Mathematics (Elementary)
Elementary mathematics items focus on number sense, the progression from concrete to abstract, and the diagnostic use of student work. You will often be asked to identify the misconception behind an error and select the intervention that addresses it.
Content anchors
- Counting and cardinality, place value, and the base-ten system underpin all computation.
- Operations: meanings of addition/subtraction (join, separate, compare, part-part-whole) and multiplication/division (equal groups, arrays, area, comparison).
- Fractions: part-whole, number line, equivalence, comparison; a fraction is a number, not two whole numbers.
- Decimals, proportional reasoning, algebraic thinking (patterns, expressions, equations), geometry, measurement, and data.
- The Standards for Mathematical Practice-reasoning, modeling, precision, using tools-apply to students with disabilities too.
Instructional design
- CRA sequence: concrete manipulatives (base-ten blocks, fraction tiles, counters, ten-frames, number lines), then representational drawings, then abstract symbols. Move on only when the student is fluent at each level.
- Schema-based instruction for word problems: teach problem types (total, difference, change, groups) rather than key-word hunting.
- Worked examples, error analysis, math interviews, and explicit vocabulary instruction (sum, difference, product, quotient, denominator).
- Progress monitoring: computation CBM, concepts and applications CBM, and error-pattern analysis.
Worked example
A student computes 45 − 27 = 22. The pattern-subtracting the smaller digit from the larger in each column-shows the student has not constructed regrouping. The teacher builds 45 with base-ten blocks, trades a ten for ten ones, removes 27, and connects the action to the written algorithm, then checks with addition. The error is conceptual, so reteaching the procedure alone will not fix it. Similarly, a student who says 1/5 > 1/3 because 5 > 3 needs fraction tiles and a number line to see that more parts means smaller parts.
Traps
Teaching the algorithm before the concept; assuming fluency equals understanding; relying on key-word strategies that fail on multi-step problems; treating a calculator as always a modification (it depends on the construct measured); and ignoring the language demands of word problems for students with reading or language disabilities.