Wednesday 16 February 2022

Sir Francis Galton (16 Feb 1822 - 17 Jan 1911)

Today marks the 200th anniversary of the birth of Sir Francis Galton. He lived from 16 February 1822 to 17 January 1911.

Sir Francis Galton
Image from Wikipedia

He was a cousin of Charles Darwin, and like many educated Victorian gentlemen, dabbled in many different areas of learning. Among Galton's interests were anthropology, genetics including eugenics (the idea of improving the human race by genetic selection), exploration, geography, invention, meteorology and statistics. He was knighted in 1909. He actually invented the term eugenics, and is considered the first person to use the expression "nature versus nurture".

Galton's wide range of interests led him to publication of over 340 papers and books during his lifetime. But it is his mathematical contributions, particularly in the field of statistics, that we are interested in. He created the statistical concept of correlation and widely promoted regression towards the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, and introduced the use of questionnaires and surveys for collecting data on human communities, which he used in his genealogical studies and his analyses of the behaviour of people, anthropometrics.

So Galton's contributions were considerable and, in some cases, revolutionary. But some of his ideas were also highly controversial, even for the time in which he lived. He attempted to draw up a 'Beauty Map' of the British Isles, and for this he classified passing girls into three categories: attractive, indifferent and repulsive, surreptitiously making pin-pricks in paper stored in his pocket as a means to build a database.

Eugenics itself is today considered a brutal concept. The practices involved in favouring certain families deemed genetically superior was practised by some governments during the early years of the 20th century, resulting in deprivation and a loss of human rights for millions.

Galton's book, Hereditary Genius (1869), was one of the first scientific attempts to study genius and greatness. It demonstrates some of the statistical techniques that Galton would develop through his life, but it is steeped in language that is far from acceptable in today's scientific literature, such as "idiots" and "imbeciles".

He founded psychometrics (the science of measuring mental faculties) and devised a method for classifying fingerprints that was later used in forensic science. He also conducted research on the power of prayer, concluding it had none, because of the fact that those prayed for lived no longer than those not prayed for.

Finally Galton was a pioneer of scientific meteorology. He devised an early weather map, proposed a theory for the formation of anticyclones, and was the first to establish a complete record of short-term climatic phenomena on a European scale.

Francis Galton was truly a great British eccentric, whose ideas were often controversial, but whose contributions to statistics and science in general cannot be ignored.

Monday 31 January 2022

Happy 50th Birthday to the HP35

Finally! I've got my hands on an HP 35 calcula...
The HP-35 scientific calculator
Image via Wikipedia

1st February 1972 marked a birth that would revolutionise mathematics education across the world. This was not, this time, the birth of a great mathematician, but of a machine. The first scientific hand-held calculator (HP-35) was introduced to the US market, and later to the UK. Costing $395, it was made by Hewlett-Packard and its name came from the fact that it had 35 keys.

The HP-35 measured 79 x 147 x 34 mm, pretty chunky by today’s standards, but ultra-sleek in its day, when a computer still filled a small room. It ran on rechargeable batteries, and its electronics used several integrated circuits. ‘Scientific’ meant the calculator was able to perform logarithmic and trigonometric functions with one keystroke. It featured a red LED display which could give scientific notation up to 10 digits, with 2 digits for an exponent (power of 10).

The price was reduced several times, eventually to $195 in the US. But this was, of course, still too high for the HP-35 to become a mainstream part of classroom teaching. Production of the HP-35 was stopped in Feb 1975, 3 years after its launch. 300,000 units had been sold.

The numbers and functions for calculations were entered in "Reverse Polish Notation". This would seem very strange to today’s maths students, since the operator always appears last: a calculation such as 3+5 was performed by typing “3 5 +”. It avoided the need for parentheses or an "=" key.

Further models from HP followed. The introduction of the HP-35, its descendants and similar scientific calculators by Texas Instruments soon brought about the demise of the slide rule in the classroom.

The appearance of the calculator gave rise to the “should we, shouldn’t we?” debate in the media and among educators, with some feeling strongly that the introduction of a calculating device would bring about a decline in students’ own calculating abilities. These arguments had little real relevance in the early days, since the cost, fragility, and short battery life of these early machines meant the calculator had limited use in the classroom, and was certainly not available to every pupil.

