Learning the meanings, relations and procedures used in arithmetic. A correct answer can arise through different strategies, so assessment should combine outcomes with informative observations of the process.
Use of different procedures across problems or occasions. Variation can be informative about learning rather than mere error; a single correct answer or verbal report may not identify the underlying strategy.
A procedure for solving an arithmetic problem, including counting, retrieval and decomposition. Different strategies can coexist and be selected according to the problem and available knowledge.
Reexpressing a calculation through useful numerical relations, such as . Decomposition can connect practiced facts with conceptual knowledge instead of requiring direct retrieval of every whole problem.
Starting from a known quantity and counting only the added units. The strategy requires a link between the starting number and its represented quantity, rather than simply omitting words from a memorized sequence.
For addition of nonnegative natural numbers and , start at and count further units. Commutativity guarantees the same sum while minimizing the increments among the two possible choices of starting addend.
Representing the relevant collections and counting the combined items to obtain a sum. Count-all can be accurate while requiring more increments and tracking than a suitable counting-on strategy.
Coordinating ordered number words or symbols with items or successive units. Reciting the sequence is distinct from counting a collection and from understanding what its final count represents.
A constraint on valid counting, such as stable ordering of number words, one-to-one correspondence with items and the cardinality principle. Understanding and coordinating these constraints should be distinguished from rehearsing a count routine.
The principle that the final number word in a valid count represents the cardinality of the whole collection. A child's knowledge of this relationship is different from merely repeating the last word after being prompted.
A description of which exact number-word meanings a child demonstrates in tasks such as giving a requested collection. Early knowledge of particular small quantities can precede a general use of the counting system; task context and errors matter when assigning the description.
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