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Posts from the ‘KS5 (VI Form)’ Category

Book Review: Plato’s Shadow

September 18th, 2009


Neel Burton, Plato’s Shadow – A Primer On Plato

Academic texts try to appeal to specific readerships. Though Plato’s Shadow has merit, this reviewer is left wondering who it was written for. It works best as a reference book of sorts, since it contains easily-read summaries, each of between two and twenty pages, of all Plato’s dialogues. Each précis is faithful to the original text and provides the reader who is unfamiliar with any dialogue a clear account of what is to be found there. The author also devotes the first forty pages to a useful account of the historical context of Athens and its relations with other city-states, and to a discussion of Pre-Socratic Greek thought and the place of Socrates in the dialogues which follow. A final introductory chapter also looks at scholarly views of when Plato’s works were written, in what sequence, and with what connection to each other.

A student encountering Plato on a Philosophy or a Classics course would undoubtedly benefit from having this book to reach for as a preliminary step before reading one of the dialogues for the first time. A general reader would also find this a useful reference book because of the way it treats each dialogue separately – something you don’t usually find in such a short and accessible paperback.

However, to call this “A Primer On Plato”, as the author does, is misleading. Anyone trying to understand Plato’s thought won’t find much help here. Nothing is done to point the reader to where Plato is specifically exploring metaphysical, ethical, epistemological, political, etc. themes. This book cries out for an index; both the student and the general reader are likely to want help in finding where Plato talks about The Sun Metaphor, or Forms, or Diotima. The occasional attempt is made to enhance understanding by the use of an illustration; this makes most sense in the Meno and Republic dialogues, though in the latter it is The Cave which is illustrated rather than The Divided Line, which almost every other book about Plato rightly and helpfully presents as a diagram.

This text is a welcome addition to a shelf of reference books, but it shouldn’t be seen as a general introduction to Plato’s thought.

John Gray – An Appreciation

July 27th, 2008


My dad taught Philosophy and Sociology all of his professional life, and in his retirement continues to study and think about these subjects. He recently gave a talk about the work of John Gray to the Erasmus Darwin Society in Lichfield, Staffordshire.

John Gray is currently Professor of European Thought at the LSE, and has been an outspoken and controversial academic throughout his career. He has written about a great breadth of topics, but the thread of thought that ties his work together is his rejection of our contemporary belief in the progress of mankind.

The prepared text of my dad’s overview of Gray’s views is an excellent introductory text, with a good bibliography pointing towards further reading. I would strongly recommend this text to students as an overview of his thought.

John Gray, An Appreciation

Exact Sine and Cosine Values

May 1st, 2008


Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.

I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.

For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!

The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.

Physics Phun

March 5th, 2008


Phun is a free two-dimensional physics sandbox for Windows.

A video of it in action can be found on this You Tube. Unfortunately they don’t yet have a Mac version, so I haven’t been able to try it out myself, but the videos looks stunning.

This has fantastic potential educational value for physics and maths, but in the same way that the Geometer’s Sketchpad does – it is easy to see the potential, but rather more difficult to harness it.

There must be some middle road between the openness of this sort of ‘sandbox’, which for university students and older computer literate school students has tremendous educational value, and something more rigid that allows more nervous or younger students to engage with the simulations it offers constructively. The problem is, what is that road?

Flat Earthers – Thinking about Limits

December 4th, 2007


I was recently invited to do an IQ test. One of the questions was as follows: “You walk five miles north, five miles west, then five miles south. How far are you from where you started?” The answer that they were looking for was 5 miles.

Perhaps our use of maps convince us of this logic; a logic based on Cartesian 2-dimensional geometry. Unfortunately, we do not live on a Cartesian plane!

How should we understand this IQ test question when we correctly consider that we live on a sphere? A good way to consider this is to think about what happens at the two poles!

The South Pole

Start at the south pole. Travel five miles north. Travelling west is to travel parallel to the equator, so then when travelling five miles west, you get no further from the pole. Then travel five miles south. You arrive where you started, with no distance between where you started and where you are now.

The North Pole

The north pole example is more difficult to imagine, and some may think there’s a trick here. If you start five miles and a bit south of the north pole, move five miles north then five miles west in many tight circles around the pole ending up exactly opposite to where you were when you originally arrived near the pole. Then move five miles south. More or less, you are now ten miles away from where you started.

The trick here is the ‘bit’ which ensures two things: firstly that you can in fact travel west; secondly that having travelled five miles west you finish up exactly opposite where you were when you started moving west. It might be argued that if you ended up exactly at the pole then you would be unable to move west at all. The ‘bit’ ensures that there is a trivially small circle around the pole that you can travel west around. It is also necessary to end up exactly opposite where you started to maximise the resultant difference between the two starting positions. If you imagine the bit as a radius of a circle around the pole, then it can be calculated as any r such that 5 miles = (2n+1)*pi*r where n is a positive integer.


