Helping a Struggling Math Student: Creating their own Number and Place Value Systems

What better way could the children investigate ‘what numbers are’ than by designing their own number system, followed by a number board which illustrated their own place value system?  It was, as ever, very interesting to see how each of my children’s minds work at understanding something.  T11 understood and seemed to put together a logical yet fairly complex number system (far right); L10, obviously understood and did really well designing her own simple, creative and logical number system (far left); C10 designed a very complex and convoluted number system (she would be by far my most creative and quirky child so it didn’t really surprise me!).  The problem was it showed no logic, so it was difficult for her, never mind me, to work out how the larger numbers would be expressed and the very thought of how her addition board would look like frazzled both of our brains.  She took a peak at her siblings ideas and consequently scrapped her first attempt and replaced it with an incredibly logical and easy to use number system of her own (middle):

The second part of their assignment was to fill out a hundreds square to double-check their system made sense and was simple to use.  This proved to be very interesting, especially concerning T11’s number system.  First he struggled to tell me what base it was in.  After studying it for a while, he realised it was not in any base, as the higher value numbers simply were given different symbols.  He was upset, thinking he had done it wrong.  However, it absolutely wasn’t wrong (the task was to design his own number system, not specifically to any base), but he was soon to discover it wasn’t going to be very easy to use.  As he filled in his number square it became apparent that as the numbers got higher the symbols required to represent the numbers got more plentiful, and he struggled to fit them into the square.  After the third line he came to me exclaiming something had to change!

He rewrote a second number system, more based on our own and to the base 10.  Really all he did was add a zero label to his own number systems and built up double and triple figure numbers using place value notation rather than a different symbol.  This was such a great lesson for him (and the girls) because it showed them why a number system with a base value and therefore a corresponding place value system worked much more efficiently than one without.

The girls wrote out a very clear and simple number board and found their own number systems very easy to use:

I then had them carry out a few addition and subtraction sums in their own number system language.  This they did with ease.

You might wonder why on earth I had them complete such an arbitrary activity.  The answer, apart from checking their general understanding of place value and number systems in general, lies in the importance of realising that our numbers are simply labels, just like the Babylonian wedges, the Egyptian dashes and pictographs were.  This would be particularly useful for L10 whose fear lies not in the maths but in the numbers.  I wanted to demystify them for her and allow her to see them for what they are – representational tools designed to help her with calculations rather than hinder her.

At the end of this activity, I asked each of the children the question I had set them.  What is a number?  I was looking for some indication that they now understood that a number was a diagrammatic representation of a value.

T11 immediately told me that they were symbols which represented values.  The girls told me they were symbols which showed amounts.  Bingo!

Could they apply their knowledge of the base ten to create a base four using a number system I designed for them?  Well, that’ll be for my next post!

Before I go, I just wanted to share something that made me smile this week.  I over heard the children talking whilst they were doing their chores.  They didn’t know I was listening.  L10 was saying how much she was looking forward to maths and figuring out the next thing ‘Mummy had for her’.  C10 exclaimed that she thought L10 hated maths.  To which L10 replied ‘only the sum maths, mummy’s maths is fuuuuunn!!’

Oh yes, there was one happy mama that night!