents, especially those who are Black or Latino, were not aware how to enroll their students in advanced classes due to a lack of communication in many districts.

In addition, implicit bias, which exists in all demographic groups, may influence teachers’ perceptions of the behaviour and academic potential of students, and therefore their subsequent recommendations. Public school teachers in the U.S. are far less racially and ethnically diverse than the students they serve.

Dallas ISD’s policy overhaul aimed to foster inclusivity and bridge educational gaps among students. Through this initiative, every middle school student, regardless of background, was enrolled in honours math, the pathway that leads to taking Algebra 1 in eighth grade, unless they opted out.

Flipping the switch from opt-in to opt-out led to a dramatic increase in the number of Black and Latino learners, who constitute the majority of Dallas students. And the district’s overall math scores remained steady. About 60% of Dallas ISD eighth graders are now taking Algebra 1, triple the prior level. Moreover, more than 90% are passing the state exam.

Efforts spread

Other cities are taking notice of the effects of Dallas ISD’s shifting policy. The San Francisco Unified School District, for example, announced plans in February 2024 to implement Algebra 1 in eighth grade in all schools by the 2026-27 school year.

In fall 2024, the district will pilot three programs to offer Algebra 1 in eighth grade. The pilots range from an opt-out program for all eighth graders—with extra support for students who are not proficient—to a program that automatically enrolls proficient students in Algebra 1, offered as an extra math class during the school day. Students who are not proficient can choose to opt in. Nationwide, however, districts that enroll all students in Algebra 1 and allow them to opt out are still in the minority. And some stopped offering eighth grade Algebra 1 entirely, leaving students with only pre-algebra classes. Cambridge, Massachusetts—the city in which Bob Moses founded the Algebra Project—is among them.

Equity concerns linger

Between 2017 and 2019, district leaders in the Cambridge Public Schools phased out the practice of placing middle school students into “accelerated” or “grade-level” math classes. Few middle schools in the district now offer Algebra 1 in eighth grade.

The policy shift, designed to improve overall educational outcomes, was driven by concerns over significant racial disparities in advanced math enrollment in high school. Completion of Algebra 1 in eighth grade allows students to climb the math ladder to more difficult classes, like calculus, in high school. In Cambridge, the students who took eighth grade Algebra 1 were primarily white and Asian; Black and Latino students enrolled, for the most part, in grade-level math.

Some families and educators contend that the district’s decision made access to advanced math classes even more inequitable. Now, advanced math in high school is more likely to be restricted to students whose parents can afford to help them prepare with private lessons, after-school programs or private schooling, they said.

While the district has tried to improve access to advanced math in high school by offering a free online summer program for incoming ninth graders, achievement gaps have remained persistently wide.

Perhaps striking a balance between top-down policy and bottom-up support will help schools across the U.S. realize the vision Moses dreamed of in 1982 when he founded the Algebra Project: “That in the 21st century every child has a civil right to secure math literacy—the ability to read, write and reason with the symbol systems of mathematics.”

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

Credit of the article given to Liza Bondurant, The Conversation

My intention with this article is to give an intuitive and non-technical introduction to the field of evolutionary algorithms, particularly with regards to optimisation.

If I get you interested, I think you’re ready to go down the rabbit hole and simulate evolution on your own computer. If not … well, I’m sure we can still be friends.

Survival of the fittest

According to Charles Darwin, the great evolutionary biologist, the human race owes its existence to the phenomenon of survival of the fittest. And being the fittest doesn’t necessarily mean the biggest physical presence.

Once in high school, my lunchbox was targeted by swooping eagles, and I was reduced to a hapless onlooker. The eagle, though smaller in form, was fitter than me because it could take my lunch and fly away – it knew I couldn’t chase it.

As harsh as it sounds, look around you and you will see many examples of the rule of the jungle – the fitter survive while the rest gradually vanish.

The research area, now broadly referred to as Evolutionary Algorithms, simulates this behaviour on a computer to find the fittest solutions to a number of different classes of problems in science, engineering and economics.

The area in which this area is perhaps most widely used is known as “optimisation”.

Optimisation is everywhere

Your high school maths teacher probably told you the shortest way to go from point A to point B was along the straight-line joining A and B. Your mum told you that you should always get the right amount of sleep.

And, if you have lived on your own for any length of time, you’ll be familiar with the ever-increasing cost of living versus the constant income – you always strive to minimise the expenditures, while ensuring you are not malnourished.

Whenever you undertake an activity that seeks to minimise or maximise a well-defined quantity such as distance or the vague notion of the right amount of sleep, you are optimising.

Look around you right now and you’ll see optimisation in play – your Coke can is shaped like that for a reason, a water droplet is spherical for a reason, you wash all your dishes together in the dishwasher for a reason.

Each of these strives to save on something: volume of material of the Coke can, and energy and water, respectively, in the above cases.

So, we can safely say optimisation is the act of minimising or maximising a quantity. But that definition misses an important detail: there is always a notion of subject to or satisfying some conditions.

You must get the right amount of sleep, but you also must do your studies and go for your music lessons. Such conditions, which you also have to adhere to, are known as “constraints”. Optimisation with constraints is then collectively termed “constrained optimisation”.

After constraints comes the notion of “multi-objective optimisation”. You’ll usually have more than one thing to worry about (you must keep your supervisor happy with your work and keep yourself happy and also ensure that you are working on your other projects). In many cases these multiple objectives can be in conflict.

Evolutionary algorithms and optimisation

Imagine your local walking group has arranged a weekend trip for its members and one of the activities is a hill climbing exercise. The problem assigned to your group leader is to identify who among you will reach the hill in the shortest time.

There are two approaches he or she could take to complete this task: ask only one of you to climb up the hill at a time and measure the time needed or ask all of you to run all at once and see who reaches first.

That second method is known as the “population approach” of solving optimisation problems – and that’s how evolutionary algorithms work. The “population” of solutions are evolved over a number of iterations, with only the fittest solutions making it to the next.

This is analogous to the champion girl from your school making to the next round which was contested among champions from other schools in your state, then your country, and finally winning among all the countries.

Or, in our above scenario, finding who in the walking group reaches the hill top fastest, who would then be denoted as the fittest.

In engineering, optimisation needs are faced at almost every step, so it’s not surprising evolutionary algorithms have been successful in that domain.

Design optimisation of scramjets

At the Multi-disciplinary Design Optimisation Group at the University of New South Wales, my colleagues and I are involved in the design optimisation of scramjets, as part of the SCRAMSPACE program. In this, we’re working with colleagues from the University of Queensland.

Our evolutionary algorithms-based optimisation procedures have been successfully used to obtain the optimal configuration of various components of a scramjet.

There are, at the risk of sounding over-zealous, no limits to the application of evolutionary algorithms.

Has this whetted your appetite? Have you learnt something new today?

If so, I’m glad. May the force be with you!

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

Credit of the article given to Amit Saha