By the early 1980s, those deterrents began to decline. Solar-powered scientific calculators began to appear, with hard cases, costing around £20. The UK exam boards bowed to the inevitable and made them a standard piece of equipment in O-Level and A-Level mathematics. The calculator was here to stay.

Later, the first hand-held calculators appeared that could graph functions. Like their simpler counterparts, these calculators were too expensive to be widely adopted when they first appeared, but today many schools encourage their use, although they are still not all tolerated within examinations.

Nowadays, the line between a calculator and a handheld computer is very blurred. Some calculators such as the TI-89, the Voyage 200 and HP-49G are able to differentiate and integrate functions, solve differential equations and run word processing software. Other handheld devices can, of course, connect to the Internet. Whether such devices will ever be used in our exams, only time will tell.

An emulation of the HP-35 is available for the Apple iPad.

Thursday 24 August 2017

GCSE Results 2017

GCSE results are out today, with a fragmented set of exams now being sat across the UK. In England, new GCSE courses have been designed to be more challenging. Overall across the UK, the pass rate has dropped across the full range of subjects, from 66.9% to 66.3% of entries. A pass under the old system was a C grade; under the new system of grades from 1 to 9 (with 9 the best), a 4 is considered a standard pass.

In England the pass rate for maths, one of the new tougher exams, it dropped slightly from 71.4% to 70.7%. The proportion of entries receiving the top grades (A/7 or above) has also fallen, to 20%, down 0.5 percentage points on last year, the lowest since 2007.

However, exam boards appear to have worked the grade boundaries to ensure the distribution of grades is broadly consistent with last year's. For example, on the higher tier maths paper, a grade 4 pass would have been achieved with just 18% of the overall marks. For a grade 9, 79% mark was required. For a grade 7, the equivalent of an A, candidates needed just over half marks.

Students and teachers have complained about a lack of textbooks and practice papers in the run-up to the summer exams.

In Northern Ireland and Wales, exams are still graded A* to G. In Northern Ireland, female students continue to outperform their male counterparts at this level, with an 8.1% difference in the pass rate (A*-C). One in ten entries are now awarded the top A* grade here.

Thursday 17 August 2017

A-Level Results Day 2017

A-Level results are out today in England, Wales and Northern Ireland. This year's results have shown an increase in the number of A and A* awards given, the first rise in 6 years.

26.6% of entries from boys were awarded these top grades, with 26.1% of girls' entries (overall 26.3%, up 0.5%). This is first time in 17 years that the boys have been ahead at this level of education.

In England, 13 A-Level subjects were reformed this year (with mathematics and the rest to follow). Changes include assessment based entirely on a final exam, eliminating coursework entirely. There was a fall in the top grades awarded in these subjects. Another change means that pupils now have to decide from the outset whether to take AS or a full A-Level because the two qualifications have been decoupled. This has resulted in a number of pupils continuing to their second year, who would previously have stopped after the first year, resulting in a larger number of less able pupils taking the full A-Level. A lack of text books for the new syllabuses and a lack of past papers may be other contributing factors.

For those who didn't make the grades they required for university, there is a very good chance they will be able to take a place through clearing this year. University applications from the UK and European Union countries have fallen compared with last year and there is a demographic dip in the number of 18 year olds.

In Northern Ireland, 30.9% of entries were awarded the top two grades, but girls are still ahead. Indeed they increased the gap to 6.5% with 33.3% of girls and only 26.8% of boys getting the top two grades.

Tuesday 4 April 2017

John Napier (1550–1617)

Napier's "Bones", a set of wooden rods used for muliplication
If you have studied logarithms, you have John Napier to thank for their discovery. He died 400 years ago today.

John Napier was a Scottish landowner, mathematician, physicist, and astronomer, but is probably best known as the discoverer of logarithms.

Before calculators became widespread in the second half of the 20th century, logarithms were widely used to perform multiplication of numbers, particularly large numbers. Logarithms were looked up in "log tables".

Logarithms have many uses and come into many areas of modern maths, and if you have studied beyond GCSE level, you will be very familiar with them. The natural logarithm (log base e) is sometimes called the Napierian logarithm, although Napier didn't explicitly work with base e.

Napier also invented "Napier's bones", which were a set of wooden rods inscribed with digits, and were another tool for performing multiplication. Napier was also one of the earlier mathematicians to make frequent use of the decimal point in his work and helped to popularise this idea.

