Should We Follow Japanese Maths Teaching?

Improving mathematics standards in the US has been a common goal since the Common Core State Standards were introduced. Unfortunately if we use the National Assessment of Educational Progress figures there have been no improvements in maths at all during the last few years.

This has pushed American educational experts to look overseas for some inspiration, and they think they’ve found some answers in the way the Japanese teach maths. Unlike the traditional US strategies which focus on memorizing, the Japanese method will focus on solving mathematical problems.The method is Sansu arithmetic and it actually aligns quite neatly with the US Common Core so it wouldn’t actually be that difficult.

It’s actually somewhat ironic that the method adopted by the Japanese was actually first identified in the USA. The National Council of Teachers of Mathematics investigated this method in the 1980s however it was never officially adopted.And so, some 40 years on our children are still learning mathematics by memorising from simple sheets.

The Japanese instantly recognized the benefits to students of being able to create their own solutions and methods for solving problems.  It means that in classes, you can create a context and student will learn much more easily than simply learning by rote from a sheet of paper.  A similar method has been tried in some European countries including in Germany, article on BBC about German education on BBC iPlayer, access here.  The focus is to learn and interest students by a concept called hatsumon (addressing a concept through questioning). If it taught well, students will see their learning benefit them in real life situations.Which also helps promote elements of self confidence and some enthusiasm for the subject.

Lessons are created by individual teachers who then try the lesson in front of students and other teachers including a professor. The lesson is then discussed with the teacher so that feedback can be given and modifications made if necessary.If there is no feedback teachers are forced to make their own decisions on the quality of their own lessons.

These might seem like simple, common sense steps but unfortunately they are missing in many Western classrooms.There are many critics of the Common Core standards despite being endorsed by most educational organisations in the United States. One of the issues though is that teachers are given very limited training in the method.

A shortcoming which we’ve seen before. In the 1960s, there was a big push to introduce ‘new maths’ to push onto the space age. There was much enthusiasm but little change simply because nothing was invested in training the teachers in new methods.

There is a feeling that maths teaching in the US should change but a lack of direction and funding to implement this change. Japanese teachers get much more support whenever new methods are implemented across any area of education.Which in turn is usually reflected in the scores of students in international comparison tables.

There is no reason to accept that American students should be worse at maths. However, the fact is that a global economy demands certain skills and mathematics is at the top of that list.

Further Reading

Mathematicians and the Lottery

Do mathematicians play the lottery, it’s a difficult question as I suspect many would be hesitant to admit it. Although ly one thing is certain, mathematicians do get asked how they should play the lottery.   Even though you don’t usually see an endless parade of maths professors walking off with  the jackpot.

However although a guaranteed win strategy is unlikely, most forms of gambling have sensible tactics which can at least maximize your chances of winning (although minimizing losses is probably nearer the mark).    So can you use statistics to predict which numbers to choose?   Again unfortunately this is unlikely and any essay entitled how to win the lottery should be treated with a large amount of caution.

The safest mathematical response to the question of how to increase your chances of winning is this – buy more tickets.   No-one can dispute  that the more tickets you have the greater your chance of walking away with the jackpot.    Although these lotteries are never meant to be a proper gambling choice, the chance of winning is so remote that you would have to buy an awful lot of tickets to be in with even a reasonable chance.    The ticket is more a chance to dream than a realistic one of winning having said that over 1500 people have won more than a million pounds in the UK lottery for instance.

The UK lottery requires you to pick 6 numbers from a choice of 1 to 49.  You win a prize if you match three of those numbers,   the more numbers you match the more you will win.  The jackpot is normally paid out for all six numbers although this will be shared by anyone else also picking these numbers.   There is an additional bonus number which can be used if you get the first 5 which also is likely to pay out a big prize.

We can assume that most national lotteries are completely fair and random, after all there is no incentive to cheat as all the money is paid out.  Most of the European countries have lotteries and most you can play online, you may need to change your IP address like this to an Italian, Spanish or French one though to get access.

Of course with the lottery we are plainly in the field of probability and statistics, because it’s ultimately all about chance.   However if you’re aim is to maximise your chance of winning a large jackpot (rather than just picking the correct numbers), then there’s an important fact to remember.

The amount of the jackpot a winning ticket will benefit from depends on how many people picked those numbers.   Here there is a fundamental fact that will help you narrow down your choices slightly.  To maximise your chances of ‘not sharing’ then you need to avoid a certain group of numbers as much as possible.   The reason is that many millions of people base their numbers on birthdays, so the numbers 1-31 will be much more likely to selected than those of 32-49.

