• A Deep Math Dive into Why Some Infinities Are Bigger Than Others

    A Deep Math Dive into Why Some Infinities Are Bigger Than Others

    Simple mathematical concepts such as counting appear to be firmly anchored in the natural process of thinking. Studies have shown that even very young children and animals possess such skills to a certain extent. This is hardly surprising because counting is extremely useful in terms of evolution. For example, it is required for even very simple forms of trading. And counting helps in estimating the size of a hostile group and, accordingly, whether it is better to attack or retreat.

    Over the past millennia, humans have developed a remarkable notion of counting. Originally applied to a handful of objects, it was easily extended to vastly different orders of magnitude. Soon a mathematical framework emerged that could be used to describe huge quantities, such as the distance between galaxies or the number of elementary particles in the universe, as well as barely conceivable distances in the microcosm, between atoms or quarks.

    We can even work with numbers that go beyond anything currently known to be relevant in describing the universe. For example, the number 1010100 (one followed by 10100 zeros, with 10100 representing one followed by 100 zeros) can be written down and used in all kinds of calculations. Writing this number in ordinary decimal notation, however, would require more elementary particles than are probably contained in the universe, even employing just one particle per digit. Physicists estimate that our cosmos contains fewer than 10100 particles.

    Yet even such unimaginably large numbers are vanishingly small, compared with infinite sets, which have played an important role in mathematics for more than 100 years. Simply counting objects gives rise to the set of natural numbers, ℕ = {0, 1, 2, 3, …}, which many of us encounter in school. Yet even this seemingly simple concept poses a challenge: there is no largest natural number. If you keep counting, you will always be able to find a larger number.

    Can there actually be such a thing as an infinite set? In the 19th century, this question was very controversial. In philosophy, this may still be the case. But in modern mathematics, the existence of infinite sets is simply assumed to be true—postulated as an axiom that does not require proof.

    Set theory is about more than describing sets. Just as, in arithmetic, you learn to apply arithmetical operations to numbers—for example, addition or multiplication—you can also define set-theoretical operations that generate new sets from given ones. You can take unions—{1, 2} and {2, 3, 4} becomes {1, 2, 3, 4}—or intersections—{1, 2} and {2, 3, 4} becomes {2}. More excitingly, you can form power sets—the family of all subsets of a set.

    Comparing Set Sizes

    The power set P(X) of a set X can be easily calculated for small X. For instance, {1, 2} gives you P({1,2}) = {{}, {1}, {2}, {1, 2}}. But P(X) grows rapidly for larger X. For example, every 10-element set has 210 = 1,024 subsets. If you really want to challenge your imagination, try forming the power set of an infinite set. For example, the power set of the natural numbers, P(ℕ), contains the empty set, ℕ itself, the set of all even numbers, the prime numbers, the set of all numbers with the sum of digits totaling 2021, {12, 17}, and much, much more. As it turns out, the number of elements of this power set exceeds the number of elements in the set of natural numbers.

    To understand what that means, you first have to understand how the size of sets is defined. For the finite case, you can count the respective elements. For instance, {1, 2, 3} and {Cantor, Gödel, Cohen} are of the same size. If you wish to compare sets with numerous (but finitely many) elements, there are two well-established methods. One possibility is to count the objects contained in each set and compare the numbers. Sometimes, however, it is easier to match the elements of one set to another. Then two sets are of the same size if and only if each element of one set can be uniquely paired with an element of the other set (in our example: 1 → Cantor, 2 →Gödel, 3 →Cohen).

    This pairing method also works for infinite sets. Here, instead of first counting and then deriving concepts such as “greater than” or “equal to,” you follow a reverse strategy. You start with defining what it means that two sets, A and B, are of the same size—namely, there is a mapping that pairs each element of A with exactly one element of B (so that no element of B is left over). Such a mapping is called bijection.

    Similarly, A is defined to be less than or equal to B if there is a mapping from A to B that uses each element of B once at most.

    After we have these notions, the size of sets is denoted by cardinal numbers, or cardinals. For finite sets, these are the usual natural numbers. But for infinite sets, they are abstract quantities that just capture the notion of “size.” For example, “countable” is the cardinal number of the natural numbers (and therefore of every set that has the same size as the natural numbers). It turns out that there are different cardinals. That is, there are infinite sets A and B with no bijection between them.

    At first sight, this definition of size seems to lead to contradictions, which were elaborated by the Bohemian mathematician Bernard Bolzano in Paradoxes of the Infinite, published posthumously in 1851. For example, Euclid’s “The whole is greater than the part” appears self-evident. That means if a set A is a proper subset of B (that is, every element of A is in B, but B contains additional elements), then A must be smaller than B. This assertion is not true for infinite sets, however! This curious property is one reason some scholars rejected the concept of infinite sets more than 100 years ago.

