#### Action Stories

Stories involving number in context, illustrated by the manipulation of concrete materials, eg the teacher asks “If there were three cars in the shed (indicates 3 blocks together), and one drove away (move one block away), how many would be left in the shed? (indicates remaining blocks)”. Alternatively, teachers tell stories while students model actions, or get students to both tell stories and model actions.

#### Additive Thinking

Students who have not yet developed multiplicative thinking will tend to approach multiplication calculations using addition as their main strategy. They may count a grouped collection of objects by ones, without reference to the group structure. An intermediate step between additive and multiplicative thinking is repeated addition through stress counting or skip counting, and double counting (eg a student may solve 3 x 4 by counting 1, 2, 3, **4**, 5, 6, 7, **8**, 9, 10, 11, **12 **while keeping track of the number of ‘groups’ of four on their fingers).

#### Area Model

See ‘Partitioning’

#### Arrays

A rectangular (including square) arrangement of drawn or physical objects, ie a collection which has been arranged into rows and columns. Arrays are particularly useful for helping to build multiplicative thinking and an understanding of multiplicative commutativity in students.

#### Base 10

If a numbering system had to provide a unique designation to each number without using some sort of pattern system, it would be quickly become unworkable as numbers increased. Base 10 systems employ a pattern of ten to describe quanitities. The most widespread Base 10 system in use today effectively consists of ten initial number names. When numbering collections larger than ten, each count of ten is unitised. So a count of one more than ten, is designated "ten and one", two more than ten is "ten and two" and so on until the second count of ten is reached and the number becomes "two tens", then "two tens and one" etc. When a count of ten tens is reached, it is generally given a new unit designation ("Hundred" in English).

Different languages have different ways of encoding, and sometimes of obsfucating, this pattern. In English for example, the word 'eleven' (designating "ten and one") does not fit linguistically into any pattern.

The current Base 10 system is used in conjunction with a positional or Place Value system allowing an efficient written expression of quantities by using symbols for the numbers one to nine with an additional symbol for zero. Not all Base 10 systems are positional however. The Grecian Ionic system for example had an additional set of names/symbols for the 9 multiples of 10 (10-90) and again for 100 to 900 so that a Place Value system for writing numbers was unneccesary.

#### Builds on from known

Students use a known fact as a basis for further calculation, for example if a student knows the total for 3 fours, but has to calculate the total for 6 fours, they use the known fact (3 fours are 12) and double it to get the answer for 6 fours (24).

#### Collection

A specific group of items or objects. It can include both physical objects such as counters or printed elements such as dots. The word *collection* is often used to distinguish between a *number of items* as opposed to a written or symbolic number or *numeral*.

#### Combinations (Cartesian product problems)

Combinations problems involve the pattern ‘x of these for each of those’, eg “If Alice has two pairs of shorts and three t-shirts which all go together, how many different ways could she wear them?” The answer would be 6; three t-shirts for each pair of shorts.

#### Commutativity

The idea that when either multiplying or adding two or more numbers, the order in which they are multiplied (or added) doesn’t matter, eg 5 x 3 = 3 x 5 and similarly 5 + 3 = 3 + 5. Note: If more than one ** type** of operation is used in an equation, the order

**matter, eg (3 x 5) + 4 ≠ (5 + 4) x 3**

*does*#### Compares (and orders)

Compares two quantities in terms of size, eg determining which of two numbers (or representations of numbers) is bigger. Ordering is where several numbers (or representations) are arranged in order of size – generally from smallest to largest.

#### Concealed

Students move from counting concrete markers such as counters (perceptual), to being able to determine a total where some of the items are concealed (figural or figurative), eg a student is told that there are 5 counters in a sealed container, or they are shown 5 counters which are then covered with a piece of card. They are then given 6 more counters and have to determine how many “altogether”

#### Concrete Materials

Also known as Manipulatives, concrete materials are tangible objects used to model ideas and operations in maths instruction. They can assist in the process of concept development by allowing the physical expression of number and maths processes. This allows students to construct understandings experientially and from first principles which prepares them for a move to more abstract forms of expressing and manipulating number. Examples of concrete materials include counters, unifix cubes and MAB (Multi-Arithmetic Base) blocks.

