An Amazing Mathematical Story!

I've done a lot of research and reading over the summer about maths as I wanted to find a definite way forwards for the girls to suit them and their strengths. Eve is dyscalculic and struggles with the rote learning and formulaic side of maths but excels at the visual-spatial side of things and Faith is a whizz at formulae and mental maths but isn't so confident applying it logically to solve problems.... I have been plucking up the courage to dive in with some problem solving and today just so happened to be the right time!

I was unpacking a package of maths cubes and the girls began playing.......Eve made a hexagon and Faith made a square and I asked them to predict how many pieces they would need to make a second layer around the first one.... Eve had used six and predicted twelve for the second round... a good guess and based on sensible reasoning! She in fact used 18, and then thought that the third round would use 24, it in fact used 30 pieces to Eve's surprise!

I introduced the idea of recording what she was investigating on paper and gave her a notebook that I'd got ready a few weeks ago for just this purpose!To my amazement, she began to write..... and wrote and wrote and wrote, not only did she write about it, she turned her information into a maths mystery story for someone else to work out! For a dyslexic and dyscalculic child to jump enthusiastically into the two things she struggles with most was just magical to watch!

Halfway through I stopped her and we chatted about the emerging pattern in her investigation, she needed a little prompting to see it as she got cross with me for asking her things that she didn't know.... this is her stock response to many new situations in maths, she panics, shuts down mentally and gets upset, refusing to allow herself to think, automatically assuming that she won't know the answer! With a little patience and coaxing she talked me through it, telling me that the gap between 6 and 18 was 'two sixes' and between 18 and 30 was also 'two sixes' and that she thought that the gap between the next two would also be 'two sixes and therefore the next layer would need 42 pieces to complete.

She had got a bit tangled up in the idea of it being linked to the six times table but it was useful to be able to link it to the number of sides that a hexagon has. Faith had found a pattern in her squares linked to the four times table and I helped them to see that there were four sides to the square and there was a link there also. I'm looking forwards to suggesting that the girls investigate a triangle in the same way, and seeing if they pick up that it has three sides and that there might be a pattern to look for linked to threes.......

What IS Living Maths and How Can I Use it?

Living maths is attributed to Charlotte Mason as she referred often to 'living' books. She actually termed it the 'living teaching' of maths and it has morphed into living maths along the way. I'm really interested in it as a concept for teaching maths at home as the other CM aspects that I've adopted have worked so beautifully well. Recently I've been researching quite a bit about living maths, what that means and how to go about it and this post is a round-up of the information that I've found out and some thoughts about that!

How I came to be here.......
I've been dabbling in it for the last year and have gone on a bit of a journey.... I started out buying a stack of maths story books that I found from the living maths book list written by Mama Jenn that tallied with the maths skills outlined in the National Curriculum. I put together a few activities for each book and we read it every day and did a different activity that illustrated the maths concept in the book. The girls really enjoyed it and it was very effective for concepts such as symmetry and even place value but when I came up against multiplication, I got stuck!! That prompted me to buy a book for teachers to explain to ME what children needed to learn about maths and how they best understood it. That was a good move and I felt much more confident but it lead me on to realising that I couldn't 'teach' some of the concepts in one session and expect it to sink in. Some skills had to be practised over and over like reading does and that meant daily maths........

............which brought me back full circle to where I'd started, wondering how to do that without resorting to a dry boring workbook based curriculum that could completely turn the girls off maths. My age-old curriculum issues arose again as I looked around - they jumped here, there and everywhere - in the name of daily practise but without any seeming logic or continuity, they didn't have explanations, and they all went about it in a different order. Some used manipulatives which I was keen on but didn't fit with my philosophy on how I want the girls to experience learning. Most of them relied on the memorisation and retention of 'maths facts', Eve finds this kind of rote learning nigh on impossible which rules that route out!

