Mathematics: Ages Nine to Twelve
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Mathematics Overview
The Children’s House provides children with a strong foundation on which to build the more advanced mathematical understandings they gain in Montessori environments prepared for children over the age of six. In the Children’s House children work with a variety of concrete materials embodying mathematical concepts. When the hand and the mind work together, young children are able to absorb concepts, sometimes without the need for words.
Each Montessori mathematics lesson, in both the Children’s House and in environments prepared for children over six, has two aims, one indirect and one direct. The direct aim is the immediate purpose that is obvious when the concrete materials are presented to the children. The indirect aim, often a more abstract aim, is achieved when children work with the concrete materials independently. The indirect aim prepares children for more advanced concepts they will meet later in the curriculum. In the Montessori view, when children work first with concrete materials and move at their own pace toward abstraction with paper and pencil, or towards mental arithmetic, they internalise concepts at a deeper, more lasting level than if they had memorised them by rote.
Beyond the age of six children continue to learn mathematical principles through the manipulation of concrete materials but they do not want to be tied to concrete materials and strive towards the freedom of working at the abstract level. The concrete materials become stepping-stones only, keys to open the door to abstraction whenever a child is ready. By generalising from their experience with the concrete materials, children over the age of six are able to work out mathematical formulae and definitions for themselves. Montessori teachers do not offer help too early but wait until children have had a chance to work with a problem and come up with a few solutions for themselves. Children also encounter problems to use in their independent work in the form of command cards prepared for all areas of the mathematics curriculum.
In the study of mathematics at this age, children use quantities with precision. They also reason using logic and abstract patterning based on observation and imagination. Children work with measurements, patterns, sequences and mathematical relationships, applying these concepts to practical projects. They learn that the creative potential of mathematics is as great as its potential to order and to provide understanding. In the Montessori view, constructing a ‘mathematical mind’ in this way is the birthright of all children.
The Montessori mathematics curriculum extends children aged from six to twelve further than is normally expected of children at this age and stage. For example, from the age of six children, explore and practise operations with the culture’s system of numeration, the decimal system. Once children have mastered this system, usually by the age of nine, they are ready to examine other number systems, with bases other than ten, from both a mathematical and a historical perspective. Children may even perform operations in number systems with bases other than ten. Extension activities of this type allow children to exercise their mental capacities as well as consolidate and reinforce their existing understanding.
As children progress through the Montessori mathematics curriculum, they learn to make connections between concepts. For example, they learn to apply compound multiplication to fractions and decimals, or progress from the four operations of addition, subtraction, multiplication and division to squaring and cubing, and to solving square roots and cube roots.
History of Mathematics 3.MA.010
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore stories about the history of numbers and mathematics
.02 Research ancient number systems
First Nations Australians
Babylonian
Egyptian
Mayan
Chinese
Hindu-Arabic
Roman
.03 Explore the history of the number system used in our culture: the decimal system
.04 Explore and research history of mathematical invention and discovery
.05 Study ancient and modern mathematicians
.06 Study ancient and contemporary numeration systems other than the decimal system
＋ Materials and Activity
Memorisation: review and extension 3.MA.020
Knowledge, Skills and Understanding
Typically, children will:
.01 Review and consolidate knowledge of number facts (addition, subtraction, multiplication, division)
.02 Develop automaticity, speed and accuracy of number fact recall and application to mastery level
.03 Understand the connection between multiplication and division as inverse operations and use this to develop families of number facts
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Multiples of Numbers 3.MA.030
Knowledge, Skills and Understanding
Typically, children will:
.01 Complete, review and consolidate knowledge of multiples
＋ Materials and Activity
LCM and GCF (HCF) 3.MA.040
Knowledge, Skills and Understanding
Typically, children will:
.01 Find lowest common multiple (LCM)