John Jones

Author of Watching BBC Abroad

One Variable Hypergeometric Functions

For more than 200 years some of the greatest mathematicians in history have studied the hypergeometric functions in their one variable context.   You’ll find them covered in all sorts of mathematical works ranging from Euler to Riemann. However it was Barnes and Mellin who initially studied the integral representations whilst Goursat investigated their special properties.

It can be difficult to classify the different type of hypergeometric functions however there are three main categories:

  • Functions defined as integrals.
  • Solutions to systems of differential equations.
  • Functions to series whose coefficients satisfy specific recursion properties.

Of course readers are probably familiar with the Mellin Barnes integral which defines these specific integrals. One variable functions have been understood for many years but mathematicians have now developed these approaches to cover multiple variables which of course yield slightly different results.    There are some interesting programmes on the UK Open University covering this, which may be available online – this link helps access the BBC since it was blocked abroad.

So while the study of the single variable functions is well over two centuries old, the study of the multivariate function is more recent. There has been something of a resurge in interest in the study of hypergeometric functions particularly in the last few years. The interest mainly is around the connections between hypergeometric functions and other areas of mathematics.

These include interesting areas such as investigating connections with such areas as algebraic geometry, symmetry and number theory for example. One of the key developments has come in the 80s and 90s through the work of Kapranov and Zelevinsky among others. In recent years this has been expanded by Saito, Sturmfels and Takayama.

Further Reading

The Breakthrough of Irrational Numbers

Many centuries ago, Pythagoras was one of the first people to make the claim that the Universe was governed by numbers.  However at the time, numbers were understood to be whole numbers and ratios of these numbers (fractions).   However mathematics had a surprise in store for this famous mathematician – there are numbers which don’t come under either of these two categories.    They are called irrational numbers and they can be difficult to comprehend as they cannot be written down as decimals or even as recurring decimals.

For example you can in fact easily define an recurring decimal in a relatively straight forward way.   Even infinitely recurring decimals can easily be expressed as a fraction if needed.  However if you try and express and irrational number as a decimal you end up with a number which continues indefinitely without a regular or consistent pattern. The concept of these ‘irrational numbers’ was a huge breakthrough for mathematicians who were trying to look beyond whole numbers and their associated fractions.   They were considered a discovery though as Leopold Kronecker stated in the 19th century – “God made the integers, all the rest are the work of man.”

The most famous irrational number is of course, Pi.  In schools it is usually approximated to a more friendly 3.14 or 3 1/7 however the true value of Pi is nearer 3.14159265358979323846 however even this is only an approximation of it’s true value.   In reality Pi can never actually be written down exactly as the decimal places go on forever without and consistent pattern.  It still useful though and these numbers are frequently used as RNG seed numbers for things like encryption and ciphers like this Smart DNS software here.

