Category: mathematics

Whether you are sharing a cake or a coastline, maths can help make sure everyone is happy with their cut, says Katie Steckles.

One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

When splitting between two parties, you might know a simple rule, proven to be mathematically optimal: I cut, you choose. One person divides the cake (or whatever it is) and the other gets to pick which piece they prefer.

Since the person cutting the cake doesn’t choose which piece they get, they are incentivised to cut the cake fairly. Then when the other person chooses, everyone is satisfied – the cutter would be equally happy with either piece, and the chooser gets their favourite of the two options.

This results in what is called an envy-free allocation – neither participant can claim they would rather have the other person’s share. This also takes care of the problem of non-homogeneous objects: if some parts of the cake are more desirable, the cutter can position their cut so the two pieces are equal in value to them.

What if there are more people? It is more complicated, but still possible, to produce an envy-free allocation with several so-called fair-sharing algorithms.

Let’s say Ali, Blake and Chris are sharing a cake three ways. Ali cuts the cake into three pieces, equal in value to her. Then Blake judges if there are at least two pieces he would be happy with. If Blake says yes, Chris chooses a piece (happily, since he gets free choice); Blake chooses next, pleased to get one of the two pieces he liked, followed by Ali, who would be satisfied with any of the pieces. If Blake doesn’t think Ali’s split was equitable, Chris looks to see if there are two pieces he would take. If yes, Blake picks first, then Chris, then Ali.

If both Blake and Chris reject Ali’s initial chop, then there must be at least one piece they both thought was no good. This piece goes to Ali – who is still happy, because she thought the pieces were all fine – and the remaining two pieces get smooshed back together (that is a mathematical term) to create one piece of cake for Blake and Chris to perform “I cut, you choose” on.

While this seems long-winded, it ensures mathematically optimal sharing – and while it does get even more complicated, it can be extended to larger groups. So whether you are sharing a treat or a divorce settlement, maths can help prevent arguments.

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*Credit for article given to Katie Steckles*

By rapidly analysing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviours.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper, published in the Proceedings of the National Academy of Sciences, they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

“Computer algorithms are increasingly dominant in scientific research, a practice now broadly called ‘AI for Science,'” Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

“However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so.”

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

“On a philosophical level, our work explores the interplay between algorithms and mathematicians,” Elimelech and Kaminer explained. “Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants.”

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

“The ‘conservative matrix field’ is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad,” Elimelech and Kaminer explained. “In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

“Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications.”

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

“The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space),” Elimelech and Kaminer said. “Finding non-trivial matrix fields that are also conservative is challenging.”

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

“Each sequence represents a path hidden in the conservative matrix field,” Elimelech and Kaminer explained. “From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC, an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible.”

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a “virtual lab.” Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

“Such hypotheses, or conjectures, are what drives mathematical research forward,” Elimelech and Kaminer said. “The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing.”

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project’s early days and relied on contributions by a large network of volunteers.

“Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research,” Elimelech and Kaminer said. “We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website. This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper.”

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

“Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant,” Elimelech and Kaminer said.

“Another example is the Riemann zeta function, central in number theory, with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians.”

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

“Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise,” Elimelech and Kaminer added. “Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics,” inspired by the long history and impact of user facilities for experimental physics.

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Credit of the article given to Ingrid Fadelli , Phys.org

The purpose of education is to ensure that students acquire the skills necessary for succeeding in a world that is constantly changing. Self-assessment, or teaching students how to examine and evaluate their own learning and cognitive processes, has proven to be an effective method, and this competence is partly based on metacognitive knowledge.

A new study conducted at the University of Eastern Finland shows that metacognitive knowledge, i.e., awareness of one’s cognitive processes, is also a key factor in the learning of mathematics. The work is published in the journal Cogent Education.

The study explored thinking skills and possible grade-level differences in children attending comprehensive school in Finland. The researchers investigated 6th, 7th and 9th graders’ metacognitive knowledge in the context of mathematics.

“The study showed that ninth graders excelled at explaining their use of learning strategies, while 7th graders demonstrated proficiency in understanding when and why certain strategies should be used. No other differences between grade levels were observed, which highlights the need for continuous support throughout the learning path,” says Susanna Toikka of the University of Eastern Finland, the first author of the article.

The findings emphasize the need to incorporate elements that support metacognitive knowledge into mathematics learning materials, as well as into teachers’ pedagogical practices.