    For example, the set of even numbers E = {0, 2, 4, 6, …} is a proper subset of the natural numbers ℕ = {0, 1, 2, …}. Intuitively, you might think that the set E is half the size of ℕ. But in fact, based on our definition, the sets have the same size because each number n in E can be assigned to exactly one number in ℕ (0 →0, 2 →1, 4 →2, …, n →n/2, …).

    Consequently, the concept of “size” for sets could be dismissed as nonsensical. Alternatively, it could be termed something else: cardinality, for example. For the sake of simplicity, we will stick to the conventional terminology, even though it has unexpected consequences at infinity.

    In the late 1800s, German logician Georg Cantor, founder of modern set theory, discovered that not all infinite sets are equal. According to his proof, the power set P(X) of a (finite or infinite) set X is always larger than X itself. Among other things, it follows that there is no largest infinity and thus no “set of all sets.”

    An Unresolved Hypothesis

    There is, however, something akin to a smallest infinity: all infinite sets are greater than or equal to the natural numbers. Sets X that have the same size as ℕ (with a bijection between ℕ and X) are called countable; their cardinality is denoted ℵ0, or aleph null. For every infinite cardinal ℵa, there is a next larger cardinal number ℵa+1. Thus, the smallest infinite cardinal ℵ0 is followed by ℵ1, then ℵ2 and so on. The set ℝ of real numbers (also called the real line) is as large as the power set of ℕ, and this cardinality is denoted 20, or “continuum.”

    In the 1870s, Cantor ruminated over whether the size of ℝ was the smallest possible cardinal above ℵ0—in other words, whether ℵ1 = 20. Previously, every infinite subset of ℝ that had been studied had turned out to be either as large as ℕ or ℝ itself. This led Cantor to what is known as the continuum hypothesis (CH): the assertion that the size of ℝ is the smallest possible uncountable cardinal. For decades, CH kept mathematicians busy, but a proof eluded them. Later, it became clear their efforts had been doomed from the start.

    Set theory is extremely powerful. It can describe virtually all mathematical concepts. But it also has limitations. The field is based on the axiomatic system formulated more than 100 years ago by German logician Ernst Zermelo and elaborated by his German-Israeli colleague Abraham Fraenkel. Called ZFC, or Zermelo-Fraenkel set theory (C stands for “axiom of choice”), the system is a collection of basic assumptions sufficient to carry out almost all of mathematics. Very few problems require additional assumptions. But in 1931 Austrian mathematician Kurt Gödel recognized that the system has a fundamental defect: it is incomplete. That is, it is possible to formulate mathematical statements that can neither be refuted nor proved using ZFC. Among other things, it is impossible for a system to prove its own consistency.

    The most famous example of undecidability in set theory is CH. In a paper published in 1938, Gödel proved that CH cannot be disproved within ZFC. Neither can it be proved, as Paul Cohen showed 25 years later. It is thus impossible to solve CH using the usual axioms of set theory. Consequently, it remains unclear whether sets exist that are both larger than the natural numbers and smaller than the real numbers.

    Cardinality is not the only notion to describe the size of a set. For example, from the point of view of geometry, subsets of the real line ℝ, the two-dimensional plane (sometimes called the x-y plane) or the three-dimensional space can be assigned length, area or volume. A set of points in the plane forming a rectangle with side lengths a and b has an area of ab. Calculating the area of more complicated subsets of the plane sometimes requires other tools, such as the integral calculus taught in school. This method does not suffice for certain complex sets. But many can still be quantified using the Lebesgue measure, a function that assigns length, area or volume to extremely complicated objects. Even so, it is possible to define subsets of ℝ, or the plane, that are so frayed that they cannot be measured at all.

    In two-dimensional space, a line (such as the circumference of a circle, a finite segment or a straight line) is always measurable, and its area is zero. It is therefore called a null set. Null sets can also be defined in one dimension. On the real line, the set with two elements—for example {3, 5}—has a measure zero, whereas an interval such as [3, 5]—that is, the real numbers between three and five—has a measure two.

    Negligible Sets

    The concept of a null set is extremely useful in mathematics. Often, a theorem is not true for all real numbers but can be proved for all real numbers outside of a null set. This is usually good enough for most applications. Yet null sets may seem quite large. For example, the rational numbers within the real line are a null set even though there are infinitely many of them. This is because any countable—or finite—set is a null set. The converse is not true: a subset of the x-y plane with a large cardinality need be neither measurable nor of large measure. For example, the entire plane with its 20 elements has an infinite measure. But the x axis with the same cardinality has a two-dimensional measure (or “area”) zero and thus is a null set of the plane.

