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Hands on Heads on: The effective use of mathematics manipulative materials

A Mathematics Manipulatives

Continuum

Geoff White Paul Swan Linda Marshall

Mathematics materials likely to be found in a primary school

What is a mathematics manipulative? A manipulative material is an object(s) that can be handled by an individual in a sensory manner during which process, conscious and unconscious thinking will be fostered. Consequently, a mathematics manipulative object(s) would lead to an awareness and development of concepts and ideas linked with mathematics and they would most likely be purpose designed. The members of the research team sought an all-embracing definition, but such a definition eluded them. With careful consideration, it was decided that there are tools (such as a calculator), teaching tools (demonstration models) and teaching aids (fraction charts); generally these were not considered as mathematics manipulative materials. Within the general definition, structured and unstructured mathematics manipulative materials were recognised. Needless to say, throughout the education literature many definitions of mathematics manipulative materials have been offered.

Mathematics materials… Abacus • looped (2, 3, 6 loops) • three-prong (different styles but same principle) Attribute Blocks and support materials Beads & Threading Laces Bead Frame (Counting frame) Centicube and support materials such as baseboards

Hands On: Heads On A Mathematics Manipulatives Continuum

Clocks - variety; both digital and analogue Classroom Kit (see page 14) Coloured rods (Cuisenaire) Construction Materials (ensure comprehensive collections of a few varieties; eg. Knex, Mobilo, Interstar, etc) Counting materials Cotton reels, cubes (variety), popsticks, counting sticks Cubes - 1 cm, 2 cm Wood or plastic (See Unifix, Multilink (Gain), Centicube, etc) Dominoes - a variety of types Fractions - a wide variety; eg. fraction games & fraction cakes. Fraction Kits - eg. Wainwright Fraction Kits

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Geoboard - 5 x 5, 10 x 10 pins

Tangrams

Geometric models (solid) and hollow, such as, Power Solids

Tessellation Pack

Johnstone Number Line

Sorting Materials - farm animals, dinosaurs, frogs, teddies etc.

MAB (Multibase Arithmetic Blocks) or Base 10 blocks and a variety of support material Matrix Games ‘What’s in the Square?’, ‘What Else is in the Square?’ and similar products Measurement Materials (see page 13) Mira (Geoflector, Geo mirror) Mirrors Number Board (1-100) - variety Pattern Blocks Peg board and pegs (variety) Puzzles: geometric, pentomino, soma cube, tangram etc Scales • Kitchen scales, bathroom scales • Balance Scales • Pan Balance Spring • Scales/Balance Assorted mass sets (see also page 13)

The Brick

Unifix cubes - (Constructo cubes, Simfit, Stack cubes) Unit Blocks (Project Blocks) Welford Blocks

2D to 3D Shapes Interlocking shapes create a net which form a solid. There is a large number of materials in this genre. Here are some examples: Polydron, Lokko, Klikko, Clixi, GeoFit, Geo Shapes and Mini Geofix ( a smaller version) A Range of Games Many schools have a wide variety of games available, stored in both classrooms and storerooms. It is recommended that students benefit from learning the skills of a few good games rather than being introduced to a wide range.

School-wide use of mathematics manipulatives

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athematics manipulatives do not teach on their own; rather they open pathways to learning.

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The Mathematics Learning Experience T

There is a tendency to use the materials as demonstration pieces - ‘Look at this, see how it works!’. Rather, the success of their use lies in the three-way interaction between the student, the teacher and the materials. By focussing on the materials, the teacher leads the student to insights which they develop through language and interactions with the teacher and their peers. The materials act as a catalyst to learning. At the end of the process, the materials have not changed, but it is a fair assumption that the learner (and maybe the teacher) will have done so.

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evelopmental considerations are essential for success.

Experience has shown that by carefully managing the use of manipulative materials, the creation of a developmentally focussed mathematics program can be achieved.

evisit earlier experiences.

