School Years Glossary

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Abstraction – One of the principles of counting: any group of objects can be counted, regardless of individual item type. For example, one orange, two pencils and three blocks can be counted (from 1 to 6) to find the total number of items.

Algebra – A branch of mathematics in which mathematical statements are used to describe relationships between variables, using letters and other symbols to represent numbers and number quantities.

Algorithm – In mathematics, an algorithm is a set of precise step-by-step instructions for how to arrive at an answer to a given problem; a formal procedure that is usually explicitly taught.

Associative Property – In addition and multiplication, the order in which three numbers are added or multiplied does not affect the sum or product. This is not true for subtraction and division. For example, (1 + 2) + 3 is the same as 1 + (2 + 3). (See also: Commutative Property.)

Automaticity – The quick, easy, and effortless (that is, the “automatic”) retrieval of facts or procedures from long-term memory.


Base-10 System – The number system most commonly used in North America, based on grouping in tens. Ten is the base, and each place to the left is 10 times greater. For example, 100 = 10 times greater than 10. Each place value to the right of base-10 is one tenth of it (1/10); for example, 1 is 1/10, or one tenth, of 10.

Biologically Primary Knowledge - Inherent types of cognition, such as language and early quantitative competencies; usually emerge with little to no formal instruction, across all cultures.

Biologically Secondary Knowledge – Skills that build on biologically primary abilities and are cultural inventions (e.g., base-10 arithmetic).


Cardinality – One of the principles of counting and initial “how to count” rules: the value of the last number word used while counting indicates the quantity of items in the set. For example, counting “1, 2, 3” means that there are three items in the set.

Cardinal Numbers – The counting numbers (1, 2, 3, 4, 5, 6…) used to measure the size, or cardinality, of a set.

Change Problems – A type of word problem that contains some event that changes the value of a quantity. For example, “Robin has 5 pencils and Carly gives him 3 more. How many does Robin have now?”

Combine Problems – A type of word problem that describes two parts that are considered separately or in combination. For example, “Robin and Carly have 8 pencils all together. Carly has 3 pencils. How many does Robin have?”

Combining Units Strategy – A strategy used to solve arithmetic problems wherein the 100s, 10s, and units are dealt with separately. For example, solving 37 + 38 by 30 + 30 then 7 + 8.

Commutative Property – In addition and multiplication, the order in which two numbers are added or multiplied does not affect the sum or product. For example, the sum of 4 + 3 is the same as the sum of 3 + 4. (See also: Associative Property.)

Compare Problems – A type of word problem that contains two amounts to be compared for the difference between them. For example, “Robin has 5 pencils and Carly has 3 pencils. How many fewer pencils does Carly have than Robin?”

Compensating Strategy – A strategy used to solve arithmetic problems wherein the numbers are adjusted to simplify the arithmetic. For example, solving 37 + 38 as (35 + 35) + 2 + 3 = 75.

Conceptual Knowledge – Knowledge of why and how a mathematical procedure works, as well as general mathematical knowledge and understanding. For example, knowing that when counting, the last number stated represents the quantity of items in the set.

Concrete Operational Stage – One of Piaget’s cognitive stages; a period when children have a better understanding of mental operations and begin to think logically about concrete events; ages 7-12.

Counting On – The ability to start at any number in the number sequence and continue counting from that number onward. (See also: Number-after Skill.)

Counting Skills – The ability to recite numbers in order. Children may be able to recite the number words in the correct order without understanding the underlying meaning.


Data – The information used as the basis of calculation.

Digit - The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, 4093 has four digits: 4, 0, 9, and 3.


Episodic Buffer – One of the three systems in the model of working memory; this memory storage system can integrate information across domains to form visual, spatial, and verbal information with time sequencing.

Estimation – An approximation of the exact value of an operation.

Extrinsic Motivation – The effort to learn made in the hope of some type of external reward such as good grades, a teacher or parent’s praise, a sticker, etc.


Factual Knowledge – Knowledge of information that can be learned through memorization and repetition (e.g., 2 + 2 = 4), as well as memory of specific events and information.

Formal Operational Stage – One of Piaget’s cognitive stages; a period in which skills such as logical thought, deductive reasoning, and systematic planning emerge and the ability to think about abstract concepts develops; ages 12-adulthood.

Fraction - Any part of a whole, number, or group. For example, ¼ or ½.

G, H, I

Geometry – A branch of mathematics that involves the study of shapes and configurations (e.g., straight lines, circles, etc).

Heuristic Methods – The systematic strategies that one uses for problem solving. (See also: Algorithm.)

Intrinsic Motivation – The desire to learn for the sheer enjoyment, challenge, pleasure, or interest of the activity.

J, K, L, M

Language-based Phonetic Buffer – One of the three systems in the model of working memory, this system is also known as the Phonological Loop. It temporarily stores the phonological, or auditory, information of language.

Manipulatives – Objects that children can handle (manipulate) to understand and work out simple arithmetic problems; for example, beans, buttons, or blocks. Children build their understanding of math with concrete objects before they move on to abstract number concepts.