    Such “negligible” sets led to fundamental questions about the size of 10 infinite cardinals, which remained unanswered for a long time. For example, mathematicians wished to know the minimum size a set must have for it not to be a null set. The family of all null sets is denoted by 𝒩, and the smallest cardinality of a non-null set is denoted by non(𝒩). It follows that ℵ0 < non(𝒩) ≤ 20, because any set of size ℵ0 is a null set, and the whole plane has size 20 and is not a null set. Thus, ℵ1≤ non(𝒩) ≤ 20, because ℵ1 is the smallest uncountable cardinal. If we assume CH, then non(𝒩) = 20, because, in that case, ℵ1 = 20.

    We can define another cardinal number, add(𝒩), to answer the question, What is the minimal number of null sets whose union is a non-null set? This number is less than or equal to non(𝒩): if A is a non-null set containing non(𝒩) many elements, the union of all the non(𝒩) many one-element subsets of A is the non-null set A. But a smaller number of null sets (though they would not be one-element sets) could also satisfy the requirements. Therefore, add(𝒩) ≤ non(𝒩) holds.

    The cardinal cov(𝒩) is the smallest number of null sets whose union yields the whole plane. It is also easy to see that add(𝒩) is smaller than or equal to cov(𝒩) because, as already mentioned, the plane is a non-null set.

    We can also consider cof(𝒩), the smallest possible size for a basis X of 𝒩. That is, a set X of null sets that contains a superset B of every null set A. (That means A is a subset of B.) These infinite cardinals—add(𝒩), cov(𝒩), non(𝒩) and cof(𝒩)—are important characteristics of the family of null sets.

    For each of these four cardinal characteristics, an analogous characteristic can be defined using a different concept of small, or negligible, sets. This other notion of smallness is “meager.” A meager set is a set contained in the countable union of nowhere dense sets, such as the circumference of a circle in the plane, or finitely or countably many such circumferences. In one dimension, the normal numbers form a meager set on the real line, while the remaining reals, the non-normal numbers, constitute a null set.

    Accordingly, the corresponding cardinal characteristics can be defined for the family of meager sets: add(ℳ), non(ℳ), cov(ℳ) and cof(ℳ). Under CH, all characteristics are the same, namely ℵ1, for both null and meager sets. On the other hand, using the method of “forcing,” developed by Cohen, mathematicians Kenneth Kunen and Arnold Miller were able to show in 1981 that it is impossible to prove the statement add(𝒩) = add(ℳ) within ZFC. In other words, the numbers of null and meager sets that must be combined to produce a non-negligible set are not provably equal.

    Forcing is a method to construct mathematical universes. A mathematical universe is a model that satisfies the ZFC axioms. To show that a statement X is not refutable in ZFC, it is enough to find a universe in which both ZFC and X are valid. Similarly, to show that X is not provable from ZFC, it is enough to find a universe where ZFC holds but X fails.

    Mathematical Universes with Surprising Properties

    Kunen and Miller used this method to construct a mathematical universe that satisfies add(𝒩) < add(ℳ). In this model, more meager than null sets are required to form a non-negligible set. Accordingly, it is impossible to prove add(𝒩) add(ℳ) from ZFC.

    In contrast, Tomek Bartoszyński discovered three years later that the converse inequality add(𝒩) ≤ add(ℳ) can be proved using ZFC. This points to an asymmetry between the two notions of smallness. Let us note that this asymmetry is not visible if we assume CH because CH implies ℵ1 = add(𝒩) = add(ℳ).

    To summarize: add(𝒩) ≤ add(ℳ) is provable, but neither add(𝒩) = add(ℳ) nor add(𝒩) < add(ℳ) is provable. This is the same effect as with CH: it is trivial to prove that ℵ1 ≤ 20, but neither ℵ1 < 20 nor ℵ1 = 20 is provable.

    In addition to the cardinal numbers defined so far, there are two important cardinal characteristics—𝔟 and 𝔡—that refer to dominating functions of real numbers. For two continuous functions (of which there are 20 many) f and g, f is said to be dominated by g if the inequality f(x) < g(x) holds for all sufficiently large x. For example, a quadratic function such as g(x) = x2 always dominates a linear function, say f(x) = 100x + 30.

    The cardinal number 𝔡 is defined as the smallest possible size of a set of continuous functions sufficient to dominate every possible continuous function.

    A variant of this definition gives the cardinal number 𝔟, namely the smallest size of a family B with the property that there is no continuous function that dominates all functions of B. It can be shown that ℵ1 ≤ 𝔟 ≤ 𝔡 ≤ 20 holds.

    Several additional inequalities have been shown to hold between the 12 infinite cardinals we just defined. All these inequalities are summarized in Cichoń’s diagram, introduced by British mathematician David Fremlin in 1984 and named after his Polish colleague Jacek Cichoń. For typographical reasons, the less-or-equal signs are replaced by arrows.

