Date of implementation: September 2015

Updated September 2016

Reviewed: November 2019

Introduction

Encounter, Mathematical Thinking & Maths Policy

Including professional development materials

Fountaindale’s Encounter, Mathematical Thinking & Maths policy encompasses the fundamental principles that are the key to improving learning and enables pupils to make progress across all the strands.

From pre-intentional to formalised mathematical systems inc. GCSE

Encounter Mathematical

Thinking Maths

• pre-formal • semi-formal • formal

Contents Page Number(s) Description

3 Overview

5 - 6 Encounter in the pre-formal curriculum

7 - 8 Mathematical Thinking in the semi-formal curriculum

9 - 11 Maths in the formal curriculum

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Overview The focus in the pre-formal curriculum is on the development of the fundamentals of communication, emotional and social development and of learners cognitive abilities; though recognising that they are likely to remain within developmentally early intellectual parameters. Every moment and situation is regarded as an opportunity to interact and communicate.

The pre-formal curriculum is an integrated curriculum that our learners access through personalised support from adults working in those learning environments. Our learners interact with the world around through observing and exploring that world; through all seven senses. (Encounter)

One key focus within the semi-formal curriculum is on the development of thinking skills for learners with severe and complex learning difficulties. Thinking skills includes thinking scientifically and mathematical thinking.

Mathematical Thinking is assessed through the Semi Formal Route Map.

The approach used throughout the semi-formal curriculum is to develop mathematical thinking through a themed approach with focused learning opportunities that introduces content appropriate to the individual’s cognitive processing capabilities.

The formal curriculum builds upon and extends the knowledge, skills and understanding introduced in the semi-formal curriculum, and adheres to the requirements of the national curriculum. Formal learners also have opportunities to access accreditation to GCSE level.

The formal curriculum aims to develop learners’ abilities by

delivering the National Curriculum in a fun and engaging way

providing learners with meaningful, engaging contexts for the learning of maths

providing appropriate accreditation pathways and opportunities from key stage 4.

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Early Mathematical Learning

A model produced by Les Staves

Maths is assessed using The Semi Formal Route Map for learners operating below Y1 age

related expectations and Learning Ladders for Y1 – Y6 age related expectations.

Tools for learning

ProcessesMathematical experiences

Senses; attention; perception;

Motor development;

communication

Investigation; problem solving;

interpretation

Changes in quantity; space and

time

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Encounter the pre-formal curriculum

The pre-formal curriculum does not teach maths; it promotes maths through explorative play. At this stage of development there is a strong correlation with the development of mathematical thinking and problem solving with exploring the world. Play provides opportunities for our pre-formal learners to encounter the world around; to be curious. Within their play our leaners may reach, hold, explore and encounter sensory items and stimuli; experience cause and effect, develop an awareness of shapes and that things may exist even when out of sight. Research, such as that carried out by Bert van Oers, state that these early actions cannot be regarded as intentional or reflective (mathematical) so is important for the adults to react in a mathematical way; for example using a mathematical name. In the earliest stages of play (sensorimotor) exploration of the world is spontaneous through touch, smell, taste as well being aware how things look and sound; these are known as naturalistic experiences and are a major mode of learning during sensorimotor stage. It is important that our pre-formal learners are able to explore the physical world as independently as possible as this enable opportunities for communication and cognitive development. Mobility allows exploration. By exploring they begin to discover more on their own and begin to learn to think for themselves. At first our pre-formal learners will discover that some spaces are big and some are small. This eventually leads to the development of spatial sense. They also begin to learn ideas of size for as they look about they sense their relative smallness. They go over, under, and into large objects and discover the size of these objects relative to their own size. Our pre-formal learners need opportunities to grasp things to experience what fits and doesn’t fit into their hands. As their cognitive abilities develop they begin to learn about weight when they cannot always lift items of the same size. They learn about shape; some things stay put whilst other things roll away.

