
Math lecturer wins nationwide online training award – Information Middle
The USA Distance Studying Affiliation has named Jeremy Glass, a senior lecturer of arithmetic at The College of Texas at Arlington, because the Gold Winner of its 2022 Award for Excellence in Distance Studying Educating. The worldwide award is given to an impressive particular person or group of people whose instructing demonstrates extraordinary achievements.
Glass stated he strives to create a way of neighborhood in his online programs in order that college students really feel welcome and supported.
“When instructors present flexibility by means of asynchronous actions and openended due dates, there’s an inherent lack of neighborhood,” Glass stated. “My aim is to create a welcoming, protected surroundings the place college students really feel comfy to study and make errors alongside their friends.”
To construct neighborhood, Glass asks his college students to hitch a synchronous webcast every week. Underneath Glass’ steerage, college students hash out the week’s hardest math issues and homework assignments in actual time.
Greg Wells, who’s in search of a second bachelor’s diploma in laptop science and engineering, took Glass’ “Calculus I” course online. Wells stated Glass’ strategies for pupil engagement created a web based studying surroundings completely different from others he had skilled.
“He sincerely needs every of his college students to succeed,” Wells stated. “His ardour for instructing was palpable, and it contributed to an distinctive studying expertise.”
Glass has performed an essential position within the Division of Arithmetic’ efforts to enhance pupil success charges, creating online arithmetic programs and growing revolutionary educational fashions. His success in implementing these new educational strategies has been mirrored within the elevated enrollment and success of scholars in primary math programs.
Jianzhong Su, professor and chair of arithmetic, really helpful Glass for the award, citing his management position in growing revolutionary online arithmetic curriculum.
“Jeremy has performed an essential position in growing a very efficient and revolutionary undergraduate instructing methodology for online arithmetic programs,” Su stated. “His success is just not solely mirrored in elevated enrollment, but additionally within the excellent efficiency of his college students.”
Within the 202021 educational 12 months, UTA supplied practically 12,000 online programs. UTA’s total core curriculum is out there online, and the College gives absolutely online undergraduate and graduate levels in additional than a dozen disciplines.
U.S. Information & World Report repeatedly acknowledges UTA’s online applications among the many nation’s finest. In 2022, online grasp’s applications within the School of Schooling and the School of Nursing and Well being Innovation ranked within the nation’s high 60.

Math for Youngsters: Interactive Studying with Brighterly
Brighterly, a webbased math studying platform, efficiently challenges the best way arithmetic is taught. Brighterly’s tutors make studying easy and enjoyable in order that your youngsters will giggle at math jokes and playfully calculate giant numbers of their minds.
Brighterly teaches PreK5 math in 1to1 lessons the place idea is mixed with interactive actions, video games, and handson examples that assist college students excel in numbers.
How Brighterly’s Math Studying Works?
Brighterly affords personalised math programs for youths aged 311. The platform connects skilled math tutors with homeschoolers and fogeys who goal to enhance their youngsters’s math abilities. Right here is how Brighterly works:
Mother and father Order a Free Trial Lesson
At Brighterly, mother and father can enroll youngsters in personalised math lessons, resolve on the schedule, and revel in interactive math studying. The service additionally affords a free trial lesson so that oldsters can select whether or not they need to proceed working with Brighterly. This fashion, mother and father can do this service with out placing their cash upfront.
Youngsters Examine Math in Customized Math Courses
You’ll be able to schedule a demo lesson on the most handy time. You’ll obtain a Zoom invitation through SMS or e mail. As soon as the category begins, you may keep behind your little one and watch how they be taught math with a private tutor.
The tutor will test your little one’s math degree and decide the information gaps. After the category, the tutor will summarize their observations and report in your little one’s information.
If a demo lesson satisfies you, you should purchase one among Brighterly’s studying packages – studying math for youths has by no means been that clear.
How College students Study Math with Brighterly’s Tutors
At Brighterly, every class is standardized in period to an instructional hour, which is 45 minutes. This firm focuses on personalised studying and believes that youngsters ought to examine comfortably at their very own tempo and within the format they love. Due to this fact, Brighterly’s tutors deliver variety to the training course of, combining a number of lesson varieties:
Core Math Classes
Throughout core math classes, youngsters get launched to math ideas. They be taught idea and follow studied subjects.