    The intricacies of Cichoń’s diagram.
    Credit: Jakob Kellner

    There are two additional relations: Add(ℳ) is the smaller one of 𝔟 and cov(ℳ). Likewise, cof(ℳ) is the larger of 𝔡 and non(ℳ). These two “dependent” cardinals are marked with a frame in the Cichoń diagram. The diagram thus comprises 12 uncountable cardinalities of which no more than 10 can be simultaneously different.

    How Different Can Infinities Be?

    If CH holds, however, ℵ1 (the smallest number in the diagram) is equal to 20 (the largest number in the diagram), and thus all entries are equal. If, on the other hand, we assume CH to be false, then they could be quite different.

    For several decades, mathematicians tried to show that none of the less-or-equal relations in Cichoń’s diagram can be strengthened to equalities. To do that, they constructed many different universes in which they assigned the two smallest uncountable cardinals, ℵ1 and ℵ2, to the entries of the diagram in various ways. For example, they created a universe for which ℵ1 = add(𝒩) = cov(𝒩) and ℵ2 = non(ℳ) = cof(ℳ).

    This work enabled researchers in the 1980s to confirm that for all pairs of cardinals, only the relationships indicated in the diagram can be proved in ZFC. More precisely, for every labeling of the (independent) Cichoń diagram entries with the values ℵ1 and ℵ2 that honors the inequalities of the diagram, there is a universe that realizes the given labeling.

    So we have known for nearly four decades that all assignments of ℵ1 and ℵ2 to the diagram are possible. But what can we say for more than two values? Could, for example, all the independent entries be simultaneously different? Some cases with three characteristics have been known for 50 years, and in the 2010s, more universes were discovered (or constructed) in which up to seven different cardinals appeared in the Cichoń diagram.

    In a 2019 paper we constructed with Israeli mathematician Saharon Shelah of the Hebrew University of Jerusalem, a universe in which the maximum possible number of different infinite values—10, that is—appears in Cichoń’s diagram. In doing so, however, we used a stronger system of axioms than ZFC, one that assumes the existence of “large cardinals,” infinities whose existence is not provable in ZFC alone.

    While we were very pleased with this result, we were not entirely satisfied. We worked for two more years to find a solution using only the ZFC axioms. Together with Shelah and Colombian mathematician Diego Mejía of Shizuoka University in Japan, we finally succeeded in proving the result without these additional assumptions.

    We have thus shown that the 10 characteristics of the real numbers can all be different. Let us note that we did not show that there can be at least, at most or precisely 10 infinite cardinals between ℵ1 and the continuum. This was already proved by Robert Solovay in 1963. In fact, the size of the set of real numbers can vary greatly: there could be eight, 27 or infinitely many cardinal numbers between ℵ1 and 20—even uncountably many. Rather our result proves that there are mathematical universes in which the 10 specific cardinal numbers between ℵ1 and 20 turn out to be different.

    This is not the end of the story. As is usual for mathematics, many questions remain open, and new ones arise. For example, in addition to the cardinal numbers described here, many other infinite cardinalities lying between ℵ1 and the continuum have been discovered since the 1940s. Their precise relationships to one another are unknown. To distinguish some of these characteristics in addition to those in Cichoń’s diagram is one of the upcoming challenges. Another one is to show that other orderings of 10 different values are possible. Unlike in the case for the two values ℵ1 and ℵ2, where we know that all possible orders are consistent, in the case of all 10 values, we could only show the consistency of two different orderings. So, who knows, there may still be hitherto undiscovered equalities—involving more than two characteristics—hidden in the diagram.

    This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

  • City student bags gold at international math olympiad
    Math Club and Olympiad

    City student bags gold at international math olympiad

    Seventeen-year-old Pranjal Srivastava’s love for mathematics has earned him international acclaim. The class XII student of National Public School, Koramangala, who represented India at the International Mathematical Olympiad (IMO), brought laurels to the country by bagging the gold medal.

    Due to the pandemic, the competition this year was held remotely from St. Petersburg, Russia, last Monday and Tuesday and the medal announcement was made on Saturday night at the closing ceremony.

    For Pranjal, this is a remarkable achievement. He is the only student from India to win gold medals twice. Two years ago, he had won a gold at the same event. This year, he also won a bronze medal in the International Olympiad in Informatics (IOI).

    On his experience at the IMO this year, Pranjal said: “The paper was extremely difficult, especially on day one of the competition.” IMO tests four different skill sets – algebra, combinatorics, number theory, and geometry – in a mixed bag of questions over two days. The top 8% scorers are awarded the gold medal.

    Given the prestige associated with an IMO gold, this becomes an intense competition. Celebrated Field medallists, considered the Nobel Prize in mathematics like Terence Tao and Maryam Mirzakhani have been IMO gold medallists.

