Galileo Galilei

Scientist and mathematician Galileo Galilei was born on February 15th, 1564, in Pisa, Italy. A pioneer of maths, physics and astronomy, Galileo’s career had long-lasting implications for the study of science.

In 1583, Galileo was first introduced to the Aristotelian view of the universe, which was a religion-based view of how the world worked. A strong Catholic, Galileo supported this view until 1604, when he developed theories on motion, falling objects, and the universal law of acceleration. He began to openly express his support of the controversial Copernican theory, which stated that the Earth and planets revolved around the sun, in direct contrast to the doctrine of Aristotle and the Church.

In July 1609, Galileo learned about a telescope which had been built by Dutch eyeglass makers. Soon he developed a telescope of his own, which he sold to Venetian merchants for spotting ships when at sea. Later that year, Galileo turned his telescope toward the heavens. In 1610 he wrote The Starry Messenger, where he revealed that the moon was not flat and smooth, but a sphere with mountains and craters. He discovered that Venus had phases like the moon, and that Jupiter had revolving moons, which didn’t go around the Earth at all.

With a mounting body of evidence that supported the Copernican theory, Galileo pushed his arguments against church beliefs further in 1613, when he published his observations of sunspots, which refuted the Aristotelian doctrine that the sun was perfect. That same year, Galileo wrote a letter to a student to explain how Copernican theory did not contradict Biblical passages, but that scripture was written from an earthly perspective, and that this implied that science provided a different, more accurate perspective.

In February 1616, a Church inquisition pronounced Galileo as heretical. He was ordered not to “hold, teach, or defend in any manner” the Copernican theory regarding the motion of the Earth. Galileo obeyed the order until 1623, when a friend, Cardinal Maffeo Barberini, was selected as Pope Urban VIII. He allowed Galileo to pursue his work on astronomy on condition it did not advocate Copernican theory.

In 1632, Galileo published the Dialogue Concerning the Two Chief World Systems, a discussion among three people: one supporting Copernicus’ heliocentric theory of the universe, one arguing against it, and one who was impartial. Though Galileo claimed Dialogues was neutral, the Church disagreed. Galileo was summoned to Rome to face another inquisition, which lasted from September 1632 to July 1633. During most of this time, Galileo wasn’t imprisoned, but, in a final attempt to break him, he was threatened with torture, and he finally admitted he had supported Copernican theory. Privately, though, he continued to say he was correct. This ultimately led to his conviction for heresy and as a result he spent his remaining years under house arrest.

Despite the fact he was forbidden to do so, Galileo still went on to write Two New Sciences, a summary of his life’s work on the science of motion and strength of materials. It was another work that has helped cement his place in history as the world’s most pioneering scientist, even if he was not fully appreciated in his own time. Galileo Galilei died on January 8th, 1642.

What initial conclusions about the future of GCSE exams can we draw from the mountain of documents which Michael Gove and the Department for Education released last week? And who will the winners and losers be if these proposals come to pass in their current form?

There is no doubt that the new exams will be harder and more “academic”. If not a return to the degree of difficulty posed by the old O-level exams, these new outline specifications match the difficulty and depth of the current IGCSE (International GCSE) specifications set by Edexcel and the Cambridge board. The message seems to have been: take the best of the current IGCSE specs and call it a GCSE instead.

The subject advisers seem to have taken this brief quite literally in most of the core subjects. It is perhaps most clearly seen in Mathematics, a subject in which the IGCSE specifications already require a number of skills that have been beyond the scope of the GCSE Maths syllabuses for 25 years but which are fundamental to AS level Maths. These include function notation, kinematic problems, set notation, rates of change and Venn diagrams, to name but a small sample of topics. There they are in the new drafts in bold print. This is IGCSE Maths by another name.

Most topics are not in bold print, implying that the boundary between what is now the GCSE Foundation and the current GCSE Higher levels is set to shift. Vectors, formerly to be found in the GCSE Higher level requirements, appear in plain text here, including the multiplication of vectors by a scalar. Some maths teachers may need to go on a refresher course to master the required skills!

Similar principles underlie the Science draft. Not only will the individual specifications require considerably more depth of study, as they do in today’s IGCSEs, but the Combined Science qualification will be the equivalent of two GCSEs, not one, just as it is today with IGCSE Science but not GCSE Science. The simple principle behind GCSE Science is to take one-third of the Biology specification, one-third of the Chemistry and one-third of the Physics, while IGCSE takes two-thirds of each of the respective individual subject specifications. The new proposals unashamedly mimic the IGCSE formula.

If this means that all candidates will now face a choice between tackling the new Double Science GCSE or leaving school without any formal recognition of their achievements in the sciences, there will be huge numbers of schoolchildren who fall in the latter category. While the old “everybody passes” philosophy of GCSE had its disadvantages, do we really want to stigmatise a whole generation as incapable of taking and passing the “simplest” of the new science specifications?

