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

Visualising numbers

During my teacher training I stumbled upon an intriguing piece of equipment - the Slavonic abacus. Though it was never part of the National Curriculum and I didn't used it in my classroom teaching practice it would have been interesting to have introduced it into an ordinary maths lesson.

The abacus has 100 beads which are subdivided into quarters. Each quarter is coloured differently to its adjacent quarters. The abacus works on the principle of 'fives', allowing us to calculate quickly by breaking numbers down into clearly identifiable chunks - so 5 rows of 5 makes 25 and 4 lots of 25 make 100 - all you need to know is your five times table - simple.

The abacus at rest

And here's how to calculate  6 x 8:

 

We can start by making 6 rows of  8 beads (these beads have been moved to the left). We now have four blocks of different coloured beads: 5 blue columns of 5 beads, 3 yellow columns of 5 beads and 1 row of 5 yellow beads plus a 'remainder' of three blue beads. Adding up all these fives makes 45 and the remaining 3 makes 48.

 

 

If you feel like playing with the abacus there's one here (you'll need Adobe Shockwave)

Understanding the clock face

I was astonished to read that Lorna Sage (a Professor of English at University of East Anglia) claimed in her autobiography 'Bad Blood' that she wasn't able to tell the time until she was 16. Hardly lacking in intellect, she said she couldn't face looking at a clock. I know how she felt. I too wasn't confident telling the time on an analogue clock until I had reached my teenage years. I later attributed it to some kind of dyslexia (I do sometimes reverse the clock face in a moment of bleary eyed exhaustion, reading 4pm as 8pm - it's mirror image).

For a long time I too couldn't bear looking at a clock face but fortunately I was saved for a while by digital watches which were very popular in the 1980s, when I was a teenager. Wearing a digital watch rather than an analogue meant that I could confidently tell the time when someone asked me (something I usually dreaded). It was entirely straight forward; the first lot of numbers before the dividing dots showed the hour and the second lot showed how many minutes had passed since the hour. Better still, you didn't have to understand what they represented - you just had to read out what you saw: 09:15 was 'nine fifteen' and no-one would be any the wiser if you didn't know what it meant. No need to convert 15 into 'quarter past' or 45 into 'quarter to'. I liked my digital watch because, to me, it represented time in the linear sense - as a journey - hour accumulating upon hour, minute upon minute. Seeing time as a number line makes it easier to know where time starts and finishes whereas a circular clock face seems to suggest infinity and makes me feel a queasy and shaky, like I'm drifting in space.

Spot the deliberate mistake above - well I did say I have trouble with the concept of time!

But my biggest obstacle to understanding a clock face was that it was supposed to do two things; the numbers 1-12 represented the hours (1am, 2am, 3am etc) but they also increments of five minutes; so depending on which hand was pointing to a number (the short or long? I would  often muddle them up) 1 could represent 1am or 1pm, or it could mean 'five past'. How could numbers represent two things at the same time? It wasn't right, I hated that it wasn't right and I switched my brain off entirely.

I realise that it was only when I learnt to see the clock face solely as representing 60 minutes, and each number as representing increments, or fractions of that 60 minutes, that I felt comfortable telling the time. All I had to worry about was where the small hand was pointing to make sense of those 60 minutes and what part of the 'number line' they fell on. This led me to thinking about how I could design a clock face to represent minutes and hours in a way that was more pleasing to me. I will write about this, and show my drawings, in my next blog post.

The point of this blog...

Maths can be a difficult subject and I think it's a pretty safe bet that it has caused most of us problems at one time or other, more often than we'd like to admit. Maths can be hard and abstract and downright dry. Who wants to do maths? To be honest many of us would rather not if we don't have to.

I remember being five years old and doing my first sum. I had already learnt to count and write numbers in sequence, and they made sense to me in terms of finding out the number of things. But when the teacher wrote 5 + 2 = ? she may have written a load of symbols on the board. Why was she writing the two after the five, surely it can before two? What were those two little lines, one above the other. And even if I could do this sum, what was the point of it? Yes, what was the point?? I could, of course see the point of reading - words made up stories and reading stories was fun. Somehow I liked words more than those devilish little numbers.

Later I did, somehow, learn to do maths - addition, subtraction, multiplication, long multiplication (eventually, and after bursting into tears repeatedly in frustration) but I have particularly horrific memories of trying to learn the time and this will be the subject of my next blog.