A common myth affects the use of mathematics manipulatives in the classroom: once exposed to a material, children have learned all that material has to offer. A person interacts with a material with a different mind-set at different times. These different reactions widen the experience the learner develops. For example, ideas fostered by the use of Pattern Blocks in early childhood classes will later translate into formal fractions concepts in the middle and upper primary classes. Likewise, formal symmetry ideas will have been demonstrated practically as students experiment with the Pattern Blocks. In summary, it is more effective to start an idea with children and carry it through.

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hat is started must be continued.

The planned use of manipulatives provides the basis for a continuum of experiences for the student throughout the school. Frequently, the child-centred early childhood experiences are abandoned too rapidly and are replaced by an almost one-way process: teacher to student. Unfortunately, the three way interaction no longer seems appropriate.

Hands On Heads On: A Mathematics Manipulatives Continuum

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Categories of mathematics manipulatives

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ey Materials

Many mathematical concepts can be developed and strengthened by the frequent use of these materials. It is suggested they ‘visit’ a classroom for 3 weeks every term. Appropriate manipulative materials are suggested on the sequence presented on the following page. (Note: Experience has shown that a three week time span is appropriate to maintain students’ interest.)

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otated or shared Materials

Substantial quantities of a material, housed in adequate containers, are circulated amongst the classrooms on a pre-planned basis. In the primary school, experience has shown that three weeks is a most satisfactory period of time and these items may visit on more than one occasion per year and be suitable for several classes; for example, children will benefit from a regular ‘exposure’ to Geoshapes in Years 4-9. Materials need to be both readily available (not in some distant storeroom) and stored in portable containers.

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lassroom Kit

One of the main problems mentioned by teachers is the need to have certain items on hand at all times. As well, classrooms need to be ‘fitted out’ with some vital teaching support items, such as, analogue and digital clocks, thermometers, an accurate balance scale (not a kitchen scale) and suitable weather or observation charts. All of these materials play a part in the mathematical experience. A suggested list of suitable items is listed later.

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aterials kept in the Mathematics Storeroom

Usually, these materials are items or collections required for special presentations or they are too bulky to be housed in a classroom. These items could include Johnstone number lines and some measurement materials such as trundle wheels. A detailed list of materials is provided so classroom teachers can plan for the appropriate use of certain materials. The aim is to rely on the storeroom as little as possible.

1. Plan sufficient quantities for groups of 4-6 children. 2. Purchase a lot of a little: rather than a little of a lot. 3. When increasing or replacing stock, ensure the same quality and style of product is obtained. (Keep a copy of the invoice in order to identify items.)

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Hands On Heads On: A Mathematics Manipulatives Continuum

A suggested mathematics manipulatives continuum

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Kindergarten

• Children need to have constant access to sand and water play, this could continue into Pre-primary and Year 1/2 environments. • In the playground, children have access to a wide variety of kinaesthetic skill activities at all times. Consider this an essential for any Early Childhood program. • Unit (Project) Blocks are a key material for a sound Early Childhood experience: children need constant access to this material. (Each Early Childhood teaching area could allocate a dedicated space for Block Play.) • We must be careful not to over-expose children to the same materials.

Key Materials

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The Brick

Unifix

Rotated Materials Pattern Blocks • What’s in the Square?

Pre-Primary

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Year 1

Year 2

3-4

• (Concepta) Begrippen • Construction materials Number, Pattern, Working Mathematically

Additional Brick components may be introduced.

Use the blocks only. As the children gain experience a range of support material will be introduced.

Coloured Rods

Different aspects of this material will be featured at various levels.

• Johnstone number lines • Attribute Blocks • Geoshapes/Polydron • Geoboards

MAB (Base 10) • Geoshapes/Polydron

Years 3 & 4

• Tangrams, Pentominoes etc • Fraction Kits

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Years 5, 6, 7

Becomes a Rotated Material at this level.

Before introducing MAB, children will experience trading activities with Place Value boards and Popsticks.