 

Children who are educated at home are unfairly denied access to examinations, according to a report in today’s Daily Telegraph.

This is not really a new story. Some home-schooled candidates have always found it time-consuming and awkward to locate suitable exam centres to take their exams for GCSE or A-level. But the Commons Education Committee has now said that it is “not reasonable” that some young people are struggling to sit national tests. It calls for a duty to be placed on councils to provide access to examination centres.  It also asks for examinaton fees to be met from public funds. We are happy to endorse all those proposals!

As Graham Stuart, the Committee’s Chairman, says: “Everyone else gets to take GCSEs and home-educated children should do so [for free] as well.”

These are welcome sentiments and time will tell whether they lead to genuine change. But they ignore the more fundamental problems of GCSE examination entry for the home-schooled, namely the need for all candidates to produce controlled assessments (coursework) in most of the main subjects, including English, History, Geography, languages and all the science subjects. Controlled assessment (as the government has defined it) is not possible for home-learners on distance learning programmes like the ones we offer, so, whether the costs are met or not, our students simply cannot take GCSE exams in these subjects.

Four years ago, Oxford Home Schooling fought hard to preserve GCSEs which were genuinely accessible to home learners, but the government rejected all our pleas. As a result, most of our students now take International GCSEs (IGCSEs) rather than GCSEs, an equally valid alternative but confusing for many families. And now those IGCSEs must call themselves “certificates”, not GCSEs, which also leads to marginalisation and confusion.

There are moves afoot to re-introduce exams (at age 16) which do not require controlled assessment, possibly based on the current IGCSEs, and such a development would be welcome news to thousands of home learners and their families.

A report in the Sunday Times on 29 July was headlined ‘State Schools ditch GCSE for tougher exam’. The exams in question were International GCSEs, or IGCSEs, as set by Edexcel and CIE.  As the report indicated, IGCSEs are being adopted not just by private schools but by a wide range of “ordinary” state schools, dissatisfied with the current GCSE qualifications. But are IGCSEs really harder than GCSEs?

At Oxford Home Schooling, we offer a broad range of both GCSEs and IGCSEs, so we are in a good position to evaluate the strengths and weaknesses of each. In some subjects like Mathematics, we offer both and it is not difficult to compare the two specifications, point for point, to see which is harder. There is no doubt that for both Foundation and Higher level students, IGCSE Maths is more demanding. There are a number of sub-A-level topics, like calculus and matrices, which appear on the IGCSE specifications but not on the GCSE while certain skills which are designated as Higher level at GCSE (negative powers, etc) appear at Foundation for IGCSE.

It is harder to make the comparison in most other subjects because the list of topics to be covered may look similar at first glance – the real difference comes in the breadth and depth of coverage of each topic; IGCSE students are expected to have covered quite a lot more ground. I would estimate that there is typically an extra 20%-30% to learn for each IGCSE as compared with its GCSE equivalent. There is no doubt that if an IGCSE student successfully covers the whole of the topic content in the specification, he or she will be in a far stronger position, knowledge-wise, to go on to A-level.

But the GCSE providers, like AQA, would argue that coursework is a crucial part of most GCSEs, and GCSE candidates must spend a significant part of their study time planning and producing coursework. With the introduction of controlled assessment for coursework, this is now a more rigorous and fair process. As a result, they argue, it is reasonable that the topic content is thinner. Insofar as coursework encourages independent research, it too can be said to provide a reasonable preparation for A-level. But the truth is that coursework skills are a bit too similar from one subject to another, with the same set of boxes to be ticked by candidates and examiners. In too many schools, the requirement for coursework across a broad range of subjects has come at the expense of “real” subject knowledge and this is why so many schools are switching to IGCSE.

When we ask whether IGCSE exams are tougher, what many of us really want to know is whether it is harder to gain a grade C (say) in IGCSE than in GCSE. Just because there is more detail on the specification does not mean that fewer people will get a high grade. The number of candidates being awarded a grade C remains an arbitrary decision. A GCSE board might set the bar at 70% (in theory) while an IGCSE board might set it at 40%, not that this happens in practice. Because of fluctuating grade-boundaries, it is hard, or impossible, to say for sure whether you have a better chance of collecting 10 A’s (or A*’s) by sticking with GCSEs.

The take-up of IGCSEs would be much faster but for the suspicion that IGCSE candidates are not yet being given the credit they deserve, i.e. it is harder to gain a grade C in the IGCSE, yet grade Cs are still being treated equally by employers and higher education institutions.  But the evidence is very hard to evaluate in this respect, for the reasons I have suggested.