.02 Find greatest (highest) common factor (GCF/HCF)
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Divisibility 3.MA.050
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore, formulate and recognise rules for establishing divisibility of numbers
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
All Operations 3.MA.060
Knowledge, Skills and Understanding
Typically, children will:
.01 Review and consolidate knowledge of operations with whole numbers
.02 Master operations with whole numbers
.03 Perform complex operations with whole numbers
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Multiplication 3.MA.070
Knowledge, Skills and Understanding
Typically, children will:
.01 Extend skills in mental multiplication
.02 Develop skill in cross multiplication
.03 Demonstrate proficiency in mental calculation
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Division 3.MA.080
Knowledge, Skills and Understanding
Typically, children will:
.01 Complete and master concept and process of distributive division with multi-digit divisors
.02 Complete and master concept and process of group division with multi-digit divisors
.03 Develop skill in division with zero in dividend and divisor
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Fractions 3.MA.090
Knowledge, Skills and Understanding
Typically, children will:
.01 Review and consolidate knowledge of fractions
.02 Build mastery in the use of fractions
.03 Review and consolidate knowledge of equivalence
.04 Review and consolidate ability to reduce fractions to lowest terms
.05 Review equivalent fractions and variation in notation for equivalent fractions
.06 Explore the following types of fractions and corresponding notation:
proper and improper fractions
mixed numbers
unlike fractions
.07 Find the lowest common multiple of unlike denominators
.08 Review adding and subtracting fractions with same denominators
.09 Add fractions with unlike denominators
.10 Subtract fractions with unlike denominators
.11 Multiply a whole number by fraction
.12 Multiply a fraction by a fraction
.13 Divide a whole number by a fraction
.14 Divide a fraction by a fraction
.15 Divide fractions using group division
.16 Perform complex operations with fractions
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Decimals 3.MA.100
Knowledge, Skills and Understanding
Typically, children will:
.01 Build knowledge of decimals to millionths (0.000001)
.02 Use notation for decimals to millionths (0.000001)
03 Add and subtract decimals using concrete materials
.04 Add and subtract decimals abstractly, using estimation, rounding and digital tools to check
.05 Multiply and divide decimals, using concrete materials to solve
.06 Multiply and divide decimals by powers of 10 without materials, using estimation, rounding and digital tools to check
.07 Multiply and divide decimals abstractly, using estimation, rounding and digital tools to check
.08 Explore repeating and non-repeating decimal numbers
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Percentage 3.MA.110
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore the concept of percentage and compare to fractions and decimals
.02 Apply knowledge of percentages to solve practical problems s
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Powers of Numbers 3.MA.120
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand the concept of powers of numbers
.02 Express any number as a power of 10
.03 Sequence powers geometrically
.04 Use exponential notation
.05 Use powers in operations
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Squaring 3.MA.130
Knowledge, Skills and Understanding
Typically, children will:
01. Review and consolidate knowledge of squaring
02. Extend knowledge and application of squaring
03. Perform operations with squares (addition, subtraction, multiplication, division, combined operations)
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Squares of Binomials and Trinomials 3.MA.140
Knowledge, Skills and Understanding
Typically, children will:
.01 Review and extend squaring of binomials (a+b)2 and trinomials (a+b+c)2
.02 Use binomial and trinomial representation and notation to expand squares of 2- and 3-digit numbers
.03 Find products of binomial and trinomial squares:
units only
tens and units
.04 Consolidate and extend notation of the squares of 2- and 3-digit numbers represented as binomials and trinomials
.05 Identify and manipulate the terms of expanded binomial (a+b)2 and trinomial (a+b+c)2 expressions
.06 Write and solve equations that include squares of numbers and sums of binomials and trinomials squared
＋ Materials and Activity
Square Roots 3.MA.150
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and work with the concept of square root of a number
.02 Determine the square root of a 2- and 3-digit number represented as a binomial or trinomial
.03 Find square roots abstractly
＋ Materials and Activity
Cubing 3.MA.160
Knowledge, Skills and Understanding
Typically, children will:
.01 Build knowledge of the concept of the cube of a number
.02 Cube successive numbers using concrete material
.03 Perform cubing calculations