Centicube

• There will be special collections of materials, such as Mira, which are used infrequently but are important experiential materials.

CLASSROOM KITS: A kit of appropriate material is allocated to each classroom (initially, a teaching block). Contents of the kit will vary according to the classroom level. Every room could be fitted out with an analogue and digital clock, large thermometer, an accurate balance scale and a form of magnification. A larger collection of calculators may be allocated; one per teaching unit or kept in the storeroom. Carefully consider whether an item needs to be on hand at all times. Use materials to provide variety in the classroom. Hands On Heads On: A Mathematics Manipulatives Continuum

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The Brick The Brick is a component of The Little Architect Construction Set, a collection of different plastic shapes designed to construct house-type models. Designed by Fab Caponio of Moorabbin, Melbourne, Victoria; The Brick is moulded in high-grade plastic and is very durable. Some additional pieces from the construction set, such as end-caps, half-bricks and faces may be added to make an even more versatile manipulative. First introduced to schools in 1980, the successful use of The Brick has been demonstrated in many early childhood classrooms, but a wide range of mathematical ideas may be generated from its use. It is recommended that children right through the primary school have a chance to explore with the material.

Appropriate for use at all levels of Primary School Allow children to become familiar with the material; do not ‘push’ formal mathematical ideas. Children will be familiar with The Brick, but free play/exploration needs to be fostered still.

The Brick can be used in a similar fashion to Unifix. However, in contrast to Unifix, The Brick may be joined in a variety of ways, providing a 3-dimensional attribute to the materials.

Unifix can be joined in a linear fashion only.

Easy joining techniques, provide the possibility of creating a range of different shapes.

Hands On Heads On: A Mathematics Manipulatives Continuum

Create classification activities after ‘What’s in the Square?’ has been introduced.

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Students create a wide range of patterns and designs; many of which have a strong mathematical structure

Tower structures provide an experience in 3D modelling.

Creating various ranges of colour combinations (2, 3, and 4 colours).

Experience has shown that upper primary students enjoy using The Brick, making it one of the most versatile manipulative materials available.

Requirements: 4000 pieces per class/group

Unifix In the early 1950s, the invention of polyethylene plastic enabled a wide range of teaching materials to be designed and manufactured. The Unifix cube is modelled on a wooden block in which a hole through the centre enabled a child to thread a number of them on a lace. This ‘joining’ principle was changed to an interlocking joiner which allowed any number of the cubes to be joined end to end. Possibly to match the ‘ten-ness’ of the number system, ten colours were moulded. A well-developed support material program is offered; included are 1-100 Number Line tracks, Grids, Place Value Trays and Number Indicators. This system is modelled on the work of the famous American early childhood educator, Catherine Stern, who published Children Discover Arithmetic in 1947.

Appropriate for use at all levels in Early Childhood Unifix materials are especially suitable for use in developing: Number awareness Four operations

While young children may be not aware of any mathematical formula, the search for and discovery of a rule for building different ‘stacks’ arouses enthusiasm. Try these: • Using the same three colours (say, red, green and white) each time, how many different 3 -stacks can be made? • Now, use four colours: how many different stacks can be made?

Pattern making

Allow children time to work at this problem. Usually, we make this challenge early in the week and suggest to the children that all solutions be finished by (say) 11:30 am on Friday. Gathering around the ‘solution table’ the children can review all the submitted ideas. Without providing a definitive solution to these, the next challenge is set.

Early algebra ideas

• Using 2 colours each time, how many different stacks of three can be made?

Unifix Support Materials

Developing mathematics with Unifix® by Swan and White is a definitive teaching guide.

Hands On Heads On: A Mathematics Manipulatives Continuum

Using Unifix cubes with trading activities.

You will find the children’s pondering exciting! Note: many ideas can be reinforced using The Brick; likewise, many ideas developed with The Brick can be reinforced using Unifix cubes.