I came to a few conclusions about my thinking on maths and where we are now:

• We can't 'do it all' just like we can't with science and history either but I also don't want there to be glaring gaps that will hamper later maths for the girls.
• The girls have very good practical maths skills using time, weighing and measuring, handling money etc.
• They learn by themselves - I know this as they come running in to tell me that 3x4 is 12 or that 70-20 is 50..... but they aren't sure of the terminology and what written symbols mean.
• The living maths that we have done so far has sunk in and been retained and used, most notably place value.
• I believe that expertise builds up and stems from truly understanding the subject and seeing it's links to other aspects of life and how it can be used.
• I want the security of feeling like we have a logical sequence and I'm not just attacking random subject matter and from our literacy approach I know that little and often builds nicely into confidence and understanding.
• I want the flexibility to scoot ahead or to stop and expand on things that confuse or enthuse the girls.

What I found out about Living Maths........

With all this in mind I launched into my research determined to find the perfect fit once and for all! Some useful resources that I found on the way were the Living Maths Yahoo group, Living Learning Lists from the blog 'everyday snapshots', a fantastic article on squidoo entitled Charlotte Mason on Math, another by the same author called Transitioning to Living Maths and a vast amount of thought provoking information here. Because 'living books' has become a term synonymous with a Charlotte Mason style education, it seems natural to assume that 'living maths' means that maths is taught purely from books alone. Having dabbled in this method myself I am strongly questioning whether it is enough as after all, she didn't advocate teaching children the mechanics of reading by simply reading to them, although very very important, so how can we teach the mechanics of maths by reading about it to them? There must be more to it!

Charlotte Mason didn't use the term 'Living Maths', rather she referred to the 'Living Teaching' of maths. This suggests to me that she believed that maths should be brought alive for the children, and she didn't rule out any particular ways of doing that. Her recommendations followed a pattern of using manipulative materials followed by visualising those materials and methods with mental arithmetic and then moving onto written maths problems once mental arithmetic was secure. Problems should challenge the child without overwhelming them, this is important when doing maths from a CM perspective as she placed importance on accuracy and getting things right first time without retrying it, it will be up to me to ensure that I pitch things at the right level for them. Charlotte Mason believed in a solid foundation of the basics, she disapproved of a child bashing through sums and problems without a true grasp of what they were trying to do. My favourite principle of a CM maths program is that you shouldn't 'overteach', she didn't want parents to get in the way of a child's learning, this is fantastic because it gives me confidence that a well chosen pathway and well-chosen materials will do the job nicely without me having to battle my own maths insecurities to facilitate it! 

How I am approaching Living Maths.....

I found that Charlotte Mason's approach to maths wasn't that far removed from what I thought it would be and how I am planning on approaching it already. I have chosen a four fold approach this year influenced by where the girls are in their maths development, what has worked previously and what I think I can manage with a newborn and a toddler!:

1. I have chosen The Dyscalculia Toolkit by Ronit Bird as our main spine. It is designed for children aged 7-14 and is aimed at supporting children with learning difficulties in maths..... HOWEVER I think it is a brilliant scheme for all children and it fits really well with Charlotte Mason's explanation of the living teaching of maths as it focuses on maths facts but without a heavy reliance on memorisation - it shows logical ways to get there by reasoning rather than rote learning. It also places an emphasis on understanding backed up with concrete experience of using manipulatives before moving on to mental working out and then written problems. From my personal wish list, it ticks the boxes of providing a logical sequence of small incremental steps that can be tackled at a rate that suits my children. It isn't designed to be a curriculum, but rather a 'pick and choose' selection of activities and games but as I want to ensure that the basics of addition, subtraction, multiplication and division are secure by the end of this year, I am going to use it as a curriculum, set-up in much the same way as I use The Reading Reflex. I will post more about how I'm doing that another time! 
2. I have purchased the first book in the Life of Fred series and will be trying this out with the girls, I've read a range of reviews on it but think we need to try it out to see how it suits us. It will fit nicely into our book basket but I'm unsure of whether the 'jump about' nature of the stories will be sufficient as a standalone curriculum in the long term, although it does follow it's own sequential course building up to high school level. The tone of the stories isn't my cup of tea and there doesn't seem to be much material to allow practise and repetition which over the next year might not suit ME as I don't want to be having to prep extra things to go along with it!!
3. The third and final aspect that I'm really looking forwards to from a personal interest point of view are living books on maths. I was inspired by this living math through history curriculum and very nearly bought it if it wasn't for a review that praised it highly but said it was pretty much a series of book lists and you needed to do many hours of prep and organisation yourself to utilise it. As they are kind enough to make their book lists freely available I chose instead to trawl through them and choose some interesting books that will tie maths to the history and science that we are doing this year, making it 'live' in terms of linking all the subjects together as a whole, which was another of my personal requirements. I am particularly looking forwards to these as my maths experience in school was limited to formulae and processes with very little enjoyment! I'm going to intersperse these with some of the living maths books that we already own, utilising the method we've found successful before of tying practical activities to them to teach concepts such as time, fractions, measurement and more.
4. Be mindful of taking advantage of opportunities to measure, weigh, tell the time, use calendars etc on an informal, real-life basis

