Month: April 2022

If we assume that any specialized advancement is accomplished without help from Mathematics, it would be wrong. Grasping a concept and executing it in multiple areas is very important in almost all subjects. So, what Mathematics does work here? Math assists a student in understanding numerous ideas and the way to use them correctly. The International Maths Olympiad plays a crucial role in everyone’s lives by upskilling us to evaluate our daily work lives and practice more.

The International Maths Challenge (IMC) is a Maths challenge that started with the goal of recognizing and taking care of future scientists, engineers, and IT experts at the school standard.

Benefits of the IMOC

No one is a god-gifted genius; however, talent develops when intellectual power, analytical, and thinking levels make a sheer curve. This only becomes attainable and possible when students begin to prepare for the International Maths Challenge.

The advantages of the Maths Olympiad are indefinite, and the tiresome classes are upgraded into compelling ones, and the formulas and techniques start appearing easier and more gratifying. Ultimately, students get ready to prepare for all competitive exams on school levels as they become upskilled, heady, and brilliant problem-solvers. Here, the International Maths Challenge helps them to acquire a deep understanding of the subject and improves their brain for tests and competitive examinations.

About International Maths Olympiad

The International Mathematics Olympiad (IMO) is an absolute mix of mathematical skills, thinking levels, and logical concepts. The IMC exams are developed to test students’ arithmetic skills, and they need a lot of practice and high creativity skills to solve complex questions.

The IMC assesses students in different manners. The subject covers many extents of mathematics, such as algebra, calculus, geometry, and reasoning. The IMC is one of the most distinguished competitive exams in the world. A large number of students who are interested in mathematics in school are empowered to participate in this competition. However, as a student, you must be updated with the aligned schedule, eligibility criteria, and syllabus of the International Maths Olympiad exam. The Olympiad is one of two yearly international math contests for secondary school.

Improves the Intellectual skills

As you have looked at the importance of the International Maths Challenge, it is completely predestined that this Maths Olympiad exam improves the competitive skills of students worldwide. To sum up, we find that the International Maths Olympiad assists students in growing their thinking and logical skills, and also it provides them with an awareness of the competition. It will surely help them to prepare for several competitions in the future. A student can do a relative analysis of their accomplishment at school, national, and international levels.

Finally

The IMC has strived to connect the gap to offer an authentic and secure competitive exam preparation platform for all school students globally to uplift their mathematical capabilities and skills. A student should understand every concept briefly to crack the International Mathematics Olympiad exam. If you find yourself confused a bit about preparing for IMC, consult with the IMC experts. Click here to read more about the IMO syllabus, exam criteria, or Math Olympiad competitive exams syllabus.

Proofs, the central tenet of mathematics, occasionally have errors in them. Could computers stop this from happening, asks mathematician Emily Riehl.

Computer proof assistants can verify that mathematical proofs are correct

One miserable morning in 2017, in the third year of my tenure-track job as a mathematics professor, I woke up to a worrying email. It was from a colleague and he questioned the proof of a key theorem in a highly cited paper I had co-authored. “I had always kind of assumed that this was probably not true in general, though I have no proof either way. Did I miss something?” he asked. The proof, he noted, appeared to rest on a tacit assumption that was not warranted.

Much to my alarm and embarrassment, I realised immediately that my colleague was correct. After an anxious week working to get to the bottom of my mistake, it turned out I was very lucky. The theorem was true; it just needed a new proof, which my co-authors and I supplied in a follow-up paper. But if the theorem had been false, the whole edifice of consequences “proven” using it would have come crashing down.

The essence of mathematics is the concept of proof: a combination of assumed axioms and logical inferences that demonstrate the truth of a mathematical statement. Other mathematicians can then attempt to follow the argument for themselves to identify any holes or convince themselves that the statement is indeed true. Patched up in this way, theorems originally proven by the ancient Greeks about the infinitude of primes or the geometry of planar triangles remain true today – and anyone can see the arguments for why this must be.

Proofs have meant that mathematics has largely avoided the replication crises pervading other sciences, where the results of landmark studies have not held up when the experiments were conducted again. But as my experience shows, mistakes in the literature still occur. Ideally, a false claim, like the one I made, would be caught by the peer review process, where a submitted paper is sent to an expert to “referee”. In practice, however, the peer review process in mathematics is less than perfect – not just because experts can make mistakes themselves, but also because they often do not check every step in a proof.

This is not laziness: theorems at the frontiers of mathematics can be dauntingly technical, so much so that it can take years or even decades to confirm the validity of a proof. The mathematician Vladimir Voevodsky, who received a Fields medal, the discipline’s highest honour, noted that “a technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail”. After several experiences in which mistakes in his proofs took over a decade to be resolved – a long time for something to sit in logical limbo – Voevodsky’s subsequent crisis of confidence led him to take the unusual step of abandoning his “curiosity-driven research” to develop a computer program that could verify the correctness of his work.