Requirements: 2000 pieces per class/group 7

Pattern Blocks In the early 1960s, a group of scientists, physicists, mathematicians and teachers collaborated to create the Elementary Science Study. Some of the scientists who developed the original atomic bomb resigned when they realised the terrifying power it released. Rather than continue, they put energy into encouraging children to ‘think’ so that the bomb would never be used again. At the same time, it may be considered ‘as an offspring of the confidence and optimism of those in the science community’. Of course, this is a simplified version, but many innovations were created and appropriate class lessons were developed. Interestingly, some of the creations would be classified as mathematics material today, but they served the same purpose - encouraging children to think. Pattern Blocks are one such creation. The acceptance by the primary school population was almost immediate and they are produced in most westernised countries.

Appropriate for use at all levels of Primary School It is recommended that children experience Patterns Blocks at all levels of the primary school. As the children gain experience, new insights will emerge as they manipulate the material. When developing some measurement concepts, such as area, use the blocks very sparingly. They were not designed for this purpose.

Hands On Heads On: A Mathematics Manipulatives Continuum

A classroom table or arrangement of two desks to allow seating for four children, allows a wide variety of activities and solutions to be carried out in the classroom at the same time. Children can share ideas while the teacher can easily check the wide range of thinking. We suggest a small but firm board is used on which to build models - they can be moved at the end of a session.

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When using Pattern Blocks, always have the camera on hand. Some of the creations might never appear again!

Design and Pattern: a never ending range of possibilities… Symmetry, Tessellations Geometry: angles, shapes Measurement concepts Number concepts

Pattern Blocks could be sorted into individual sets, so the children can take a set home. We have found that parents enjoy exploring the blocks.

Developing mathematics with Pattern Blocks by Swan and White is a definitive teaching guide. Reflections in a hinged mirror.

Requirements: 2000 pieces per class/group (8 sets of 250 pieces)

Coloured Rods (Cuisenaire) In 1953, Professor Caleb Gattegno (Cattegno) met George Cuisenaire, an innovative Belgian teacher. Over the previous quarter of a century, Cuisenaire was using, with success, a range of coloured sticks to aid the teaching/learning of arithmetic to children. Immediately, Cattegno saw the potential of these rods and a year later, in Britain, he established the Cuisenaire Company to distribute the material. Within a short time, Cuisenaire rods were used throughout the western world. One might say, “The rest is history!” Today, countless boxes, crates and cartons of Cuisenaire rods lie idle in storerooms or, as in many cases, have been dumped or destroyed. A new generation of teachers has little or no knowledge of the rods. It will be in the interests of the achievement of the students to find the rods and revive their appropriate use. New knowledge and understanding of how children learn mathematics can justify the revival of this most versatile material that gave children a rich concrete experience in mathematical exploration.

Appropriate for use in Junior Primary to Lower Secondary School

Coloured Rods are especially suitable for use in developing: Number awareness Fractions understanding Number bonds Partitioning

Fraction Ideas

Requirements: 1 container (250 pieces) per 3-4 students; that is, 8 containers per class group

Hands On Heads On: A Mathematics Manipulatives Continuum

Early Algebra Ideas

It is time we re-visit Cuisenaire Rods in order to give children a rich concrete experience in discovering mathematical ideas. (It was not the Cuisenaire rods that were unsuccessful!)

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MAB - Base Ten Blocks MAB (Multibase Arithmetic Blocks) were introduced to Australian teachers by Professor Zoltan Dienes as part of the Adelaide Project of 1960-2. Initially, students were introduced to blocks in Bases 3, 4, 5 and 6. Dienes argued, quite correctly, that these smaller denominations were easier for young children to handle because he was concerned that the students understood the Number System before being introduced to Base Ten. Around this time, the Mathematics Education craze which came to be known as “The New Maths” swept the country – unsuccessfully. Embedded in this new program was a heavy emphasis on work in bases other than 10 and converting them to Base Ten. This proved impractical and was abandoned. Other than Base 10 blocks, MAB in bases other than 10 was abandoned. Generally, today Base Ten blocks have been reduced to a material primarily used for demonstration purposes. A wide range of support materials are marketed and include magnetic MAB and place value arrows. The inherent developmental features of the blocks are ignored, both in print and practice.

