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 - 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!!!

More toddler maths

With this activity Idris explored one to one correspondence and capacity as well as some patterning and a touch of probability! This activity was created to build on an interest that Idris showed in this egg box from the recycling pile. I watched him take it out and experiment with putting other objects inside and trying to close the lid. Initially I gave him a box of coloured cotton spools to put in and out of the box, but they didn't seem to interest him too much. Later on in the day I found a tube of Christmas tree decorative bells and they appealed to him much more! There is no picture of this set of equipment all pretty and ready to go as Idris was far too eager to explore it!

 He spent quite a bit of time putting them one at a time into the sections of the egg box. He always built a square of nine, even when he filled them up in a different order!

 He added some of the cotton spools in after a while. The metal tubs that you can see are 3 sets of differently sized sauce and dipping pots that I picked up in the pound shop. They fitted pretty well into the sections on the egg box and Idris experimented with how many bells would fit into each pot and whether he could stack the pots on top of each other.

 If he was really careful he could balance all ten bells in the largest of the pots. His fine motor control, hand strength and finger dexterity got a great workout with these as he was holding two or three at a time in one hand and rolling them through his fingers to place them in a cup.

 He started an impromptu game of hide the bell and was delighted when we hid one under a cup and moved them all around and he had to guess which one the bell was under! He was very good natured about this age old magician's trick and loved the pantomime of looking under each one and shaking his head no when there was nothing there!

After losing interest in hide the bell he concentrated on stacking different combinations of pots on top of each other. This is an enduring interest of his. He likes to try and stack all sorts of things of all shapes and sizes.

Easy Make-at-home Maths Games

I have been trying out some maths games from the book 'Games For Math' by Peggy Kaye. It fits superbly with my 2012 promise to myself to stop procrastinating being a perfectionist and start doing things right now without over planning and missing the moment! This book fits with that because the games can all be made with close to hand household materials, often simply paper and pens, no photocopying, colouring, laminating etc. You could make them pretty and permanent if you wanted to, but I have chosen to just DO them!!

This game is called 'double it'. You pick two number cards and one instruction card and move your piece around the snail board according to the answer. I tweaked it slightly to have various forms of the same instruction such as add/plus and minus/take-away/subtract. It aims to build knowledge of basic number facts. My girls are confident calculating these, currently using a range of techniques to get the answer, some 'in their head' and some on fingers etc. I thought this would help them as they don't always link the words with the actions. As I predicted, minus caused some confusion!

I have one 'happy maths' child and one more reluctant maths child and I think I made the spiral a bit too long for the reluctant maths child. The game lasted about 20 minutes and she was unhappy for at least ten minutes of that, but cheered up at the end as she won. A shorter spiral (and better counters that didn't keep falling over!) would make this more successful! Like many areas of home educating, I am constantly making little changes to help things run more smoothly!

Toddler Maths

Idris stunned me yesterday by easily building this huge tower!! He confidently placed one unit block on top of another, using his other hand to steady the tower when needed. I helped with a few balance adjustments as it got higher but he wasn't upset when it fell over, he laughed and rolled around on the blocks then built another. I digress....... about halfway up, I handed him a longer block and he put it on and turned for the next one - I handed him the original size and he turned to put it on........ then stopped, frowned, said 'na' and took the long one off, dropped it on the floor, did a happy wiggle then put the same sized one on top and turned for the next. It made me think back to some Penny Tassoni training I did recently where she had put a blue duck with yellow ones and noted that a toddler took it out of the bucket, showing that toddlers take notice of groupings - an early mathematical observation. It got me thinking what else I have seen Idris do that comes under the 'early maths' heading.

Here he is sitting with a metal teapot from his metallic treasure basket and he went straight for the tin that would fit inside it and popped it in and out, opening and shutting the lid - he seems to be very accurate when he chooses things to fir inside something else. He knows what will fit and what won't without using trial and error - showing that he can accurately judge size/capacity at a very young age!

This is him again exploring a similar concept to the one above with the teapot...... this is the treasure tub that came about after my post on him threading beads into plastic bottles - my parents put together a collection of plastic plumbing pipes and different threads of beads, a funnel, a few jars and bottles and a denture pot! This treasure tub is fab as it's in a round plastic bird seed tub and is easy to carry around - it has been to the bathroom many evenings at bath time! It was in the bath that I first noticed that Idris was selecting exactly the right tube to create a 'nest' of tubes with each one getting smaller. I was fascinated!

Idris finds plenty of opportunities to explore maths outdoors as well, as he carries all sorts of objects, heavy and light, and plays with his beloved collection of balls of all shapes and sizes! When you think about it, there are so many ways for toddlers to explore mathematical concepts, have a close look next time your toddlers are exploring things and see what maths they are applying without even knowing it!

Living Maths Curriculum - Symmetry

We kicked off with symmetry as a starter to our 'Living Maths' curriculum. I have chosen to follow this because it includes both of the girls at the same time and can easily be adapted to different depths of understanding. It also really suits Eve's preferred learning style and makes our days fun and interactive. The final bonus is that we can carry on easily even when Idris needs to sit down and feed or cuddle as we can do it gathered around the sofa with lap trays!!

We started with the book 'Let's Fly a Kite' by Stuart J. Murphy. I read the book to the girls and we talked about the pictures. They giggled at the story and identified with the siblings that bicker occasionally! We read the book everyday and did corresponding activities to build their understanding of symmetry. Above is one of the leaf collages that were made after a walk in the park examining which leaves were symmetrical and which weren't.

