trajectories, a term that emphasizes the sequential and direct route from one skill level to the next. The sources of a teaching-learning path are: (1) the subject matter being taught—what skills and knowledge provide the foundation for later learning, and (2) what is achievable/understandable for children at a certain age given their prior knowledge. Teaching-learning paths also provide a basis for targeting the curriculum, assessing children’s progress along the path, and adapting their instruction to help children make continued progress.

Mathematizing refers to reinventing, redescribing, reorganizing, quantifying, structuring, abstracting, and generalizing concepts and situations first understood on an intuitive and informal level in the context of everyday activity into mathematical terms. This process allows children to create models of situations using mathematical objects or actions and their relationships to solve problems, including the use of increasingly abstract representations.

Measurement refers to the process of determining the size of an object with respect to a chosen attribute (such as length, area, or volume) and a chosen unit of measure (such as an inch, a square foot, or a gallon).

Morphological marker refers to the word element that signifies quantity, such as whether the word is singular or plural. For example, the s on the end of dogs, which indicates that the word is plural, is the morphological marker. The term quantifier morphology is used interchangeably with morphological marker.

Number competencies refer equally to both the knowledge and skills concerning number and operations that can be taught and learned.

Number sense refers to the interconnected knowledge of numbers and operations. It is a combination of early preverbal number sense and the increasingly important influence of experience and instruction.

Numeral refers to the symbol used to represent a number.

Numerosity refers to the quantity of a set.

Object file system refers to the representation of each object in a set comprised of very small numbers, but no representation of set size. For this form of representation, the objects in a small set are in 1-to-1 correspondence with each mental symbol. Thus, a set of three items is represented as “this” “this” “this” rather than “a set of three things.”

One-to-one (1-to-1) correspondence refers to correspondence between two collections if every member of each collection is paired with exactly one member of the other collection and no members of either collection is unpaired or is paired with more than one member.

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