＋ Materials and Activity
Cube Root 3.MA.170
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and work with the concept of cube root of a number
.02 Determine the cube root of 2- and 3-digit numbers represented as a binomial or trinomial
.03 Find cube roots abstractly
＋ Materials and Activity
Algebra 3.MA.180
Knowledge, Skills and Understanding
Typically, children will:
.01 Find unknown values in numerical equations involving addition and subtraction
.02 Identify, explain and create addition and multiplication patterns using algorithms
.03 Create and use algorithms to experiment with factors, multiples and divisibility
.04 Find unknown values in numerical equations involving multiplication and division
.05 Use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems, choosing operations and efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation
.06 create and use algorithms involving a sequence of steps and decisions that use rules to generate sets of numbers; identify, interpret and explain emerging patterns
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Positive and Negative Numbers 3.MA.190
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand the concept of integers
.02 Explore positive and negative numbers in different contexts (temperature, elevation, financial)
.03 Use positive and negative numbers in operations
.04 Locate points in the 4 quadrants of a Cartesian plane; describe changes to the coordinates when a point is moved to a different position in the plane
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Rational and Irrational Numbers 3.MA.200
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore the concept of rational and irrational numbers
.02 Recognise and use notation for rational and irrational numbers
.03 Perform calculations with rational and irrational numbers
.04 Use mathematical modelling to solve practical problems involving natural and rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and efficient calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, justifying the choices made
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Data and Probability 3.MA.210
Knowledge, Skills and Understanding
Typically, children will:
.01 Review and extend knowledge of data collection, representation and interpretation
.02 Identify and extend patterns in data and number
.03 Use vocabulary for talking about data and data representation: set, average, mean-median-mode; quantity-category, continuous-discrete; cluster-outlier; sample-census, spreadsheet, database, distribution, frequency, range, statistics
.04 Construct simple line graphs to represent continuous change and use vocabulary for talking about the parts of line graphs (axes, horizontal axis, vertical axis, scale, point, plot)
.05 Conduct statistical investigations, collect and interpret data, choose relevant displays and communicate results. Build a repertoire of data collection and representation techniques and tools, including tables and matrices, bar graphs and pie graphs, flowcharts, electronic database and spreadsheet
.06 Interpret and compare data sets for ordinal and nominal categorical, discrete and continuous numerical variables using comparative displays or visualisations and digital tools; compare distributions in terms of mode, range and shape
.07 Identify statistically informed arguments presented in traditional and digital media; discuss and critique methods, data representations and conclusions
.08 Describe possible everyday events and the possible outcomes of chance experiments and order outcomes or events based on their likelihood of occurring
.09 Conduct repeated chance experiments including those with and without equally likely outcomes
.10 Recognise that probabilities lie on numerical scales (e.g., of 0-1 or 0%-100%) and use estimation to assign probabilities that events occur in a given context, using fractions, percentages and decimals
＋ Materials and Activity
＋ Links to Australian Curriculum (ACv9)
Digital Systems 3.MA.220
Knowledge, Skills and Understanding
Typically, children will:
.01 Identify and explore a range of digital systems with peripheral devices for different purposes, and transmit different types of data
.02 Collect, access and present different types of data using simple software to create information and solve problems
.03 Define problems with given or co-developed design criteria and by creating user stories
.04 Explore how digital systems use numbers and data can be represented by zeros and ones
.05 Explore the digital instructional language of coding, including algorithms involving sequencing, comparison operators (branching) and iteration
.06 Evaluate existing and student solutions against the design criteria and user stories and their broader community impact
.07 Implement algorithms as visual programs involving control structures, variables and input
.08 Select and use appropriate digital tools effectively to share content online, plan tasks and collaborate on projects, demonstrating agreed behaviours
.09 Explain the creation and permanence of their digital footprint and consider privacy when collecting user data