Mathematics: Ages Three to Six

The study of mathematics is a reflection of the human tendencies for investigation and orientation, for order and classification, for reasoning and making judgements, and for calculating and measuring. In the Montessori Children’s House, when mathematical concepts are first presented to children, they are embodied in concrete materials.

Mathematics in the Children’s House builds on and extends the exercises of practical life and the exercises of the senses, as well as the many mathematical experiences children encounter incidentally in their daily lives, including experiences with:

  • visual representation of mathematical concepts

  • pattern and order

  • problem solving

  • cardinal and ordinal numbers

  • place value

  • operations (addition, subtraction, multiplication, division)

  • fractions

  • relations e.g., placement of objects, spatial patterns, one-to-one correspondence of objects and two-dimensional shapes

  • measurement e.g., length, mass, time, temperature, volume, perimeter, area

  • word problems (addition, subtraction, multiplication, division)

As they work with the exercises of the senses, children are making judgements in relation to distance, dimension, graduation, identity, similarity and sequence. Building on this foundation, the Montessori mathematics materials introduce children to:

  • counting (from 1 to 10, 10 to 90, linear 1 to 100 and 1000, and skip-counting as an introduction to multiplication)

  • place value to four digits

  • number operations (addition, subtraction, multiplication, division).

The base ten number system is represented for children in concrete form using golden beads organised so they vary simultaneously in quantity, size, mass and geometric shape. Children are also given the corresponding symbol for each quantity. In this way, children experience relations between the hierarchies of the system in multiple ways. Using this material in active and enjoyable games, children learn to add, subtract, multiply and divide. This material supports the heightened sensitivity for number children tend to experience around the age of four.

The progression of the Montessori mathematics curriculum in the Children’s House follows a five-step sequence.

1.  Introduction of Concrete Materials: (The quantity is presented in isolation.)

Concepts are presented in a concrete form children can manipulate. Children are given accurate language to talk about the concrete impressions. Only after they have experienced the concrete material are they given the symbolic mathematical notation.

2.  Introduction of Symbols: (The visually recognised symbols are offered in isolation.)

When the child is comfortable with the concrete representation and the oral language, mathematical symbols are introduced.

3.  Association of the Concrete with the Symbols: (Accurate language is the link.)

Only after the child has completed the first two steps are the concrete materials and symbols combined.

4.  Practice

After being presented with new information children need the opportunity for repetition. Children are offered a way to practise using and remembering this new knowledge and integrating it with what is already known. They have the opportunity to build and consolidate the knowledge through use of the materials, until it becomes automatic.

5.  Self-Assessment

Children are given exercises to affirm or verify their own understanding and knowledge, and to establish whether they are ready for the next exercise.

Children work through the Montessori mathematics exercises and games in a supported and incremental way. There is no pressure for children to move onto a new topic before mastering the one they are working on. Students are free to progress through the mathematics curriculum following their own interests, so not all children complete the activities in the same sequence or in the same timeframe.

An overview of the mathematics curriculum is represented in the following diagram:

overview of the mathematics curriculum1024_1.jpg

Jump to section:

1.MG.010 Language of mathematics

1.MG.020 Mathematical concepts: indirect preparation

1.MG.030 Quantities and symbols 1 to 10

1.MG.040 Decimal system

1.MG.050 Language of Numbers larger than 10

1.MG.060 Counting: continuation

1.MG.070 Operations

1.MG.080 Expanding the decimal system: beyond 1000

1.MG.090 Memorisation

1.MG.100 Geometry

1.MG.110 Algebra

1.MG.120 Time and sequence

1.MG.130 Fractions

Language of mathematics 1.MG.010

 Knowledge, Skills and Understanding

Typically children will:

01. Understand mathematical terms, for example, long/short, more/less, share

＋ Materials and Activity

Activities include:

  • everyday tasks
  • preparing food
  • gardening
  • singing and finger plays.

Resources are the sets of objects in the environment.

  • I spy game

＋ Links to the EYLF & Australian Curriculum (V9)

AC9EFLA08
AC9MFM01

EYLF 4.4.1

Mathematical concepts: indirect preparation 1.MG.020

 Knowledge, Skills and Understanding

Typically children will:

01. Know materials have a purpose

02. Understand beginning, middle and end

03. Have established work habits

04. Experience mathematical concepts incidentally, informally and/or indirectly e.g., spatial awareness, one-to-one correspondence, geometry, classification, measurement, set, order, seriation, sequence, matching, grading

＋ Materials and Activity

Activities include:

  • movement activities for development of gross and fine motor skills
  • practical life exercises
  • exercises of the senses
  • language/communication activities
  • singing and finger plays
  • classifying, sorting, matching, sequencing and grading
  • everyday tasks (one-to-one correspondence of, for example, sock to foot, cutlery and place on table setting)
  • preparing food, including measuring, sequencing actions
  • puzzles, construction, stereognostic bags.

Resources are:

  • the sets of objects in the environment.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFSP01
AC9SFH01
EYLF 1.2.1

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EYLF 1.2.1

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EYLF 4.2.1

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EYLF 4.2.1
EYLF 5.3.2

Quantities and symbols 1 to 10 1.MG.030

 Knowledge, Skills and Understanding

Typically children will:

01. Learn to recognise, sequence and count numbers 0 to 10

02. Experience and understand the function of ‘zero’ as an empty set and as a place-holder

03. Experience 9 as a counting limit

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • numbers to 10 games and exercises.

