The article’s authors, Sol Garfunkel and David Mumford, argue strongly in favour of a maths curriculum that exists in relation to the science, engineering, finance (and so on) that it serves every day, and they say why it’s so important.

Garfunkel and Mumford make an analogy with languages. Learning French teaches both abstract grammatical skills and practical language, but you will never have the chance to use Latin to buy a loaf of bread. In this respect – they argue – the abstract skills that you might gain from a dead language are wasted if they aren’t learned in a context that you can use.

This chimes with recent suggestions from a review into maths education in the UK (led by the lovely Carol Vorderman) that our maths curriculum pre-A-level should be broken up into ‘practical’ and ‘formal’ maths.

In this curriculum every student takes practical maths, arming them for a life choosing between mortgages and interpreting medical test results. More able and motivated students can opt to take the additional formal, and more abstract, maths module.

Mumford and Garfunkel say that the new US Common Core State Standards, which I’ve helped Whizz align its maths lessons to, is “highly abstract”, and “simply not the best way to prepare a vast majority of high school students for life.” But I’m not sure they’ve got the point of that (or any) curriculum. A curriculum should tell teachers and students what they need to study, not necessarily how, or even why, except maybe to put a subject in context.

It’s the good teacher who draws the abstract and the practical together to greatest effect. Not every teacher – especially at primary level – may be confident enough to link the two, but by giving the teacher the opportunity to do this, rather than prescribing how to apply ‘practical’ maths, an ‘abstract’ curriculum can be as relevant as the teacher and her students want it to be.

Maths is often called the science of patterns. And if chemistry really comes alive when we blow something up, then maths should come alive when we use it to, say, spot the ‘golden ratio’ in Chartres Cathedral (see Marcus du Sautoy’s BBC Series, The Code).

But it’s a two-way street. Project Euler, which I wrote about last month, trains budding programmers by getting them to solve maths puzzles. Garfunkel and Mumford say that practical skills, like learning to code, are more useful than abstract, but in the case of Project Euler the abstract is successfully used to teach the practical, and with some success.

I’ll concede that the principle outlined by Mumford, Garfunkel, Vorderman, et al – of the difference between the maths you use to calculate change and the maths you use to describe fractal geometry – makes perfect sense. But it may be a false distinction.

Whizz Education US company president, Ben, has a skill which I envy – he can intuitively understand the relationships between numbers. For me, it’s not quite so easy, but I wouldn’t want to be denied this insight into abstract maths because of a lack of natural ability; I can still appreciate Chartres Cathedral.

To return to the NY Times’ authors’ analogy, I took Latin and Ancient Greek at GCSE – two subjects that are, in themselves, utterly useless in the real world. But I was enriched by learning them. I could spot some of the hidden histories and meanings in words with ancient roots, and read street signs in modern versions of the ancient Greek alphabet.

If creativity lies in making connections between apparently unrelated subjects then ‘pure’, or theoretical, maths must be vital for creative young science minds, wherever they are. To assume that they should only use maths that tells them how to calculate compound interest (a truly vital skill) is to assume that they will get no pleasure from discovering a subject for its own sake. And that is sad.

At Maths-Whizz we try to instil a sense both of the practical and the abstract – and with our 1200+ maths lessons there’s ample opportunity for the young Whizzer to link the two. As Hilary and Steve, two of our expert founding mathematicians, once pointed out to me – put a pound sign in front of a sum or subtraction with decimal values and it suddenly becomes easier.

So, here’s to producing mathematicians who know that the abstract and the practical are two sides of the same coin. It’s a coin that can teach them as much about pi – that most wonderful of irrational numbers – as it can about the price of a loaf of bread.

But, for your delayed edification and delight, a couple of topical nuggets about that loveliest of irrational numbers.

First, from kottke.org, celebrating pi day with a great fact:

…this is my new favorite fact about pi: we have calculated pi out to over 6.4 billion digits but only 39 of them are needed to calculate the circumference of a circle as big as the universe “with a precision comparable to the radius of a hydrogen atom”. (via @santheo)

And, second, a musical number (pardon the pun) devoted to pi, and based on its shapely digits. Listen, and enjoy:

And that’s that!

So it’s a good thing they were halfway decent at maths…

[click the pic to see the whole cartoon]

Trigonometry to rule the world

For those whose trigonometry skills are behind (or ahead) of them, an explanation: SOH-CAH-TOA is a useful tool for remembering Sine, Cosine, and Tangent functions. These are ratios which help describe right-angled triangles.

So there it is: learn maths, and rule the world… or not.

As with the legendary Ramanujan and his disappearing numbers, if you haven’t heard of Mandelbrot and his set, you need to get reading.

Mandelbrot gave his name to a famous fractal ‘set’, whose image has adorned thousands of student bedrooms, science labs, and psychedelic paraphernalia, but he has contributed a huge amount to maths in particular, and science in general.