Here’s a starter of a few hundred decimal places for Pi-

3.141592653589793238462643383279502884197169399375105820974944592307816406286 208998628034825342117067982148086513282306647093844609550582231725359408128481 117450284102701938521105559644622948954930381964428810975665933446128475648233 786783165271201909145648566923460348610454326648213393607260249141273724587006 606315588174881520920962829254091715364367892590360011330530548820466521384146 951941511609433057270365759591953092186117381932611793105118548074462379962749 567351885752724891227938183011949129833673362440656643086021394946395224737190 702179860943702770539217176293176752384674818467669405132000568127145263560827 785771342757789609173637178721468440901224953430146549585371050792279689258923 542019956112129021960864034418159813629774771309960518707211349999998372978049 951059731732816096318595024459455346908302642522308253344685035261931188171010 003137838752886587533208381420617177669147303598253490428755468731159562863882 353787593751957781857780532171226806613001927876611195909216420198938095257201 065485863278865936153381827968230301952035301852968995773622599413891249721775 283479131515574857242454150695950829533116861727855889075098381754637464939319 255060400927701671139009848824012858361603563707660104710181942955596198946767 837449448255379774726847104047534646208046684259069491293313677028989152104752 162056966024058038150193511253382430035587640247496473263914199272604269922796 782354781636009341721641219924586315030286182974555706749838505494588586926995 690927210797509302955321165344987202755960236480665499119881834797753566369807 426542527862551818417574672890977772793800081647060016145249192173217214772350 141441973568548161361157352552133475741849468438523323907394143334547762416862 518983569485562099219222184272550254256887671790494601653466804988627232791786 085784383827967976681454100953883786360950680064225125205117392984896084128488 626945604241965285022210661186306744278622039194945047123713786960956364371917 287467764657573962413890865832645995813390478027590099465764078951269468398352 595709825822620522489407726719478268482601476990902640136394437455305068203496 252451749399651431429809190659250937221696461515709858387410597885959772975498 930161753928468138268683868942774155991855925245953959431049972524680845987273 644695848653836736222626099124608051243884390451244136549762780797715691435997 700129616089441694868555848406353422072225828488648158456028506016842739452267 467678895252138522549954666727823986456596116354886230577456498035593634568174 324112515076069479451096596094025228879710893145669136867228748940560101503308 617928680920874760917824938589009714909675985261365549781893129784821682998948 722658804857564014270477555132379641451523746234364542858444795265867821051141 354735739523113427166102135969536231442952484937187110145765403590279934403742 007310578539062198387447808478489683321445713868751943506430218453191048481005 370614680674919278191197939952061419663428754440643745123718192179998391015919 561814675142691239748940907186494231961567945208095146550225231603881930142093 762137855956638937787083039069792077346722182562599661501421503068038447734549 202605414665925201497442850732518666002132434088190710486331734649651453905796 268561005508106658796998163574736384052571459102897064140110971206280439039759 515677157700420337869936007230558763176359421873125147120532928191826186125867 321579198414848829164470609575270695722091756711672291098169091528017350671274 858322287183520935396572512108357915136988209144421006751033467110314126711136 990865851639831501970165151168517143765761835155650884909989859982387345528331 635507647918535893226185489632132933089857064204675259070915481416549859461637 180270981994309924488957571282890592323326097299712084433573265489382391193259 746366730583604142813883032038249037589852437441702913276561809377344403070746 921120191302033038019762110110044929321516084244485963766983895228684783123552

and it keeps on going!

Euclid was one of the first mathematicians to really tackle the issue of irrationality in numbers in the 10th volume of ‘Elements’ where he tried to prove that there could be a number which couldn’t be expressed as a normal fraction.   He however started with something different from Pi and used the square root of 2, ie the number which when multiplied by itself is equal to 2.  His proof involved first assuming it could be written as a hypothetical fraction and then started to simply the expression.

However during his proof he demonstrated that unlike a normal fraction which can be simplified to a single form.  The fraction which was representing the square root of two could be simplified over and over again, in fact an infinite number of times, basically it could never be simplified.   This was his proof that this number was indeed irrational and the hypothetical fraction he used, could not in fact exist.

John Steadway

Using Baye’s Theorem

Baye’s theorem is usually one of the easiest ways to calculate probabilities as long as you have sufficient information about related conditions. It can be considered a style of understanding the way probability is affected by introducing a new variable or condition. So you need to take care that you fully understand the conditions when using it to calculate probabilities. Keep in mind when using the theorem that the entire probability of all potential x needs to be equal to 1.

The theorem can subsequently be used to find out the level of belief in the hypothesis using the experimental data. When you have ever come across Bayes’ theorem, you likely know it is a mathematical theorem and there is a solution possible. Bayes’ theorem is often used in medical statistics for instance in trials to proves that even if an individual tested positive in a particular scenario. It is certainly now a crucial tool for statisticians and scientists, as well as many people working with probabilities in all sorts of industries. In all of these cases,an understanding of the theorem is an excellent tool for all sorts of statistical work. Bayes’ theorem integrates well with helping to prove or disprove hypothesis, as long as you should consider all the subsequent conditions.

Another area it is used is in the assessment of risk. It is of course a useful way to gain a little insight into possible risks by using Bayes’ to obtain some probability data concerning the event . John Bayes’ was a famous mathematician who published much work particularly in the areas of calculating reverse probability by utilizing conditional probability.

This is the key to understanding this theorem – that you are basically trying to discover the probability that T is true whilst supposing that another piece of evidence is true. Think of a deck of cards which contains 52 individual cards. You can work out the probability easily before a card is drawn however after the calculation is different as there are less cards and of different values. Too bad this type of question isn’t asked in science it’s covered well on the BBC Maths Bytesize site – you’ll need a BBC iPlayer proxy to access from outside the UK.

The difference in the past equation results from the truth of using smart adjustment. When cards are drawn from the pack the maths continually changes as long as they are not replaced or put back. Nonetheless, the fact that it’s possible to describe decision making behaviour with a mathematical function proves that folks utilize some rules or behave irrationally.

Effective evidence is an issue of the level to which an individual’s total evidence for H is dependent upon her opinion about E.  Regarding the Bayesian strategy, the proof is more complicated. The simplest way is often to put all these values in a table which can make it simpler to visualize the potential conditional choices.