Self-assessment and understanding of one’s own learning help to face new challenges

Metacognitive knowledge helps students not only to learn mathematics, but also more broadly in self-assessment and lifelong learning. Students who can assess their own learning and understanding are better equipped to face new challenges and adapt to changing environments. Such skills are crucial for lifelong learning, as they enable continuous development and learning throughout life.

“Metacognitive knowledge is a key factor in learning mathematics and problem-solving, but its significance also extends to self-assessment and lifelong learning,” says Toikka.

In schools, metacognitive knowledge can be effectively developed as part of education. Based on earlier studies, Toikka and colleagues have developed a combination of frameworks for metacognitive knowledge, which helps to identify students’ needs for development regarding metacognitive knowledge by offering an alternative perspective to that of traditional developmental psychology.

“This also supports teachers in promoting students’ metacognitive knowledge. Teachers can use the combination of frameworks to design and implement targeted interventions that support students’ skills in lifelong learning.”

According to Toikka, the combination of frameworks enhances understanding of metacognitive knowledge and helps to identify areas where individual support is needed: “This type of understanding is crucial for the development of metacognitive knowledge among diverse learners.”

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Credit of the article given to University of Eastern Finland

Not content with shapes in two or three dimensions, mathematicians like to explore objects in any number of spatial dimensions. Now they have discovered shapes of constant width in any dimension, which roll like a wheel despite not being round.

A 3D shape of constant width as seen from three different angles. The middle view resembles a 2D Reuleaux triangle

Mathematicians have reinvented the wheel with the discovery of shapes that can roll smoothly when sandwiched between two surfaces, even in four, five or any higher number of spatial dimensions. The finding answers a question that researchers have been puzzling over for decades.

Such objects are known as shapes of constant width, and the most familiar in two and three dimensions are the circle and the sphere. These aren’t the only such shapes, however. One example is the Reuleaux triangle, which is a triangle with curved edges, while people in the UK are used to handling equilateral curve heptagons, otherwise known as the shape of the 20 and 50 pence coins. In this case, being of constant width allows them to roll inside coin-operated machines and be recognised regardless of their orientation.

Crucially, all of these shapes have a smaller area or volume than a circle or sphere of the equivalent width – but, until now, it wasn’t known if the same could be true in higher dimensions. The question was first posed in 1988 by mathematician Oded Schramm, who asked whether constant-width objects smaller than a higher-dimensional sphere might exist.

While shapes with more than three dimensions are impossible to visualise, mathematicians can define them by extending 2D and 3D shapes in logical ways. For example, just as a circle or a sphere is the set of points that sits at a constant distance from a central point, the same is true in higher dimensions. “Sometimes the most fascinating phenomena are discovered when you look at higher and higher dimensions,” says Gil Kalai at the Hebrew University of Jerusalem in Israel.

Now, Andrii Arman at the University of Manitoba in Canada and his colleagues have answered Schramm’s question and found a set of constant-width shapes, in any dimension, that are indeed smaller than an equivalent dimensional sphere.

Arman and his colleagues had been working on the problem for several years in weekly meetings, trying to come up with a way to construct these shapes before they struck upon a solution. “You could say we exhausted this problem until it gave up,” he says.

The first part of the proof involves considering a sphere with n dimensions and then dividing it into 2ⁿ equal parts – so four parts for a circle, eight for a 3D sphere, 16 for a 4D sphere and so on. The researchers then mathematically stretch and squeeze these segments to alter their shape without changing their width. “The recipe is very simple, but we understood that only after all of our elaboration,” says team member Andriy Bondarenko at the Norwegian University of Science and Technology.

The team proved that it is always possible to do this distortion in such a way that you end up with a shape that has a volume at most 0.9ⁿ times that of the equivalent dimensional sphere. This means that as you move to higher and higher dimensions, the shape of constant width gets proportionally smaller and smaller compared with the sphere.

Visualising this is difficult, but one trick is to imagine the lower-dimensional silhouette of a higher-dimensional object. When viewed at certain angles, the 3D shape appears as a 2D Reuleaux triangle (see the middle image above). In the same way, the 3D shape can be seen as a “shadow” of the 4D one, and so on.  “The shapes in higher dimensions will be in a certain sense similar, but will grow in complexity as [the] dimension grows,” says Arman.