Above I had the girls group the alphabet bean bags according to whether they had a line of symmetry or not. They found this a little tricky as a fair amount of visualisation was required. After that the girls drew the lines on their own sheet of letters - I introduced the idea of multiple lines of symmetry as we did this.

Here the girls enjoyed making symmetrical designs on butterflies. I did a quick demonstration about how to make the designs symmetrical before they started. Not all of the designs were entirely successful but they provided a good starter for discussing how they were or weren't symmetrical. I think that it deepened their understanding of the concept and definitely re-enforced the vocabulary!

 This was our first activity of the week - we used the Montessori insets for design and traced then cut them out before folding them along a line of symmetry. The girls found this easy to do. An extension to this would be exploring the multiple lines of symmetry that exist in the shapes. I plan to cycle back round to symmetry when we have covered the other topics and increase the level of difficulty next time round!

I built my Living Maths book list using Mama Jenn's living math book list as a spring board. Once I started searching, Amazon threw up many more options for me to choose from and I was able to select books that are really lovely picture books in addition to being useful maths starters (Get it? In addition to? he he!!). I am choosing a mixture of outdoor or physical activities, games, mental arithmetic and worksheet or art based activities to re-enforce the concept without being samey and boring!

Exploring Fractions

We began chatting about fractions the other day when Eve was cooking with me. Her task was to chop each peeled potato into four before popping it in the pan. It provided a great opportunity to explore wholes, halves, quarters and eighths. I was pleasantly surprised (as I often am) about how competent Eve was with fractions after not having 'done' any for a while. We talked through the numerator and the denominator and how we know which number to put where. She wrote them out on some scrap paper from the recycle bag as we went! We touched on equivalents as well a little.

This morning I pulled out these fractions cards that I printed and laminated ages ago and Eve played around with them with her Dad, matching equivalents and putting the correct written fraction with the correct image. My only criticism of these cards is that they don't have all the section lines drawn in, for example one third has one third in green then two thirds in white with no dividing line. However, it is nothing a sharpie marker can't fix so not a problem for long!!

I think I will continue the fraction theme this week as Eve seems keen to continue. I have a few more resources ready to use:

• Fraction wall to rebuild.
• Half, quarter and third worksheets.
• Colourful fractions worksheet.
• Fraction pizzas.

I found a few more sites with plenty of resources and worksheets to choose from:

• Teaching Ideas.
• Super Teacher Worksheets.
• Project Happy Child.

That should keep you all busy!

Exploring Shapes with Pattern Blocks

I saw this idea being used on The Homeschool Den, and the printables come from PreKinders. Faith has never been very interested in the pattern blocks but she took a liking to the patterns for animals that I printed out.

Here Eve is combining shapes to build triangles. Faith later worked on this one and found lots of very inventive ways to combine the shapes and it was a learning curve for her. Whereas Eve intuitively knew which shapes would fit, Faith adopted trial and error to find out.

We learned some shape vocabulary along the way..... trapezium and parallelogram (or palallemogram of you are Faith!) were the new ones for the girls.

Tackling maths planning.

Well, as is becoming typical of me, after looking at several options for a maths curriculum, none seemed to fit my daughter better than the Montessori method that I am training in, but I took a look at what the National Curriculum expects of children in Year One. It made interesting reading because I had two versions to compare; the original curriculum from 1998 and the revised edition from 2008.  To my surprise, two thirds of what used to be expected of Year 2(6-7) children has been added to the Year 1(5-6yrs) syllabus!  It makes for a very long list and here it is..... my intention is to link back to this list as the year goes by so that in the future this post will contain links to 'how to' posts and resources lists for you to use.  I will be using my Montessori files, making my own resources and hopefully making use of some offerings from the Internet too, although we don't do a lot of paper based learning at the present time!

Be familiar with numbers 11-20
Extend to counting to 100 and beyond
Read an write numbers 1-20 then 100+
Position numbers 0-20 in sequence
Order one and two-digit numbers on a number line and square.
recognise place value (0 as the place holder)

Recognise sequences of odd and even numbers to 30+
Know patterns of multiples 2, 5 and 10.
Number patterns (+ and -), predicting the next number.

Doubling and halving numbers as an inverse relationship
Know number bonds to ten
know two and ten times table.
Know doubles of numbers to ten
Know halves of even numbers to twenty

Addition and Subtraction
+ done in any order
- can mean 'take away' or 'the difference between'
- the inverse of +
use = sign

Record calculations in number sentences
Solve 'missing number' problems
Understand x as repeated +
Mental methods for working out ten plus a single digit
+/- multiples of ten within two digit numbers
Choose a calculation method to solve whole number problems (money, measures etc)

Use lists/tables/charts to sort and classify information
Discuss it and explain results.

Describe mathematical features of 2D and 3D shapes (triangles/rectangles/circles/cubes/cuboids/hexagons/pentagons/cylinders/pyramids/cones/spheres)
Create 2D and 3D shapes

Reflective symmetry in 2D shapes and patterns
Recognise translations and rotations.
Recognise right angles
Understand angle as a measure of turn using whole turns, half turns and quarter turns.

Observe, visualise and describe positions, directions and movements using common words.

Estimate size of objects and order them by comparison.
Put events in chronological order
Compare and measure objects using standard and non-standard units of measurement.
Compare durations of events using standard measurements of time.
Estimate, measure and weigh objects.