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.