The new teachers: people who inspire me

I often imagine that at some point I’ll go back into paid teaching in some capacity but before I do, I want to have a different perspective on learning and of teaching than the one I came away with after my teacher training. There are a few people out there who are doing it differently...  

I recently read about Taiwanese ShaoLan’s new Chinese language book, Chineasy. The basic Chinese characters are incorporated into large, colourful and imaginative yet simple graphics, each illustrating its own meaning. These characters form the building blocks for more words and phrases, for example, the pictogram for woman, repeated twice, means ‘argument’. Fascinating culturally, as well as linguistically.    

I’ve already mentioned him, but there’s no harm in doing it again; writer and educator, Daniel Tammet explores role of the imagination in perception and learning in his TED talk on ‘Different ways of knowing’. Daniel, who identifies himself as an autistic savant, gives a fascinating insight into how he uses leaps of the imagination to perform complex mathematical calculations. As he says ‘different kinds of perceiving create different kinds of knowing and understanding’ and ‘our personal perceptions are at the heart of how we acquire knowledge’.

Finally, Benny Lewis, National Geographic’s Traveller of the Year, spent 11 years on the road, picking up various short-cuts and unconventional learning techniques and claims that it’s possible to become fluent in a new language within three months. A self-confessed mediocre school student, he attributes his success in learning languages to swapping coursework for conversation. Describing language books as ‘generally written by people with PhDs in linguistics’ and not transferrable to the real world, he is full of practical tips to aid the memory such as using obscure associations, for example ‘gare’ (French for station) reminds him of Garfield the cat, so he pictures the cartoon character running for a train.

Proof that whatever you’re learning, from maths, Cantonese to a new language, imagination goes a long way to helping you get there.

Making sense of alegbra

Why is it that when something becomes 'school-work' it ceases to be interesting? Ever had the experience of opening your books after your GCSEs and thinking 'actually this is quite interesting'... If only we could access that curiosity before the exam rather than after.

Recently I took a question from a maths GCSE paper:

Q: If I borrow £400 and pay back compound interest at 5% who much interest do I pay over three years?

A: £400 x 1.05 cubed = £463.05 

Interest over 3 years = £63.05

The formula works and I have no difficulty applying it, but I don't fully understand how it works.  To me it's abstract, a foreign language. Attempts to translate it into words fail me. Compound interest (or exponential growth) is difficult for me visualise - growth on growth - it's too much for my mind to encompass. 

I was quite fascinated though. What does 1.05 cubed actually represent? When I tried to convert it into 400 + 5% and then cubed it (i.e. 420 x 420 x 420) I knew the answer would be too large, even before I tried to calculate it. I was on the wrong track. Then I googled 'order of operations'. Apparently if there is anything to be cubed it has to be done first before progressing with the rest of the calculation. 1.05 cubed only seems to have meaning in relation to £400 in the sense that it's a multiplier - my intuitive understanding is that this formula represents the trajectory of exponential growth. But that's as far as I got. 

How can exponential growth be represented visually? A graph might work but I wonder whether other senses could be involved? A plane taking off came to mind - not only can you see it climb but you also hear the noise increase - the plane goes faster and faster and the sound becomes louder and louder. 

Over to you...

A month has passed and I've explored a tiny corner of the vast world maths from a playful perspective. But I'm more interested in other people's maths stories.

What did you hate about maths and what did you love? Who were the best maths teachers and who were the worst and why?

What kind of clock face would you design?

How would you represent the number 4? Would you knit it, crochet it or make a mosaic?

It's over to you...

Representing odd numbers

My next line of enquiry took me to the 5 times table. I experience 5 very differently from 4. As I mentioned in my previous blog entry 'Numbers are beautiful', to me 5 is yellow/orange, sparky and alive. I thought for a while how I could represent 5 as a shape and a star immediately seemed appropriate (obviously one with 5 points) so I printed and cut out 10 yellow stars and arranged them in different ways. A pyramid of 10 stars, to me, perfectly represents the 5 times table. As the numbers grow, the shape expands from one star  ever outwards, representing the expansive, extrovert nature of 5. I created some pictures, below. You'll see that in each picture I've replaced some of the yellow stars with white ones to show products of the 5 times table, for example 6 white stars represents 6 x 5, a 'trapezium shaped' group of 4 stars represents 4 x 5, and so on.

What I like about this arrangement is that it represents 10 x 5, the final product of the 5 times table but at the same time all the products of the 5 times table can be represented, depending on how the stars are arranged.

1 x 5 = 5

6 x 5 = 30

4 x 5 = 20

Representing even numbers

In the past few weeks I have been exploring times tables. We roll them off parrot fashion:

1 x 4 = 4
2 x 4 = 8
3 x 4 = 12

and so on, but these black symbols are hard to learn. After exploring the Slavonic abacus I wondered if there was another way of experiencing them; actually viewing them as tangible, physical products rather than marks on the page.