Having identified these shapes, mathematicians now hope to study them further. “Even with the new result, which takes away some of the mystery about them, they are very mysterious sets in high dimensions,” says Kalai.

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*Credit for article given to Alex Wilkins*

Using the tools of computational complexity, researchers have discovered it is impossible to figure out whether certain Super Mario Bros levels can be beaten without playing them, even if you use the world’s most powerful supercomputer.

Figuring out whether certain levels in the Super Mario Bros series of video games can be completed before you play them is mathematically impossible, even if you had several years and the world’s most powerful supercomputer to hand, researchers have found.

“We don’t know how to prove that a game is fun, we don’t know what that means mathematically, but we can prove that it’s hard and that maybe gives some insight into why it’s fun,” says Erik Demaine at the Massachusetts Institute of Technology. “I like to think of hard as a proxy for fun.”

To prove this, Demaine and his colleagues use tools from the field of computational complexity – the study of how difficult and time-consuming various problems are to solve algorithmically. They have previously proven that figuring out whether it is possible to complete certain levels in Mario games is a task that belongs to a group of problems known as NP-hard, where the complexity grows exponentially. This category is extremely difficult to compute for all but the smallest problems.

Now, Demaine and his team have gone one step further by showing that, for certain levels in Super Mario games, answering this question is not only hard, but impossible. This is the case for several titles in the series, including New Super Mario Bros and Super Mario Maker. “You can’t get any harder than this,” he says. “Can you get to the finish? There is no algorithm that can answer that question in a finite amount of time.”

While it may seem counterintuitive, problems in this undecidable category, known as RE-complete, simply cannot be solved by a computer, no matter how powerful, no matter how long you let it work.

Demaine concedes that a small amount of trickery was needed to make Mario levels fit this category. Firstly, the research looks at custom-made levels that allowed the team to place hundreds or thousands of enemies on a single spot. To do this they had to remove the limits placed by the game publishers on the number of enemies that can be present in a level.

They were then able to use the placement of enemies within the level to create an abstract mathematical tool called a counter machine, essentially creating a functional computer within the game.

That trick allowed the team to invoke another conundrum known as the halting problem, which says that, in general, there is no way to determine if a given computer program will ever terminate, or simply run forever, other than running it and seeing what happens.

These layers of mathematical concepts finally allowed the team to prove that no analysis of the game level can say for sure whether or not it can ever be completed. “The idea is that you’ll be able to solve this Mario level only if this particular computation will terminate, and we know that there’s no way to determine that, and so there’s no way to determine whether you can solve the level,” says Demaine.

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*Credit for article given to Matthew Sparkes*

What does “equals” mean? For mathematicians, this simple question has more than one answer, which is causing issues when it comes to using computers to check proofs. The solution might be to tear up the foundations of maths.

When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal.

While this argument has been quietly simmering for decades, a recent push to make mathematical proofs checkable by computer programs, called formalisation, has given the argument new significance.

“Mathematicians use equality to mean two different things, and I was fine with that,” says Kevin Buzzard at Imperial College London. “Then I started doing maths on a computer.” Working with computer proof assistants made him realise that mathematicians must now confront what was, until recently, a useful ambiguity, he says – and it could force them to completely redefine the foundations of their subject.

The first definition of equality will be a familiar one. Most mathematicians take it to mean that each side of an equation represents the same mathematical object, which can be proven through a series of logical transformations from one side to the other. While “=”, the equals sign, only emerged in the 16th century, this concept of equality dates back to antiquity.

It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics. Set theory deals with collections, or sets, of mathematical objects, and introduced another definition of equality: if two sets contain the same elements, then they are equal, similar to the original mathematical definition. For example, the sets {1, 2, 3} and {3, 2, 1} are equal, because the order of the elements in a set doesn’t matter.

But as set theory developed, mathematicians started saying that two sets were equal if there was an obvious way to map between them, even if they didn’t contain exactly the same elements, says Buzzard.

To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren’t equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism.

Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn’t technically the same concept of equality that most of us are used to. “These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well,” says Buzzard.

Having two definitions for equality is of no real concern to mathematicians when they write papers or give lectures, as the meaning is always clear from the context, but they present problems for computer programs that need strict, precise instructions, says Chris Birkbeck at the University of East Anglia, UK. “We’re finding that we were a little bit sloppy all along, and that maybe we should fix a few things.”

To address this, Buzzard has been investigating the way some mathematicians widely use canonical isomorphism as equality, and the problems this can cause with formal computer proof systems.