I haven't worked out yet how I will fit these different aspects in and there is much more to say on the subject of living maths books and the Dyscalculia resource book, but they will have to wait for another post or two, especially as I am so spoiled for choice with the book lists that I can't decide what to get! There are things I would like to bring into our maths that I simply don't have the time and capacity to achieve at the moment. I keep reminding myself that I can't do it all and neither do I have to..... we can try new things another year and I think what we have for this year is rich and varied enough! In the meantime, you can browse through my Charlotte Mason Pinterest board and my Maths Musings Pinterest board for more ideas and inspiration on teaching maths in a CM way!

Living Maths Curriculum - Multiplication

I am so glad that I bought the Mathematics explained book to help me understand how children learn maths and what they need to know to help them make connections between ideas! I had a simplistic view of multiplication before I began reading but a much better understanding now!! I used the information I read to put together some activities and ideas to bring multiplication to life for the girls and make it understandable, non-threatening and fun! Here goes the theory bit; 

If children know a range of ways to tackle multiplication increases the ability of a child to handle multiplication problems. There are two ways to approach it:

• Repeated aggregation structure - language used is "so many sets of/lots of" and "How many altogether". An important use of aggregation is in shopping, for example buying 8 tins of tomatoes at 31p each, how much does it cost altogether?
• Scaling Structure - increasing a quantity by a scale factor, language used is "doubling", "trebling", "so many times bigger/longer than", "so many times as much as/as many as". Comparisons are important in scaling operations, for example I paid 50p for my drink, Daddy paid three times as much, how much did he pay?

Commutative Law can be used to make problems easier if children understand it. The law says that a x b is the same as b x a so in practise, 5 x 14 is the same as 14 x 5. As children are usually more able to handle multiples of 5 than multiples of 14, working out 14 lots of five is easier for them than the daunting prospect of five lots of fourteen.

Many children find it helpful to hand ideas on an image to help with remembering and understanding them. Multiplication can be associated with the 'rectangular array' which means that 3 x 4 can be represented visually on graph paper as three squares up and four squares along. This leads naturally on to using multiplication to find out the area of a rectangle in the future!

Children need to have concrete, hands on working of multiplication secure before they move onto mental arithmetic techniques.

With all this in mind I put together a few ideas to try alongside our living maths book called 'Anno's Mysterious Multiplying Jar'

We played a game from Peggy Kaye's book 'Games for Maths' which comes under the 'repeated aggregation' format of multiplying. This game was popular, and quick using scrap paper, pens and a dice. You roll it once and draw that many circles then roll it again and draw that number of dots inside each circle. The player with the highest number of dots wins.

This game was also from her book and was useful for introducing the 'commutative law' idea as the dice roll this time gave two side of a rectangle which was completed using the squared paper then the area was calculated by multiplying the sides. It showed the commutative law idea as I pointed out to Eve that is was easier to count in two's which she knew confidently, than struggle with counting in fours and get it wrong!

 The skip counting in this lead to us making a skip counting bracelet with flower beads and pipe cleaners for practising counting in threes. Having the bracelet on prompted them to count in threes quite often and progress was made in learning them. We also used colouring sheets from 'The Multiplication Tables Colouring Book' which the girls enjoyed as colouring the numbers in a pattern that were multiples of two for example, revealed a picture of something, such as a mouse. There is a multiplication table which asks them to colour in the multiples of two to reveal the pattern of numbers that are multiples of that number. This was the first time they had come across the idea of 'multiples of' and the language associated with that and it caused a false start or two but was generally successful. 

