The Ghosts of Departed Quantities and the difficulties of teaching and learning caculus

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus.  The infidel was probably Halley (of comet fame) or Newton.

If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man.


Quotes include:

“Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply’d infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever”


“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”

Some sympathy for the thesis is gained by Berkeley’s examination of tangent reasoning:

“Therefore the two errors being equal and contrary destroy each other; the first error of defect being corrected by a second error of excess. ……. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold mistake you arrive, though not at Science, yet at Truth. For Science it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means.”

The student of sixth form level mathematics who is eager to see how this is resolved must continue their path to mathematical enlightenment by studying the \epsilon-\delta Analysis of Cauchy, Riemann and Weierstrass.





Euler and the Properties of the Second Derivative

The summer examination season sees pupils searching for maximums and minimums on their text papers.

This long-standing pursuit was initiated by the likes of Newton and Leibniz in their calculus.

It can be all too difficult to think about for some:

“our modern Analysts are not content to consider only the Differences of finite Quantities: they also consider the Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum.”

Bishop Berkeley, The Analyst, 1734

The example above is one in which Euler demonstrates the geometrical significance of the first and second derivatives.

Note that the Point of Inflection is where f''(x)=\dfrac{d^{2}y}{dx^{2}}=0 and is a change from upward to downward convex curvature, or vice versa.  There is no need for f'(x)=\dfrac{dy}{dx}=0 too.





Yachting and similar triangles

A yacht has two parallel masts which are both perpendicular to a level deck.  One mast is 12 m high, the other is 8 m.  Stays are rigged from the head if each mast to the foot of the other.  Find the perpendicular height of the crossing point above the deck.


Finding the height of the crossing point is a good similar triangles challenge for bright maths pupils.  In fact that the distance between the two masts is not given introduces two variables and makes this a subtle challenge.


I find the independence of ‘h’ with the distance between the masts intriguing.


The linear combination of a sine and a cosine is itself a sine wave

A linear combination of two functions, f(x) and g(x) is a sum involving constant multiples of the functions.  That is,

a f(x)+b g(x)

where a, b \in \mathbb{R}.

So, in the case of \sin x and \cos x, we would have,


It is a slightly surprising fact that the linear combination of two sine waves is itself a sine wave.  The set of sine waves is closed under linear combination.

The in the featured animation, a\sin(x) is green and dotted and b\cos(x) is red and dotted.  The resulting linear combination is the continuous blue line.  The value a is set to 2.5, whereas the value b is animated.

This principle occurs in A level maths, Core 3, and is responsible for many long and complex questions.

Exdexcel Formula Sheet lays Pooh Traps for Further Mathematics Pupils

FP3 integration standard forms could lead unwary pupils and teachers into a Pooh Trap. The following issue arises each year as the finer points of this course get sharpened.

When consulting the Edexcel formula sheet to find the integral,

\displaystyle \int \dfrac{1}{\sqrt{a^{2}+x^{2}}} \textnormal{d}x

two anti-derivatives are given,

\textnormal{arsinh}\left(\dfrac{x}{a}\right)\ \ \ \textnormal{and}\ \ \ \ln\{x+\sqrt{x^{2}+a^{2}}\}

Some might be tempted to infer that these functions are equal, this is the Pooh Trap, because whilst they both differentiate to \dfrac{1}{\sqrt{a^{2}+x^{2}}} they are not equal.

Play with exponential functions and quadratic equations give us the logarithmic form of \textnormal{arsinh}\ x which is also represented in formula books.

\textnormal{arsinh}\ x=\ln\{x+\sqrt{x^{2}+1}\}

\textnormal{arsinh} \left(\dfrac{x}{a}\right)=\ln\left\{\dfrac{x}{a}+\sqrt{\left(\dfrac{x}{a}\right)^{2}+1}\right\}=\\\ln\left\{\dfrac{x+\sqrt{x^{2}+a^{2}}}{a}\right\}=\ln\left\{x+\sqrt{x^{2}+a^{2}}\right\}-\ln a
The two anti-derivatives differ by the constant \ln a.