In particular, the work of Alexander Grothendieck, one of the leading mathematicians of the 20th century, is currently extremely difficult to formalise. “None of the systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol,” says Buzzard.

The problem has its roots in the way mathematicians put together proofs. To begin proving anything, you must first make assumptions called axioms that are taken to be true without proof, providing a logical framework to build upon. Since the early 20^th century, mathematicians have settled on a collection of axioms within set theory that provide a firm foundation. This means they don’t generally have to use axioms directly in their day-to-day business, because common tools can be assumed to work correctly – in the same way you probably don’t worry about the inner workings of your kitchen before cooking a recipe.

“As a mathematician, you somehow know well enough what you’re doing that you don’t worry too much about it,” says Birkbeck. That falls down, however, when computers get involved, carrying out maths in a way that is similar to building a kitchen from scratch for every meal. “Once you have a computer checking everything you say, you can’t really be vague at all, you really have to be very precise,” says Birkbeck.

To solve the problem, some mathematicians argue we should just redefine the foundations of mathematics to make canonical isomorphisms and equality one and the same. Then, we can make computer programs work around that. “Isomorphism is equality,” says Thorsten Altenkirch at the University of Nottingham, UK. “I mean, what else? If you cannot distinguish two isomorphic objects, what else would it be? What else would you call this relationship?”

Efforts are already under way to do this in a mathematical field called homotopy type theory, in which traditional equality and canonical isomorphism are defined identically. Rather than trying to contort existing proof assistants to fit canonical isomorphism, says Altenkirch, mathematicians should adopt type theory and use alternative proof assistants that work with it directly.

Buzzard isn’t a fan of this suggestion, having already spent considerable effort using current tools to formalise mathematical proofs that are needed to check more advanced work, such as a proof of Fermat’s last theorem. The axioms of mathematics should be left as they are, rather than adopting type theory, and existing systems should be tweaked instead, he says. “Probably the way to fix it is just to leave mathematicians as they are,” says Buzzard. “It’s very difficult to change mathematicians. You have to make the computer systems better.”

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*Credit for article given to Alex Wilkins*

Human society includes various minority groups. However, it is often difficult to know whether someone is a minority member simply by looking at the person, as minority traits may not be visually apparent (e.g., sexual orientation, color vision deficiency). In addition, minorities may hide their minority traits or identities. Consequently, we may have been unaware of the presence of minorities in daily life. Probabilistic thinking is critical in such uncertain situations.

The people with whom we interact in our daily lives are typically a group of several dozen individuals (e.g., a school class). How do we judge the probability of including at least one minoritymember in such groups? For example, how does a school teacher estimate the probability of having a minority in the class?

Cognitive psychology states that humans often make unrealistic judgments about probabilities, such as risk. So, do we also misperceive the probability of minority inclusion in a group or can we accurately assess the probability through heuristics or knowledge?

Associate Professor Niimi of Niigata University demonstrates that people unrealistically underestimate such probabilities. The study is published in the Journal of Cognitive Psychology.

First, the researchers examine how the probabilities are computed mathematically. If the prevalence of the minority in question is 0.03 (3%) and the group size is 30, the probability of including one or more minority members in the group is one minus the probability that all 30 members are NOT the minority.

Because the probability that one person is not a minority is 0.97, the probability of minority inclusion is given by 1– (0.97)³⁰ (if there is no other information). The computer tells us that the result is 0.60 (60%). When the minority prevalence is 7%, it increases to 89%. These mathematical probabilities appear to be higher than those of naive intuition.

Indeed, most respondents estimated probabilities far below the mathematical probabilities. The second image shows examples of the questions and results. Approximately 90% of the respondents estimated below-mathematical probabilities, and the majority of the estimates were lower than 10%. This underestimation was repeatedly observed under a variety of conditions (online worker and student samples, revised wording, etc.).

Why are the probabilities of minority inclusion underestimated? Is this a result of prejudice or stereotyping against minorities? The answer was “No.” The same underestimation occurred even when minorities unlikely to be associated with negative stereotypes were used (e.g., people with absolute pitch and fictional minorities). Of course, the mathematical calculations cannot be performed mentally. No wonder the respondents’ estimates were inaccurate.