My explorations started with the four times table and my attempt to express it pictorially resulted in a trip to Foyles where I bought a copy of the 'Geometrix Colouring Book' by Jennifer Lynn Bishop - a bit of a gem. I  was immediately drawn to the Mandala design on the front cover which shows a series of squares, growing ever larger and overlaid on top of each other. To me a square perfectly represents 4 (it has 4 corners) and therefore two squares, one inside the other represent 8, and so on. My rather crude attempt to represent the 4 times table is shown below (sorry the pencils weren't up to it). In a corner of each of the squares I've shown the  number of times 4 has been multiplied and in the opposite corner I've shown the product. In fact I could probably have got away without showing the product as the square grows by one 'layer' each time, for example 2 x 4 is represented by a square with two layers, 3 x 4 with three layers, and so on. Each square is coloured in shades of red which to me captures what 4 looks like to me (synaesthetically).

A second bit of colouring took me on another journey though I'm not sure what to make of this picture or really whether this line of enquiry is worth following, but I thought I'd share it anyway. This time I arranged the products of the 2, 4, 6 and 8 times tables in a grid. Each coloured square represents 'one', for example '10' is shown as 10 yellow squares split into two lots of five, arranged in a right angle. Alongside the 2 x table square, the products of the 4, 6 and 8 times tables are shown where they overlap with the 2 times table. Naturally, each band shows at wider intervals, the bigger the number. Each number has been coloured the way I see it synaesthetically. I don't think this really worked - it looks confusing close up but somehow it's strangely pleasing at a distance!

And finally a bit of fun... Part of the four times table crudely drawn by me in paintbox. The products of 4, along with the colours they represent, are shown in the bottom left hand corner.

Numbers are beautiful

In my first post I described how at school I had imagined that numbers were 'dry' and 'pointless' entities but I have only recently become aware that it's not how I really experience them at all. I have a type of sensory perception called synaesthesia (see the wikipedia entry for more information http://en.wikipedia.org/wiki/Synesthesia) and an aspect of this is perceiving numbers as having personalities and colours. In my mind numbers are quite delicious and fun:

• 2 is blue, cuddly, solid, stable, loving and female, 
• 3 is a vibrant orange/yellow, very physical and gregarious and male 
• 4 is a deep red/maroon, male and very rigid and conservative 
• 5 is a similar colour to 3 and also very gregarious but in chatty, hyperactive way rather than in the physical way that 3 is. It is also androgynous. 
• 6 is similar to 2 and is also blue and female, very kind and sensible but not cuddly and more aloof 
• 7 green and male and an oddball - a thinker - part scientist and part magician  
• 8 is male and shares similar qualities to 4, and is the father of 4 (its also maroon) but is much more aggressive and ambitious with the attributes of a businessman
• 9 is androgynous and a very soft pale brown. It is peace-loving and very benign and spiritual 

If I'd shared the above with Pythagorus 2000 years ago or more, I might not have been viewed as particularly unusual. Pythagorus believed that numbers represented the very essence of spiritual truths and some claim that he was the originator of numerology. I'm not sure I have a view on numerology and it can hardly be described as 'maths' in today's terms, but reading at the descriptions above, I can see certain patterns. 'Sensible, conservative' numbers are even and all 'gregarious' or 'more unusual numbers' are odd. 4 and 8 had similar 'personalities' and 4 is a factor of 8. Four is quite literally square as a square has four corners. Five can be represented as a star shape with five points.   

I share my synaesthesia with another mathematician, Daniel Tammet (although as an autistic savant he is in another league to me when it comes to calculating). Daniel uses his synaesthesia to do complex mathematical calculations and in this TED talk he describes how imagination and aesthetic judgements guide and shape the process of learning - fascinating!

Reconstructing the clock face

In my last blog post I wrote about why the clock face had made me queasy as a child and how I often wondered why the minutes couldn't be separated out from the hours instead of having to share the digits of the clock face. It would have been satisfying if we'd had the chance to design our own clock faces at school, perhaps as an art project or just for fun. Our individual clocks would have told our teachers a lot about how we saw time and where our blocks to understanding were. We might also have learnt a bit more about time in the process of our explorations.

I recently had fun looking at alternative clock faces http://www.boredpanda.com/cool-and-creative-clocks/ and drew a few of my own. I succeeded in discovering just about the most impenetrable way of telling the time - by measuring the angle between the hour and the minute hand - not terribly practical because 180 degrees could be six o/clock or half past twelve!

After a bit more messing around I settled on a design that seem to meet my objectives (at least I hope it would have been pleasing to my 13 year old self):

Quarter to six

• The clock face doesn't have hands so no confusion over which is the minute or the hour hand
• It's possible to trace the progress of the hours as the time is represented a solid line - no hands floating in space
• The clock face has no numbers, apart from showing the most recent hour - so no clutter and confusion.
• The minutes are represented as solid fractions which are displayed inside the 'hour' line. Half past, quarter past, quarter to are easy to see at a glance but to avoid confusion, subtle delineation between the five minute intervals can be added making it possible to read five past, 10 past, 20 past and 25 etc. 
  
• Minutes and hours are shown in different colours