Mandelbrot, who died of cancer last Thursday aged 85, was one amongst many for whom the Nazis’ loss was Britain’s, and America’s, gain. His family fled occupied Poland before WWII and young Benoit landed up in the USA.

Whilst a research mathematician for IBM Mandelbrot produced the 1980 book The Fractal Geometry of Nature, which illustrated such fractals with fascinating graphics. The Mandelbrot Set has since become one of the most famous images in modern science, instantly recognisable and fascinating.

You can get a taste for the Mandelbrot Set with this wonderful video. Zoom from the classic multicoloured bubble view deep into apparently random nooks and crannies to reveal another, tiny, set. Carry on zooming to see another set, and yet more, demonstrating one of the beautiful features of fractal images – self-similarity.

Self-similarity is everywhere, when you look. It is the same way that vast lines on the ocean seen from a passenger jet look similar to the wave sets visible from a few hundred metres, which look similar again to ripples in the water seen from a few feet. Look closely enough at many natural objects and you might see this self-similarity.

Even the humble Romanesco broccoli looks as though it was devised by a mathematician.

Self-similar Romanesco broccoli

In fact, Mandelbrot used the ‘complicated and yet simple’ broccoli to illustrate his point at a charming TED talk he gave earlier this year, weaving in the ‘roughness’ of mountain ranges and the fractal qualities of lungs.

This is fascinating stuff for yours truly. The equally humble fern is another great example of self-similarity, first described by British Mathematician Michael Barnsley, and animated in a computer program which the God of Whizz remembers with some fondness running on his venerable 8086 PC.

But Mandelbrot’s famous 1967 paper – ‘How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension‘ – showed how apparently trivial observations in geography can prompt quite deep questions.

In the 1967 paper Mandelbrot studied earlier measurements of coastlines which had found that their length increased with shorter measuring increments. For example, the coastline of Britain measured with a 200km stick is 2,400km; measured with a 50km stick the coast gains another thousand km.

Mandelbrot, in effect, noted that such coastlines exhibit a kind of self-similarity – they can behave like fractions. The further into a coastline that you look, the more detail you see; and the coast of Britain is wiggly enough to ensure there’s lots to look at. The seminal paper paved the way for Mandelbrot’s later observations about fractals in nature.

We can only give you a tiny taste of Mandelbrot’s work and the influence it has had on our understanding of nature and natural processes, so why not mug up on the great man and carry on reading…

[via the Register]

Rather than serve only to be hooked, sliced, or shanked into the near distance, chased by a volley of insults, hundreds of red and blue golf balls have been put towards a magnificent three-dimensional Sierpinski Triangle (or tetrahedron, in this instance).

A use for golf balls - maths fractal pyramid

To those otherwise unversed in Sierpinski’s Triangle (also known as a ‘gasket’), it’s a beautifully elegant fractal.

See how successively small triangular ‘subtractions’ from the main triangle produce a lovely, almost threadlike, fractal pattern. The rules for creating this are simple, and can be repeated ad infinitum.

Sierpinski Triangle (Wikimedia)

(Wikipedia)

You don’t even have to start with a triangle to end up with a triangular-shaped Sierpinski fractal. Try it!

Or, better, try the ‘Chaos Game‘, an exotic-sounding name for something you can play with just pencil, paper, ruler, and die:

(Wikipedia)

[Via Make Blog]

As comedy maths puzzlers go, this one is a little forced, but nevertheless fun to share. I confess it took The God of Whizz a few goes to figure out that Pi = ‘pie’ (look at the scribbly reflected numbers).

Despite this neat coincidence, I somehow suspect that sharing this fancy formula in an exam or test won’t get you extra credit…

(via RedEye Puzzler)

[Props to Keat, via BoingBoing]

Now carry on.

Some of them look a little contrived, but nevertheless almost give you the idea of the numbers that lie just below the surface of everything. And that The God of Whizz likes.

(via Kottke.org)

With the help of a pen, paper, ruler, set-square and some patience, you can make your very own honeycomb-like Voronoi Diagram (via Metafilter).

“Woo”, I hear you cry, “why on earth would I want to do that?”
Besides the simple pleasure of learning maths and general principles of shape, space and measurement (which Whizz Education can help you with), a Voronoi Diagram might help you understand more about the world. The God of Whizz will give you three good reasons:

It is used in derivations of the capacity of a wireless network.

In climatology, Voronoi diagrams are used to calculate the rainfall of an area, based on a series of point measurements. In this usage, they are generally referred to as Thiessen polygons.

Voronoi diagrams are used to study the growth patterns of forests and forest canopies.

Voronoi diagrams are also used in computer graphics to procedurally generate some kinds of organic looking textures.

In robot navigation, Voronoi diagrams are used to find clear routes. If the points are obstacles, the edges of the graph will be the routes furthest from those obstacles.

If you’re a Key Stage 3, or high school, teacher, why not explore Voronoi diagrams with your students?