The problem was why the estimates were not random, but strongly biased toward underestimation. Even if one does not know how to calculate it, one may have learned from daily experience that the probability of inclusion is much higher than the prevalence (e.g., the probability of including a woman in a group of randomly selected 100 individuals should be greater than 50%). However, the present results suggest that most people are unfamiliar with the concept of probability of inclusion and do not know how to think about it.

Further analysis revealed that the major source of underestimation was the use of heuristics, such as ignoring group size and reporting prevalence, or calculating the expected value of the number of minorities. Although most heuristics were erroneous, some yielded relatively reasonable estimates (e.g., assuming a high probability if the expected value exceeded one).

Underestimating the probability of minority inclusion may lead to the misconception that minorities are irrelevant in our daily lives. However, there was one promising finding in the present study.

When the respondents were given the mathematical probability of minority inclusion, their attitudes changed in favour of inclusive views about minorities compared to conditions in which mathematical probability was not given. Knowledge may compensate for cognitive bias.

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Credit of the article given to Niigata University

For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what WEcall narrow mathematics—a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, “Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics,” Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by ‘math-ish’ thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what WEcall “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book WEshare an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they’re more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. WEthink we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and We still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

WEwonder if people consider the physical representations more appropriate for younger kids.

That’s the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they’re drawing, they’re sketching, they’re building models, and nobody says that’s elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense.

They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

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Credit of the article given to Stanford University

Mathematicians are celebrating a 1000-page proof of the geometric Langlands conjecture, a problem so complicated that even other mathematicians struggle to understand it. Despite that, it is hoped the proof can provide key insights across maths and physics.

The Langlands programme aims to link different areas of mathematics

Mathematicians have proved a key building block of the Langlands programme, sometimes referred to as a “grand unified theory” of maths due to the deep links it proposes between seemingly distant disciplines within the field.

While the proof is the culmination of decades of work by dozens of mathematicians and is being hailed as a dazzling achievement, it is also so obscure and complex that it is “impossible to explain the significance of the result to non-mathematicians”, says Vladimir Drinfeld at the University of Chicago. “To tell the truth, explaining this to mathematicians is also very hard, almost impossible.”

The programme has its origins in a 1967 letter from Robert Langlands to fellow mathematician Andre Weil that proposed the radical idea that two apparently distinct areas of mathematics, number theory and harmonic analysis, were in fact deeply linked. But Langlands couldn’t actually prove this, and was unsure whether he was right. “If you are willing to read it as pure speculation I would appreciate that,” wrote Langlands. “If not — I am sure you have a waste basket handy.”

This mysterious link promised answers to problems that mathematicians were struggling with, says Edward Frenkel at the University of California, Berkeley. “Langlands had an insight that difficult questions in number theory could be formulated as more tractable questions in harmonic analysis,” he says.

In other words, translating a problem from one area of maths to another, via Langlands’s proposed connections, could provide real breakthroughs. Such translation has a long history in maths – for example, Pythagoras’s theorem relating the three sides of a triangle can be proved using geometry, by looking at shapes, or with algebra, by manipulating equations.

As such, proving Langlands’s proposed connections has become the goal for multiple generations of researchers and led to countless discoveries, including the mathematical toolkit used by Andrew Wiles to prove the infamous Fermat’s last theorem. It has also inspired mathematicians to look elsewhere for analogous links that might help. “A lot of people would love to understand the original formulation of the Langlands programme, but it’s hard and we still don’t know how to do it,” says Frenkel.

One analogy that has yielded progress is reformulating Langlands’s idea into one written in the mathematics of geometry, called the geometric Langlands conjecture. However, even this reformulation has baffled mathematicians for decades and was itself considered fiendishly difficult to prove.

Now, Sam Raskin at Yale University and his colleagues claim to have proved the conjecture in a series of five papers that total more than 1000 pages. “It’s really a tremendous amount of work,” says Frenkel.

The conjecture concerns objects that are similar to those in one half of the original Langlands programme, harmonic analysis, which describes how complex structures can be mathematically broken down into their component parts, like picking individual instruments out of an orchestra. But instead of looking at these with harmonic analysis, it uses other mathematical ideas, such as sheaves and moduli stacks, that describe concepts relating to shapes like spheres and doughnuts.

While it wasn’t in the setting that Langlands originally envisioned, it is a sign that his original hunch was correct, says Raskin. “Something I find exciting about the work is it’s a kind of validation of the Langlands programme more broadly.”

