# Foundation Phase Mathematics

The Foundation Phase Mathematics Curriculum and Assessment Policy Statement (CAPS) provides teachers with a definition of mathematics, specific aims, specific skills, the focus of content areas, the weighting of content areas, recommended resources for the Foundation Phase Mathematics lessons, suggested guidelines on supporting learners with barriers to learning Mathematics, mental mathematics and enhancing the teaching of early numeracy skills in Grade R.

## What is mathematics?

Mathematics is a language that uses symbols and notations for describing numerical, geometric and graphical relationships. It is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. It helps develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making.

## Specific aims

The teaching and learning of Mathematics aim to develop the following in the learner:

• critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations;
• confidence and competence to deal with any mathematical situation without being hindered by a fear of Mathematics;
• a spirit of curiosity and a love of Mathematics;
• appreciation for the beauty and elegance of Mathematics;
• recognition that Mathematics is a creative part of human activity;
• deep conceptual understanding to make sense of Mathematics; and
• acquisition of specific knowledge and skills necessary for:
• the application of Mathematics to physical, social and mathematical problems,
• the study of related subject matter (e.g. other subjects); and
• a further study in Mathematics.

## Specific Skills

To develop essential mathematical skills, the learner should:

• develop the correct use of the language of Mathematics;
• develop number vocabulary, number concept and calculation and application skills;
• learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained;
• learn to investigate, analyse, represent and interpret information;
• learn to pose and solve problems; and
• build an awareness of the critical role that Mathematics plays in real-life situations, including the learner’s personal development.

## The focus of Content areas

Mathematics in the Foundation Phase covers five content areas. Each content area contributes to the acquisition of specific skills. The table below shows the general focus of the content areas and the particular focus of the content areas for the Foundation Phase.

### Numbers, operations and relationships

Development of number sense that includes:

• the meaning of different kinds of numbers;
• the relationship between different kinds of numbers;
• the relative size of other numbers;
• representation of numbers in various ways; and
• the effect of operating with numbers.

The number range developed by the end of Grade 3 includes whole numbers to at least 1 000 and common fractions. In this phase, the learners’ number concept is developed through working with physical objects to count collections of objects, partition and combine quantities, skip count in various ways, solve contextually (word) problems, and build up and break down numbers.

• Counting enables learners to develop number concepts, mental mathematics, estimation, calculation skills and recognition of patterns.
• Number concept development helps learners learn about numbers’ properties and develop strategies that can make calculations easier.
• Solving problems in context enables learners to communicate their thinking orally and in writing through drawings and symbols.
• Learners build an understanding of basic operations of addition, subtraction, multiplication and division.
• Learners develop fraction concepts through solving problems involving the sharing of physical quantities and by using drawings. Problems should include solutions that result in whole number remainders or fractions. Sharing should involve not only finding parts of wholes but also finding parts of collections of objects. In this phase, learners are not expected to read or write fraction symbols.

### Patterns, Functions and algebra

Algebra is the language for investigating and communicating most of Mathematics and can be extended to studying functions and other relationships between variables. A central part of this content area is for the learner to achieve efficient manipulative skills in algebra. It also focuses on the:

• description of patterns and relationships through the use of symbolic expressions, graphs and tables; and
• identification and analysis of regularities and change in patterns and relationships
•  that enable learners to make predictions and solve problems.

In this phase, learners work with number patterns (e.g. skip counting); and geometric patterns (e.g. pictures).

Learners should use physical objects, drawings and symbolic forms to copy, extend, describe and create patterns.

• Copying the pattern helps learners to see the logic of how the pattern is made.
• Extending the pattern helps learners to check that they have properly understood the logic of the pattern.
• Describing the pattern helps learners to develop their language skills.
• Focussing on the logic of patterns lays the basis for developing algebraic thinking skills.
• Number patterns support number concept development and operational sense built in Numbers, Operations and relationships.
• Geometric patterns include sequences of lines, shapes and objects but also patterns in the world. In geometric patterns, learners apply their knowledge of space and shape.

### Space and shape (Geometry)

The study of Space and Shape improves understanding and appreciation of the pattern, precision, achievement and beauty in natural and cultural forms. It focuses on the

• properties, relationships;
• orientations, positions; and
• transformations of two- dimensional shapes and three-dimensional objects.