    “Becoming an IMO champ is possible only if you have a deep love for the subject and are ready to invest in exploring it,” Pranjal said. He acknowledged the constant support and encouragement received from his parents and school. Pranjal’s immediate goal is to pursue mathematics in higher studies and in the long term, he may pursue algorithms or computer science.

  • Vestavia students form Alabama Youth Math Club
    Math Club and Olympiad

    Vestavia students form Alabama Youth Math Club

    The Alabama Youth Math Club (AYMC), recently founded by Vestavia middle school and high school math team members, aims to promote math throughout Alabama through volunteering and raising money to support math education.

    This summer, they directed three fundraisers for the Vestavia math teams, selling sticky rice dumplings, qingtuan, and crafts at Hometown Supermarket and Oriental Pearl to commemorate the Chinese Dragon Boat Festival. AYMC members worked together on everything from preparing food to advertising and selling. The club raised around $4,000. They believe this is the first student-led off-campus club to raise money for the Vestavia Hills Math Team. 

    The AYMC also held a week-long math camp for kids to prepare for the school year and math competitions and to inspire a lifelong love of math, with about 70 students attending. The camp was split into four classes: fun math (lower elementary level), Continental Math League (upper elementary level), AMC 8 and Mathcounts (middle school level), and AMC 10 (high school level). The classes are meant to prepare for nationally recognized competitions. Each class was taught by successful math team members who planned the material, explained the problems, and organized activities themselves. 

    More recently, they invited the IMO (International Mathematics Olympiad) head coach, Po-Shen Loh, to speak at Pizitz on August 2nd about Math in Games, Strategy, and Invention. A crowd of about100 gathered, consisting of math team members, parents, and teachers. Professor Loh talked about game theory, graph theory, and how they relate to fighting COVID. 

    The club is currently in the process of talking with the Alabama Department of Education on ways to support math and science education in Alabama. 

    For more information, visit: www.birminghammathclub.org

    — Submitted by Tina Gao

  • Anil Kapoor Was Always a Math Whiz, Ram-Lakhan’s Iconic Song Has Now Proved It

    Anil Kapoor Was Always a Math Whiz, Ram-Lakhan’s Iconic Song Has Now Proved It

    What can possibly be the connection between a tough mathematical problem and a Bollywood song? Well, a lot, if you come to think of it. Because the same math problem is now making actor Anil Kapoor go viral on social media because of his cult movie Ram Lakhan’s iconic number “One Two Ka Four, Four Two Ka One, My Name Is Lakhan.” The actor is harking back on his wildly popular song number because of its stellar mathematic problem-solving ability! Say what? Yes!

    It all started a while back when film director Anubhav Sinha posted a video explaining a complex mathematical problem known as the ‘Collatz conjecture’. The video explained the equation and Sinha tagged Kapoor and the film’s director Subhash Ghai, saying, “The code has been cracked after so many years. Anil Kapoor, please see this. Subhash Ghai.”

    Kapoor then quote tweeted the video and wrote, “No matter how complicated the math problem I always know the answer…#watchtilltheend. “

    The video ends with a clip of Kapoor from the song itself, singing and dancing to the One, Two Ka Four chorus, thereby hilariously hinting how the song had already explained the equation all those years ago.

    So what is the mathematical equation all about? It is termed as the ‘Collatz conjecture’ which is a complex math problem and the clip is an attempt to explain the same.

    “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture had said, according to Quanta Magazine.

    The idea is to pick a number and if it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. The new number will also go through the same rules. While conjecture says that some numbers may come down to 1 and others go off to infinity, Collatz’s prediction said that if one started with a positive whole number and run the procedure for long, it all comes down to 1 and eventually a loop of 1, 4, 2, 1, 4, 2, 1, on and on forever.

    Fans reacted to the video with some hilarious responses:

    However, whether or not the puzzle has ultimately been solved, one thing is certain, Kapoor knew about this all those years ago. The song proves it.

    Read all the Latest News, Breaking News and Coronavirus News here

  • Shanghai lauded as ‘a cradle for modern math education’

    Shanghai lauded as ‘a cradle for modern math education’

    Shanghai lauded as 'a cradle for modern math education'

    Ti Gong

    A flag with signatures of famous maths educators from around the world is showned at the 14th International Congress on Mathematical Education.

    A flag with signatures of famous mathematics educators from around the world was displayed at Shanghai’s East China Normal University on Monday night, kicking off the 14th International Congress on Mathematical Education.

    It is the first time for the world’s largest academic conference on maths education, held once every four years, to take place in China since it was initiated in 1969.

    It was scheduled to be held at Shanghai Convention and Exhibition Center of International Sourcing from July 12 to 19 last year, but was postponed by a year due to the COVID-19 pandemic.

    The seven-day conference will involve about 500 workshops and other activities with over 2,000 people, including mathematics educators, curriculum developers, mathematicians, researchers and resource producers, from 129 countries and regions attending online or offline.