“It’s the first time we have a really complete understanding of one corner of the Langlands programme, and that’s inspiring,” says David Ben-Zvi at the University of Texas, who wasn’t involved in the work. “That kind of gives you confidence that we understand what its main issues are. There are a lot of subtleties and bells and whistles and complications that appear, and this is the first place where they’ve all been kind of systematically resolved.”

Proving this conjecture will give confidence to other mathematicians hoping to make inroads on the original Langlands programme, says Ben-Zvi, but it might also attract the attention of theoretical physicists, he says. This is because in 2007, physicists Edward Witten and Anton Kapustin found that the geometric Langlands conjecture appeared to describe an apparent symmetry between certain physical forces or theories, called S-duality.

The most basic example of this in the real world is in electricity and magnetism, which are mirror images of one another and interchangeable in many scenarios, but S-duality was also used by Witten to famously unite five competing string theory models into a single theory called M-theory.

But before anything like that, there is much more work to be done, including helping other mathematicians to actually understand the proof. “Currently, there’s a very small group of people who can really understand all the details here. But that changes the game, that changes the whole expectation and changes what you think is possible,” says Ben-Zvi.

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*Credit for article given to Alex Wilkins*

A centuries-old maths problem asks what shape a circle traces out as it rolls along a line. The answer, dubbed a “cycloid”, turns out to have applications in a variety of scientific fields.

Light reflecting off the round rim creates a mathematically significant shape in this coffee cup

Sarah Hart

The artist Paul Klee famously described drawing as “taking a line for a walk” – but why stop there? Mathematicians have been wondering for five centuries what happens when you take circles and other curves for a walk. Let me tell you about this fascinating story…

A wheel rolling along a road will trace out a series of arches

Imagine a wheel rolling along a road – or, more mathematically, a circle rolling along a line. If you follow the path of a point on that circle, it traces out a series of arches. What exactly is their shape? The first person to give the question serious thought seems to have been Galileo Galilei, who gave the arch-like curve a name – the cycloid. He was fascinated by cycloids, and part of their intriguing mystery was that it seemed impossible to answer the most basic questions we ask about a curve – how long is it and what area does it contain? In this case, what’s the area between the straight line and the arch? Galileo even constructed a cycloid on a sheet of metal, so he could weigh it to get an estimate of the area, but he never managed to solve the problem mathematically.

Within a few years, it seemed like every mathematician in Europe was obsessed with the cycloid. Pierre de Fermat, René Descartes, Marin Mersenne, Isaac Newton and Gottfried Wilhelm Leibniz all studied it. It even brought Blaise Pascal back to mathematics, after he had sworn off it in favour of theology. One night, he had a terrible toothache and, to distract himself from the pain, decided to think about cycloids. It worked – the toothache miraculously disappeared, and naturally Pascal concluded that God must approve of him doing mathematics. He never gave it up again. The statue of Pascal in the Louvre Museum in Paris even shows him with a diagram of a cycloid. The curve became so well known, in fact, that it made its way into several classic works of literature – it gets name-checked in Gulliver’s Travels, Tristram Shandy and Moby-Dick.

The question of the cycloid’s area was first solved in the mid-17th century by Gilles de Roberval, and the answer turned out to be delightfully simple – exactly three times the area of the rolling circle. The first person to determine the length of the cycloid was Christopher Wren, who was an extremely good mathematician, though I hear he also dabbled in architecture. It’s another beautifully simple formula: the length is exactly four times the diameter of the generating circle. The beguiling cycloid was so appealing to mathematicians that it was nicknamed “the Helen of Geometry”.

But its beauty wasn’t the only reason for the name. It was responsible for many bitter arguments. When mathematician Evangelista Torricelli independently found the area under the cycloid, Roberval accused him of stealing his work. “Team Roberval” even claimed that Torricelli had died of shame after being unmasked as a plagiarist (though the typhoid he had at the time may have been a contributing factor). Descartes dismissed Fermat’s work on the cycloid as “ridiculous gibberish”. And in response to a challenge from Johann Bernoulli, Isaac Newton grumpily complained about being “teased by foreigners about mathematics”.

