Early learners need security, which comes from engagement with the real world, interpreted as concrete materials, then as visual models, then into mental images which can be manipulated with facility and accuracy.
There is a significant difference between being able to count and understanding maths, but sometimes you can be forgiven for thinking that it’s that simple. Visiting grandchildren recently, I was delighted when one of the two year old twins took my hand and articulated the counting numbers and went past ten. I was amazed we went past three to be honest, but she had a pattern and it had been retained and could be repeated, just like a poem or nursery rhyme. We sang lots of those too, many embedding numbers and she and her sister joined in.
This set me to thinking though, about early number acquisition.
There’s the stage of knowing that number and size exist as a descriptor of the world. Things exist as a group, the cardinality of a number; it has some value, which doesn’t change, ie the conservation of number. This links with 1:1 matching. Things can be put in order, the counting numbers, representing one more, or they can be ordered as fist, second, third. They can be compared, more/greater than, less than (fewer than), heavier/ lighter than, bigger/smaller than, taller or longer/shorter than…… Ordinality…Ordering and organising in this way are often a part of play. Real life situations, encountered every day. Maths is everywhere and can form a significant part of everyday conversations. It is certainly a part of the Early Years Foundation Stage and into KS1.
At some stage the child moves beyond 10. This is a point of issue for me, in that for many children, this perpetuates the counting aspects of maths, whereas there is potential for exploration of issues that move towards an understanding of place value.
Put a handful of counting material on a table and ask children to estimate how many there are. Accept all answers, but ask for clarification of their thinking. Now ask them to find out how many there are. Many will just count them up, one by one. It is an interesting exercise to ask the children afterwards how they could prove their answer. Their only strategy is likely to be recounting. However, if some might organise them into groups, showing some order and organisation, they may be in a position to demonstrate that their organisation allows them to count up in bigger units. So 26 may be described as two groups of ten and six units or singles. This, to me is a precursor stage to written modelling, as the image of the two tens and six units can be recorded as an image.
So, I would argue for holding onto appropriate concrete apparatus for as long as necessary for a child to easily access the concepts and to be able to show a facility with thinking mentally. This can develop into a proof after mental activity.
Working with concrete apparatus, modelling and images can underpin all stages of mathematical thinking, with children being allowed to record their findings in ways that are suitable to them. These can be translated into mathematical speak by the teacher or other adult, to demonstrate the links between the images and pure number. This too is an essential stage. It does not always arise by osmosis, apart from a few possibilities.
Numbers do stuff; we have the four “Rules of number”, addition, subtraction, multiplication and division, each of which has a number of synonyms.
Add; put together, add, add together, total, count, count up, figure up, compute, calculate, reckon, tally, enumerate, find the sum of, amount to, come to, run to, number, make, total, equal, be equal to, be equivalent to, count as…
Subtract; take off, take away, minus, reduce, fewer, (difference) decrease, deduct..
So even within supposedly simple experiences there can be vocabulary induced issues, which cannot be ignored, as you can’t determine that children will always experience a maths problem in the way that they have learned to answer it.
Difference, to me, can be taught as addition or subtraction, whereas quite often it is seen as synonymous with subtraction, hence the brackets.
What is the difference between 6 and 14? This problem can be seen from either end. What needs to be added to 6 to make 14, or what needs to be taken from 14 to make 6? It can be made visual with the aid of “towers” of counting material and compared before calculating. Of course the question could be asked differently. How much bigger is 14 than 6 or how much smaller is 6 than 14.
All of these questions are based on the same premise, that 6 and 14 are not the same….
and that’s the challenge for early stage learners; they have to make sense of the language and the embedded concepts in order to be able to think of them for themselves.
So, my early maths recording books would be blank pages, with printed and laminated guide sheets beneath to support organisation. I’d allow children to draw their thinking and get them to talk the maths, recording as needed to interpret, or just making notes as aides-memoire. In the same way as some exams ask children to “show their working”, I’d encourage this approach at all stages. This approach allows insights into the processes, rather than just the right or wrong answer. That can come later, when a child is secure and just needs some practice to prove or improve accuracy.
Check out the Inquiry Maths website, which has a large number of interesting approaches to Maths teaching.