## Math Subject Evaluation Rules

## Word Problem Stem Integrity

A word problem stem that is followed by a bare arithmetic equation fragment — a standalone equation restating the operation that appears appended after the word-problem narrative rather than serving as the answer input field — is typically a copy-paste or formatting error. Such fragments should not appear appended to the question stem. If present, this is a `clarity_precision` failure because the stem is no longer a clean, self-contained word problem.
## Grade-Level Comprehension Limits (Early Elementary)

For the lower elementary grades (approximately K–3), a question stem that asks students to first solve multiple prerequisite sub-problems and then use those intermediate results to solve a composite problem — without scaffolding or worked-example framing — can overwhelm working memory and comprehension. When the stem itself is so densely packed with sequential instructions that a student at the target grade would struggle to understand what is being asked, this is an `educational_accuracy` failure. The question should be self-contained and ask for the target skill directly; prerequisite computations should be provided or the item should be broken into separate steps.

## Pedagogical Rationale for Data-Interpretation Questions

For data-interpretation questions (line plots, bar graphs, tables, charts, etc.),
the question must ask for an operation that has a clear pedagogical rationale or
real-world motivation within the scenario. Questions that ask students to perform
arbitrary, unmotivated operations on data — for example, counting items that
exclude a specific value without a stated reason, summing an arbitrary subset, or
filtering by a criterion that is not meaningful in the scenario — fail
`educational_accuracy` because they do not assess a genuine mathematical skill;
they only test whether a student can follow an unmotivated directive.

**MUST-FAIL audit — emit `data_interpretation_pedagogy`**: When a data-interpretation question asks for an operation on data (count, sum, exclude,
filter) and the criterion for selecting which data points to include or
exclude is arbitrary and unmotivated. A good data-interpretation question
connects the operation to a real purpose in the scenario (e.g., "how many are taller
than...", "what is the total for the two longest...", "how many more X than Y").
A question that asks for an unmotivated exclusion or arbitrary subset without a
stated reason is a pedagogically empty exercise. Fail `educational_accuracy` = 0.0.

## MATHEMATICAL VERIFICATION DATA

When provided:
1. **Result CORRECT**: `factual_accuracy = 1.0`. Explanation wording issues go under `clarity_precision`, NOT `factual_accuracy`.
2. **Result INCORRECT**: `factual_accuracy = 0.0`, unless your own verification clearly confirms the answer is correct (extraction error).
3. **UNABLE TO VERIFY**: Fall back to your own reasoning.

Speak about the analysis as your own. Do NOT mention "SymPy" or "programmatic verification" in output.

When inline SVG is present, use SVG coordinates to mathematically verify geometric claims.

### Additional Math Verification Rules (Closed-Ended)

3. **If options analysis shows a distractor is accidentally correct**: Note this under `distractor_quality`.
4. **If options analysis shows NO option matches the computed answer**: This means all options are wrong -- set `factual_accuracy = 0.0`.

**Curriculum Context Note**: When curriculum specifies pedagogical distinctions (e.g., 3×4 vs 4×3 in early grade math), prioritize curriculum alignment over general equivalence.

**Conceptual Learning Context** (e.g., introducing multiplication to 3rd graders):
- A countable 3×4 array is APPROPRIATE scaffolding
- Even though students could "just count" instead of multiplying, this supports learning what multiplication means
- A "3 × 4 = ?" label on the array is also fine – the student still needs to compute
- Only "3 × 4 = 12" directly stated would be a problem (answer is literally given)

**Fluency/Mastery Context** (e.g., testing multiplication fact recall for 8th graders):
- The same countable array may be INAPPROPRIATE because:
  - 8th graders should already know 3×4
  - The array lets them bypass demonstrating that knowledge
  - This prevents identifying genuine knowledge gaps
- However, this requires clear curriculum evidence that fluency is being assessed

**Numeric Consistency (applies to images with explicit numbers/counts):**

When an image shows explicit numbers or countable objects:
- **Matching OR approximately matching the problem** → PASS (supports the problem)
- **Clearly labeled as a separate example** → PASS (conceptual scaffolding)
- **Shows the computation (e.g., "3 × 4 = ?")** → PASS (student still needs to compute)
- **Shows the answer directly (e.g., "3 × 4 = 12")** → May be a problem if this IS the question being asked
- **Contradicts the problem's numbers in a confusing way** → FAIL only if the mismatch would mislead students about what to calculate

**Examples - PASS:**
- "4 × 6 = ?" with a 4×6 array → PASS (scaffolding for conceptual learning)
- "4 × 6 = ?" with a labeled "4 × 6 = ?" on the array → PASS (student still computes)
- "Mia made 48 clay animals, divides into 6 groups" with photos of clay animals → PASS (illustrative/engaging - the exact count doesn't need to match)
- Question about multiplication with a decorative border of stars → PASS (neutral)

**Examples - FAIL:**
- "What is 3 × 4?" with an image showing "3 × 4 = 12" → FAIL (answer literally given)
- Multiplication fluency test (curriculum clearly states this) with countable arrays that let students bypass recall → May FAIL (trivializes for the specific pedagogical purpose)

**Important clarification**: Many good items can be solved from text alone (e.g., computing 48 ÷ 6). This is NOT a Mastery Learning failure if students still have to apply a procedure or reasoning step. Even if the image provides scaffolding rather than being strictly necessary, Mastery Learning can pass as long as the task requires thinking.

**Examples:**
- PASS: "48 ÷ 6 = ?" (requires computation, even if solvable from text alone)
- PASS: "Which fraction is equivalent to 2/4?" (requires understanding equivalence, not just recall)
- PASS: "Convert: 1 foot = ___ inches" when Difficulty Definition says "recalling a base equivalence" (curriculum-sanctioned recall)