Additional Reading

Gromov’s Theorem

There isn’t any official necessary textbook for this program. It’s a history that’s probably so long as geometry’s. There’s a huge literature on growth prices, leading up to Gromov’s theorem. I’ve given a concise overview of each product, but you will discover a better overview included in each product at the beginning.
Mathematics, on the opposite hand, is cumulative. Nonetheless, this isn’t so, by and big, for mathematics. Valid mathematics that was done previously is still valid, and frequently still intriguing and useful. Number theory is among them. There’s a great reason such generalizations are worthwhile to make. This is called the Gap conjecture. In the end, the axioms specify there is a scalar product between any 2 elements, which causes a scalar.

The inner product is extremely important, since it isn’t only an algebraic construct, but additionally it provides the space a topology. This might sound intimidating, but the pieces are in fact simple and illuminating. No, it’s not a standard Chinese dish. We are also going to review a little projective geometry. This variety hints this notion is important from several points of view.

Sometimes, there could be an exceptional function that’s the solution, and one would love to have the ability to recognize such scenarios. I will attempt to explain the fundamental induction argument. We’ll construct the most well-known example, the Grigorchuk group. But a few of the outcomes are strikingly different in the complicated case. This outcome is striking for a number of reasons. Assume that we’re managing a massive number N of material particles and that the particles are extremely close to one another.

But things can get complicated quickly. The thought that the universe could be understood when it comes to geometry is an old one. Obviously, as just remarked, we can’t really start to explain the concepts within this brief space. It can be that these two apparently contradictory facets of mathematics aren’t unrelated.

Physical situations continue to be modeled, just as they were in Newton’s work, with respect to differential equations. Another issue that is more prominent with complicated functions is that some pure functions even as easy as the intricate square root or the organic logarithm could possibly be multiple-valued. Thus, we aren’t likely to attempt to summarize here what seem to be the main open questions. We’ll survey these interconnected topics within this talk.

This program has turned out to be quite challenging. It’s likewise an infinite torsion group. The fundamental group structure comes from the accession of vectors. As luck would have it, are other flat surfaces you’ll be able to consider that aren’t the torus. I want to provide some observations on a few details of the mechanics that may be useful.  Conservation of angular momentum actually is an important concept in celestial mechanics.

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Schrödinger’s equation

V limitations of the online browser, partial derivatives aren’t explicitly indicated. The derivation isn’t included within this brief story line. A complete derivation is provided in Lanczos. This very first derivation wasn’t published. It’s always fruitful to search for invariants under transformation. Furthermore, it is crucial that the wave function should have just an individual value at any certain point, as it corresponds to the probability of locating the particle at that point. In this instance the wave function could possibly be utilised to predict the relative likelihood (i.e. the probability) of each one of the probable outcomes.

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A bit of linear algebra is a little price to cover untangling all of this. Liner equations with a couple of variables have an infinite quantity of solutions. The probability of locating changes exactly as the rate of the probability of locating an electron does.  There’s an interesting example featured in the documentary – History of Maths, which is accessible on a few media streaming sites – this should help access from outside the UK – using a proxy.

The Schrodinger equation takes a number of different forms, based on the physical circumstance. It shows how the quantum wave function changes over time. It’s almost enjoy the equations are attempting to inform you a story. It’s simple to work out this equation. The solution of both of these equations is beyond the reach of this class. This previous equation is in an incredibly significant dimension, hence the solutions aren’t simple to visualize. It is in a very high dimension, so that the solutions are not easy to visualize.

Below it’s a slightly rewritten form. If, on the opposite hand, you assume it’s in a mix of each of the probable states it can be, you are going to be correct.” The wave function is a mix of all the feasible wave functions which exist,” says Martell. In the event the system isn’t conservative, it’s still a constant, but not the complete energy. The waveform analysis procedure is often hard and confusing. You’re not predicted in order to do this transformation. This relationship is called the dispersion relation.

A belief is just one more method of viewing the world. You’re helping confirm your belief, whether or not your belief is true or not. Considering the simple fact that, mathematically speaking, relativity theory and quantum theory are not just distinct from one another, but in addition oppose one another, Dirac’s work could be thought of a fruitful reconciliation between both theories.

Yet Schrodinger’s interpretation couldn’t explain quantum tunnelling. It is simply the conventional Copenhagen Interpretation of the outcomes of them that isn’t right.
Do not forget that low momentum usually means a very long wavelength. In many conditions, an electron will behave as an easy, easy-to-quantify particle. It’s known as the electron. When you look at it in 1 way it seems as a particle. If you differentiate velocity with regard to time, then you’re measuring acceleration. The angle does not seem in Eq. The genuine motion of this completely free particle provides the least average kinetic energy.

