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 and investigation

• 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 children apply mathematical thinking to real-life scenarios—such as sharing snacks or comparing lengths—they begin to use mathematics to make sense of the world, laying the groundwork for future modelling skills. Informal investigations, such as counting classroom objects, measuring ingredients, or noticing variations, foster an intuitive sense of data collection, variability, and uncertainty, forming a basis for later statistical investigations.

As they work with the exercises of the senses, children are making judgements in relation to distance, dimension, graduation, identity, similarity and sequence. These experiences support early development of pattern recognition, abstraction, and logical reasoning, building the foundations of their computational thinking.

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).

These foundational skills are taught using Montessori materials that make use of manipulatives, physical patterns, and structured repetition. 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. This material supports the heightened sensitivity for number children tend to experience around the age of four. Using this material in active and enjoyable games, children learn to add, subtract, multiply and divide and provide an early introduction to computation as a process—where operations and transformations are applied to mathematical objects to produce meaningful outcomes.

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. This step begins by bridging the gap between visual/physical representation and abstract notation, preparing for future algorithmic understanding.

3.  Association of the Concrete with the Symbols: With accurate language as the link, children begin to make connections between the materials and their symbolic representations—supporting early mathematical communication and conceptual generalisation.

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. Repetition strengthens fluency, pattern recognition, and logical sequencing—skills that support the future use of algorithms and computational strategies. Materials are self-correcting, so that children receive immediate feedback that allows them to learn from mistakes directly instead of having to wait for the teacher to correct their work.

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. They 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 on which they are working and 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.

While digital tools are typically not central to the 3–6 environment, Montessori sensorial and mathematical materials function as manual computational tools, helping children visualise and manipulate mathematical operations and develop the foundations of computational thinking. Later, this hands-on understanding supports purposeful engagement with digital platforms and simulations.

Children progress through the mathematics curriculum at their own pace. The Montessori method respects each child’s natural developmental trajectory, allowing for deep engagement, personal interest, and mastery before advancement. This also means children build mathematical proficiency and process skills in an integrated, interconnected, and experiential way—preparing them not just for academic success but also for real-world problem-solving and participation in a technologically complex society.

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

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