Invisible Numbers

There is a brilliant programme on BBC Radio 4 called “More or Less”. It is presented by Tim Harford, a financial journalist, who probes the numbers bandied about by the media. For instance, he recently delved into the UK Covid stats to get to the true picture. He explains everything so well that, for the half hour that the programme lasts, I actually believe that I understand numbers.

Occasionally he poses puzzles for the listeners. A recent puzzle went something like this: add 28 to 50-something (I’m sorry, I can’t recall the exact sum). The point was, people wrote in with what seemed, to me, to be unnecessarily complicated solutions. There was talk of carrying numbers over and other technical terminology. Surely, I thought, you just add 20 and then 8?

I remembered then that, when I was at school, my parents were told that children without sight thought of figures differently to sighted children.

So: is this true?

I don’t remember much of the maths we did at my infants’ school. We did do some but I can only really remember sheets of numbered squares. To help us learn our times tables we had to colour in, say, every third square or every ninth square. I loved colouring and all that stays with me now is a memory of the fun I had choosing the colours to use. The numbers were a secondary matter as far as I was concerned.

I also remember buying a Ladybird addition and subtraction book which taught you to do sums by putting numbers above and below lines. I know I enjoyed learning how to do this and completing the book but I can no longer recall how to do sums in this way.

I first went to a school for the blind when I was seven and three quarters. (I was very proud of knowing my precise age!) I still had residual vision and was given something called a Colour Factor. This was a box with bars of different colours and lengths representing the numbers from 1 to 12. The figure one was a small white cube rather like a sugar lump. The number two was pink and twice the size. You played around with these bars until you discovered that pink and light blue equalled yellow. Again, as I loved colours, I really enjoyed playing with the Colour Factor. I don’t think many of my classmates had enough sight to make use of this piece of apparatus. It had limited educational value but it looms large in my memory.

After that I moved on to sums brailled on red card. When you finished each card, you went to the teacher and got the next one, thus getting a good feeling of making progress. There must have been some teaching involved. I can’t have magically known how to do all the types of sums on the card, but, again, the cupboard is bare. I can’t recall the teaching, I just remember the cards.

It was at this point in my education that I started missing lessons through having to spend long spells in hospital and in sick bay.

When I went to Chorleywood College, my secondary school, I got on all right with numbers to start with. In the days before electronic talking calculators, we used abacuses. I loved mine and still use it for addition and subtraction although I have forgotten now how to do division and multiplication. Incidentally, when I visited Russia in the 1990s and stayed in a town on the Russian-Chinese border, I was delighted to find that many of the shopkeepers still used abacuses to tally amounts rather than electronic tills.

Another fun maths activity at school was creating geometric shapes. We had tactile graph paper laid out on rubber mats. Following the teacher’s instructions, we would count, say, five squares along the bottom row and, say,  six from the left edge, and then fix a drawing pin at said point. We would follow further instructions and determine the location of the next pin. When all the drawing pins were inserted, we joined them up with an elastic band and, hey presto, there was an interesting shape!

At some point we tackled matrices, which involved writing figures in squares on our Perkins braillers. I don’t recall what we did with said figures but, whatever it was, I did manage to do it.

I wasn’t too bad at algebra in the beginning but at some point it left me behind.

We didn’t have to take O-level maths because it was understood that it was a difficult subject for visually-impaired children and I never sat the exam. Some girls did go on to do A-level maths and even studied it at degree level, but they were few and far between.

I left school with the conviction that I was useless at numbers. Looking back, I believe now that, at some point, the gaps left by my earlier absences from class had undermined my ability to keep up.

As an adult, I did get some confidence back by doing the numbers problems on the TV game show “Countdown”. Using the four stand arithmetic operations of addition, subtraction, multiplication and division, you have to make a specified large number from a random collection of six small numbers, none of which is larger than 10. I couldn’t necessarily do the sum in the 30 seconds allowed on the show but I started to get there if I gave myself time and didn’t panic.

So, back to my original question: is it true that children without sight think of numbers differently to sighted children?

My considered answer would be yes, but it depends on what age you were when you lost your sight.

My experience was that children who lost their sight after or around the age of 11 still thought of calculations in terms of carrying numbers and putting numbers above and below lines. But if you lost your sight at a very young age, like I did, you almost had to start off all over again and relearn arithmetic from scratch.

I don’t know for sure but I think there is something inherently visual about Maths and you have to have a certain kind of mind to overcome this if you can’t see. I expect that if I Googled enough I would find learned research papers on this very topic…

…But life is too short and, besides, it’s time for a tea break. (Yes, I can tell the time – with a tactile watch, of course!)