I have used two books to go over the concept of place value. The girls are certain that they do understand. I am not so sure. Whilst they do usually get long addition, subtraction and multiplication correct, there are occasions where it seems they have forgotten everything they have been taught. This tells me they may understand, but they haven’t internalised that understanding. An Elephant in the Classroom author explains how a child, at the point of being taught, can seemingly understand the concept being taught and can readily replicate the method they were shown and arrive at the correct answer. The teacher, who assumes they have done their job correctly, moves on. Later that same student is unable to do similar calculations. She puts this down to the fact that students are rarely allowed to play with concepts, to discuss them, to obtain other students angles on each concept and therefore they are unable to fully internalise it to the point of being able to apply the concept to even a differently worded or more complicated problem. So it was internalisation I was aiming for. This would take time. And you know what? It did. But it was well worth the time taken. I’m thinking they have now internalised place value and the base 10 system.
The first book I used was this one:
This was a good book, and in general I enjoy all the books in this series. However, I have to admit to being left wanting more. It didn’t really tell the girls anymore than they already knew. In it’s defence it was probably written for students younger than my twins, but for me it was back to the drawing board. I happened upon the following book by pure chance and all my children were spell bound from the very first moment and it was me that constantly had to put the breaks on and refuse to read ‘just one more chapter’. Go and beg, steal or borrow it, you won’t regret it!
And this wonderful book? It is called ‘How to Count Like a Martian’:
This is essentially a book about place value, but it takes it’s reader through the number systems of all the societies, ancient and modern of the world, questioning if each has a place value and what base it is in. I have advanced level maths and can you believe I really didn’t understand what base 10 was. I knew and understood that our system is a decimal system that works on a system of 10’s but I really didn’t know what that meant. A bit like being able to operate a computer but not really understanding how it works, I was able to utilise our base 10 system with ease, but never really understood how it worked. Enter this book.
This is surely a gem, and should be part of every child’s reading list, whether they use maths curriculum or not. I now understand the Babylonians base 60 system and the Mayan base 20 system. When we covered both these civilisations I avoided teaching these in any great depth because I didn’t know what the history books were talking about!
I read this to all three older children and had T11 join in the whole exercise. C10 really struggled all the way along, especially when we reached the abacus method of counting. This was important because without realising it we too use the abacus way of counting and allot the same place value to our numbers. This was a key subject which they had no depth of understanding. It was wonderful to see the confusion on their faces replaced with clarity. I am so enjoying this way of teaching the children because I am learning right beside them. We all had trouble understanding the Mayan way of counting but together we got there…in the end!
The way the book was written reminded me of a song of my youth, ‘Walk like an Egyptian.’ Much to the amusement of my children I kept bursting into (characteristically very out of tune) song, about counting like an Egyptian. We all agreed that a great assignment would be to use the tune of this song and sing about the different methods of counting. Off they went to compose their very own mathematical song. My intention is to collate all the songs, taking the best from each and write a final rendition we could all use as a memory tool. We’re still perfecting them at the moment, but I will share when we are done!
One thing I am making a concerted effort to do, is to get the children to think. I don’t mean necessarily mathematically, I mean think about the reality of what they are learning. In order to do this, I am needing to switch the way I do things. Instead of me teaching, telling and then asking for replication, I am trying to ask one big question which will sum up the purpose of whatever we are learning at the time. My one big question for number systems, place value and base systems? What is a number?
Having read the above book, the children then went about designing their own number systems. But that’s for next weeks post!