skills/k12-check-for-understanding/references/math.md

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# Math — k12-check-for-understanding

Your full instruction set for the build (SKILL.md Step 4).

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## 1. Aspect of rigor

The standard's own verbs settle it — *understand*, *explain*, *represent* → **conceptual**;
*fluently*, *compute*, *solve* → **procedural**; *use … to solve real-world problems* →
**application**; spanning two → **Mixed**, saying which item covers which. Label it in the meta
block; it drives item format, representation, and what a good distractor looks like.

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## 2. Map documented errors to stem features

Build this map from the errors prioritized in `references/learning-commons-kg.md` before writing any
item. For each: *what would a student reasoning this way do here, and what must the stem contain to
give them the chance?*

| Error pattern | Prerequisite gap or on-grade? | Stem feature needed |
|---|---|---|
| [pattern] | [which] | [what must appear in the stem] |

Internal working; it appears in neither file. **A distractor invented after the stem was
written comes out merely wrong rather than plausible**, and discriminates nothing.

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## 3. Item design

Write 1–3 items; one that discriminates beats three that don't.

| Rigor | Format |
|---|---|
| **Conceptual** | MC, multiple select, or CR — whichever reveals the understanding rather than the computation |
| **Procedural** | CR is primary; at most one multiple-choice item |
| **Application** | CR, with the student's approach and reasoning visible |

- **Representation.** Conceptual K–8: at least one item includes or describes a diagram, number line,
area model, or tape diagram — an all-symbolic conceptual item measures procedure instead.
Procedural: at least one item shows work as a student would — a partially worked problem, a worked
example with an error to find, or a show-your-work prompt. Any representational form the confirmed
component names is the one to use.
- **Number values** stay in the range the standard and grade expect: above it a conceptual check
becomes an arithmetic obstacle course, below it a student can just count.
- **Discrimination.** At least one item lets a student reach the correct answer by a faulty method,
or exposes partial understanding in a "correct" response.
- **Every item reveals thinking, not just correctness.** MC: every distractor maps to a named error
from the map. CR: the prompt explicitly asks for the explanation, justification, or work — *"How do
you know?"* at grade 3, *"Explain how you decided"* at grade 6.
- **Figures state in words what they show** — "4 groups of 3 dots each", not "dice"
(`references/output.md` has the markup). **And what a picture cannot say, the stem says:** *"Which
expression shows the number of groups times the size of each group?"* is settled by figure plus
stem; *"Which expression matches the picture?"* is not, because factor order is a convention the
picture cannot carry.

### Multiple-choice items

- Each distractor corresponds to one specific error from the map; prefer documented ones.
- At least one reflects a **prerequisite gap** and at least one an **on-grade confusion**, so
responses route to different next steps.
- The key is not distinguishable by format, length, or specificity.
- **A distractor is a place a real student's reasoning lands** — working this stem's numbers, the
error arrives at that choice. Checked at `references/verification.md`, on the written files.

### Student-facing language sits at the grade's reading level

Above-grade wording puts a second question inside the first, and the student lands in a distractor
reporting a misconception they don't have. Applies to every string a student reads: stems, choices,
directions, figure `<title>`/`<desc>`.

- **Vocabulary.** Terms this grade's instruction teaches stay — *numerator*, *denominator*, *equal
parts* belong in a grade 3 item. Every other word is the one the grade already has: *split into 4
equal parts*, not *partitioned into fourths*; *find*, not *determine*. Standard wording is written
for adults, so the meta block quotes it verbatim for the teacher and the item says the same idea in
the students' words.
- **Sentences.** One idea each, actor first, present tense, no idioms. **K–2** about 8 words; **3–5**
about 12 and at most one subordinate clause; **6–8** about 15; **9–12** length comes from the
mathematics. A sentence past its band carries something the mathematics didn't ask for — split it.
A familiar context costs no reading; an unfamiliar one must be understood first.

**Stems and choices are student-facing**: no annotation, symbol, label, bolding, or formatting that
identifies the key, references an error pattern, or names a next step. Drafting them clean once makes
the handout safe to hand over unedited.

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## 4. Teacher-facing notes

For each item:

1. **What each response reveals** — one row per choice for MC, the expected response types for CR,
describing the **observable behaviour**: what the student writes, draws, or circles, not an
inferential label. "Adds the denominators as well as the numerators" is usable; "doesn't
understand fraction addition" is not.
2. **The next step.** Name which kind it is from the coherence map — **prerequisite gap** (prior-grade
content not yet secure) or **on-grade confusion** (this standard's own demand) — then make it
concrete from the retrieved guidance: the specific sub-skill, representation, or task type. A
prerequisite-gap row routes to a **named prior standard code**, never "review earlier work."
3. **What to look for across responses** — two or three sentences on whether a shared pattern points
to one prior-grade gap several students can all be routed to, versus scattered on-grade confusions
each needing different handling. **If two rows route alike, the data isn't working.**

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## Meta-block content (templates: `references/output.md`)

Six rows, in this order:

- **`Standard`** — the verbatim statement retrieved in Step 2, never paraphrased, with its code.
- **`Grade`** — the grade the items are written for.
- **`Rigor`** — Conceptual / Procedural / Application / Mixed.
- **`Learning goal`** — two sentences: what a student who has this can do, scoped to the confirmed
focus, and how that shows up in these items.
- **`Prerequisite standards`** — the 1–3 most relevant backward-progression standards, code plus a
short gist, and the same codes the guide routes prerequisite-gap responses to.
- **`Leads to`** — the 1–2 most relevant forward-progression standards, by code.
- **Copyright:** never reproduce retrieved problem text, contexts, or teacher notes verbatim, and
never name the curriculum, publisher, or research source.

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SHA-256: a2a73b86b8a6bc00e9d08d4ab4dc377cbb553387a9a7aa02bf2e57e104f5f2ef