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Mathematics achievement is attained performance on specified mathematical learning objectives. Its meaning depends on the content and scoring of the assessment: proficiency in counting, arithmetic and geometry need not form an identical profile. A total score can summarize performance while concealing which skills changed.

An informative intervention therefore measures more than success on its practiced activity. In Geetha B. Ramani's study with Robert S. Siegler, children played either a linear numbered board game or a corresponding color board game. Assessment included counting, numeral identification, numerical magnitude comparison and number-line estimation, rather than only skill at winning the game.^[ref]

Improvement on an assessment is evidence about that outcome in the studied population. Claims about broader mathematical understanding require appropriate generalization tests and a comparison design that separates the intervention from maturation, repeated testing and other experiences.

Numeral identification is recognition or naming of written numerals. When shown 7 and asked its name, a child maps a visual symbol to a conventional numerical label. This differs from counting seven objects or choosing whether 7 is larger than 4: related components of numeracy place different demands on the learner.

Separate assessments make those differences visible. Geetha B. Ramani and Robert S. Siegler asked preschool children to name numerals as one of four numerical tasks before and after a number board game intervention. A naming score therefore measures performance under that task's symbols and response rules; it does not by itself establish understanding of every property of the represented numbers.^[ref]

Numerical magnitude comparison is a task in which a person decides which of two represented numbers is larger or smaller. The inputs might be written numerals, spoken number names or sets of objects. Choosing 8 rather than 3 as the larger number tests a numerical relationship; simply naming the symbol 8 instead tests numeral identification.

A task records both the decision and, when appropriate, reaction time. These outcomes depend on the representation and instructions, so successful comparison of two familiar digits need not establish equal skill with unfamiliar notation. Geetha B. Ramani and Robert S. Siegler assessed comparison of pairs of numerals separately from naming and counting in their number board game experiment.^[ref]

An operational definition specifies how a variable is measured or manipulated in a particular investigation. Defining stopping performance by a specified stop-signal task, scoring rule and estimation procedure makes the observation reproducible enough to inspect. The broader psychological construct of response inhibition need not be identical to that one procedure.^[ref]

For example, “number of correctly named digits from 1 to 10 under these instructions” operationally defines a numeral identification score. It leaves open whether the same child can compare quantities or count objects. An operational definition makes the measurement concrete; evidence about what that measurement represents is a separate question of construct validity.