For home-schoolers and distance learners in general, there are a number of reasons why IGCSEs are not the harder proposition. The GCSE coursework system was unfairly loaded in favour of school-based candidates, so those involved in home schooling have something much closer to a level playing-field with the final exam-only IGCSEs. It is also true that the home-educated find it almost impossible to comply with the controlled assessment requirements of modern GCSEs. So, in general, it is easier (and cheaper!) to sit an IGCSE examination if you are an independent candidate.  There is no clear evidence that home learners will emerge with lower grades if they take IGCSEs and, on the other side of the coin, there is little doubt that they will actually learn more and be in stronger position to go on to A-level, etc.

 

 

Einstein was born in Germany in 1879 and, other than a short period spent studying in Switzerland, he lived there until 1933. He moved to the United States of America due to the rise to power of the Nazis under Adolf Hitler.

Why is he famous?

Many people use the word Einstein as a way of describing a genius. His most important work was the theory of general relativity which caused a huge sea change in our understanding of physics.

Unusual facts

Einstein studied for a mathematics and physics teaching diploma but struggled to find a job after graduation. He ended up spending some time working as a clerk in a patent office where he was passed over for promotion on at least one occasion. He was also turned down at one point when he applied for a job as a professor of mathematics.

In 1933 the Nazis organized mass burnings of books written by Jews and works by Einstein were among those set alight.

Einstein’s first wife was also a mathematician and some people have suggested that she may have made contributions to his works, although no evidence has been found of this.

Einstein had a huge respect for Isaac Newton and apparently had a picture of him on his wall. Newtonian Mechanics is now seen as a special case within Einstein’s theory of general relativity. People argue about which mathematician was the greater genius – there is even a Facebook page devoted to the argument.

Links to modern mathematics

Whilst much of Einstein’s work is way beyond Key Stage 3 and GCSE level you may well have heard of his famous equation E = mc² which links mass, the speed of light and energy. It is often used in exercises on rearranging formulae and substitution. You first need to understand algebra if you are to go on to understand the work of Albert Einstein.
Debby Gill

Tutor

For more information on Oxford Home Schooling’s home study courses please contact a student adviser on 0800 0 111 024.

In todays distance learning Maths blog an Oxford Home Schooling tutor looks at the life and work of Isaac Newton.

Isaac Newton is probably the most famous British mathematician. He was born in Lincolnshire in 1643 and spent most of his adult life in Cambridge.

Why is he famous?

Newton’s most famous work was a book called Philosophiae Naturalis Pricipia Mathematica or “Mathematical Principles of Natural Philosophy”. This is basically a book about how gravity acts on objects.

Unusual facts

Newton is famously supposed to have discovered gravity when an apple fell on his head. Some people say this is just a myth but you can visit trees in several places which are claimed to be “the” tree.

His work on calculus (a method for finding maxima and minima) caused him to be involved in a dispute with another mathematician called Leibniz over who invented it first. Most historians now believe calculus was actually discovered by them both independently and, while Leibniz’s notation is commonly used in Pure Mathematics today, Newton’s very different notation is often used in Mechanics and Physics.

Newton died in 1727 and was buried in Westminster Abbey. It is thought that he may have suffered from mercury poisoning in later life due to various experiments he had carried out as traces of mercury were found in his hair after his death.

Links to modern mathematics

Newtonian Mechanics is still studied today – his laws of motion are about force, velocity and acceleration. Much of it has been superseded by the work of Albert Einstein but, when studying motion on Earth for example, his laws are still very good approximations to what we see happen in real life.

Debby Gill

Tutor

For more information onhome study GCSE Maths and KS3 Maths courses please contact a student adviser on 0800 0 111 024.

In today’s distance learning Maths blog, an Oxford Home Schooling tutor looks at the life and work of Geralamo Cardano.

When and where did he live?

Cardano was born in Pavia, Italy in 1501. His mother had moved out of Milan in order to avoid the plague that was sweeping the city at that time. He spent much of his life in Italy although he did live in Scotland for a while.

Why is he famous?

Cardano wrote the first study of the probability of dice rolling, as well carrying out some ground breaking work in algebra and number theory.

Unusual facts

Cardano was a successful doctor and his services were sought by many important people. The only person to persuade him to leave Italy was the Archbishop of St Andrews who paid him a large sum of money for his services and eventually made a full recovery.

In addition to being a physician Cardano was a renowned gambler and he used his understanding of probability to great advantage at dice and card tables. He also wrote some controversial articles which caused him to be jailed for heresy for a few months.

Cardano apparently correctly predicted his own date of death although many people think he “cheated” and committed suicide in Rome on 26th September 1576

Links to modern mathematics

Writing the probability of throwing a 6 on a 6 sided dice as 1/6 may seem obvious today but Cardano was the first person to formally do so. He also did a lot of work on solving equations.