Depending on the way that it vibrates determines what type of force you believe you saw. In the current essay, energy is only a constant of the motion, based on specific conditions, but is still quite significant and useful. For an electron travelling through an electric area, as an example, the entire energy is equivalent to the kinetic energy in addition to the possible energy of the area.

Further Reading – Expat UK TV

Understanding Polynomials

Polynomials are categorized into various types. Put simply, as soon as a polynomial is represented in the shape of equation, it is called polynomial equation. They play a critical part in mathematics. Also referred to as factorization, this technique is mostly utilised in simplifying polynomials. You should currently be all set for subtracting polynomials.
There are several different ways of factoring these numbers based on their types and forms. When you bring polynomials, you are just likely to bring the like terms which are categorized according to the amount of terms and the degree. Such numbers are also classified as prime polynomials.


As a result of how polynomials obey the exact same rules are real numbers we could likewise do the exact same calculation in the fashion of elementary arithmetic, i.e., However many terms a polynomial has, it’s always essential to check for a best common factor (GCF) first. A polynomial is an expression with a power that’s an entire number. Polynomials are employed in a diverse selection of problems where they’re called as polynomial equations. Such polynomials are called reducible polynomial.

The potential things of the trinomial are the binomials which we may make out of these feasible things, taken in each probable order. If that’s the case, you can component out that common component. Inside this expression, there’s a typical element for the initial two terms. A function which consists of polynomial is known as a function. You simply count up how many variables you’ve got the exact same and compose the number before the typical variable part. The maximum value of exponents is known as degree of polynomial. It is also used online to define and protect internet connections for example some use it in VPNs to allow anonymous torrenting as this.

You may prefer the vertical method as you are accustomed to adding numbers vertically. So, since you can imagine, we’ve got a significant number of resulting terms! This time you should bring the numbers together as you’re finding the sum. In the overall form, the quantity of constants, on account of the term of level 0, is always one more than the level of the polynomial. In mathematical provisions, Hamming codes are a category of binary linear codes.

Since the level of the polynomial is the maximum level of all of the terms, Since there’s a single term, this is a monomial. The amount of a polynomial is the maximum level of the terms. It is the degree of the leading term. It is the highest degree of its terms. Since the level of the polynomial is the maximum degree of all of the terms, because there are 3 terms, this is a trinomial. Since it is the highest degree of all the terms, Make sure that you don’t fall into the trap of thinking it is always the degree of the first term. It is the greatest degree of its terms.

Finding out how to factor polynomials doesn’t have to be hard. The theory of polynomials is quite an important and intriguing portion of mathematics. Given 2 people’s ages, there are numerous mathematical relations you are able to calculate utilizing both of these numbers. Similarly the past two terms have a standard aspect. It’s important to not forget to multiply the terms along with the numbers.

Integral Calculus Primer

The study of integral calculus is perhaps best introduced by translating the word calculus which comes from the Latin phrase which means ‘small stone’. Calculus is generally split into two main sections – Differential Calculus and Integral Calculus. The former splits problems into small pieces to try and understand how it changes whereas Integral calculus joins the pieces together to try and understand the sum of the parts.

Integral calculus is used to solves a series of problems found anywhere;

  • How do you work out the area of a curve?
  • How can you work out the length of the curve?
  • How fast is something falling?
  • Where will a thrown object land?

You can use the tools covered in integral calculus to work out problems like this and many others. The core idea to remember though is they are all about ‘changes’ – changes in speed, distance, height and so on. The below video demonstrates what is meant by the area under the curve and how you can use simple rectangles to define the area. It included simple examples using a spreadsheet program to calculate – it is a very well done and important introduction to integral calculus.

There are lots more great examples that you can find both on YouTube and on the various educational sites, if you can’t get access based on your location try this which shows you a VPN that isn’t banned by the majority of sites like Netflix and others.
Integral calculus is used to assign values to functions specifically to try and calculate changes in area, volume and similar concepts that can be described by adding small changes together.

There are two main operations which are used here – integration and it’s inverse differentiation. This is no new mathematical concept, indeed the first principles of integration were determined by no less than Isaac Newton and Gottfried Leibniz in the end of the 17th Century. The concept and practice have of course been further developed up to the present day by people like Bernhard Riemann who produced the first comprehensive mathematical definition.

Further Reading