In a definite integration, this constant is added and then taken away making no difference as to which anti-derivative the student uses. In a particular solution to a differential equation, the evaluation of the integral using boundary condition would lead to two different constant values.

The worse case is that a student takes the integration standard from and reads it as the logarithmic form of the inverse of \sinh x. I have seen this happen.

support for school mathematics exams

I have developed a database driven website which shares past papers, markschemes and my own ‘write outs’ for most Edexcel modules in mathematics and further mathematics:

jped Exam Bucket

I use this mostly as a resource for my own teaching; using it I can lay my hands of exam questions and solutions of different types very quickly.  My students and pupils find it useful too because it shows how to do exam questions using the techniques taught in class; there is an important difference with mark schemes here.

The UK mathematics exam system is in a state of flux at the moment.  I see this is as a great opportunity to review and refresh all my teaching material.  I look forward to generating new material to support a revised school exam system.

depression near Greenland – logarithmic spirals in nature

Logarithmic spirals frequently occur in nature.  Is this such a manifestation?

The general equation for a logarithmic spiral is as follows.

r=ae^{b\theta} &s=4

Changing the variables a &s=2 and b &s=2 produces spirals of different qualities.  The a &s=2 is really an enlargement scale factor but the b &s=2 controls how the spiral grows per revolution.

If b=0.2 is more representative of an ammonite then b=0.75 seems to be our Greenland depression.

Wouldn’t it be nice to take the equations for atmospheric fluid dynamics and show this explicitly.  Unfortunately this is beyond the scope of this blog: here we have circular motion on the surface of a rotating sphere in an elliptical orbit … . The way to go with this would be to take the basic baratropic equations and then perform a scale analysis to disregard ‘small’ terms.  At this point the pure mathematician get frustrated with approximations and goes into a sulk.  We are left with contemplating the beautiful images though…

third century wisdom of Diophantus gives us the essential truth about teaching and learning mathematics

The Google facsimile of the 1621 edition of Diophantus’s Arithmetica lays out, in original Greek and Latin translation, the essentials for success in teaching and learning mathematics.  Also included is the translation of the second half by Heath.

When I first studied and learned to love mathematics I was beset by friends and family who stridently claimed that this field of study was closed to them because of some biological, or other, reason.  This always seemed to undervalue my own effort in getting off the first page.

Teachers of mathematics cannot be permitted to admit that nature is able to restrict the ability to learn the subject but the oriental wisdom of the student making their way to the master is a precondition for success when things seem difficult; in tales of martial arts the student must somehow prepare themselves to learn.


  1.  Mathematics appears rather difficult if one is not familiar with it.
  2. Students need energy, eagerness and enthusiasm.
  3. When such motivation is backed up by good teaching rapid learning results.

In school we struggle because of item 1 and bear our hearts to gain item 2, being so often rebuffed.  I suppose that we get paid for 1 and 2 because, when item 3 is achieved the job is it’s own reward.

Before the beginning – Euclid’s Common Notions

This preamble from Euclid’s Elements is where mathematics education still goes wrong, even after 2000 years or so.

Sound notions of what equality is are required as a bedrock of algebra.  Dynamic approaches of teaching equation solving, in which terms and numbers move and change sign, seem to work at first but lack the simplicity of mathematical logic and create all sorts of problems when randomly, but apparently sensibly, applied.

Conic sections in the sand give shipwrecked philosophers good hope


Conic sections in the sand give shipwrecked Socratic philosopher, Aristippus, good hope.

I wonder if this ancient tale, in which the mathematics of planetary motion some 2000 years prior to it’s application in space travel, is used to signify the presence of intelligent life and therefore hope of salvation has some echo in our modern era.

What obscure and impenetrable modern theorems could be left on a 21st century beach to do the same job?