Remember: Mathematics is an

essential part of

communication. English and

maths use similar processes and complement each other in

practical communication. For

example, understanding ‘object permanence’ is needed to

comprehend quantities and

changes. (Les Staves)

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We, as practitioners, need to be aware that restricted mobility since birth will have restricted that development of those mathematical concepts neuro-typical developing children experience. It also restricts for this neuro-diverse group opportunities to think mathematically and develop their problem solving skills.

When physical support is required strategies such as hand—under—hand and hand—under—elbow ensure the learner is still able to engage in the activity in an independent way.

Movement in water also removes some of those barriers; it enables our pre-formal learners to experience free play that similar supports the development of mathematical concepts. Social communication time along with play, sensory and Halliwick swims provide opportunities for our pre-formal learners to develop body awareness.

Assessment is through Routes for Learning.

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Mathematical Thinking in the semi-formal curriculum

The semi-formal curriculum develops that early mathematical thinking our pre-formal children develop intuitively by using their bodies and senses to a more organised/formal approach to mathematical thinking. The semi-formal curriculum enables learners to develop mathematical thinking that acts as pre-cursor to knowledge, formalising and developing mathematical language and use of formalised mathematical systems and abstract ideas. It is designed to encourage mathematical graphics (Worthington & Carruthers – Research Uncovers Children’s Creative Mathematical Thinking 2003); that is mark making (scribbles); personal or invented numerals; drawings or tallies; icons such as dots; numerals and early written maths as prerequisites to abstract mathematical writing. As well as these desired goals, maths in the semi-formal curriculum is about developing attitudes or dispositions (curiosity, desire to experiment, challenge and share ideas). Therefore, maths involves asking questions, probing for answers, conducting investigations and classifying. Maths becomes a way of thinking and trying to understand the world.

As our semi-formal learners progress maths will increasingly involve ordering, patterning, counting and naming numbers, and using non-standard measurements. They are beginning to acquire fundamental mathematical concepts through active involvement with their environment. By providing opportunities for them to explore

we are providing them with opportunities to actively construct their own knowledge. These experiences can be naturalistic, informal or structured; naturalistic or spontaneous are where the learner controls choice and action; in informal experiences, the learner chooses the activity and action, but adults intervene at some point; and in structured experiences, the adult chooses the experience for the learner and gives some direction to the child’s action.

‘The mathematical mind is

rooted in the human visual,

tactile and motor systems.

Counting is based on touch and

movement, geometry is visual.’

Ian Stewart

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Mathematical concepts, mathematical language and formalised mathematical systems are introduced at appropriate stages matched to the learner’s cognitive processing capabilities, particularly as the learner move towards a more formal curriculum. In the semi-formal curriculum we would expect to see most experiences being structured experiences being provided through the themes and topics. Opportunities for informal experiences may be provided through play sessions. This learning should be meaningful to the learner. It remains important for adults to continue react in a mathematical way during play interactions in order to develop mathematical meaning. For our semi-formal learners there is a natural integration of fundamental concepts and learning fundamental process skills. When fundamental mathematics concepts—comparing, classifying, and measuring—are applied to problems, they are referred to as process skills. These mathematical concepts are necessary to solve some problems. The other process skills—observing, communicating, inferring, hypothesizing, and defining and controlling variables—are equally important for solving problems in both mathematics and science.

Our encouragement of inquiry through problem solving does not expect our semi-formal learners to discover everything for themselves; rather, our learners should be supported on relating new mathematical knowledge/concepts to previously learned knowledge and to experiential learning. This helps our learners to have a consistent picture of their environment; the physical world they live in.

