Diagnostic assessment records: synthetic records and judgments for one fraction-addition item

Goal and scope

For the item Q-FRAC-01, 'compute 3/4+1/6 and write the answer as a fraction a/b with integers a, b and b≠0', first compute the expected result e exactly, then state the judgment rule R_val (a response matches when its value equals e) and the stricter rule R_low (equal value and lowest terms). Use them to judge three synthetic records: D1 with response 4/10, D2 with response 22/24, and D3 with response 22/24 but an empty rule field; and show that the label 'no match' on D1 supports no conclusion about a real student's ability or error cause.

Treats only the single-item record D(q,r) defined by the MKO: the four fields item identity, expected mathematical result, observed synthetic response and judgment rule, and the label an explicit rule computes from them. Every response is synthetic data generated for this exercise (MKO assumption a1), and labels describe only agreement with the explicit rule (assumption a2). No multi-item scale, validity, reliability or educational inference is built; the canonical MKO and its English edition are not changed.

Givens and symbols

• frac_rules
• eq_congruence
• zero_product
• mul_inverse_nonzero
• function_def
• contrapositive
• numeric_evaluation

• G1 — Record definition (MKO): a diagnostic record D=(q,e,r,R) consists of an item identity q, an expected mathematical result e, an observed synthetic response r and a judgment rule R. The label Λ(D) is one of 'match', 'no match' and 'incomplete'; when any field is missing the only permitted label is 'incomplete' (MKO, first paragraph).
• G2 — Item Q-FRAC-01: compute 3/4+1/6 and write the answer as a fraction a/b, where a and b are integers and b≠0.
• G3 — Synthetic records: D1 has response r1=4/10 and rule R_val; D2 has response r2=22/24 and rule R_val (compared also under R_low); D3 has response 22/24 and an empty rule field. Every response is synthetic data generated for this exercise and involves no real learner (MKO assumption a1).
• G4 — Rule R_val: a response a/b matches ⇔ a/b equals e as a rational number. Rule R_low: a response a/b matches ⇔ a/b=e, b>0 and gcd(|a|,b)=1 (lowest terms).
• G5 — MKO assumption a2: a label describes only whether the response agrees with the explicit mathematical rule; an incorrect result is given no psychological or causal interpretation.
• G6 — Two hypothetical synthetic generators (procedures, not people): Gen_A, given an item p/q+s/t, outputs (p+s)/(q+t); Gen_B outputs the entries of the preset list [4/10, 7/12, 11/12] in order, so its first call outputs 4/10.

• D=(q,e,r,R) — a diagnostic record: item identity, expected result, response, judgment rule
• e — the expected mathematical result, here 11/12
• r_1,\ r_2 — the synthetic responses 4/10 and 22/24
• R_{\mathrm{val}},\ R_{\mathrm{low}} — the two explicit judgment rules
• \Lambda(D) — the label of record D
• \mathrm{Gen}_A,\ \mathrm{Gen}_B — two hypothetical synthetic response generators