    At the opening ceremony held at the indoor stadium of East China Normal University on Monday night, Li Qiang, Party secretary of Shanghai, stressed in his speech the importance of mathematics in urban development.

    He said mathematics, as the oldest and most active science of mankind, was a fulcrum and cornerstone for all other sciences.

    “Nowadays, mathematics has been applied in all human activities as a leading driving force for a new round of reforms in science and technology as well as in industries,” he said. “Shanghai is a window to international exchanges in mathematics, a cradle for modern mathematics education and a test land for innovation.”

    He pointed out that Shanghai is speeding up its efforts in the construction of “five centers,” namely global centers for economy, finance, trade, shipping and science and technology innovation, and improving its capacity and core competitiveness.

    “To achieve scientific and technological innovation and economic development, we need to pay more attention to the development of mathematics to enhance its support for urban development as a driving force,” he said. 

    “We should enhance the support for the cutting-edge exploration of mathematics research, the innovation and development of mathematics education and the extensive application of mathematic science so that mathematics can better lead innovation and development and help create a better life.”

    He said Shanghai will create a more flexible environment to encourage mathematicians dedicated in basic mathematical theory research and contribute more wisdom to expand the boundary of mankind’s mathematical knowledge.

    He called for innovation in math teaching approaches to enable young people to feel the charm of maths and develop expertise. He also called for the provision of solid support for the development of emerging industries, such as integrated circuits, biomedicine and artificial intelligence, and offering powerful momentum for the city’s digital transformation with new mathematical theories, technologies and methods.

    “I hope this conference will be an opportunity for all of us to hear the thoughts and wisdom of all guests, for all participating countries to exchange experiences in math education, for young scholars and students to get their enlightenment from the math masters,” he said. 

    “I also hope it will help widen and deepen international cooperation in mathematics, making new contributions to science and technology innovation in the world and development of the civilization of mankind.”

    Weng Tiehui, vice minister of education, said China has been attaching great importance to international exchanges in math education and called for enhanced international cooperation.

    Shanghai lauded as 'a cradle for modern math education'

    Ti Gong

    The 14th International Congress on Mathematical Education kicks off on Monday.

    Carlos Kenig, president of the International Mathematical Union, said that research and education of mathematics shouldn’t and couldn’t be separated, therefore he wished mathematicians and math educators would keep close cooperation to benefit each other.

    Frederick K. S. Leung, president of the International Commission on Mathematical Instruction and an endowed professor at the University of Hong Kong, said the conference has played an important role in promoting  excellence and tolerance in research and practices in mathematics education.

    He said the diversity of cultures can enrich research and practice and people need to respect different cultural traditions.

    Qian Xuhong, president of East China Normal University, welcomed all participants from around the world and said the university took the conference as an opportunity to further develop its mathematics education.

    According to Qian, the university has a long history of math education and has made great contributions to it in China. “One Lesson One Exercise,” the famous Shanghai math learning-aid book series, has gone beyond China and entered more than 400 British schools.

    Top awards in mathematical education – the Klein Award, Freudenthal Award and Castelnuovo Award – were presented to outstanding educators and researchers during the conference opening ceremony.

    French mathematician Cédric Villani, 2010 winner of the The Fields Award, delivered a plenary lecture on the topic of “Mathematics in Society” after the opening ceremony.

    Gu Lingyuan, a Shanghai math teacher and professor at East China Normal University, has delivered a plenary lecture to introduce the famous 45-year math teaching reform led by him, known as the “Qingpu Experiment,” to the audience.

    Teachers and students from Shanghai High School will show the changes in math teaching and learning during the pandemic on Thursday.

  • Math whiz, student surveyor, star-gazer – School News Network
    Math Club and Olympiad

    Math whiz, student surveyor, star-gazer – School News Network

    Name: Shambhabi Gautam
    School: Forest Hills Eastern High
    Jam: STEM, especially math and astronomy

    Forest Hills — This Eastern High School sophomore, a STEM standout and astronomy buff, developed a math survey this year to learn what areas her peers struggle with most. Programs to address what she found have already started.

    How old were you when this became something you wanted to pursue, and what’s the story there? “My love for math started around fourth grade,” Shambhabi said. “I was one of the fastest kids in class to finish this online game called Extra Math, (and) that was when I really discovered my thrill of solving problems.” Enrolling in the after-school Kumon learning program “definitely took my math skills to the next level, and definitely influenced my high school career a lot. 

    Her bottom line where math is concerned: “Getting a problem and finding the answer is really satisfying. It makes me feel super-accomplished when I come up with a finite answer, like when other students finish a book or a drawing.”

