Doug Harmer delves into the historical archives to reveal a contemporary rationale to avoid your maths teaching in the early years of a child’s education being merely an exercise in memory retention.
I would like to share an aspect of subject knowledge in maths that has had a profound effect on both my own teaching and my involvement in the professional development of others. As part of my research for my Masters in Mathematics Education I explored the ‘Origin of Numbers’. At the time (fairly recently), its relevance to the teaching and learning of mathematics, especially in the Early Years and Key Stage One was not as palpable for me as is it now in my role as a Teaching and Learning Adviser. Providing opportunities for Continuing Professional Development (CPD) and working with Senior Leadership Teams, Middle Leaders, teaching staff and parents/carers has bought the ‘Origin of Numbers’ to the forefront of my thinking when considering effective pedagogy in maths. This is grounded on the notion of number and its definition. The online Oxford dictionary describes it as:
‘Noun - An arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations.’
When considering this definition the notion of ‘numeral’ has particular significance. In the same dictionary it is described as:
‘Noun’ 1. A figure, symbol, or group of figures or symbols denoting a number. 1.1 A word expressing a number.’
The key phrase highlighting the difference between the terms is ‘arithmetical value’. In the historical process of creating ‘Numbers’, the connection to ‘Numerals’ was explicit and unambiguous. Our current number system is based on Hindu-Arabic numerals which is a set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the decimal number system. They originated in India in the 6th or 7th century and were introduced to Europe around the 12th century. They entered common use from the 15th century to replace Roman numerals. The significant difference between Roman numerals and Hindu-Arabic numerals was the inclusion of the symbol to represent ‘0’ in a ‘place value’ system where the same numeral had a different value to represent a power of ten dependent on its position in a number. The Roman numeral system was additive where, for example, ‘L’ and ‘D’ are always worth 50 and 500 respectively.
Although there is some scepticism in ‘mathematical circles’ on the rationale it is commonly believed that the creation of the Hindu-Arabic numerals was based on what was already known about angles. As the graphic shows ‘0’ has no acute or obtuse angles whereas ‘1’ has one angle and ‘2’ has two angles etc. Over time it became easier to write the numerals with curves and also leave out some straight lines. Subsequently, this resulted in the original clear correlation between ‘Numbers’ (value) and ‘Numeral’ (symbol) being lost. Even if the rationale that numbers were not created using angles is considered, the obvious impact on teaching and learning prompts the pertinent question: How does a learner understand that the numeral ‘2’ has a value of ‘2’?
The representation of the symbol ‘2’ certainly does not have any concrete connection to its value and this is where my understanding of the ‘Concrete, Pictorial, Abstract’ (CPA) process was significantly enhanced. For a learner to understand the concept of ‘arithmetical value’ that value needs to be made explicit. This could be in the form of manipulatives - counters or cubes etc. initially which are linked to pictorial representations e.g. pictures of apples and then created by the learner in the form of ‘mark-making’ e.g. circles, dots or crosses etc. The movement to the abstract, seeing or writing a symbol e.g. ‘2’ to represent the ‘arithmetical value’ of two counters or apples is then made for the learner to gain conceptual understanding and to realise ‘2’ as a number.
The issue for me has been that the step to the abstract is all too often taken too early. If I see number bonds or times tables being chanted and repeated by children my first reaction is to wonder if they have the conceptual understanding. In many situations this has sadly been the case, where they do not, and the process becomes merely an exercise in memory retention where children memorise the names of symbols or numerals. Memorisation of facts has its place in mathematics education, but not to the detriment of learners gaining relational understanding which can be clearly internalised through the use of the CPA approach.
When working in situations where rote learning is at the forefront I have usually been able to address the situation through CPD in which I always begin with the ‘Origin of Numbers’ by asking the question ‘When is a number not a number?’ to allow the participants to understand the clear correlation between ‘Numeral’ (symbol) and ‘Number’ (value). The reaction to this has quite often been dramatic where the clarification has had a profound effect on the pedagogical skills of both individuals and groups.
In some jurisdictions in mathematics education, Finland, Sweden and Singapore for example, the links to using the rationale I have described here to facilitate effective maths pedagogy, are as dramatic. In the early years of their education the names of numerals (symbols) are used but not written by children. This is to ensure that the aforementioned conceptual understanding is embedded through concrete and pictorial approaches before entering into the abstract domain.
Hopefully, the sharing of the information in this blog will have the same impact on your teaching and learning of mathematics or the way that you think about how children learn mathematics effectively, especially in the early years of a child’s education; as it did on me.
For further reading please see Deborah Mulroney’s blog ‘Ever thought about what comes before counting?’
Rachel Rayner’s blog: The CPA approach
Jerome Bruner's ‘Three Modes of Representation’
Richard Skemp’s relational and instrumental understanding