Napier's birthplace, Merchiston Tower in Edinburgh, is now a part of Edinburgh Napier University, which is named after him.

Napier died in 1617 at his home at Merchiston Castle from the effects of gout, but his revolutionary work on what became a key part of modern mathematics will ensure he is remembered for a long time to come.

Thursday 25 August 2016

GCSE Results 2016

GCSE results across England, Wales and Northern Ireland this year have suffered a significant fall. This is in marked contrast to last week's A-Level results, which showed a great deal of stability when compared with recent years. GCSE results peaked in 2011 after a long period of "grade inflation", and have been falling year on year ever since.

Of the five million results being released today, the proportion of entries being given A* to C grades has fallen from 69% to 66.9%, an unprecedented fall of 2.1%. The proportion of top A* and A grades was down overall by 0.9% to 13%. Awards of the very top A* grade have also fallen slightly, down from 6.6% to 6.5%.

One reason for the fall in the overall pass rate is that more pupils in England are re-taking GCSE English and maths, following measures by the government to attempt to ensure that all pupils reach a grade C in these subjects. As a result there were re-sits for tens of thousands who did not reach a C grade last year. Among over 16s, GCSE entries were up by 380,000, up roughly 25% on last year.

By subject, there were falls in the pass (A*-C) rate in maths (down 2.3%), English (down a worrying large 5.2%) and in other subjects.

But even without the re-sit figures, there was a fall in the results of 16 year olds, with the proportion getting A* to C falling by 1.3%. It is thought that a move to an alternative qualification, the iGCSE, for many able students, may account for some of the drop.

While the overall results were downwards, in Northern Ireland the proportion of passes increased slightly to 79.1% and top A* grades rose to 9.3%. In Wales, the level of A* to C passes remained at 66.6%, with A* grades rising slightly to 6.1%.

Girls continue to do better than boys at GCSE. The A*-C rate for girls was 71.3% compared with 62.4% for boys, a gap some would describe as worrying, or even unacceptable. The percentage of entries gaining A* grades for girls was 7.9% and 5.0% for boys.

Next year will start to see the phasing in of a radical change to the way that GCSEs are graded. The new GCSE exams will be graded from 1 to 9 (with 9 the highest), rather than the current A* to G.

Thursday 18 August 2016

A-Level Results Day 2016

Hundreds of thousands of teenagers in England, Wales and Northern Ireland are receiving their A-level and AS results today. The A-level results show a stable pattern compared with last year.

Maths remains the nation's most popular choice of subject, as it was last year, and increasing numbers are also opting to take Further Maths at A-Level.

The proportion of top grades (A and A*) is 25.8%, down by 0.1% on last year. The overall pass (grades A*-C) rate of 98.1% remained the same. Northern Ireland remains the region with the biggest proportion of top grades, 29.5%.

Girls fared better than boys, once again, with 79.7% of girls getting a pass grade, compared with 75% for boys. Boys are getting more A* grades (8.5% compared with 7.7% for girls), although this gap between the very top-performing girls and boys has narrowed for the first time in five years. The overall level of A* grades (8.1%) has been falling now for 2 years.

According to Ucas, the universities' admissions service, 424,000 university places have been offered to hopeful students, which is up by 3% on the same time last year. But many places are reportedly still available through clearing, including at leading universities and for highly sought-after courses, such as medicine.

The increase in the number of places still available is mainly due to two factors: a roughly 2% fall in the number of school-leavers and the removal of the cap on the number of places universities in England can offer.

AS levels are being "decoupled" from being part of A-levels - and this year's figures show a 13.7% drop in entries for the AS course. Previously the AS has carried with it the option of continuing on to a full A-Level, but this will now cease to be the case: if you register for an AS, this is the qualification you will get.

Teachers and head teachers' leaders have warned that, although the overall results appear to show stability when compared with recent years, individual schools and pupils are facing some unpredictable outcomes.

The current cap of £9000 for university tuition fees is being removed, so students starting at England's universities in the autumn could face higher fees. Exeter University has been the first to announce that it will increase fees to £9,250 for all current and new students.

With the prospect of increasing fees, school leavers may opt not to go to university at all. Financial services firm PwC says that it has had a 20% increase over two years for new recruits of those leaving school with A-levels.