    As for her love of astronomy, that came out of a visit with her dad to Veen Observatory in Lowell Township when Shambhabi was in sixth grade. She still remembers “the little lights” that lined the uphill path they traveled after dusk, and getting to look through telescopes at Venus and Mars. “It was just amazing to me that we can see things that are so far away. There’s much more to astronomy than dots in the sky.”

    Shambhabi next to her sign on the grounds at Eastern High to honor all varsity tennis players (courtesy)

    A few related accomplishments: This year, her Science Olympiad team took a regional competition first-place award in astronomy — their project was on galaxy evolution — and third place in designer genes. “That’s more genetics and how DNA is replicated,” Shambhabi explained. She also was accepted to work under Michigan State University research professor Wolfgang Kerzendorf. Her task: “analyzing stellar data we got from telescopes. We were studying a supernova remnant called Cas A. We’ve been working on that for a few months. I feel like, right now, I’m just starting my journey in research.”

    Eastern High Principal Amy Pallo is particularly effusive about Shambhabi’s math survey conducted this year. Its origins: in ninth grade, Shambhabi explained, her math teacher handed back a quiz she scored quite well on, but many of her classmates did not. 

    “That really bugged me, seeing other people struggle. I wanted (to create) a platform where people didn’t feel shy about saying where they struggled, and felt comfortable getting help.”

    Pallo helped connect her with all Eastern High math teachers, who asked their students in February to take the survey Shambhabi created. Resources and online and in-person assistance — often from students who volunteered to help — is already underway. 

    Is there a teacher or teachers who have had a big impact on your involvement in this? Taking chemistry and AP biology and physics classes this year, thanks to her counselor, Mitchell Blink. “He let me take pre-calculus as a freshman; he believed in me and knew I could handle it.” Though she admits with a laugh, “He was a little skeptical at first.” 

    When it comes to influential math teachers, she said, “Honestly, all the ones I have had so far, but Mr. (Dan) Morley in particular believed in me as a freshman and helped me start a math club and let me take charge. He knew I was not only a good student academically, but I could be a responsible leader and get things done.”

    Do you plan to pursue this professionally? “I definitely know that I am going to go into STEM, because I love science. I just know that I am going to use my math skills and apply them to genetics and biology. I see myself leading research projects, whether that is in space or in our cells.”

    The biggest lesson you have learned from your involvement in this is… The importance of looking at the bigger picture, but also (as in research), of studying individual responses and categorizing general trends. 

    “I also realized I have to persevere, and run through trial and error. The first proposal will never be perfect.” And finally: “self-motivation is key to doing anything, especially during a time like COVID. As the internet gets more integrated into society the possibilities get greater, but it also means distractions get greater too.”

    Other hobbies/interests: Shambhabi plays flute in Eastern’s wind ensemble, and is a drum major and leader. She’s also a junior black belt in karate, and is on the varsity tennis team. “And I’m a big watcher of anime.” 

    Speaking of: If you walked into your school building to theme music, what would the song be? The theme music from the anime “Fairy Tail.” “It has sentimental value,” she said. “It was the first song I learned to play on the piano, and the first song I played at the talent show at my middle school.”

  • Two key neurotransmitters linked to math ability

    Two key neurotransmitters linked to math ability

    A child presses the buttons of a calculator with a pencil to solve problems in a math book.Share on Pinterest
    Researchers have found a link between two key neurotransmitters and math ability. Bob Rowan/Getty Images
    • An individual’s math abilitymay have links to levels of two chemical messengers — gamma-aminobutyric acid (GABA) and glutamate — in the brain, a study suggests.
    • To determine this, scientists measured the levels of these neurotransmitters in children and adults and correlated them with test scores.
    • Children who were good at math were likely to have higher GABA and lower glutamate levels in their brains.
    • Meanwhile, the reverse was true for adults: lower GABA and higher glutamate levels reflected greater math ability.
    • The findings suggest that neurotransmitter levels in the brain might predict future math ability.

    Could today’s math professors and arithmetic geniuses have been born with a biological advantage?

    Seeking to explore this possibility, a new study set out to find whether an individual’s math ability was associated with concentrations of two key neurotransmitters involved in learning.

    The researchers, led by Roi Cohen Kadosh, professor of cognitive neuroscience, and George Zacharopoulos from the University of Oxford in the United Kingdom, looked into GABA and glutamate levels in the brain to see if these neurotransmitters could predict mathematics ability for the future.

    GABA and glutamate are both naturally occurring amino acids that have complementary roles: the former inhibits or reduces the activity of neurons or nerve cells in the brain, while the latter makes them more active. Their levels fluctuate across the lifespan.

    “We focused on GABA and glutamate as it is known that these neurotransmitters are key players in neuroplasticity, learning, and cognition. We chose mathematical ability as it is a complex cognitive skill that takes years (if at all) to gain real expertise. This combination made the experiment really interesting as we could see how GABA and glutamate are involved in a complex cognitive skill that takes years to mature,” said Dr. Kadosh, speaking to Medical News Today.

