Mathematics
The Mathematics domain tests early number and operations, fractions, measurement, geometry, and data, plus the reasoning practices that hold them together: making sense of problems, justifying claims, modeling, using tools strategically, and attending to precision. Items often show a student's written work and ask you to name the error and choose the next instructional step.
Number and place value
- Counting has two jobs: producing the sequence and attaching cardinality. A child who recounts after being asked "how many?" has not yet attached cardinality to the count.
- Ten is the organizing unit. Composing and decomposing 10, then 100, underpins every later algorithm.
- Regrouping is trading, not "carrying a one." Language should say "trade one ten for ten ones."
Operations and problem structures
- Addition and subtraction situations: join, separate, part-part-whole, compare. Multiplication and division: equal groups, arrays, area, comparison, fair share.
- The equal sign means "same value as," not "compute here." True/false sentences like 8 = 5 + 3 build that meaning.
- Word-problem meaning comes from the situation structure, not from keywords.
Worked example
A student solves 52 − 27 and writes 35. Reasoning: 7 − 2 = 5 and 5 − 2 = 3, so digits were subtracted without regard to place value. The correct reasoning decomposes 52 as 40 + 12, subtracts 20 and 7, and gets 20 + 5 = 25. Diagnose by having the student build 52 with base-ten blocks and remove 27, trading one ten for ten ones; then connect the blocks to the written algorithm. The error is conceptual, so re-explaining the steps louder will not fix it.
Fractions, measurement, geometry
- Fraction meanings: part-whole, measure, quotient, operator; the whole must be the same size across comparisons.
- Length is measured by iterating a unit with no gaps or overlaps.
- Area counts square units; perimeter counts linear units.
- Shapes are defined by attributes (number of sides, right angles), not by orientation or size.
Traps that cost marks
- Keyword strategies ("in all" means add) that break on compare problems.
- Rehearsing "carry the one" with no place-value language.
- Confusing area and perimeter, or counting perimeter units as squares.
- Treating manipulatives as a reward after the algorithm instead of the source of meaning.
- Assuming more digits means a larger number.