GlossaryAddend A number to be added. In 8 + 6 = 14, 8 and 6 are addends, and 14 is the sum.

Additive task A generic label for tasks involving what adults would regard as addition. The label “additive task” is used to emphasise that children will construe such tasks idiosyncratically, that is, differently from each other and from the way adults will construe them.

Algorithm A step-wise procedure for carrying out a task, e.g. standard written algorithms for adding and subtracting.

Array A rectangular grid of rows and columns, often used as a setting for multiplication.

Arithmetic rack Also see Rekenrek. An abacus-like instructional device consisting of two rows of 10 beads. In each row the beads appear in two groups of five, using two different colours for the beads.

Associative principle The operation of addition is associative because , when any three numbers are added, the order of performing the two addition operations does not affect the sum. Multiplication is also associative, but subtraction and division are not.

Backwards number word sequence BNWS A regular sequence of number words backward, typically but not necessarily by ones, for example, the BNWS from ten to one, the BNWS from eighty-two to seventy-five, the BNWS by tens from eighty-three.

Base ten A characteristic of numeration systems and number naming systems whereby numbers are expressed in a form that involves grouping by tens, tens of tens, and larger powers of ten.

Basic Facts Combinations or number bonds for addition, subtraction, multiplication or division.

Centuple A multiple of 100, e.g. 200, 1400. This is distinguished from a century which means a sequence of 100 numbers, e.g. from 267 to 366.

Colour coding Using colour to differentiate different parts of an instructional setting, e.g. a collection of 6 red counters and a collection of 4 green counters.

Commutative Addition is said to be a commutative operation because, for any two numbers, the order of adding does not change the result, for example, 6 + 8 = 8 + 6. Multiplication is also a commutative operation, whereas subtraction and division are operations that are not commutative.

Compensation strategy A strategy that involves first changing a number to make an easier calculation, then compensating for the change.

Counting – back –from A strategy used by children to solve Removed Items tasks, for example 11 remove 3 – “eleven, ten, nine – eight”. Also referred to as counting-off-from or counting-down-from.

Counting – back- to Regarded as the most advanced of the counting-by-ones strategies. Typically used to solve missing subtrahend tasks, e.g. have 11, remove some, and there are eight left – “eleven, ten, nine – three”. Also referred to as counting-back-to.

Counting-by-ones Initial or advanced arithmetical strategies which involve counting-by-ones only. Examples of initial count-by-one strategies would be perceptual and figurative counting which involve counting-from-one . Examples of advanced count-

by-one strategies would be counting on, counting back and counting-down-to.

Counting-on An advanced counting-by-ones strategy used to solve additive tasks or missing addend tasks involving two hidden collections. Can also be referred to as counting-up.

Counting-up-from An advanced counting-by-ones strategy used to solve additive tasks involving two hidden collections.

Counting-up-to An advanced counting-by-ones strategy used to solve missing addend tasks.

Decade or decuple A multiple of ten

Decrementing Decreasing a number typically by one or more ones, tens, hundreds or some combination of these.

Difference See Minuend.

Digit The digits are the ten basic symbols in the modern numeration system, that is 0, 1, ….9

Distributive principle The principle that multiplication distributes over addition and subtraction, as does division. For example, (7-5) x 3=(7x3) – (5x3)

Dividend In a division equation such as 29 divided by 4 = 7 r1, 29 is the dividend, 4 is the divisor and, 7 is the quotient and 1 is the remainder.

Domain Used to refer to a broad area of arithmetical learning such as addition and subtraction, or place value.

Empty number line A simple arc or line which is used by children and teachers to record and explain mental strategies for solving problems.

Facile Used in the sense of having good facility, that is, fluent or dexterous, for example a facile counting-on strategy.

Factor If a number F, multiplied by a whole number gives a number M, we call F a factor and M a multiple. In 3 x 9 = 27, 3 is a factor of 27, and 27 is a multiple of 3.

Figurative Figurative thought involves re-presentation of a sensory-motor experience that is a mental replay of a prior experience involving seeing, hearing, touching and so on. Figurative counting may be figural, in which visualised items constitute the material counted, motor, in which movements constitute the material counted, or verbal, in which number words constitute the material which is counted.

Five-wise pattern A spatial pattern for a number in the range 1 to 10 made on a ten frame. The five-wise patters are made by progressively filling the rows, top row first.

Flashing Briefly displaying some aspect of an instructional setting, e.g. a ten frame.

Formal algorithm A standard written procedure for calculating with multi-digit numbers that relies on the conventions of formal place value, e.g. column based procedures for addition or chimney sums.

Forward number word sequence FNWS A regular sequence of number words forward, typically but not necessarily by ones, e.g. the FNWS from one to twenty, the FNWS from eighty-one to ninety-three, the FNWS by tens from twenty-four.

Incrementing Increasing a number typically by one or more ones, tens, hundreds or some combination of these.

Jump strategy A category of mental strategies for 2-digit addition and subtraction. Strategies in this category involve starting from one number and incrementing or decrementing that number by tens or ones.