In this phase, learners focus on three-dimensional (3-d) objects, two-dimensional (2-d) shapes, positions and directions.

• Learners explore properties of 3-d objects and 2-d shapes by sorting, classifying, describing and naming them.
• Learners draw shapes and build with objects.
• Learners recognise and describe shapes and objects in their environment that resemble mathematical objects and shapes.
• Learners describe the position of objects, themselves and others using the appropriate vocabulary.
• Learners follow and give directions.

### Measurement

Measurement focuses on selecting and using appropriate units, instruments, and formulae to quantify characteristics of events, shapes, objects, and the environment. It relates directly to the learner’s scientific, technological and economic worlds, enabling the learner to:

• make sensible estimates; and
• be alert to the reasonableness of measurements and results.

In this phase, the learners’ concept of measurement is developed by working practically with different concrete objects and shapes, learning the properties of length, capacity, mass, area and time.

• Learners measure the properties of shapes and objects using informal units where appropriate, such as hands, paces, containers, etc.
• Learners compare different quantities by using comparative words such as taller/shorter, heavier/lighter etc.
• Learners are introduced to standard units such as grams, kilograms, millilitres, litres, centimetres, metres.
• Activities related to time should be structured with the awareness that learners’ understanding of the passing of time should be developed before they read about time.

The fifth content area is data handling (Statistics).

## Weighting of content areas

The weighting of mathematics content areas serves two primary purposes: firstly, the weighting gives guidance on the amount of time needed to address the content within each content area adequately; secondly, the weighting advises on the spread of content in assessment. The weighting of the content areas is not the same for each grade in the Foundation Phase.

Foundation Phase Mathematics forges the link between the child’s pre-school life and life outside the school on the one hand and the abstract Mathematics of the later grades on the other hand. Children should be exposed to mathematical experiences in the early stages that give them many opportunities “to do, talk and record” their mathematical thinking.

The amount of time spent on Mathematics has a decisive impact on learners’ mathematical concepts and skills. However, the activities learners engage in should not be “keep busy” activities but should be focused on mathematics as outlined in the curriculum.

## Suggested guidelines for classroom management

All the time allocated to Mathematics on a single day should be considered as one period. during the Mathematics period, the following should usually happen:

• Whole class activity
• – Mental mathematics
• – Consolidation of concepts
• – Classroom management (allocation of independent activities, etc.)
• small group teaching
• – Counting
• – Number concept development (oral and practical activities)
• – Problem-solving (oral and practical activities)
• – Written recording
• – developing calculating strategies (oral and practical activities)
• – Patterns
• – Space and shape
• – Measurement
• – data Handling
• independent work

Learners practise and consolidate concepts developed in whole class and small group teaching.

Whole class activity: where the focus will be mainly on mental mathematics, consolidation of concepts and allocation of independent activities for at least 20 minutes per day at the start of the Mathematics lesson. Mental mathematics will include brisk mental starters such as “the number after/before 8 is; 2 more/less than 8 is; 4+2; 5+2, 6+2” etc. During this time the teacher will also work with the whole class to determine and record (where appropriate) the name of the day, the date, the number of learners present and absent, and the nature of the weather. During this time, the teacher can also consolidate concepts that are a little challenging. Also noteworthy is that the teacher should assign the class their general class activity and independent activities that they do on their own while she gets on with the small group focused sessions.

Small group focused lessons: are most effective when the teacher takes a small group of learners (8 to 12) who have the same ability with her on the floor or at their tables while the rest is engaged in independent activities. The teacher works orally and practically with the learners, engaging in such activities as counting, estimation, number concept development and problem-solving exercises, and activities concerning pattern, space and shape, measurement and data handling, which should be carefully planned for. To reinforce learning, written work (workbook, worksheet examples, work cards etc.) should form part of the group session where possible. Learners should have writing materials (class workbooks, etc.) available for a problem-solving activities. The group sessions should be very interactive, and learners should be encouraged to “do, talk, demonstrate and record” their mathematical thinking.

Teachers should take care not to underestimate the slower learners; they should also be stretched. It is easier to match the difficulty level of the work to the learners if the group the teacher is working with is of approximately equal ability. However, mixed ability groups can work well for construction, measurement and patterning, sorting activities, or games.

