Mastery through the Masters

This is a guest post from one of our Education Consultants, Andy Tidswell. A big thank you to Andy for his contribution. To find out more about Andy please see his bio at the bottom of his post.

pascal

We are generally the better persuaded by the reasons we discover ourselves than by those given to us by others.

This quote from the French mathematician Blaise Pascal (1623 – 1662) strikes home for me the importance of HOW we teach maths. Getting children to understand and MASTER maths concepts can seem to be a complex and difficult process and teachers are often looking for a magic formula (pun intended) which means their children can ‘perform’ all the skills required for their particular age group.

I choose the word perform deliberately as this is sometimes the way we visualise what makes a successful mathematician – a person who has a range of ‘tricks’ which enable them to answer tough questions quickly. The Human Computer Shakuntala Devi (1929 – 2013) was well known for her ability to solve complex puzzles rapidly. She demonstrated her skills around the world and even featured in the Guinness Book of Records, performing feats such as calculating the cube root of 61,629,875 and the seventh root of 170,859,375, or for multiplying two random 13-digit numbers (7,686,369,774,870 × 2,465,099,745,779) mentally. (The answers by the way are 395, 15 and 18,947,668,177,995,426,462,773,730 respectively.) But, she argued in her books, such as ‘Figuring – The Joy of Numbers’ (1977), that these skills are based on a true understanding of numbers and how they work.

Look at the following number

274,207,281 – 1

What is it? What kind of number is this? Why is it interesting? How was it found? What would it look like if it was written out fully?

It means multiply 2 by itself 74,207,281 times then subtract 1, which makes it an ODD number with the key being the ‘subtract 1 from a power of 2’. More significantly, however, it is the largest Prime Number yet found, discovered in January 2016. It is a 22,338,618 digit number and if you printed it with each digit measuring 1mm it would be 22.34km long! It is also known as a Mersenne Prime, named after the French monk Marin Mersenne (a colleague of Pascal) who discovered this method of locating Primes. This short BBC News video shows you some interesting facts about the number, whilst Matt Parker of Numberphile shows off his 3 volume printed copy here.

Clearly this is a complex concept for primary age children, however huge numbers are exciting and we can certainly get children exploring the foothills of this particular mountain.  Perhaps the best way to explore Primes is by making use of the Sieve of Eratosthenes, which enables children to discover the primes for themselves.

Eratosthenes was a Mathematician from the third century BC who became the Librarian at the Library of Alexandria. His ‘Sieve’ enables you to filter out multiples of different numbers; magically leaving the prime numbers behind.

There are various ways of doing this in class but here’s an example from the excellent NRICH site which explains the process. This lovely interactive tool also allows you to further explore the process in a variety of ways.

Providing a stimulating model, image, video or even number at the start of learning is often a good way to get children intrigued by maths and to enable them to bring to the surface all kinds of mathematical possibilities before exploring these ideas in structured tasks. Displaying the largest prime number could be such an example, or the video of the digits being displayed or the video of the three volume printed copy.

This simple video (made with Photostory 3 using the work of various artists), opens up a wealth of discussion about the properties of shapes before tackling further investigations and explorations.

The work of M.C. Escher is full of possibilities, not least the exploration of tessellation.

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Notice the hexagonal arrangement? This is the key to the construction of this piece, as this series of images shows:

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Children can easily construct exciting patterns made from squares or rectangles by creating card templates, cutting out small parts from the template and re-attaching them on opposite edges with sticky tape. The new template can then be used to produce an Escher style pattern. This image gives a simple example:

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You can however create really complex pieces, such as Escher’s piece Pegasus:

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You can view the animation of the Pegasus piece here.

The German Renaissance painter Albrecht Durer’s etching ‘Melancholia’ contains an amazing 4×4 magic square.

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magic square

As in all magic squares, the totals of the numbers in the rows, columns and diagonals make a ‘magic constant’ – in this case 34. Durer’s Square, however, sees the magic constant appearing in lots of ways, all of which have describable patterns. For example, the 4 corner numbers 16, 13, 4 and 1 or the 4 central numbers 10, 11, 6 and 7. Amazingly this image was made in 1514, which is recorded by the central two numbers on the bottom row. This square can provide HOURS of exploration for children, whilst giving them LOTS of practise on mentally adding four 2-digit numbers.

This animated image of the fractal pattern of Sierpinski’s Triangle has lots of interesting properties which can be explored. This includes constructing and repeating patterns of equilateral triangles, exploring symmetry, securing understanding of the properties of triangles, exploring repetition and number patterns, exploring infinite sequences….

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Finally to return to Pascal. Here is part of his famous triangle

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Children can be provided with the following image to explore and help them to understand how Pascal’s Triangle works.

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Once they have discovered and continued the pattern there are a wealth of other patterns and possibilities to explore; including finding the triangular numbers (1,3,6,10…) and explaining how these are made. The triangular numbers also record the total number of numbers in Pascal’s Triangle when each new row is added!

Other patterns and sequences include the Square numbers, the Tetrahedral numbers, the powers of two (links back to our largest Prime!), the Fibonacci Sequence, prime number patterns and more. This page on the fantastic ‘Maths is Fun’ website explains much of this.

However, when you simply colour in the odd and even numbers in contrasting colours you get… Sierpinski’s Triangle!

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In this video, from the people at Numberphile, Professor Edward Frenkel discusses ‘Why People Hate Maths’. He argues that we should show, explore and teach maths through the amazing ideas and beauty of the works of the Masters of Maths, just as we may teach art by exploring Picasso and Monet or music through Mozart and the Beatles. Some of the ideas I have set out above are possible starting points for this ‘Mastery through the Masters’ approach. At the very least, though, getting children to explore ideas and asking them questions which enable them to ‘dig deeper’ is at the heart of quality maths teaching.

 


Andrew Tidswell Biographyandrew tidswell

Andrew is a freelance trainer and consultant based in Leeds. He has broad teaching experience, working with pupils from Foundation to KS4. He continues to teach children in various capacities, delivering a series of visiting ‘Maths Master Classes’ – which provide stimulating extended investigations based on the lives and work of mathematicians and artists from history. He also delivers creative hands on sessions in Coding, Animation, Digital Video and Digital Audio, using both PCs and iPads.

He provides a wide range of consultancy and training services both independently and on behalf of Discovery Education. These cover product specific training, the use of digital media, iPads, Coding and the innovative use of Interactive Whiteboards.

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