#### Counting

Counting is a complex activity which involves three main understandings; that numbers are named in a fixed sequence, that one number is counted per object and that the last number counted names the count. Initially children tend to believe that counting is just about a number sequence (ordinal concept), however they eventually realise that is also about naming a specific quantity (cardinal concept).

#### Counting On

When adding two numbers, students start with the larger (or sometimes the trusted) number and count on from this to find the total, eg to add 12 and 3, the students would start counting from twelve and count on three more (ie “thirteen, fourteen, fifteen”). Counting On is not reccomended as a strategy for adding more than 3 as it becomes inefficient and error prone for larger numbers. Part-part-whole knowledge related strategies are better for efficient addition.

#### Counting Back

When subtracting a small number (no more than 3) from a larger one, students would start with the larger and count back from this to find the difference. For example to subtract 3 from 15, students would start at 15 and count backwards 3 (ie "fifteen, fourteen, thirteen") perhaps keeping a tally of the count using their fingers. Counting Back is not reccomended as a strategy for subtracting more than 3 as it becomes inefficient and error prone for larger numbers. Part-part-whole knowledge related strategies are better for efficient subtraction.

#### Doubles and Near Doubles

Once students have achieved a knowledge of Doubles (or 2x facts) they can be encouraged to use this understanding to facilitate rapid mental addition. The expression 7+8 for example can be conceptualised as "Double 7 and one more" or "Double 8 and one less". The use of Doubles and Near Doubles is therefore an efficient mental addition strategy.

#### Double Count

An early multiplication strategy where a student stress or skip counts at the same time as keeping track of the number of ‘counts’, typically using their fingers, eg for 3 x 5 the student may count 5, 10, 15 out loud while marking off 1, 2, 3 simultaneously on their fingers.

#### Group Structure (Using the Group Structure)

When finding the total of a collection that has been arranged in groups, such as counting the students in a classroom when they are all sitting four to a desk, it helps to use groups, eg stress or skip count by fours, or count the number of desks and multiply by four. The group structure is noticed and used to make the calculation more efficient. Students who have developed multiplicative thinking are more likely to use group structure.

#### Halving

See Partitioning - Halving Strategy

#### Hidden

See Concealed

#### Identify (to)

When a number is verbally named, the student can pick out the correct numeral from a list or collection of possible candidates. This should not be confused with naming a numeral.

#### Known

When used in the sense of counting on from a 'known' number, the implication is that the student 'trusts' the number that they are counting on from - that is they have reached the stage in their development where they have determined the number of a collection through counting, subitising or part-part-whole understanding and so are happy to count on from this number to find a total. It is not necessary for them to count the total starting from one because they have confidence in the number that they are counting on from. (see also trusted)

#### Locates

Students identify the relative position of a number on an empty number line where the interval has been defined, eg a student is presented with a number line representing 0 to 100 where the interval between is unmarked. They are then asked to indicate the position of (locate) 48 for example. Locating a number requires an application of mental partitioning and an understanding of the relative magnitude of a given number.

#### Make-all/Count-all

Students find the total of two collections by combining them together (at least mentally) and counting the resulting collection, starting from one, by ones. There is assumed to be no attempt at partitioning or part-part-whole strategy using of the initial group structure to aid the addition.

#### Make to Ten

By memorising the pairs of whole numbers that add to make ten, as well as their part-part-whole understanding of digits, students are able to use efficient strategies to find a total when adding. For example, when adding 12 and 9, the student would first add 8 to make to ten, then add the remainder (1) to quickly give a total of 21

#### Matches Numerals to Collections

Students begin to recognise that number names and numerals are not just a random sequence to be memorised, but that each numeral and number name describes a specific collection size. Students demonstrate their increasingly sophisticated understanding of number when they are able to pair numerals with corresponding collections.

