by Chris Clayton (September 2020)
I was reminded of the early foundations in the teaching of mathematics when I attended a seminar on subitising for early years. At the seminar, the book Making Numbers: Using manipulatives to teach arithmetic, by Griffiths, Back & Gifford (2016) was recommended. While the book is aimed at EYFS and KS1, as I started to read it I realised the impact it could have on mathematics teachers at KS2 and KS3.
It will certainly have an impact on my own work.
The book makes reference to research and children’s understanding and offers four key principles. As I absorbed the authors’ key points it struck me that later in a child’s school career, we often forget the basis upon which we are building.
Key principle 1: Be explicit about the different types of number
Cardinal (for example, recognising sets of three, etc)
Ordinal (numbering in terms of order)
Numeral (symbols for number)
In Mathematics Explained for Primary Teachers (2014, p3), Derek Haylock writes about the importance of all teachers really understanding the mathematics they teach and being able to explain it clearly to the children they teach. Following on from this, I believe that we all need to understand the foundations of number and build on them in each lesson.
The question I am asking myself is: do I teach and reinforce children’s understanding of the difference between numerals and numbers as well as the cardinal and ordinal aspects of number in an explicit way to ensure a clear understanding?
Key principle 2: Use the child’s visual memory to support the key learning as well as the learning needed to develop new skills
In 2018, Dr Sue Gifford from the University of Roehampton wrote: “Young children also have powerful visual memories and some may find it easier to remember images than words”.
I see the importance of this in building on what the child already knows, especially in this time of closing the gap. We all need our memory refreshed.
Key principle 3: The importance of subitising
According to Claire Gerrard, there are two types of subitising:
Perceptual Subitising: being able to identify how many are in a group, without counting. This is linked to cardinality.
Conceptual Subitising: being able to see numbers within numbers. (Eg 6 is made of 4 and 2). This is sometimes referred to as ‘composition’.
Perceptual subitising helps to develop understanding of cardinality and quantity. Conceptual subitising could underpin a child’s first steps in algebra.
Research is consistent in regard to the fact that subitising may provide the basis for understanding counting principles, such as cardinality, as well as arithmetic ideas. It is important to reinforce this and to start from the principle of what is already known.
Key principle 4: Teach without targets
Ask the children to go A.P.E.:
As well as using important mathematical skills, this could also help ensure that the child does not develop misconceptions.
Give every child a cup of tea with every maths lesson. Imagine that the cup needs to be filed with care, skills, security and attention. Some children will have a full mathematical cup most of the time or will know how to refill it. Children get nervous when they do not understand, so we need to help children to fill their cup! Part of this is showing them how to apply and understand.
I am challenging myself to make this more explicit and build on what the children already know. This year I will be reminding the young people in my class what they have learnt and will support them to recall and apply.