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!
Thank you Claire… I can say two things with great certainty on this beautiful Tuesday morning: 1) I have no clue what you are saying and hence I should perhaps invest in that Martian Math book and 2) I shall spend the rest of this day hearing The Bangles go on about Walking like an Egyptian… in my head. Thank you. 😉
You should think yourself lucky. In my house the children are singing it ALL DAY LONG singing about counting like a Babylonian, counting like an Egyptian, counting like a Mayan……you get the picture. And they don’t know the tune!
That is hysterical! 😉 Right you are there… considering myself VERY lucky. The Egyptians were more than enough… Bu most certainly an excellent example of multi-sensory learning. Now you just need to get them to dress up at the same time!
I’ll be singing Walk Like an Egyptian too! 😀
Looks like a great book – I’ll be getting my hands on a copy – thanks!
I was listening to a book about the history of the internet yesterday. It was talking about binary and I was thinking how I’d like to get a better understanding of differently based number systems to help my kids understand them – this will be especially relevant to my computer-obsessed son.
This book includes binary! It is such a great book!
Sounds like a really good book, but it seems to be going for astronomical prices at the moment. 🙁 Good to see that the children are understanding place value in an indepth way. 🙂
Strange that I’ve been assigned a different ID without my knowledge…
I just checked on Amazon. Oh my goodness!!! I didn’t pay that! I only bought mine a month or so ago. I’m going onto my amazon account to see how much it cost. I’ll let you know.
It only cost me £10.20. They’ve gone up a bit haven’t they?!
I would love to get a copy of the latter book but agree with comment above Very expensive on Amazon at the moment! Where did you get your copy and if you don’t mind me asking, how much did you pay? Thanks
I just checked on my Amazon account and I bought the book about three weeks ago for £10.20 included p&p. Believe me, the book is good but I wouldn’t have spent a couple of hundred pounds on it!!
Yes I don’t know why it should have shot up in price so much. Will have to keep hunting alternative sources.
Pea struggles at math, and not being a homeschooler, it is hard to find the time or convince her to do more schoolwork after school hours on top of the large pile of math homework she already has. But the more I read these posts, I’ve come to realize that maybe she and I need to go back to the early math as well, and hopefully that will translate to better understanding of what she is doing at school… I will be using these posts for help!
That’s kind of what I’m hoping too. They have been through all the work meant for their age group and there is a lot they can do, but there is also something missing which is hindering them more and more as they get older. So we’re just trying to find out what it is. I do wish I’d taught like this from the start though, because I can really see how much better it is for them.
Your comment above made me laugh…I can only imagine three different children singing that song with different words and their own melodies! That’s worth video taping, ha ha. Thank you for the book recommendation, it is going on our must-read list.
I hope you can find one at a decent price, they seem to have gone up a bit since I bought mine just a few short weeks ago…..
Thank you-that is really useful. What age do you think is best for the Sir Cumference books? I was wondering about them for my almost 7 year old.
You know, I think it probably depends on the child. Language wise I would say six or seven upwards. Mathematically probably depends on the child. This one is a simple illustration of organising numbers of people in a logical way. My five year old would understand. The books on circumference and ratios and pythag, she would probably struggle with mathematically but still enjoy the essence of the story. HTH
As I read this I was thinking about the Stuart J Murphy Math Start books, you’d probably love them, though they’re intended for younger than your girls are.
When I was in college I took math for elementary teachers, and my professor had us all learn how to add, subtract, multiply, and divide in “alphabitian” (base 4), it was really fun to me.
One of the exercises I had the children do was utilise a shape number system I did in base 4. I wanted to check they fully understood. You’re right, it’s fun!
I agree with everything you have said. I love this series of brilliant posts. Have you seen my posts about playing with other bases? http://homeschooljournal-bergblog.blogspot.com/2012/04/place-value-introduction-counting-game.html
Thanks for the reference to your post. I enjoyed it. You are very creative!