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learning theories

CPA: using Cuisenaire to support pupils to develop fractional understanding

Louisa Ingram is a primary mathematics adviser for HfL

Identifying Fractions

To begin with, pupils need to become familiar with assigning a value to a rod and finding the fractional value of the other rods. A good starting point is to find the value of the white rod as this then allows you to find the value of all other rods. When the brown rod equals 1; the white rod is one eighth. Compared to dark green, the white rod’s value is one sixth. Against blue, it is one ninth and against orange one tenths etc. You can then start to apply this such as assigning the brown rod a value of 2. Through this you can also draw attention to fractions such as which rod is one half, one quarter, one third the length of etc. Continue reading “CPA: using Cuisenaire to support pupils to develop fractional understanding”

The ‘CPA’ Approach

Rachel Rayner is a primary mathematics adviser for HfL

One of the most fundamental learning theories to be implemented within any mastery classroom is the ‘CPA’ (Concrete, Pictorial, and Abstract) approach. It was first proposed by Jerome Bruner in 1966 as a means of scaffolding learning. The psychologist believes that the abstract nature of learning (and this is especially true in mathematics) is a “mystery” to many children. It, therefore, needs to be scaffolded by the use of effective representations. He saw that, when pupils used the CPA approach, they were able to build on each stage towards a fuller understanding of the concepts being learnt and, as such, the information and knowledge were internalised to a greater degree. This allowed the teacher to build upon this secure learning. Bruner, and others, demonstrated that each stage of the approach acts as a scaffold for subsequent and connected learning. Continue reading “The ‘CPA’ Approach”

Making Every Question Count

Charlie Harber is the Deputy Lead Adviser for the HfL Primary Mathematics Team.

Do pages and pages of repetitive questions deepen our children’s understanding?

Teaching to mastery requires a mind shift on many different levels and presents many challenges (differentiation, whole class teaching, culture and ethos to name but a few). It is underpinned by a range of theories… You can’t explore mastery for long without encountering ‘Variation’ theories. There are two distinct variation theories (conceptual and procedural) which develop symbiotically to grow deep conceptual understanding in learners. These theories are used widely in the high performing East and South East Asian countries. They are exploited skilfully by teachers to expose the underlying structures of mathematics and allow children to ‘self-discover’. Continue reading “Making Every Question Count”

Mathematical Voices

Rachel Rayner is a primary mathematics adviser for Herts for Learning.

How many times have you heard the following?

‘I only really understood maths once I started teaching it.’

We all recognise the importance of subject knowledge in teaching any subject. Many of our schools in Hertfordshire have been engaging with our advisory team to discover what that really means for mathematics. Together, we have wrestled with the fact that the subject knowledge we were taught ourselves may not have translated into the deep conceptual knowledge and understanding we are committed to exploring alongside our charges. That leaves us with personal knowledge gaps that we need to fill. Continue reading “Mathematical Voices”

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