Appropriate for use in Middle Primary to Junior Secondary School

Introduce this material after extensive Tactile Trading experience and introduction to Base 10 using bundling with pop sticks and Place Value charts Develop a common terminology throughout the school.

Base Ten Blocks are especially suitable for use in developing: • Four operations (After a thorough Trading experience) • Value relations eg. If a long has a value of 100; what is the value of the cube? • Decimal system • Decimal fractions (use with Coloured Rods to develop Percentage ideas)

Hands On Heads On: A Mathematics Manipulatives Continuum

The effectiveness of Base Ten Blocks is lost if they are used purely as a demonstration piece and are taken out of the hands of children.

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Appropriate containers contain 40+ Flats, 40+ longs and about 50 minis. This will provide enough for a group of 4 children. Important Tip: Make sure all your materials are the same dimensions. Some schools have mixed imperial with metric MAB and this causes problems when the children match up the pieces.

Developing mathematics with Base Ten by Swan and White is a definitive teaching guide.

Centicube (Centifit) The increase in interest in Mathematics Education in the 1960s encouraged the development of a wide range of teaching materials. In Britain, E.S. Perry, home of the Osmiroid pens, created the quality suite of Osmiroid educational materials. The Super Beamer Balance was designed for use by young children and Centicube, the 1 cm³ interlocking cube which weighed 1 gram (near enough), was moulded. For years, it was promoted as an essential ‘hands-on’ item for use in the teaching of mathematics. Because of its success, manufacturers in several countries copied it but tight patent laws beat such opposition until the mid-1980s. Now, there are several look-a-likes but, unfortunately, most are incompatible with the original and other brands of 1 cm cubes. There is nothing more frustrating than working with materials that are not compatible. So, warning, when your school chooses a 1 cm cube, make sure the same model is purchased when you re-supply, or you may simply start over again. A wide range of printed support material may be found in ‘older’ mathematics resources collections.

Appropriate for use in Upper Primary to Junior Secondary School Centicube is especially suitable for use in developing: • Spatial Investigations • 3D modelling to isometric drawings spatial visualisation

These 1 cm³ plastic cubes are suitable for creating three-dimensional shapes. Weighing a gram (with a 5% tolerance), the mathematical attributes of these cubes may be applied to many measurement topics. These are not really suitable for use in Early Childhood classrooms. It is a fair comment that, initially, fingers become sore after using the cubes. It is recommended that students are permitted to play freely with them.

• Measurement activities: volume, capacity, mass, etc. • Pre-algebra challenges

Work out how many Centicube cubes are used to make each model. Is there a rule which would work if the base was 12 x 12, not 10 x 10?

2 Bar - 3 Bar Investigations By joining a 2 bar and a 3 bar; create as many different shapes as possible.

Requirements: Centicubes come in containers of 1000 cubes. In most circumstances, one container for 4 to 6 students will be adequate. There is a wide range of support materials for ideas.

Hands On Heads On: A Mathematics Manipulatives Continuum

See inside a cube

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Shared Materials (Early Childhood)

D Mobilo

evelopmental Play (Construction- type materials)

Lego or Coko

Construction (Clever) Sticks

Interstar

Kapla Blocks

and a range of extras

Junior Engineer

These are samples of the many construction kits available. Limit the choice but make sure you have good quantities of each item.

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idening the range of mathematical experiences Geoboards

A small set of simple geoboards (5x5 pins) will provide students with geometric shape experiences.

Pegboards

Colour Cubes

Beneficial for creating number patterns and bonds.

Ideal for additional patterning experience.

These are listed as samples of suitable materials: preference may be shown for other materials, especially those already on hand.

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orking Mathematically

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Concepta & similar products

What’s in the square? Introduce matrix logic at an early stage

Attribute Blocks

and support materials. Children need to experience this material on a regular basis.