Knowledge A collective term for all of what the child knows about number. Knowledge is shown by instant recall, rather than having to use a strategy.

Micro-adjusting Making small moment-by-moment adjustments in interactive teaching which are informed by one’s observation of child responses.

Minuend In subtraction of standard form, e.g. 12 – 3 = 9, 12 is the minuend, 3 is the subtrahend and 9 is the difference. Thus the difference is the answer obtained in subtraction, the subtrahend is the number subtracted and the minuend is the number from which the subtrahend is subtracted.

Missing addend A subtractive task posed in the form of addition with one addend missing, e.g. 12 and how many make 15? In NZ maths these are known as change unknown problems.

Missing subtrahend A task where the minued and difference are known, e.g. 12 - ? = 8. In NZ maths these are known as change unknown problems.

Multiple see Factor

Multiplicand The number multiplied, e.g. in 8 x 4 = 32, 8 is the multiplicand, 4 is the multiplier and 96 is the product.

Non-canonical The number 64 can be expressed in the form of 50 + 14. This form is referred to as a non-canonical (non-standard) from of 64. Knowledge of non-canonical forms is useful in addition, subtraction and so on.

Non-count-by-ones A class of strategies which involve aspects other than counting-by-ones and which are used to solve addition and subtraction tasks. Part of the strategy may involve counting-by-ones but the solution also involves a more advanced procedure. For example 6 + 8 is solved by saying 6 + 6 = 12, then 13, 14.

Notating Writing which relates to numbers and numerical reasoning.

Number A number is the idea or concept associated with how many items in a collection. We distinguish between the number 24 – that is, the concept, the spoken or heard number word twenty-four, the numeral 24 and the read or written number word twenty-four. These distinctions are important in understanding children’s early numerical strategies.

Number word Number words are names or words for numbers. In most cases in early number, the term “number word” refers to the spoken and heard names for numbers rather than the read or written names.

Numeral Numerals are symbols for numbers, e.g. “4” and “28”.

Numeral identification Stating the name of a displayed numeral. The term is used similarly to the term “letter identification” in early literacy.

Numeral recognition Selecting a nominated numeral from a randomly arranged group of numerals.

Numeral sequence A regularly ordered sequence of numerals, typically but not necessarily a forward sequence by ones, e.g. the numerals as they appear on a numeral track.

Numeral roll An instructional device consisting of a long strip of paper containing a sequence of numerals increasing from left to right. They may have a slotted card to use as a window, so that one numeral only is displayed.

Numeral track An instructional device consisting of a sequence of numerals and for each numeral, a hinged lid which can be used to screen or display the material.

Numerosity The numerosity of a collection is the number of items in a collection.

Pair-wise pattern A spatial pattern for a number in the range 1 to 10 made on a ten frame, made by progressively filling the columns from the left (useful to show doubles and near doubles).

Part-whole thinking The ability to conceive simultaneously of a whole and two parts, e.g. conceiving of 10 and also of the parts 6 and 4. This means children do not need to rely on counting-by-ones to add and subtract.

Partitioning An arithmetical strategy involving partitioning, or breaking up, a number into two parts without counting, e.g. partitioning 6 into 5 and 1.

Partitive division A division equation, such as 15 divided by three, that involves distributing items into a 3 groups (sharing), as opposed to quotitive division, where items are sorted into groups of 3 (counting the number of groups).

Perceptual Involving direct sensory input, usually seeing, but may also refer to hearing or feeling. Thus perceptual counting involves counting items seen, heard or felt.

Product see Multiplicand

Quinary The use of five as a base in some sense. The rekenrek may be regarded as a quinary-based instructional device.

Quotient see Dividend

Quotitive division see Partitive Division

Rekenrek An abacus-like instructional device consisting of two rows of 10 beads. In each row the beads appear in two groups of five, using two different colours for the beads.

Scaffolding Actions on the part of a practitioner to provide support for a child’s learning during interactive teaching.

Screening A technique used in the presentation of tasks which involves placing a screen over all or part of the setting, e.g. placing a card over counters.

Split strategy A category of mental strategies for 2-digit (or larger) addition and subtraction. Strategies in this category involve splitting the numbers into tens and ones and working with them separately before recombining them.

Strategy A generic label for a method by which a child solves a task, e.g. counting from one, or adding through ten.

Subitising The immediate, correct assignation of a number word to a small collection of perceptual items.

Subtractive task A generic label for tasks involving what adults would regard as subtraction. The label is used to emphasise that children will construe such tasks idiosyncratically, that is, differently from each other and from the way adults will construe them.

Subtrahend see Minuend.

Sum see Addend

Temporal sequence A sequence of events that occur sequentially in time, for example, sequences of sounds or movements.

Ten frame A 2 x 5 rectangular array which is used to support children’s thinking about combinations to 10.

Transforming strategy A strategy that involves simultaneously changing two numbers to make an easier calculation, e.g. 17 + 38 is transformed to 15 + 40, 83 – 18 is transformed to 85 – 20.