An amazing property of the cycloid was discovered by Christiaan Huygens, who designed the first pendulum clock. Pendulums are good for timekeeping because the period of their motion – the time taken for one full swing of the pendulum – is constant, no matter what the angle of release. But in fact, that’s only approximately true – the period does vary slightly. Huygens wondered if he could do better. The end of a pendulum string moves along the arc of a circle, but is there a curved path it could follow so that the bob would reach the bottom of the curve in the same time no matter where it was released? This became known as the “tautochrone problem”. And guess which curve is the solution? An added bonus is its link to the “brachistochrone problem” of finding the curve between any two points along which a particle moving under gravity will descend in the shortest time. There’s no reason at all to think that the same curve could answer both problems, but it does. The solution is the cycloid. It’s a delightful surprise to find it cropping up in situations seemingly so unrelated to where we first encountered it.

When you roll a circle along a line, you get a cycloid. But what happens when you roll a line along a circle? This is an instance of a curve called an involute. To make one, you take a point at the end of a line segment and roll that line along the curve so it’s always just touching it (in other words, it’s a tangent). The involute is the curve traced out by that point. For the involute of a circle, imagine unspooling a thread from a cotton reel and following the end of the thread as it moves. The result is a spiralling curve emerging from the circle’s circumference.

When a line rolls along a circle, it produces a curve called an involute

Huygens was the first person to ask about involutes, as part of his attempts to make more accurate clocks. It’s all very well knowing the cycloid is the perfect tautochrone, but how do you get your string to follow a cycloidal path? You need to find a curve whose involute is a cycloid. The miraculous cycloid, it turns out, has the beautiful property that it is its own involute! But those lovely spiralling circle involutes turn out to be extremely useful too.

A circle with many involutes

My favourite application is one Huygens definitely couldn’t have predicted: in the design of a nuclear reactor that produces high-mass elements for scientific research. This is done by smashing neutrons at high speed into lighter elements, to create heavier ones. Within the cylindrical reactor cores, the uranium oxide fuel is sandwiched in thin layers between strips of aluminium, which are then curved to fit into the cylindrical shape. The heat produced by a quadrillion neutrons hurtling around every square centimetre is considerable, so coolant runs between these strips. It’s vital that they must be a constant distance apart all the way along their curved surfaces, to prevent hotspots. That’s where a useful property of circle involutes comes in. If you draw a set of circle involutes starting at equally spaced points on the circumference of a circle, then the distances between them remain constant along the whole of each curve. So, they are the perfect choice for the fuel strips in the reactor core. What’s more, the circle involute is the only curve for which this is true! I just love that a curve first studied in the context of pendulum clocks turns out to solve a key design question for nuclear reactors.

We’ve rolled circles along lines and lines along circles. Clearly the next step is to roll circles along circles. What happens? Here, we have some choices. What size is the rolling circle? And are we rolling along the inside or the outside of the stationary one? The curve made by a circle rolling along inside of the circle is called a hypocycloid; rolling it along the outside gives you an epicycloid. If you’ve ever played with a Spirograph toy, you’ll almost have drawn hypocycloids. Because your pen is not quite at the rim of the rolling circle, technically you are creating what are called hypotrochoids.

A cardioid (left) and nephroid (right)

Of the epicycloids, the most interesting is the cardioid: the heart-shaped curve resulting when the rolling circle has the same radius as the fixed one. Meanwhile, the kidney-shaped nephroid is produced by a rolling circle half the radius of the fixed one. Cardioids crop up in the most fascinating places. The central region of the Mandelbrot set, a famous fractal, is a cardioid. Sound engineers will be familiar with cardioid microphones, which pick up sound in a cardioid-shaped region. You might also find cardioid-like curves in the light patterns created in coffee cups in some kinds of lighting. If light rays from a fixed source are reflected off a curved mirror, the curve to which each of those reflected rays are tangent will be visible as a concentrated region of light, called a caustic. It turns out that a light source on the circumference of a perfectly circular mirror will result precisely in a cardioid!

Of course, in our coffee cup example, usually the light source isn’t exactly on the rim of the cup, but some way away. If it were very far away, we could assume that the light rays hitting the rim of the cup are parallel. In that situation, it can be shown that the caustic is actually not a cardioid but another epicycloid: the nephroid. Since a strong overhead light is somewhere between these two extremes, the curve we get is usually going to be somewhere between a cardioid and a nephroid. The mathematician Alfréd Rényi once defined a mathematician as “a device for turning coffee into theorems”. That process is nowhere more clearly seen than with our wonderful epicycloids. Check them out if you’re reading this with your morning cuppa!

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*Credit for article given to Sarah Hart*