We can facilitate this process in several ways through asking questions that are relevant to the learner. Our semi-formal learners can experience inquiry through structured problem solving activities that builds on informal experiences previously encountered. The driving force behind problem solving is curiosity—an interest in finding out. The challenge we face is to create an environment in which problem solving can occur. Research shows that structured problem solving and reflective play are important factors in the development of mathematical thinking. Assessment is through the Semi Formal Route Map

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Maths the formal curriculum Maths in the formal curriculum is based on the national curriculum and is designed to develop and extend mathematical thinking as well as expanding mathematical concepts, mathematical language and formalised mathematical systems. Learners whose cognitive processing levels equates to Y1 expectations onwards will follow the national curriculum guidance; a summary of key aspects is shown below. Most formal learners at Fountaindale will be operating between Y1 and Y6 expectations; if a learner progresses beyond these expectations refer to the national curriculum documentation. Learners within the 14-19 age range will follow appropriate accredited maths courses. Some younger learners may be working within the upper end of the P scales; they will also follow the national curriculum matched to their cognitive processing levels.

The national curriculum for maths aims to ensure that all pupils:

become fluent in the fundamentals of mathematics, including through varied and

frequent practice with increasingly complex problems over time, so that pupils

develop conceptual understanding and the ability to recall and apply knowledge

rapidly and accurately

reason mathematically by following a line of enquiry, conjecturing relationships

and generalisations, and developing an argument, justification or proof using

mathematical language

can solve problems by applying their mathematics to a variety of routine and non-

routine problems with increasing sophistication, including breaking down problems

into a series of simpler steps and persevering in seeking solutions

.

Formal learners working at age related expectations for Y1 – Y2 the principal focus of

mathematics teaching is to ensure that pupils develop confidence and mental fluency with

whole numbers, counting and place value. This should involve working with numerals,

words and the 4 operations, including with practical resources [for example, concrete

objects and measuring tools].

At this stage, pupils should develop their ability to recognise, describe, draw, compare and

sort different shapes and use the related vocabulary. Teaching should also involve using a

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range of measures to describe and compare different quantities such as length, mass,

capacity/volume, time and money.

By the end of age related expectations for Y2, pupils should know the number bonds to 20

and be precise in using and understanding place value. An emphasis on practice at this

early stage will aid fluency.

Pupils should read and spell mathematical vocabulary, at a level consistent with their

increasing word reading and spelling knowledge .

Formal learners working at age related expectations of Y3 and Y4 the principal focus of

mathematics teaching is to ensure that pupils become increasingly fluent with whole

numbers and the 4 operations, including number facts and the concept of place value. This

should ensure that pupils develop efficient written and mental methods and perform

calculations accurately with increasingly large whole numbers.

At this stage, pupils should develop their ability to solve a range of problems, including

with simple fractions and decimal place value. Teaching should also ensure that pupils

draw with increasing accuracy and develop mathematical reasoning so they can analyse

shapes and their properties, and confidently describe the relationships between them. It

should ensure that they can use measuring instruments with accuracy and make

connections between measure and number.

By the end of age related expectations for Y4, pupils should have memorised their

multiplication tables up to and including the 12 multiplication table and show precision and

fluency in their work.

Pupils should read and spell mathematical vocabulary correctly and confidently, using their

growing word-reading knowledge and their knowledge of spelling

Formal learners working at age related expectations of Y5 and Y6 the principal focus of

mathematics teaching is to ensure that pupils extend their understanding of the number

system and place value to include larger integers. This should develop the connections that

pupils make between multiplication and division with fractions, decimals, percentages and

ratio.

At this stage, pupils should develop their ability to solve a wider range of problems,

including increasingly complex properties of numbers and arithmetic, and problems

demanding efficient written and mental methods of calculation. With this foundation in

arithmetic, pupils are introduced to the language of algebra as a means for solving a variety

of problems. Teaching in geometry and measures should consolidate and extend

knowledge developed in number. Teaching should also ensure that pupils classify shapes

with increasingly complex geometric properties and that they learn the vocabulary they

need to describe them.

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By the end of age related expectations for Y6, pupils should be fluent in written methods

for all 4 operations, including long multiplication and division, and in working with

fractions, decimals and percentages.

Pupils should read, spell and pronounce mathematical vocabulary correctly

Assessment is through the Learning Ladders (national curriculum expectations for Y1+)