Mastery-Oriented Goals – Students with mastery-oriented goals seek to master that which they are learning, focusing on their own achievement and attributing their success to effort. These students tend to challenge themselves and persist in the face of difficulty.

Math Anxiety – An emotional reaction, ranging from mild apprehension to fear or dread, in academic and everyday situations that deal with numbers.

Mental Number Line – A mental representation of numbers and relative magnitudes; requires the ability to visualize and abstract number, to order numbers by quantity, to locate a given number along an imaginary line, and to generate any portion of the number line that may be required for problem solving.

Metacognitive Knowledge – Also known as Metaknowledge; the knowledge of how, when, and why to use specific strategies or resources; what an individual knows about his or her own thinking.

Metacognitive Regulation – How one’s knowledge is used to regulate and control one’s own thinking.


Number-After Skill – The ability to state the next number in the counting sequence when one starts at any number. For instance, knowing that five is the next number after four without needing to count up from one. (See also: Counting On.)

Number Line – 1A horizontal line on which numbers are written in order from left to right. Once children have understood ordinality, they can look at the line and see that numbers farther to the right represent larger quantities than those on the left. As children learn about counting and about the concept of number itself, they start to generate a mental picture of the number line.

Number Sense – The understanding of number and operations; encompasses three subcomponents: 1) knowing about and using numbers; 2) knowing about and using operations; and 3) knowing about and using numbers and operations in computational settings.

Numeracy – A broad term that includes knowledge of number, arithmetic, procedures, problem solving, and measurement.

Numerosity – An approximate sense of number that babies as well as non-human animals (e.g., rats, lions, primates) have.

Numerosity Accumulator – The posited mechanism in the brain that accumulates the number of a small group of objects, sounds, or events and compares it to an internal knowledge of number, usually only for one to four items.


One-to-One Correspondence – One of the principles of counting and initial “how to count” rules: one, and only one, word can be assigned to each counted object. For example, an item in a set that has been assigned “3” cannot also be assigned “5”.

Order-Irrelevance – One of the principles of counting: items in a set can be counted in any sequence and still reach the same total. For example, counting from right to left, left to right, or in no particular sequence at all will result in the same total number of items.

Ordinality – At its most basic level, the concept of more and less; develops to an understanding that higher numbers are associated with more items, and lower numbers with fewer items.

Ordinal Number – The number that refers to place or position (e.g., 1st, 2nd, 3rd).


Pattern – Any repeated design or recurring sequence. For example, the sequence 1, 2, 3, 1, 2, 3 or the flowers on wallpaper are both patterns.

Performance-Oriented Goals – Students with performance-oriented goals are focused outward, comparing their performance and learning to that of others; they tend to attribute success to ability, avoid challenging themselves, and give up when dealing with a difficult problem.

Place Value – The value of a digit based on its position in a number. For example, in the number ‘12’ the digit ‘1’ is in the ‘tens’ position and the digit ‘2’ is in the ‘ones’ position, which indicates that 1 ten + 2 ones = 12).

Procedural Knowledge – Knowledge about how to complete and activity or task, including the motor sequences and skills needed. For example, knowing how to solve the problem 2 + 3 by continuing to counting on from 3 to get the sum – “4, 5.”

Probability – The relative frequency with which an event will occur; expressed by the ratio of the number of actual occurrences to the total number of possible occurrences.

Proportionality – Refers to the multiplicative relationships between rational quantities; the basis for rational number operations, basic algebra, and problem solving in geometry.

R, S

Self-Efficacy – The set of beliefs one holds about one’s own ability to succeed at difficult tasks.

Self-Regulation – A combination of motivation and cognitive processing; includes goal setting, planning, self-monitoring, evaluation, learning adjustments, and strategy choice.

Sequence – An ordered set of objects, numbers, shapes, etc. that are arranged according to a rule. For example, arranging dolls in order based on height, tallest to shortest.

Sequential Strategy – A strategy used to solve arithmetic problems wherein the value of the second number is counted up or down from the first number. For example, 37 + 38 is solved by 37 + 30 = 67, then 67 + 8 = 75.

Set - A collection of items that are grouped together; members of a set are called elements.

Stable Order – One of the principles of counting and initial “how to count” rules: the order in which number words are used to count objects is always the same. For example, counting in the order of “1, 2, 3” is correct, but “1, 2, 4” is incorrect.

T, U, V

Transfer – Also known as Learning Transfer; the ability to apply the skills and concepts used to solve one type of problem to another type of problem; learning can be applied beyond problems studied to both similar problems (Near Transfer) and to dissimilar problems (Far Transfer).

Visuospatial Sketch Pad – One of the three systems in the model of working memory; this system enables short term storage and manipulation of visual or spatial information.

W, X, Y, Z

Working Memory – Attention-driven control of information represented in the brain in one of three content-specific systems: the language-based phonetic buffer, the visuospatial sketch pad, and the episodic buffer.

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