Debby Gill

Tutor

For more information on Oxford Home Schooling’s home study courses please contact a student adviser on 0800 0 111 024.

In blog 6 in our series for students studying distance learning maths an Oxford Home Schooling distance learning tutor looks at the life and work of Archimedes.

When and where did he live?

Archimedes was born in Sicily in about 287 BC. He was educated in Egypt and then returned to Sicily where he lived until his untimely death in around 212 BC.

Why is he famous?

Archimedes is probably most famous for his discovery that when placed in water a floating object displaces an amount of water equal in weight to the object itself. Legend has it that he realised this stepping into his bath so he shouted “Eureka!” and ran home without his clothes. He also wrote many important mathematical works on geometry and mechanics.

Unusual facts

Archimedes is credited with the invention of the catapult and also the Archimedes Screw, which is becoming increasingly common as a water feature in children’s play areas.

A popular story tells how Archimedes died at the hands of a Roman soldier. He was working on a mathematical drawing in the sand when he was ordered to stop to meet the General. Archimedes refused saying he first needed to finish his diagram and so the soldier killed him. Some believe Archimedes’ last words were “Don’t disturb my circles.”

Links to modern mathematics

Much of Archimedes’ work is a foundation for modern mathematics. Calculations we do today involving the diameter and circumference of the circle are based on his discoveries. He developed an early approximation to the number we call pi.

Debby Gill

Tutor

To find out more about Home Education please visit our website.

For information on our maths courses visit the webpages for KS3 Maths, GCSE Maths, IGCSE Maths or A Level Maths.

Or you can contact a student adviser to discuss your home study options on 0800 0111 024.

In blog 5 in our series on famous mathematicians, an Oxford Home Schooling tutor looks at the life of Pierre-Simon Laplace.

When and where did he live?

Like Descartes, Laplace was born in France but over 150 years later in 1749. He was a member of the French Academy of Sciences and they cancelled their meeting on the day of his death in 1827 as a mark of respect.

Why is he famous?

Laplace is famous for his work in many areas of mathematics including calculus and probability as well as for his writings on what he called “celestial mechanics”, the motion of planets in the solar system.

Unusual facts

Laplace was a contemporary of Napolean and was Minister of the Interior in his government. He was dismissed after six weeks, however, and while the first edition of his writings on probability is dedicated to Napolean later editions are not.

Laplace’s brain was reportedly smaller than average! It was removed after his death and displayed in a museum for some time – yuck!

Mathematical proofs by Laplace often contained lines such as “it is therefore obvious that…” indicating that he had either mislaid that part of the proof or not quite completed it. Such an approach is not recommended for students of mathematics today!

Links to modern mathematics

Laplace was one of the first people to formalise writing on probability. He popularised basic principles such as “the probability of an event occurring is equal to the number of favourable outcomes divided by the number of possible outcomes” which we still use at Key Stage 3 and -GCSE level today. Due to his more advanced work on calculus he has a transform and an equation named after him, which you are sure to meet if you study mathematics at university.

Debby Gill

Tutor

To find out more about home education visit the Oxford Home Schooling website.

For information on our maths courses visit the webpages for KS3, GCSE, IGCSE or A Level Maths.

Or you can contact a student adviser to discuss your home study options.

In blog 4 of our famous mathematicians series an Oxford Home Schooling tutor looks at the life and work of René Descartes.

Introducing Mathematicians from History 4 : René Descartes

When and where did he live?

Descartes was born in France in 1596. He also lived in the Netherlands and Germany. Descartes is generally believed to have died of pneumonia in Sweden in 1650 although some people believe he was poisoned because of his religious beliefs.

Why is he famous?

Descartes was the first person to draw graphs to represent functions. The system of x and y coordinates we use today at Key Stage 3 and GCSE level for plotting graphs is named Cartesian Coordinates after him.

Unusual facts

You may also have heard one of Descartes’ sayings: “Cogito ergo sum” (I think therefore I am). He was a great philosopher but some of his writings got him into a lot of trouble and in 1633 the Pope placed his works on a list of prohibited books.

Links to modern mathematics

In addition to introducing coordinates Descartes was one of the first people to use a superscript to represent a power (for example the small 2 in x²). He also chose to use the symbols x, y and z to represent unknowns in algebra. It is thought that x became the default letter to use in mathematics for this purpose simply because it appears least often in written French and the printer publishing Descartes’ work therefore found it easier to use. This has had an impact on the English language as X (an unknown) appears in words such as X-ray and X-files (Bellos, 2010 Alex’s Adventures in Numberland ).

Debby Gill

Tutor

To find out more about home education visit the Oxford Home Schooling website.

For information on our maths courses visit the webpages for KS3, GCSE, IGCSE or A Level Maths.

Or you can contact a student adviser to discuss your home study options.

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