Independent activities: While the teacher is busy with the small group focused lesson, the rest of the class must be purposefully engaged in various mathematical activities that focus on reinforcing and consolidating concepts and skills that have already been taught during small group focused lessons. These independent activities should be differentiated to cater for different ability levels. Independent activities may include:

• workbook activities;
• graded worksheets/work cards for counting, manipulating numbers, simple problems in context (word problems), etc.;
• mathematics games like Ludo, dominoes, jigsaw puzzles; and
• tasks that involve construction, sorting, patterning or measurement.
• The Mathematics period should also support learners experiencing barriers to learning, enrichment activities for high flyers, assessment activities, etc.
• Both independent and small group focused lesson activities must be observed (practical, oral), marked and overseen (written recording) by the teacher as part of her informal and formal assessment activities.
• Close tracking of learners’ responses (verbal, oral, practical, written recording) in learning and teaching situations enables the teacher to do a continuous assessment, monitor learners’ progress and plan support for learners experiencing barriers to learning.

## Learners with barriers to learning mathematics

It is essential for learners who experience barriers to learning Mathematics to be exposed to activity-based learning. Practical examples using concrete objects together with practical activities should be used for a longer time than with other learners, as moving to abstract work too soon may lead to frustration and regression. These learners may require and should be granted more time for:

• completing assessment activities and tasks; and
• acquiring thinking skills (own strategies
• The number of activities to be completed should be adapted to the learner without compromising the concept and skills that are addressed.

## Mental mathematics

Mental mathematics plays a vital role in the curriculum. The number bonds and multiplication table facts that learners are expected to know or recall fairly quickly are listed for each grade. In addition, mental mathematics is used extensively to explore the higher number ranges through skip counting and by doing activities such as “up and down the number ladder”, e.g. the Grade teacher might ask the following “chained” questions: “Start with 796. Make that 7 more. Yes, it is 803. Make that 5 less. Yes, it is 798. Make that 10 more … 2 more … 90 more … 5 less …” etc. These activities help learners to construct a mental number line.

Mental mathematics, therefore, features strongly in both the counting and the number concept development sections relating to the topics Number and Patterns, and may also occur during Measurement and data Handling activities. When doing mental mathematics, the teacher should never force learners to do mental calculations that they cannot handle — writing materials and counters should always be available for those learners who may need them.

The approach to learning Mathematics should be based on the principles of integration and play-based learning. The teacher should be pro-active, a mediator rather than a facilitator. A mediator makes the most of incidental learning opportunities that arise spontaneously during a range of child-centred activities such as free play in the fantasy corner or block construction site, sand and water play activities as well as teacher-guided activities that focus on mathematical concepts such as counting, number concept development, space and shape, patterns, time and other emergent mathematics activities. Colour is not a mathematical concept but can be used to promote mathematical concepts in activities such as sorting, grouping and classifying.

All aspects of Grade R, including the classroom environment and teaching and learning practice, should promote the child’s holistic development. Development that is an integral part of emergent numeracy includes cognitive development (problem-solving, logical thought and reasoning), language development (the language of mathematics), perceptual-motor, and emotional and social development. All these aspects can be developed through stories, songs, rhymes, finger games and water play, educational toys including board games, construction and exploration activities (mass, time, capacity, measurement, etc.), imaginative play, outdoor play and “playground games”. Many kinds of games and play could include aspects of numeracy, for example, measuring during cooking or counting during shopping.

In other words, the acquisition of emergent mathematics and related mathematical concepts should, like all good teaching, adhere to the following learning principles where children move through three stages of learning, namely:

• the kinaesthetic stage (experience concepts with the body and senses);
• the concrete stage (3-d, using a variety of different objects such as blocks, bottle tops, twigs and other things in the environment); and
• paper and pencil representation (semi-concrete representations using drawings, matching cards etc.)

In the Grade R year the timetable is called the daily programme (see Figure 1) and it comprises three main components, namely:

• teacher-guided activities;
• routines; and
• child-initiated activities or free play.