    Kadosh and his colleagues not only found a link but also found that levels of these neurotransmitters switched as children grew into adults.

    Their research appears in the journal PLOS Biology.

    As part of the study, the researchers subjected 255 participants, aged 6 to university level, to two math achievement tests, 1.5 years apart, and analyzed their performances.

    They then correlated the test results with the GABA and glutamate levels in their brains.

    The children who had higher GABA levels in a region of the brain called the left intraparietal sulcus (IPS) scored higher on math tests. Conversely, those with high glutamate in the IPS had lower test scores.

    However, for adults, the scientists noticed the exact opposite.

    Those with high levels of glutamate in their brains had better scores on their math tests and those with high concentrations of GABA scored lower.

    After testing the participants twice and 1.5 years apart, the researchers found that adults with lower GABA got high marks on the first math test and they did well in the test the second time around, too.

    This longitudinal approach the scientists pursued helped them predict math ability for the future.

    The findings also show that the levels of GABA and glutamate in the brain switch later around puberty. This suggests that the role these neurotransmitters play differs during a person’s development.

    “The results that surprised us the most was that the link between GABA and glutamate and mathematical ability was switched from childhood to adulthood. It tells us that the relationship between GABA and glutamate and skill acquisition/ability is not similar across the developmental stages, and depends on our age.”

    – Dr. Roi Cohen Kadosh

    Commenting on the study, Dr. Santosh Kesari, Ph.D., neuroscientist and neuro-oncologist at Providence Saint John’s Health Center in Santa Monica, California said:

    “We do know that during brain development things change and the sensitivity of a brain region to a particular neurotransmitter can be affected as the brain develops and one matures. So, even though it is the same transmitter like GABA or glutamate, the effects can be different earlier on in development versus later on in development to how those neurotransmitters may work or affect the brain.”

    Dr. Kesari said this shift at an early age may also be a marker indicating that certain people are more prone to improving their mathematical abilities.

    The study’s authors say this switch in neurotransmitter levels during development also highlights an “unknown principle of plasticity”.

    Brain plasticity, also called neuroplasticity, is the nervous system’s ability to change and rewire its connections and structure in response to stimuli, such as learning, and experience.

    The link scientists found between plasticity and brain excitation (via glutamate) and inhibition (via GABA) across different stages of development suggests that this link is not fixed and can change over time, giving us further insight into the process of brain development.

    One shortfall of the study is the limited group of participants, which means the findings may not be generalizable to other racial or ethnic groups.

    “We do not know [if they are] at this stage. We will need to examine that. I think that the answer will be positive, although it might be that the exact brain region might differ, as the way we teach mathematics could differ from one culture to another,” said Kadosh.

    Kesari also pointed out that this study alone also does not propose a way of manipulating these neurotransmitters or how to use this information to make a decision about how better to teach people.

    Although we do have drugs that can affect levels of these transmitters, such as antidepressants, the real question is determining whether they would help earlier, later or at specific known disease states, Kesari told MNT.

    With that also comes the challenge of not affecting other functions in the brain. As these same neurotransmitters are involved in a complex network of functions, a change to any one of them could have unwanted outcomes if the balance is disrupted.

    “You cannot switch the balance of the neurotransmitters and expect only mathematical achievement to improve without potentially causing a negative effect on other issues such as anxiety, depression, mood, etc. This is something that really needs to be further thought about and tested in the future,” said Dr. Kesari.

    “[We won’t know] how this would translate into the future until someone does a study where they use a method [analyzing how] it would be neuropsychologically different earlier on versus later on,” he added.

    Kadosh said it is best to exercise caution when interpreting this data for real-life applications. Nevertheless, it has opened up an area of great potential.

    He told MNT that as their research focused on only healthy children, adolescents, and adults, it was too premature to talk about clinical implications at this stage.

    “It would be interesting in the future to see how these results look in those with learning difficulties, and whether manipulation of GABA and/or glutamate could improve learning and education,” he said.

    Dr. Kesari, meanwhile, said knowing there is an age dependency could affect how we teach or how we improve mathematical abilities in young and older kids.

    “The clinical implications [from this study] really is that how we propose to use this knowledge to improve, for instance, mathematical achievement, will vary depending on the age that you wish to try to predict an intervention that you would want,” he added.

    The next steps from this research could be to come up with better strategies to teach and help students learn math and explore the possibility of making learning interventions to improve cognition.

    On that note, Kadosh said they hope to examine the development of brain-based interventional programs in the future and added:

    “We are working on neurostimulation methods to improve learning and cognition. We examine its application in clinical (children and adults) and non-clinical populations (only adults). Previous studies have suggested that these neurostimulation methods can change GABA and/or glutamate. We hope that this will open ways to improve learning and cognition in those who need that.”