I’ve been doing research this week on mathematical development in the Early Years, which has produced three very interesting revelations for me and how I’ve always perceived mathematical development.

First of all, let me say that I was a little sad when the EYFS moved away from ‘Problem Solving, reasoning and numeracy (PSRN)’ back to ‘mathematical development’ as a descriptor for this area of learning and development. PSRN really explained that this area is not just about numbers, but how we use maths, its benefits and how children learn about maths.

Let me give you an example. Probability is the mathematical term of how likely something will occur. If you have 10 balls, nine yellow and one pink, and you randomly choose a ball it is more likely (more probable) to be yellow than pink. As adults, this is fairly intuitive – obvious even.

However – it turns out that babies as young as six months old show surprise when there is an improbable colour ball drawn out – i.e. if you drew out the only pink ball in the box in the example here (Denison et al., 2013).

The inference is that babies start to reason mathematically about their world at around six months old – they are logical.

My first revelation was that I was simply not aware that babies were so capable. I will certainly be using this information to challenge and stretch the babies in the baby room. Maybe a modified Kim’s game, where a toy is swapped for another one under the cloth to see if the babies can spot which one has changed or reproducing the experiment and drawing coloured balls from a transparent box.

My second revelation was that maths and mathematical thinking should be a creative process.

I have always regarded maths as a ‘science’ subject, diametrically opposed to ‘arts’ and creativity. However, I have now read a number of articles that have changed how I think about children’s creativity and their representations of maths (for example, Worthington, 2006).

For example, if you didn’t know how to write a fraction, how would you represent a half?

Worthington gives a great example where ‘Catherine’ represents her sister’s age of two and a half by drawing the numeral two and then adding half of the numeral two. I think this is exceptionally clever – and also made me wonder why we do represent a half like we do, half a numeral seems to make so much more sense!

This use of mathematical graphics can give a real insight into children’s creative thinking and understanding of the world. I think I have sometimes been a bit quick to encourage children to write numerals ‘correctly’ rather than investigating why they have chosen to draw them as they have. Similarly, that some marks may not be emergent writing, but representations of emergent maths. This will certainly make me more curious when discussing children’s drawings.

My third revelation this week has been about mathematical language.

I would like to think that I have always encouraged children to think mathematically, including language such as smaller, bigger, longer, heavier, slower. However, children will only learn specific, meaningful mathematical terms if they hear them in the correct context (Hutchin, 2012).

For example, not only asking “I wonder how much this jug will hold?” but including “I wonder what its capacity is?” or the “symmetry” of a pattern or the “orientation” of a shape. There is every reason to think that children will be able to incorporate such language into their vocabulary. After all, if they know there is a dinosaur called “Tyrannosaurus Rex”, the “capacity” of a jug is pretty tame!

I hope that further research will give me pause for thought, but meanwhile I will be:

1. Giving babies more credit for their advanced mathematical skills and reasoning

2. Investigating mathematical graphics more closely

3. Watching my use of mathematical language

**References**

Denison, S., Reed, C. and Xu, F. (2013) The emergence of probabilistic reasoning in very young infants: Evidence from 4.5- and 6-month-olds *Developmental Psychology*, Vol. 49 Iss 2, pp. 243-249

Worthington, M. (2006) *Creativity meets maths* Practical Pre-school

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Another interesting book about mathematical graphics is Carruthers and Worthingon (2006) “Children’s Mathematics, Making Marks, Making Meaning”. I’ve also found Sue Gifford’s book “Teaching Mathematics 3-5” very helpful.

Here in East Lothian and Edinburgh, we have a big focus on Stages of Early Arithmetical Learning (SEAL). “Teaching Number, Advancing children’s skills and strategies” is good to understand the development, but nowhere near playbased enough for me.

You might find the East Lothian work on the Maths Pyramid for Babies and Children interesting if you’ve not come across it before. http://www.edubuzz.org/curriculumforexcellence/planning-support/numeracy-support/staff-support/maths-pyramid-for-babies-and-young-children/

The link to the parent leaflet on that page is also useful.

I love that there are so many things that need to be in place before we even start with counting and number. Movement, problem solving, spatial understanding, language – those are the foundations of mathematical understanding.

I do lots of training for practitioners and teachers in developing maths and numeracy through play. I’m looking forward to reading your book when it’s finished.

Hi Catriona,

Many thanks indeed for all the information. I’ve found Carruthers and Worthington’s research to be very interesting and incredibly useful! My focus at the minute is on the under-threes, but I will have to have a look at Sue Gifford’s book to see what is in there too.

The maths pyramid is excellent, thank you.