Thinking about a problem often involves thinking around its extremes or limits. When thinking about compass bearings on a sphere, the poles offer places where their odd relationship to each other are most apparent. The IQ problem assumes that we live somewhere where the relationship between the compass bearings closely resembles the relationship between Cartesian axes. At the poles this similarity breaks down most markedly. By thinking about moving to and and from the poles, it transpires that if you move five miles north, five miles west, and five miles south, you may end up a distance of x miles from where you started, where x is such that: 0 miles <= x < 10 miles.

Now, if the IQ test was testing for this as an answer, I would have been suitably impressed!

Philosophical Footnote

The earliest known argument against the earth being flat comes from Aristotle, who argued that the shadow that the Earth casts onto the moon during lunar eclipses is always circular. The only object which casts a circular shadow irrespective of its orientation is a sphere, and since night and day convince us that the earth does not have a constant orientation with respect to the moon and the sun, the Earth must be spherical. (Aristotle De Caelo, 297b31-298a10)

Our natural instincts about the world are perhaps that it exists on a plane that is looped; the outside of a cylinder with the poles at the top and bottom of the cylinder. This is a practical simplified model of the earth because until you get into the arctic and antarctic more or less, two people moving north are moving more or less parallel to each other, and the consequences of longitude and latitude working on quite separate principles need not be considered.

We could of course change the way that north and south work, and make them akin to east and west. Perhaps the great circle through Grenwich could be the East-West equator, as in some senses it is, and we could therefore define an east and west pole, somewhere on the equator! It is an interesting thought experiment. Would our concepts be more easily understandable if we did this? Why did North and South become defined as it is?

In Our Time – Fibonacci Sequence

November 30th, 2007


In Our Time is a BBC Radio 4 presented by Melvyn Bragg. It is an intellectual talking-heads discussion programme about philosophy, science, mathematics and so on. This week the discussion was aobut the Fibonacci Sequence. The podcast can be downloaded here.

For maths teachers, KS3 and KS4 students will enjoy and learn from the discussion between Melvyn Bragg, Professor Marcus du Sautoy, and others. It is well worth a listen.

Online Marking System Farce

September 22nd, 2007


BBC News today reports that "Online marking systems ‘faulty’". Prima facie evidence is that with the growth of online marking there has been a corresponding increase of complaints about grades from teachers.

Exams are an inexact method of assessing students’ abilities. Teachers know their students thoroughly and are able to gauge with a very good degree of accuracy how they should perform in exams. When differences between expected performance and actual performance start become too widespread, then there is a problem with the examination system.

John Bangs, head of the NUT is reported as saying "They are not able to annotate the scripts by hand, there’s a time constraint and you can’t take into account youngsters who do quite a lot of writing and don’t fill in the standard box that online marking demands. So legitimately there’s a question whether or not online marking is missing some of the achievements of youngsters." There is also reported a trend towards less well paid, less well qualified examiners.

Technology has a worrying tendency to make things more uniform than they might otherwise be. Marking an exam can be a complex business, and it seems reasonable to contend that someone whose performance is good but unusual could be at a disadvantage in the new marking regime.

The German Ideology

May 16th, 2007


Some years ago my dad Alan McEachran – a teacher of Philosophy for over 25 years – wrote a few students guides to classic texts. His precis of The German Ideology in modern English is perfect as a primer to the text, and I’ve published it here for posterity and in the hope that students of this important text might find it useful.

The German Ideology

Reith Lectures 2007

May 8th, 2007


Combining morality, economics and politics, Jeffry Sachs discusses our global future and humanity’s survival.

This might be a bit beyond sixth-form students – it may be a bit beyond you or I, for that matter – but what reason for teaching students how to think critically can there be that is more important than the hope that they can think critically about these issues?

It is compelling to listen someone with such a depth of understanding of our global state of affairs to give an optimistic view of the future.

Listen to the Reith Lectures by clicking here

Promoting Discussion about Morality

March 8th, 2007


The cognitive evolution laboratory at Harvard Universtiy are currently studying morality. As part of their study they have an interesting and thought provoking online Morality quiz, which is open for anyone to take part in.

This would be an extremely useful starting point for discussion about morality with sixth form students. Some may need to be reminded that this is scientific research of course, and so flippant and stupid answers would be inappropriate. The quiz is a series of pairs of moral dilemmas, and choices of action which you are invited to distinguish between. Ideally, I would ask all the students to do the quiz in preparation for a lesson, then would use a data projector or IWB and repeat the test as a classroom exercise, requiring students to come to a consensus about each dilemma.