Resources include:

  • number rods and cards
  • sandpaper numbers
  • spindle box
  • cards and counters
  • memory game of numbers
  • games with zero.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFN01
AC9MFN02
EYLF 4.4.1

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Decimal system 1.MG.040

 Knowledge, Skills and Understanding

Typically children will:

01. Understand the concept of units, tens, hundreds and thousands

02. Make and read composite numbers.

＋ Materials and Activity

Activities include:

  • forming numbers with beads and number cards. Resources include:
  • golden bead material and number cards.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9M1N02
AC9M2N02
AC9MFN04
EYLF 4.4.1

AC9M1N02
EYLF 4.4.1

Language of numbers larger than 10 1.MG.050

 Knowledge, Skills and Understanding

Typically children will:

01. Compose and count ‘teen’ and ‘ten’ numbers to 100

02. Use the language of ‘teen’ numbers and ‘tens’ to 100

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with teen and ten numbers.

Resources include:

  • teen boards and coloured beads
  • ten boards and coloured beads.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9M1N01
AC9MFN03
EYLF 4.4.1

AC9EFLA08
EYLF 4.4.1

Counting: continuation 1.MG.060

 Knowledge, Skills and Understanding

Typically children will:

01. Count numbers 1-10, 10-19, 1-100, 100-1000

02. Recognise recurring patterns across hierarchies

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • counting games and exercises.

Resources include:

  • teen boards and coloured beads
  • ten boards and coloured beads
  • 100-chain, 1000-chain and skip counting chains (1-9) in the chain cabinet.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9M2N01
EYLF 4.4.1

AC9MFA01
EYLF 4.4.3

Operations 1.MG.070

 Knowledge, Skills and Understanding

Typically children will:

01. Experience, understand and apply the operations and their relationships (addition, subtraction, multiplication, division)

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with concrete material.

Resources include:

  • golden bead material and number cards
  • dot game
  • stamp game
  • word problems.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFN05
AC9M2A02
AC9M3A01
EYLF 4.4.1

Expanding the decimal system: beyond 1000 1.MG.080

 Knowledge, Skills and Understanding

Typically children will:

01. Recognise number patterns recurring across hierarchy ‘families’: simple, family of thousands, family of millions

02. Experience and work with large numbers to millions, consolidating earlier work with addition subtraction, multiplication and division

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with concrete material.

Resources include:

  • wooden hierarchical material
  • small and large bead frames
  • racks and tubes materials

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFA01
EYLF 4.4.1

AC9M4N02
EYLF 4.4.1

Memorisation 1.MG.090

 Knowledge, Skills and Understanding

Typically children will:

01. Memorise and apply the essential number facts for addition, subtraction, multiplication and division

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with concrete material.

Resources include:

  • snake games
  • addition and subtraction strip boards and finger charts
  • box of bead bars
  • multiplication bead board and charts
  • division bead board and charts.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFN05
AC9M1N05
EYLF 4.4.1

Geometry 1.MG.100

 Knowledge, Skills and Understanding

Typically children will:

01. Recognise and explore 2- and 3-dimensional shapes and their relationships

02. Learn and apply accurate terminology

03. Experience, discover and explore the constructive power of triangles

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with concrete material and objects in the environment.

Resources include:

  • simple puzzles
  • folding cloths
  • shapes in the inside and outside environments
  • geometry cabinet and cards
  • botany cabinet and cards
  • geometry solids and bases
  • superimposed geometric figures/graded geometric figures
  • constructive triangle boxes.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFSP01
EYLF 4.4.2

AC9EFLA08
EYLF 4.4.2

AC9MFSP01
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EYLF 4.2.1
EYLF 4.4.2

Algebra 1.MG.110

 Knowledge, Skills and Understanding

Typically children will:

01. Prepare indirectly for algebra

＋ Materials and Activity

Activities include:

  • precise and engaging demonstrations offered by the adult
  • games and exercises with concrete material.

Resources include:

  • binomial cube
  • trinomial cube
  • sensorial decanomial
  • constructive triangle boxes.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFA01
EYLF 4.2.1

Time and sequence 1.MG.120

 Knowledge, Skills and Understanding

Typically children will:

01. Experience sequence, order and routine

02. Begin to understand the concepts of today, yesterday, tomorrow, past, present, future

03. Begin to tell the time using o’clock, half past, quarter to, quarter past

＋ Materials and Activity

Activities include:

  • daily and weekly routines
  • experiencing order and sequence in the environment
  • celebrating seasons and festivals.

Resources include:

  • clocks and card material
  • calendars.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFM02
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AC9MFA01
EYLF 4.4.1

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EYLF 4.4.3

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Fractions 1.MG.130

 Knowledge, Skills and Understanding

Typically children will:

01. Experience cutting up and sharing a whole

02. Learn the language of fractions (quantities <1)

03. Begin to use operations (addition, subtraction, multiplication, division) with fractions

＋ Materials and Activity

Activities include

  • preparing food
  • sharing.

Resources include:

  • fraction circle insets and labels.

＋ Links to the EYLF & Australian Curriculum (V9)

AC9MFN06
AC9M2N03
AC9SFI01
EYLF 4.4.1

AC9EFLA08
EYLF 4.4.1

AC9M2N03
EYLF 4.4.2

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