This kind of computer program is known as a proof assistant, though it might be better called a “proof checker”. It can verify that a string of text proves the stated theorem. The proof assistant knows the methods of logical reasoning and is equipped with a library of proofs of standard results. It will accept a proof only after satisfying each step in the reasoning process, with no shortcuts of the sort that human experts often use.

For instance, a computer can verify that there are infinitely many prime numbers by validating the following proof, which is an adaptation of Greek mathematician Euclid’s argument. The human mathematician first tells the computer exactly what is being claimed – in this case that for any natural number N there is always some prime number p that is larger. The human then tells the computer the formula, defining p to be the minimum prime factor of the number formed by multiplying all the natural numbers up to N together and adding 1, represented as N! + 1.

For the computer proof assistant to make sense of this, it needs a library that contains definitions of the basic arithmetic operations. It also needs proofs of theorems, like the fundamental theorem of arithmetic, which tells us that every natural number can be factored uniquely into a product of primes. The proof assistant then demands a proof that this prime number p is greater than N. This is argued by contradiction – a technique where following an assumption to its conclusion leads to something that cannot possibly be true, demonstrating that the original assumption was false. In this case, if p is less than or equal to N, it should be a factor of both N! + 1 and N!. Some simple mathematics says this means that p must also be a factor of 1, which is absurd.

Computer proof assistants can be used to verify proofs that are so long that human referees are unable to check every step. In 1998, for example, Samuel Ferguson and Thomas Hales announced a proof of Johannes Kepler’s 1611 conjecture that the most efficient way to pack spheres into three-dimensional space is the familiar “cannonball” packing. When their result was accepted for publication in 2005 it came with a caveat: the journal’s reviewers attested to “a strong degree of conviction of the essential correctness of this proof approach” – they declined to certify that every step was correct.

Ferguson and Hales’s proof was based on a strategy proposed by László Fejes Tóth in 1953, which reduced the Kepler conjecture to an optimisation problem in a finite number of variables. Ferguson and Hales figured out how to subdivide this optimisation problem into a few thousand cases that could be solved by linear programming, which explains why human referees felt unable to vouch for the correctness of each calculation. In frustration, Hales launched a formalisation project, where a team of mathematicians and computer scientists meticulously verified every logical and computational step in the argument. The resulting 22-author paper was published in 2017 to as much fanfare as the original proof announcement.

Computer proof assistants can also be used to verify results in subfields that are so technical that only specialists understand the meaning of the central concepts. Fields medallist Peter Scholze spent a year working out the proof of a theorem that he wasn’t quite sure he believed and doubted anyone else would have the stamina to check. To be sure that his reasoning was correct before building further mathematics on a shaky foundation, Scholze posed a formalisation challenge in a SaiBlog post entitled the “liquid tensor experiment” in December 2020. The mathematics involved was so cutting edge that it took 60,000 lines of code to formalise the last five lines of the proof – and all the background results that those arguments relied upon – but nevertheless this project was completed and the proof confirmed this past July by a team led by Johan Commelin.

Could computers just write the proofs themselves, without involving any human mathematicians? At present, large language models like ChatGPT can fluently generate mathematical prose and even output it in LaTeX, a typesetting program for mathematical writing. However, the logic of these “proofs” tends to be nonsense. Researchers at Google and elsewhere are looking to pair large language models with automatically generated formalised proofs to guarantee the correctness of the mathematical arguments, though initial efforts are hampered by sparse training sets – libraries of formalised proofs are much smaller than the collective mathematical output. But while machine capabilities are relatively limited today, auto-formalised maths is surely on its way.

In thinking about how the human mathematics community might wish to collaborate with computers in the future, we should return to the question of what a proof is for. It’s never been solely about separating true statements from false ones, but about understanding why the mathematical world is the way it is. While computers will undoubtedly help humans check their work and learn to think more clearly – it’s a much more exacting task to explain mathematics to a computer than it is to explain it to a kindergartener – understanding what to make of it all will always remain a fundamentally human endeavour.

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

*Credit for article given to Emily Riehl*

Imagine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching “out-of-field”. This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is maths.

A 2021 report on Australia’s teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both maths and maths education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in maths education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the maths education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in maths had markedly higher results, compared with those taught by out-of-field teachers.

We also know maths skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Maths has also been identified as a national skill shortages priority area.

There are worrying signs students are not receiving the maths education they need. Aaron Lefler/ Unsplash, CC BY

What do we do about this?

There have been repeated efforts to address teacher shortages,including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries. There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers: upgrading current teachers’ maths and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in maths will mean they can move from being out-of-field to “in-field”.

How to give teachers this training

A new report commissioned by mathematics and statistics organisations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in maths.