The emphasis should be on using these aspects of the daily programme to promote the acquisition of emergent numeracy in a fun and spontaneous context. For example, teacher-guided numeracy learning opportunities are offered during ring time. Most rings can be given a mathematical focus. The early morning ring when children are greeted, and a roll-call is taken an opportunity for playing with numbers and, for example, counting. Other rings, such as the Mathematics ring, perceptual-motor rings, movement, music and science rings, can also provide a Mathematics focus.

Creative art activities could also have a mathematical emphasis, for example, using geometric shapes such as circles and squares to make a collage or designing a pattern to frame a picture. The weather chart, calendar and birthday rings also provide opportunities for exploring mathematical concepts. It is the teacher’s knowledge and initiative that can maximise learning potential.

• Routines where children participate actively, such as snack time, arrival, home time and toilet routines, can also be given a Mathematics focus. Children wearing red, for example, go to the toilet first (colour and ordinal number), each child gets a plate and a sandwich (one-to-one correspondence), Thandi would like a second sandwich, David doesn’t want any more. This amounts to identifying and utilising a teachable moment, in other words, being a mediator of learning.
• During free play, the teacher can promote emergent mathematics through the appropriate structuring of the free-play area. Outdoor free play such as climbing on a wooden climbing frame or riding on the cycle track might promote the acquisition of crucial mathematical vocabulary such as up/down, bottom/top, fast/slow, high/ low, etc. Sand and water play will also enhance the understanding of concepts such as mass, volume and capacity. These activities will also promote essential underpinning perceptual-motor skills, which become an inherent part of the successful acquisition in the formal school of literacy and numeracy. Examples of these skills are:
• developing an understanding of your position in space e.g. behind, in front, underneath or next to an object (this can, for example, be linked to place value in mathematics); and
• directionality and laterality (this can be linked to number and letter formation and reading from left to right).

The practice outlined above is illustrative of a Grade R approach that promotes problem-solving, logical thinking and reasoning, and education for citizenship because of its focus on cooperative learning and negotiation. By utilising teachable moments, a teacher can encourage children to reflect on their decisions and predict possibilities, e.g. whether they think a container being used in water play will hold more than another container.

By making helpful suggestions and inviting a child to think about alternative positions and ways of problem-solving, a teacher can encourage children to think more deeply about an issue and find good reasons for the choices they make. In this way, mathematical and holistic development are addressed, and critical premises underpinning CAPS are brought into play.

Assessment practices in Grade R should be informal, and children should not be subjected to a ‘test’ situation. For this reason, assessment activities have not been included in the Grade R CAPS. Each activity used for assessment should be carefully planned so that it integrates a variety of skills.

In Grade R most of the assessment takes place through observation, with the teacher recording the assessment results using a checklist. Thus, as the year progresses, a full picture of each child, complete with challenges and strengths, is gradually built. This allows for challenges to be addressed and strengths to be maximised.

A traditional, formal classroom-based learning programme that is tightly structured and ‘basics bound’ should be avoided, as it does not optimise numeracy acquisition for the Grade R child. Grade R should not be a ‘watered down’ Grade 1 class. It has its unique characteristics based on how children in this age group make sense of their world and acquire the knowledge, skills, values and attitudes that will maximise the opportunities afforded in the formal learning years.

• Counters
• Large dice
• A big counting frame
• A height chart
• Big 1 – 100 and 101 – 200 number grid posters (100 – charts)
• different number lines (vertical and horizontal)
• A set of Flard cards (expanding cards)
• Play money — coins and notes
• A calendar for the current year
• A large analogue wall clock
• A balance scale
• Building blocks
• Modelling clay
• A variety of boxes of different shapes and sizes brought from home.
• A variety of plastic bottles and containers to describe and compare capacities
• Good examples of a sphere (ball), a rectangular prism (box), cube, cone, pyramid and cylinder. The teacher can make this herself.
• A number of plastic or cardboard squares, different rectangles, circles, different triangles all of different sizes
• Mathematical games, e.g. Ludo, Snakes and Ladders, Jigsaw Puzzles, Dominoes, Tangrams etc.
• Essential for Grades R and 1:
• Areas for sand and water play
• Apparatus for climbing, balancing, swinging and skipping
• A play-shop with items to be bought with play-money
• A variety of appropriate games such as ‘what’s in a square’?
• Blocks

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