These classes are the cornerstone of the mathematics program and comprise a considerable a part of the training course of.
Enjoyable Classes
Enjoyable classes revolve round math video games and interactive actions which are designed to increase youngsters’s math abilities, entertain them, and spark their curiosity in math.
Brighterly’s tutors skillfully embed enjoyable parts into STEM actions to make learning fascinating and enhance youngsters’s motivation to be taught arithmetic.
Suggestions & Analysis Courses
Suggestions and analysis lessons are designed in order that tutors can test children’ information and decide additional studying route and subjects to be emphasised. These classes additionally enable youngsters to share their ideas on the training course of and provide ideas.
How A lot Does Brighterly’s Math for Youngsters Price?
Brighterly affords a number of math lesson plans:
8 Courses at $34 per Lesson
The smallest math package deal consists of:
 Eight 1to1 classes.
 Interactive video games and puzzles throughout classes.
 A progress report on the finish of the eighth class.
24 Courses at $31 per Lesson
The second package deal consists of all choices of the primary lesson plan and affords extra options:
 Twentyfour math lessons.
 Two information & progress analysis studies.
 Two makeup classes.
 A full section I of Brighterly Math Studying Journey.
 A course completion certificates.
48 Courses at $28 per Lesson
The third lesson package deal consists of options of the earlier plans and a few new perks:
 Fortyeight math lessons.
 Class notes despatched through e mail after your pupil completes a studying block.
 4 information & progress analysis studies.
 One parentteacher assembly.
 Sportbased studying assets with solutions.
 Section II of Brighterly Math Studying Journey.
 Versatile class schedule.
 A course completion certificates.
Math for Youngsters Online: What Technical Setup Do You Want?
College students at Brighterly examine online, and you have to:
 A private pc or a laptop computer.
 An online digital camera (no want to purchase one when you’ve got a laptop computer) and a microphone.
 A steady web connection.
Brighterly suggests learning on a laptop computer or a PC utilizing a Chrome browser for the most effective studying expertise.
Brighterly Stands out of Different Math for Youngsters Web sites
Brighterly’s math assist for youths affords college students wellrounded, diversified, personalised math training utilizing a playbased method and nurturing love for numbers and arithmetic.
Brighterly works with skilled math academics who possess an in depth background in online tutoring. Much more so, Brighterly completely selects tutors, and fogeys don’t must spend their time looking for particular academics. All of them have excellent expertise in arithmetic.
Testimonials
Carol, Linda’s mother, 2nd grade
I employed a number of particular person tutors earlier than, nevertheless it didn’t work due to a scarcity of consistency in classes. However guys from Brighterly alleviated all my fears about personalised tutorship. Linda is getting higher and higher at math.
She likes to retell math details she discovered from her tutor. I’m fascinated to see my little one getting fluent in arithmetic and can proceed working with Brighterly for certain.
Maverick, Evelyn’s dad, third grade
I’m certain I’ll guide extra classes from Brighterly. Evelyn’s completed the eighth math class, and I admit she will get higher at numbers. Plus, I obtained an in depth report from her tutor, so I see these guys are professionals.
The one factor I can point out is that it’s ideally to work within the Chrome browser if you wish to have a easy expertise. But, it’s not a giant deal as a result of my daughter will quickly outperform me in multiplication.
Flora, Charles’s mother, 4th grade
Six months in the past, I had no clue tips on how to clarify math to my little one. Charles got here residence from college, and we scratched our heads over his homework. My pal whose daughters studied at Brighterly beneficial this web site to me. I’ve had no points with this service for half a yr now.
I often have online conferences with my son’s tutors to debate his efficiency, and I see him getting higher at numbers in school. I’m planning to review with Brighterly till Charles finishes elementary college.
FAQ
Can I Cancel the Plan?
In case you consider it’s essential to cancel the subscription plant at Brighterly, you are able to do it at any second. In case you resolve to cancel the subscription in the midst of the plan, it is best to test the Fee and Refund Coverage on Brighterly’s webpage.
What Are the Minimal Age Necessities?
Brighterly offers lessons for college students aged 311. So long as you consider your child wants to review math, you may enroll them within the math programs.