After being taught how to use these materials, children will be competent to use them without a lot of teacher supervision. Important thinking skills are strengthened with their use.

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orting and classification materials

Is there an end to the list of both commercial and non-commercial materials that may be used for sorting? Regularly vary items for sorting, but have a sorting area as a permanent fixture in the classroom.

• Water play and sand play are considered a natural part of early childhood classrooms: why can’t all children in this area continue their kindergarten experience? • Unit (Project) Blocks are available in most kindergartens. Consider extending their availability to all early childhood grades. The mathematical experience is invaluable.

Buttons

Keys

Bears

Rocks

• Light boxes and mirror reconstructions will be another vital developmental play area.

Rotated/Shared Materials (Middle to Upper Primary)

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orking Mathematically

Students who have not been introduced to matrix logic principles need to become familiar with ‘What’s in the Square?’ and ‘What Else is in a Square?’. Attribute Blocks will “visit” classrooms on a regular basis. A wide variety of activities and challenges may be offered to students in upper primary classes

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Attribute Blocks

idening the range of mathematical experiences Geoboards

Mira

Geo Shapes

Tessellation

There is a wide variety of this genre of construction material, for example: Polydron, Clixi, Polyfit, Lokko, Googolplex, Klikko. Any of these materials provide a sound geometrical experience.

Tessellation pack: a variety of tessellating shapes. Mosaics: while used in the early childhood classrooms more sophisticated applications may be offered.

Constructo Straws, Orbit

A variety of geometric puzzles

As an introduction to perimeter and area as well as geometrical concepts, the geoboard is available in square, circle and isometric form.

Using straws and connectors, three dimensional shapes may be created. Orbit is more sophisticated.

Probably most appropriately used in middle primary and lower secondary classes, this is a valuable aid to teach symmetry. Also include flexible and hinged mirrors.

Pentominoes Soma Cubes Tangrams and other similar items.

Construction Materials

Think About!

• Classrooms have access to water and/or sand to support activities in mass and volume. • Pliable clay or plasticine will be accessible at all times.

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easurement Materials

Appropriate collections of measurement materials are best stored in large plastic containers that may be moved easily to a classroom. Many measurement items will be placed permanently in the classroom – see Class Kit Materials. A Checklist of Measurement Materials (with suggested minimum numbers of each item) Bow calliper x 4 Graduated calliper x 4 Metre rule x 8 (Metline Metre and Quadrat Links make a suitable substitute) Tape measure 20 m x 2 Tape measure 2 m x 10 Rulers 30 cm x 10 Trundle Wheel x 2

Balance scale x 4 Kitchen scale x 2 Bathroom scale x 2 Spring scales – a variety Masses (weights): plastic, metal



Litre Set Volumetric shapes Graduated cylinders Measuring cups and spoons Measuring jugs Displacement jug Geometric Solids

Hands On Heads On: A Mathematics Manipulatives Continuum

Some of the materials used in the Early Childhood classrooms could ‘visit’ the middle primary classes, for example: Mobilo, Kapla blocks, Gears, Knex, etc.

Access to a sand and/or water tray is recommended.

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(Suggested) Items to be included in Classroom Kits

The need for teachers to ‘search’ for manipulative materials was a common complaint. As a compromise, a Classroom Kit is suggested. Each classroom has a collection of frequently used items allocated to the room for the school year. This list is limited but it will be supported by the “Key” and “Rotated” materials which are delivered to the classroom on regular intervals.

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ower Primary Classroom Kits

Playing cards, dominoes, dice, popsticks and counters are used frequently to support the development of number understanding. A Balance Scale is always on hand: children benefit from ‘working-out’ balancing arrangements. A large Thermometer enables a constant check on the temperature. Encourage children to keep records. All children will have their own calculators. Play Money on hand means a shop can operate.

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iddle-Upper Primary Classroom Kits

Items more appropriate for older children will be included in this kit. Most of what is included in the Early Childhood kit will still be used.