Best regards

Kathy

Hello,

And thank you for an interesting article. Maths have not been my favourite subject (the only thing I *love* are equations, only because you have to discover something!) but I had to befriend maths since my son started to show some interest on the subject at a very early age. What I did was to introduce him to maths with daily life occurrences such as cooking, going to the shops and piking a certain amount of bananas , say. Music plays a very important part together with repetition and rhythm. So far he is quite capable of collecting a certain amount of objects in a conscious manner (not just repeating the numbers) and tap with his fingers a simple melody.

Hi,

Many thanks indeed for your comments.

You have some great ideas for practical activities! There’s long been a link between music and maths, for just the reasons you describe.

Thank you for sharing,

Kathy

I think it is really important to teach the vocabulary to children from an early age. I am trying to introduce mathematical songs to children of concepts that they might be aware of but haven’t been taught. David Godfrey has some great songs that cover the whole of primary including foundation. My 2 year old will be learning some of them soon too.

https://numberfun.co.uk/

Hi Josef,

Many thanks for your comment. I’ve not come across David Godfrey before, so that’s one for me to research! His site does look fun, and I love the idea of number ‘sense’ rather than rote repetition.

Best regards

Kathy

I am a Registered Childminder with a BSc in Mathematics so this subject is close to my heart. Using mathematical vocabulary from an early age must be key, something I have picked up from a Maths teacher husband who follows Jo Boaler’s research and teaching. I believe if a child knows the vocabulary involved before being taught a concept, they will understand the concept much more quickly. A wide vocabulary and also knowing that it is alright to make mistakes (don’t we often remember things we got wrong first time better?). Encourage creative thinking rather than just reciting numbers.

Hi Laura,

Many thanks for your comment. I only know Jo Boaler’s work from “The Elephant in the classroom”.

Absolutely agree that creative thinking is the basis, as well as being secure enough to make mistakes (learning opportunities).

Just out of interest, did your Maths degree include abstract concepts, where a number didn’t even appear?

Best regards

Kathy

Hi Kathy, I spent the first 5 years of this century teaching yr 1 and 2 and when I finally moved back to teaching reception in 2005 I was keen to use correct mathematically language as this was something many 7 year old struggled with. Much of my maths teaching took place outside using large scale games the children helped devise. One favourite used Community Playthings large hollow blocks and a large dice with the aim of building tall towers. The language always came back to ‘what is the ‘difference’ between the towers, who has ‘more’, who has ‘fewer’, how many etc. I also encouraged children to use their drawing skills to record and explain to others their mathematical thinking, as 7 yr olds who were able to draw any maths problems were so much better at understanding and solving them. In my role as an advisor I have sharedthe National Strategies reception maths vocabulary list with pre-school practitioners to encourage awareness and give them confidence in the language that they should be using.

Hi Jane,

Many thanks for your comments. What an interesting perspective, from year 2 back.

Mathematical graphics is a whole revelation for me, and how interesting that you can confirm that children who can draw maths problems are better at understanding them. Fascinating stuff.

I have also returned to the National Strategies documents. To be honest, I use their documentation in many areas because it is so detailed and practical. The maths one is great, especially as there is so little in the way of literature available about early maths, compared to say literacy or language.

Great to hear that you are able to cascade this knowledge to others as well!

Best regards

Kathy

Dear Kathy, reading your examples of mathematical thinking and language has brought back many memories of what happened at my nursery! Children instinctively learn maths when the learning is in context as you rightly said. There is so much learning in food when eating for instance, a piece of bread. I used to cut these in different shapes on different days. I Encouraged children to create shapes using food items and cooking. Sometimes symmetry occurred. Baking sessions are so language rich in mathematics and children enjoy taking part and saying if the bag of flour is heavy or not, how the egg feels in their hand and how many more spaces we needed to fill in the muffin tray. These were 2-3years olds.

Creating activities that are meaningful truly helps children with mathematical thinking.

Hi Carmen,

Many thanks for your comments!

Wow, yes, cooking, tons of opportunities for mathematical development and understanding size vs weight, volume, all sorts – and, of course, great motivation for getting it right for the sake of a good tasting muffin. Thanks for mentioning this, I’m going to have to do some more research on this too 🙂

Best regards

Kathy

Hello

I find this so interesting for a lot of practitioners mathematics is feared and always low on the tracking scores I try and encourage mathematics in every area as it is all around us. From mathematical language to playing in the water or even digging outside you can incorporate mathematics. I look forward to reading more.

Hi Elizabeth, many thanks for your comment.

It is a very pertinent point that a lot of practitioners lack confidence with maths – it’s often the smallest section of observations! And absolutely agree – maths is everywhere!

Best regards

Kathy