Does Brighterly Provide Offline Math Courses?
No. Brighterly affords solely online lessons for essentially the most comfort – you don’t have to choose up your youngsters or fear about getting late for the assembly. So long as it’s time for a webbased lesson, your child can change on their PC/laptop computer, and that’s it.
What Is the Length of a Class?
Brighterly’s online lessons are 45 minutes lengthy.
How Many Courses Does My Child Want?
Your little one can proceed learning at Brighterly for so long as they want. In case your child loves math lessons, and so they assist them change into higher at arithmetic, you may carry on studying with Brighterly.

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 10^{10100} (one followed by 10^{100} zeros, with 10^{100} 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 10^{100} 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 settheoretical 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 10element set has 2^{10} = 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 wellestablished 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 selfevident. 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 ℵ_{0}, or aleph null. For every infinite cardinal ℵ_{a}, there is a next larger cardinal number ℵ_{a+1}. Thus, the smallest infinite cardinal ℵ_{0} is followed by ℵ_{1}, then ℵ_{2} 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 2^{ℵ0}, or “continuum.”
In the 1870s, Cantor ruminated over whether the size of ℝ was the smallest possible cardinal above ℵ_{0}—in other words, whether ℵ_{1} = 2^{ℵ0}. 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 GermanIsraeli colleague Abraham Fraenkel. Called ZFC, or ZermeloFraenkel 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 twodimensional plane (sometimes called the xy plane) or the threedimensional 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 a ∙b. 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 twodimensional 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 xy plane with a large cardinality need be neither measurable nor of large measure. For example, the entire plane with its 2^{ℵ0} elements has an infinite measure. But the x axis with the same cardinality has a twodimensional 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 nonnull set is denoted by non(𝒩). It follows that ℵ_{0} < non(𝒩) ≤ 2^{ℵ0}, because any set of size ℵ_{0} is a null set, and the whole plane has size 2^{ℵ0} and is not a null set. Thus, ℵ_{1}≤ non(𝒩) ≤ 2^{ℵ0}, because ℵ_{1} is the smallest uncountable cardinal. If we assume CH, then non(𝒩) = 2^{ℵ0}, because, in that case, ℵ_{1} = 2^{ℵ0}.
We can define another cardinal number, add(𝒩), to answer the question, What is the minimal number of null sets whose union is a nonnull set? This number is less than or equal to non(𝒩): if A is a nonnull set containing non(𝒩) many elements, the union of all the non(𝒩) many oneelement subsets of A is the nonnull set A. But a smaller number of null sets (though they would not be oneelement 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 nonnull 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 nonnormal 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 ℵ_{1}, 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 nonnegligible 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 nonnegligible 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 ℵ_{1} = 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 ℵ_{1} ≤ 2^{ℵ0}, but neither ℵ_{1} < 2^{ℵ0} nor ℵ_{1} = 2^{ℵ0} 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 2^{ℵ0} 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) = x^{2} 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 ℵ_{1} ≤ 𝔟 ≤ 𝔡 ≤ 2^{ℵ0} 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 lessorequal signs are replaced by arrows.
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, ℵ_{1} (the smallest number in the diagram) is equal to 2^{ℵ0} (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 lessorequal 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, ℵ_{1} and ℵ_{2}, to the entries of the diagram in various ways. For example, they created a universe for which ℵ_{1} = add(𝒩) = cov(𝒩) and ℵ_{2} = 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 ℵ_{1} and ℵ_{2} 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 ℵ_{1} and ℵ_{2} 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 ℵ_{1} 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 ℵ_{1} and 2^{ℵ0}—even uncountably many. Rather our result proves that there are mathematical universes in which the 10 specific cardinal numbers between ℵ_{1} and 2^{ℵ0} 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 ℵ_{1} 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 ℵ_{1} and ℵ_{2}, 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
Seventeenyearold 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
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 studentled offcampus club to raise money for the Vestavia Hills Math Team.
The AYMC also held a weeklong 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, PoShen 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, RamLakhan’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 problemsolving 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. “

Shanghai lauded as ‘a cradle for modern math education’
Ti Gong
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 COVID19 pandemic.
The sevenday 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 cuttingedge 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.
Ti Gong
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 learningaid 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 45year 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.