Hands On Heads On: A Mathematics Manipulatives Continuum

Geometric puzzles, such as Pentominoes, Soma Cubes and Tangrams may be included.

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One or two games, such as Numero or Tantrix, are played as focus games in a classroom; that is, children learn how to play the game with skill. For symmetry work (a constant interest for children) it is recommended that Mira and a small collection of mirrors be included. Some teachers recommend that real money (especially coins!) be used instead of play money.

Digital vs analogue clocks

Many classroom teachers recommend that in each classroom both an analogue and a digital clock be installed permanently. Children will have experience of using both clocks. An idea: in one week all the time readings are spoken in analogue terms; the next week, all the times are spoken or recorded in digital terms. Also you can have the occasional week using 24-hour time. By having the two clocks sideby-side, children become efficient at reading both clocks, appreciating the different notations. Older children may become responsible for time allocations in the classroom.

A sample Key Materials Rotation Plan Staff members would receive this plan at the beginning of the school year. Early Childhood Term 1 Weeks 1-3 Weeks 4-6 Weeks 7-9 (10)

Middle Primary

Upper Primary

Pre-P

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7 Measure -ment Materials

The Brick

Unifix

Pattern Blocks

Coloured Rods

Pattern Blocks

Centifit

MAB Base 10

Pattern Blocks

The Brick

Unifix

Pattern Blocks

Coloured Rods

Measure -ment Materials

Centifit

MAB Base 10

Unifix

Pattern Blocks

The Brick

Coloured Rods

Pattern Blocks

MAB Base 10

Measure -ment Materials

Centifit

• A similar plan is created for Terms 2, 3 and 4. • Items in each rotation will depend on the equipment in the school. It is recommended that all small collections are collated into one (or two) large collections. • Older children can be made responsible for the change over of materials. • Items such as The Brick and Pattern Blocks may become Rotated/Shared materials in the Middle and Upper sections of the primary school.

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athematics manipulatives in action…

Students handle the containers of manipulative materials with ease.

Hands On Heads On: A Mathematics Manipulatives Continuum

Support materials and teacher resources may travel with the containers.

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Thank you! Many teachers and students have contributed to the creation of this booklet. The sharing of ideas and the consequent development of those ideas is happening in many classrooms; as well, children are continually making important contributions to our overall experience of the use of manipulatives in mathematical learning. Many schools have a legacy of the acquisition of manipulative materials, much of this legacy being located in storerooms. We hope this booklet will encourage you to explore the storerooms and cupboards of your school; amongst the variety of materials there will be items which will have a worthwhile place in any mathematics manipulative materials organisational plan.

As a means of helping children develop mathematical understandings, the use of concrete materials or mathematics manipulatives such as Pattern Blocks and MAB have been advocated since the early 1960’s. Arising from this notion, is the often used flimsy justification that the children are involved in “hands on” learning.

Hands On Heads On: A Mathematics Manipulatives Continuum

A research team from Edith Cowan University decided to examine the extent to which mathematics manipulatives are being used, how they are being used and the outcomes associated with their use. This document has been produced in response to some of the findings in the research. As well, based on the data collected, research papers have been presented by various members of the team in different educational conferences throughout the world.

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Many teachers participating in this research expressed interest in learning how to manage and organise mathematics manipulative materials, not only in their own classroom but also throughout the school. Hence the genesis of A Mathematics Manipulatives Continuum. The research team acknowledges the support of our research partners: the Association of Independent Schools of Western Australia (AISWA), Edith Cowan University and RIC Publications.

Edith Cowan University © 2009 All rights reserved This document may be photocopied in its entirety for educational purposes. Written by: Geoffrey White, Dr Paul Swan, Linda Marshall Graphic Design: Kinyon Design Key photography: Alisha Hughes

Further information: [email protected] [email protected]

This copy distributed to schools with the compliments of : R.I.C. Publications Pty. Ltd. www.ricpublications.com.au