#### Models

Students use concrete or visual materials to illustrate a number concept, eg the Base10 structure of a number, using stacks of 10 multilink cubes along with individual cubes to show that the structure of 34 is 3-tens and 4-ones. May also model using dedicated base-10 materials such as tens frames, MAB blocks and place-value grids. Fractional concepts can be modelled using partitioned (folded or cut) paper rectangles or circles.

#### Multiplicative Thinking

This term describes a set of related ideas a student must develop in order to be able to perform calculations involving multiplication flexibly and in a variety of contexts. These ideas include being able to see each equal group in a collection as a countable unit in itself, eg they understand that 3 x 4 refers to a group of 3 units, each of which has a ‘quality’ of four. They understand that multiplication is commutative and that it can apply to scales (‘that building is three times taller than me”) and rates (petrol costs $2:50 per litre) and that it is the opposite (inverse) of division. See also Think of multiplication and Additive thinking

#### Name (to)

Verbally state the name of a given numeral. For example, if a student sees '42', they will say: "fourty two". This skill is distinct from identifying a number (see identify).

#### Number Fact Knowledge

Through a process of understanding a concept, then repeatedly solving related problems by applying a strategy, the student comes to learn certain facts ‘off by heart’. This is not the same as rote learning, where the student may ‘parrot’ facts but does not understand where they came from or where they fit in. Times tables are a good example; first the students must understand that it is about numbering equal groups and finding the total, next they learn and practice strategies such as doubling, identifying patterns and commutativity (5 x 7 = 7 x 5) and after sufficient (possibly timed) practice, the facts become memorised.

#### Numeral

A symbolic representation of a number. Numbers can be represented as collections, as written words (ie Forty two) or symbolically, as a numeral (ie 42).

#### Orders

Arranges a set of disorganised *collections* or *numerals* (typically printed individually on cards)so that they are displayed in consecutive, ascending order.

#### Part-Part-Whole

In the context of this document, part-part-whole refers to the ability to see whole numbers in terms of their parts and their relationships with other numbers, eg to see that 8 is 6 and 2, 1 more than 7, 2 less than 10 etc. Can also incorporate Place-value deconstruction of numbers, eg 26 is 2-tens and 6 ones.

#### Patterns

Students identify and explore patterns with multiples, particularly 10, 9 and 5, using hundred charts. Once patterns have been identified, students can practice using them in order to improve their multiplication strategies.

#### Place Value

The established method of writing Base 10 numbers where the order of a digit in a number corresponds to a particular value for that digit. Specifically, the first digit (to the left of the decimal point) gives the count of "ones" (10 to the power of 0), the next digit to the left gives the count of "tens" (10 to the power of 1), the next gives the count of "hundreds" (10 to the power of 2) and so on. The number 374 for example encodes a value of 3 hundreds, 7 tens and 4 ones.

#### Place Value Partitioning

Place Value Partitioning is where a number is broken up into its place-value parts e.g. 253 is rewritten as 2 Hundreds, 5 tens and 3 ones. This skill helps students to understand the magnitude of a number and is also a vital part of many addition and subtraction strategies.

#### Reading

The act of reading involves assigning the correct meaning to written symbols. Reading in a mathematical context requires the students to accurately identify number symbols and, through their context, determine whether their meaning is ordinal (about sequence) or cardinal (about quantity) in nature.

#### Renames (Whole Numbers)

Flexibly renames numbers in terms of place value parts, eg 467 is 4 hundreds and 67 ones, 46 tens and 7 ones, or 467 ones.

#### Renames (Fractions)

Identifying like factions within ‘families’, particularly when scaffolded through area model partitioning, eg 2/5 = 4/10

#### Rounding Strategies

When estimating the total of two or more quantities, students are encouraged to round each quantity to the most appropriate place value position. Number lines are a good way for students to develop a sense of the nearest hundred (or ten, one, tenth etc) to any given value in order to round effectively.