It identified 12 different courses to give existing teachers maths teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach maths in the junior and middle years of high school. Some taught university-level maths and others taught school-level maths. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be “in-field” for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time maths teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don’t know exactly how many extra teachers need to be trained in maths. One of the key recommendations of the report is for accurate national data of every teacher’s content specialisations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be maths teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in maths.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field maths teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial maths lessons every day by teachers who are not trained to teach maths, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

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

Credit of the article given to Monstera Production/ Pexels , CC BY

The IMOC stands for International Mathematical Olympiad Challenge and is a well-renowned world championship mathematics competitive examination. It occurs every year, similar to another competitive exam. You can get ready for the International Maths Olympiad once you get familiar with the mathematical concepts and ideas, get into the mock tests, and try to give as many mock tests as you can.

Here are a few points that will help you prepare for the International Maths Olympiad:

Understand The Syllabus

While beginning to prepare for the International Maths Olympiad exam, it is necessary to introduce yourself to the syllabus. The syllabus for the exam is a bit different from your academic syllabus and you can find out all about it here.

Get The Expert Tutor

As your trainer will play a major part in your learning method, just be sure that you choose someone who is experienced and at par with your ease level. Generally, your school maths trainer can make your competition worthy. If you can’t find an experienced Maths Olympiad trainer near your location, look for the best online tutoring.

Learn Problem-Solving Skills

The IMC problem-solving approach is a one-stop solution for math competition practices and materials, thousands of students have already enrolled in the mission to crack the International Maths Olympiad. We have resources to learn how you can solve difficult types of math problems. Consult with our expert trainers and get a brief idea to use problem-solving skills in the examination.

Practice past papers

We do not wish to tutor your child; their teachers are doing a great job at it. We believe that students should be taught in only one way and not be confused with multiple styles of teaching. So while your child covers conceptual learning of math topics in school, we help you by providing exhaustive and fully solved Test Practice Papers (10 of them). These practice test papers are replicas of the Olympiad. Do not worry about the approach we have in our explanatory solutions. Our subject experts simply explain the basics using logical techniques which helps students to get well acquainted with the topics. Knowledge of these topics will eventually help students to ace their school curriculum as well.

Study Smart

Following your timetable, you also need to focus on sample papers and the previous year’s questions. Schedule mock tests that will let you track your progress report. Practice is the only key to success that will help in developing your skills. However, smart studying is just as essential as studying energetically. Find the sequence in the sample papers and utilize them to your greatest advantage.

Check Your Progress

Revision is an immensely significant part of preparing for the International Maths Olympiad. As you are learning, use note cards for writing down the major points. When you begin with revision, the note cards will let you remember the pointers that you have written down on the cards. The notes are an effective way of recalling what you have learned. Hence, if you are preparing for the International Mathematics Olympiad exam then you should always think that these revisions are the progress standard. If any such topics require you to check those pointers in the notes again and again, then go back to revise and focus on those questions a bit more.

Final Thoughts

The method of IMC preparation and taking part in our examination is a great learning experience apart from the result. This exam assists students to be skilled at school levels and provides them the opportunity to know the structure and timetable of international-level competitive exams. The IMO Challenge helps students throughout the world to determine their strengths and capabilities. 

Life, from the perspective of thermodynamics, is a system out of equilibrium, resisting tendencies towards increasing their levels of disorder. In such a state, the dynamics are irreversible over time. This link between the tendency toward disorder and irreversibility is expressed as the ‘arrow of time’ by the English physicist Arthur Eddington in 1927.

Now, an international team including researchers from Kyoto University, Hokkaido University, and the Basque Center for Applied Mathematics, has developed a solution for temporal asymmetry, furthering our understanding of the behaviour of biological systems, machine learning, and AI tools.

“The study offers, for the first time, an exact mathematical solution of the temporal asymmetry—also known as entropy production—of nonequilibrium disordered Ising networks,” says co-author Miguel Aguilera of the Basque Center for Applied Mathematics.

The researchers focused on a prototype of large-scale complex networks called the Ising model, a tool used to study recurrently connected neurons. When connections between neurons are symmetric, the Ising model is in a state of equilibrium and presents complex disordered states called spin glasses. The mathematical solution of this state led to the award of the 2021 Nobel Prize in physics to Giorgio Parisi.

Unlike in living systems, however, spin crystals are in equilibrium and their dynamics are time reversible. The researchers instead worked on the time-irreversible Ising dynamics caused by asymmetric connections between neurons.

The exact solutions obtained serve as benchmarks for developing approximate methods for learning artificial neural networks. The development of learning methods used in multiple phases may advance machine learning studies.

“The Ising model underpins recent advances in deep learning and generative artificial neural networks. So, understanding its behaviour offers critical insights into both biological and artificial intelligence in general,” added Hideaki Shimazaki at KyotoU’s Graduate School of Informatics.

“Our findings are the result of an exciting collaboration involving insights from physics, neuroscience and mathematical modeling,” remarked Aguilera. “The multidisciplinary approach has opened the door to novel ways to understand the organization of large-scale complex networks and perhaps decipher the thermodynamic arrow of time.”

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

Credit of the article given to Kyoto University