1. Mastery at Hillyfield

2. What are the characteristics of a child who is good at maths?
From the early stages onwards, children and young people should experience success in mathematics and develop the confidence to:
take risks
ask questions and explore alternative solutions without fear of being wrong
enjoy exploring and applying mathematical concepts to understand and solve problems
explain their thinking and presenting their solutions to others in a variety of ways
reason logically and creatively through discussion of mathematical ideas and concepts
become fluent, flexible thinkers able to see and make connections
Carol Dweck’s research and growth mind sets.
One of the most important things we can say to our children is “you can not do it YET!”Since few people learn in the same way and taking the above into consideration it is difficult to identify a comprehensive list of characteristics since there may be able children who display only some of these.
After much thought, here are my 5 characteristics, in no particular order.
1. Extremely curious with an intense interest in mathematics
2. Learns quickly and understands complex subject content with less practice and/or less repetition than other students.
3. Thrives on challenge and persists, leading to working at abstract levels with general rules and formulae.
4. Is creative mathematically, imaginative and has original thinking
5. Student is an able and independent learner
Nurturing able and creative students. .
Teachers should nurture the talents of students and be always conscious that understanding and progress can be fragile at times even in a more able student. able and creative mathematicians can be highly self - critical as opposed to good mathematicians who are often satisfied and pleased with their own learning. It is our duty to encourage able and creative children to be independent learners, and to enjoy a life long passion for mathematics.
It is important for the teacher to be a good role model. Teachers should demonstrate intellectual engagement with the subject themselves, be passionate about maths with a real desire to experience their students enjoying mathematics, meeting and exceeding all expectations. The teacher should establish a learning environment that is secure, purposeful, fun and positive with a focus on hard work, persistence and achievement. able children must feel that it is OK to make mistakes. Good mathematicians make mistakes. It is by making mistakes and putting them right that we make progress.
able and creative children have a right to teacher intervention and interaction with time for reflection.
Teachers should understand the student’s needs and allow time for the curious and often persistent questions. The able student should know that their thoughts and opinions are relevant and valued. There should be opportunities for extension and enrichment within the curriculum within all the phases.
©Oxford University Press 2014

3. We need to provide opportunities for children to do these.
Make up an activity that would enable the children to show that they are pattern sniffers.

4. Mastery requires knowledge
There are three forms of knowledge:
Factual: I know…( eg number bonds, times tables)
Procedural: I know how… (following rules and procedures)
Conceptual: I know why… (understanding)
The children don’t need to just know if the answer is correct, they need to know why.

5. What is mastery? Mastery in mathematics involves:
Emphasis on mental strategies
Emphasis on Problem Solving and reasoning
Deep and sustainable learning
Ability to build on something already mastered
Ability to reason about a concept and make connections to other concepts
Procedural and conceptual fluency (can’t solve problems without these)
The understanding of how and why it all works
Mastery is a continuum… mastery at a particular point of time that is sufficient mastery for that stage of learning and then built on at a later stage

6. Forms of representation in Mastery
Concrete
Hands on using ‘ real’ objects
Pictorial
Using a diagram or model to represent the situation
Abstract
The symbolic stage - a student is now capable of representing problems by using mathematical notation, for example: 12 ÷ 2 = 6
Bruner looked at how children made sense of the world and came up with 3 forms of representation:
Enactive: learning through action – becomes automatic with practice – because we have a ‘motor’ memory don’t need to keep learning each time
Iconic : learning through image making - often taken just to mean visual image but it means all the senses. Smell can evoke all kinds of memories.
Symbolic learning through language–spoken/written
B suggests that learners should progress through the stages but that
best learning takes place when all three modes of learning are used together, action imagery and conversation. .
IM uses structured imagery motivates numerals, providing a picture of the number ideas and the relationships between the number ideas. Children are actively involved in doing mathematics and also in talking about the maths that they do.
Some of the notes below are taken from MC. Singapore maths for teaching professionals.
We need a store of mental images on which to draw in order to form abstract ideas.
Some have subdivided the symbolic stage in two , spoken language and written symbol.
Our means of building concepts lie in the three modes of representing the world; the enactive, the iconic and the symbolic.
Bruner suggests that the essential points for teachers to consider are
Children’s predisposition towards learning(the will to learn an succeed is deeply ingrained)
The way in which the knowledge to be learnt is structured(teachers role is to help children verbalise what they have done, so that they can develop the required concepts
The sequence in which the knowledge should be presented.(Bruner suggests the spiral curriculum)
The motivation rewards provided(important that children feel that their learning is leading towards a goal