#### Partitioning

In this context, partitioning refers to equal division or sectioning of collections or areas resulting in a representation of fractions, or the subdividing of fractions of areas and collections to represent smaller fractions. In this sense, partitioning is a key to the conceptual understanding of fractions and decimals (see below for details).

#### Partitioning - Halving Strategy

#### Partitioning - Thirding Strategy

#### Partitioning - Combination of Halving and Thirding Strategies

#### Partitioning - Fifthing Strategy

#### Partitioning to Model tenths

Siemon, D. (2004). *Partitioning – The missing link in building fraction knowledge and confidence. *Retrieved 25th May, 2009 from Eduweb Victoria website

#### Partitioning - Place Value

The ability to break up numbers in terms of their place value parts. For example, the number 234 could be broken up as two hundreds, three tens and four ones. It could also be broken up as 23 tens and 4 ones.

#### Subitising

The ability to identify the number of items in a collection without actually counting the items. Usually practiced by displaying a collection to students for less than two seconds to and checking to see if they can state the number of items. A foundational skill enabling the use of efficient addtion strategies.

#### Supported (strategy)

When students are initially learning a new concept, it is useful to model the idea using concrete materials. A supported strategy is one which relies on using materials to support the building and application of a concept. Support materials may be counters, MAB materials, number lines, even pen and paper. The idea is that ideas are built from the concrete to the abstract. It is expected that this cycle is repeated throughout the students’ learning – that while one idea is learned and can be applied without support, the materials are modified to support the learning of the next idea.

#### Skip Counting

When finding the total number of a collection where the items are grouped, the count proceeds in multiples of the group size eg, for a group size of four, the count would sound like “four, eight, twelve, sixteen…” This method is also used to speed up a count where the items are initially ungrouped, eg the counter visually or physically makes groups the items as they count, then adds on any remainders.

#### Stress (or rhythmic) counting

As in skip counting, the group structure determines the count. In stress counting however, all numbers are named with the group multiple emphasised (stressed), or a pause inserted between groups as the sequence is recited (producing a rhythm) eg for groups of 4, stress counting would sound like “one, two, three, * four*, five, six, seven,

*… where rhythmic counting would sound like “one, two, three, four (*

**eight****) five, six, seven, eight (**

*pause**)… Stress and rhythmic counting are often seen as steps leading to skip counting.*

**pause**#### Table Strategies

Strategies that assist the efficient calculation of multiplication facts. Examples include; doubling for 2x facts, doubling plus one more group for 3x facts, doubling and doubling again for 4x facts, exploiting the pattern for 5x and 9x facts and so on. See also Number Fact Knowledge.

#### Teacher Talk

The phrase 'Teacacher Talk' is used to indicate a critical component of activity based instruction; the role of verbal teacher guidance. A common misconception around Activity Based Instruction is that simply by doing the activity, the students will learn. In fact students will learn very little from doing an activity without verbal input from a tutor or teacher; specifically, the strategic use of leading questions and explanations in a language that is readily understood by the student.

#### Think of Addition

It is often easier for students to work with a subtraction problem by reconceptualising it as an addition problem with a missing addend. A formally expressed example is when the equation 8-5=x is rearranged as 5+x=8. In practice, students will respond to a much less formal explanation. Verbally they might reason that "to find 8 take 5, I think 5 plus something equals 8 and then count on from 5 to find the answer"

#### Think Of Multiplication

When dividing, students can use known number facts (tables) to help solve division problems. For example 63 ÷ 7 can be turned into ‘7 x what? = 63’. This approach can also be used to find the closest whole number where there are remainders.

#### Trusted (or known) number

When children first begin to realise that a count is not just a sequence, but relates to ‘how many’ objects in a collection, they will not always ‘trust the count’. They are not convinced that the count of larger collections for example, will stay constant and would not be surprised if a recount of the same collection gave a different total. A trusted number is a count that the child has confidence in. Initially these will be smaller collections that have been counted by ones.