It was in reading about multiplication and researching activities to support this book that I realised just what a huge topic multiplication is and how it needs to be tackled regularly in small chunks rather than one dose, like the symmetry could be for instance. The girls enjoy multiplying and often come and tell me little snippets such as 'what four threes are' and so on. The idea of using cuisenaire rods popped up several times and that led me to another maths book which has added in a new dimension to our maths, it has also derailed my 'living maths' planning for a while as we swerved to one side to accommodate a new direction. I have got my head around it now though so will hopefully be back to posting a few more living maths explorations!!

Living Maths Curriculum - Place Value

I thought I knew what place value represented but I was interested to read about our number system's origins in Mathematics Explained for Primary Teachers. A summary of what Place Value represents is that the place in which a digit is written represents the value of a power of ten. So for example, in the number 3487, the 7 represents 7 ones, the 8 represents 8 tens, the 4 represents 4 hundreds and the 3 represents 4 thousands. Zero is a place holder, because it holds the place when there are none of that value there. So for 1001, there are two place holding zeros as there is 1 unit, no tens, no hundreds and 1 thousand.

 

I like to understand the links between the areas of maths and to do that it is useful to look at place value in terms of 'powers of ten'.

• Ten is 10 to the power of one or '10'
• One hundred is 10 to the power of two or '10x10'
• One thousand is ten to the power of three or '10x10x10'

From that you can see that each of the powers of ten is worth ten times that of the value less than it, so ten tens are one hundred, ten lots of one hundred are 1000 and so on. The reason that this is important to know is that it helps with calculations as once you have nine of a certain power, when you add one more, and you have ten of them, they are exchanged for one in the next value up. Confusingly for children who read from left to right, this is worked from right to left, so with nine units, if one more is added then it is exchanged for one ten, in the place to the left of the units. Ten tens will be exchanged for one hundred, again to the left of the tens column in place value. It is the principal on which carrying one over in addition or doing subtraction by decomposition is based.

 

Although some of these are higher level than your child will begin with, it is useful to understand it fully as an adult guiding a child through the ideas, so that you can answer any questions your child might have, and so you are fluent in it yourself!

 

I based the place value exploration on the book 'A Place for Zero, A Math Adventure'. The girls enjoyed the story and I thought that it simply explained the concepts behind Place Value. We based our activities around the Montessori golden bead material that I have. A base ten set is similar, and you can print out sets of number cards from units up to thousands from here. Another effective way of explaining place value with concrete materials is with money - £1 coins represent the hundreds, ten pences represent the tens and one pences represent the units.

 



I downloaded  Place Value worksheets one and two from the teaching ideas website, purely to use as a guide for myself, rather than as a worksheet for the girls. I wrote large numbers out in pen and asked them which number represented hundreds, which was tens, which was units and so on and got them to circle the correct number. I asked them to write down three hundred and eight, or two hundred and ninety seven and so on. The next day we did the same but in thousands.

The way we usually work our living maths is to read the book together, chat about it as we go or afterwards, then do a related activity that hopefully builds on their understanding. They really enjoy doing maths this way and I always get a cheerful 'yeah' when I suggest doing it which is so nice! The third day I introduced the Montessori golden bead material (base ten printables here to use - scroll down to find them, but they aren't as good as the real thing but better than putting it off for ever until you can get the real deal!).

 

The idea behind the Montessori materials is that you choose a number card from each of the 'powers of ten' so a thousands, a hundreds, a tens and a units, I gave the girls the individual numbers in a tray and they went across the room to the 'bank' which holds all of the golden bead material. They counted out the blocks required and stacked them into the tray.......

.... then came back and assembled them on the place value mat as shown and put the number cards together correctly before telling me the number that they had.

They enjoyed making each other fetch larger and larger numbers from the bank - this number required several trips to and fro!!!