Rules and sources

• frac_rules · prerequisite — Fraction rules (all denominators nonzero): a/c+b/c=(a+b)/c; (a/b)(c/d)=(ac)/(bd); (a/b)/(c/d)=(ad)/(bc) (also needing c≠0); a/b=(ak)/(bk) for k≠0.
• eq_congruence · prerequisite — Congruence of equality: if u=v, applying the same defined operation f to both sides gives f(u)=f(v). Operating on one side only is not an instance of this rule.
• zero_product · prerequisite — Zero-product property: in ℤ, ℚ, ℝ, ℂ (or any integral domain), uv=0 ⇔ u=0 or v=0.
• mul_inverse_nonzero · prerequisite — Inverse of a nonzero number: in ℚ, ℝ or ℂ, if u≠0 there is 1/u with u·(1/u)=1, and v/u means v·(1/u). 0 has no inverse, so v/0 is undefined.
• function_def · prerequisite — Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every a.
• contrapositive · prerequisite — Contrapositive: P⇒Q is equivalent to ¬Q⇒¬P. The converse Q⇒P is in general not equivalent to P⇒Q.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• def_intro · prerequisite — Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable.
• mul_assoc · prerequisite — Associativity of multiplication: (uv)w=u(vw).
• mul_comm · prerequisite — Commutativity of multiplication: uv=vu (in commutative number systems such as ℤ, ℚ, ℝ, ℂ).
• mul_one · prerequisite — Multiplicative identity: 1·u=u.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• case_analysis · prerequisite — Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility.
• mko_diag_record · cited — Here a diagnostic record consists of an item identity, expected mathematical result, observed synthetic response and judgment rule. The rule may label whether the response matches the expectation, but one response cannot establish a real student's ability, error cause, personality or learning outcome. · mko-diagnostic-mathematical-assessment
• gcd_lowest · prerequisite — Lowest terms: a fraction a/b with b>0 is in lowest terms ⇔ gcd(|a|,b)=1. If d=gcd(|a|,b)>1, then a/b=(a/d)/(b/d) names the same value with a smaller denominator.
• L_cross · lemma — For integers a, c and nonzero integers b, d: a/b=c/d ⇔ ad=bc.

Main derivation

\text{diagnostic record}

D=(q,\,e,\,r,\,R),\ \Lambda(D)\in\{\text{match},\ \text{no match},\ \text{incomplete}\}

\tfrac{3}{4}+\tfrac{1}{6}

\tfrac{9}{12}+\tfrac{2}{12}

\tfrac{9}{12}+\tfrac{2}{12}

e=\tfrac{11}{12},\ \gcd(11,12)=1

\tfrac{a}{b}=\tfrac{11}{12}

12a=11b

D_1=(\text{Q-FRAC-01},\ \tfrac{11}{12},\ \tfrac{4}{10},\ R_{\mathrm{val}})

D_1\ \text{complete}

r_1=\tfrac{4}{10}:\ 12\cdot 4\ \text{vs}\ 11\cdot 10

48\neq 110\ \Rightarrow\ \Lambda(D_1)=\text{no match}

\tfrac{4}{10}

\tfrac{4}{10}=\tfrac{3+1}{4+6}

D_2:\ r_2=\tfrac{22}{24},\ R_{\mathrm{val}}

12\cdot 22=264=11\cdot 24\ \Rightarrow\ \Lambda_{\mathrm{val}}(D_2)=\text{match}

r_2=\tfrac{22}{24}\ \text{under}\ R_{\mathrm{low}}

\gcd(22,24)=2\neq 1\ \Rightarrow\ \Lambda_{\mathrm{low}}(D_2)=\text{no match}

D_3=(\text{Q-FRAC-01},\ \tfrac{11}{12},\ \tfrac{22}{24},\ \varnothing)

\Lambda(D_3)=\text{incomplete}

\Lambda

\Lambda:(q,e,r,R)\mapsto\Lambda(q,e,r,R)

\mathrm{Gen}_A(\text{Q-FRAC-01})=\tfrac{4}{10}=\mathrm{Gen}_B(\text{Q-FRAC-01})

\text{the record does not determine the generating procedure}

\Lambda(D_1)=\text{no match}

\text{no claim about a real student's ability or error cause}

D_1,\ D_2,\ D_3

\Lambda(D_1)=\text{no match},\ \Lambda_{\mathrm{val}}(D_2)=\text{match},\ \Lambda_{\mathrm{low}}(D_2)=\text{no match},\ \Lambda(D_3)=\text{incomplete}

Substeps and lemmas

\tfrac{a}{b}=\tfrac{c}{d}

\tfrac{a}{b}\cdot bd=\tfrac{c}{d}\cdot bd

\tfrac{a}{b}\cdot bd=\tfrac{c}{d}\cdot bd

ad=bc

Reverse audits

• S14 · sufficient_goal_reduction · S06, S08, S09, S10, S13 — Goal G: the four labels and the scope of what they support. Sufficient condition H: the separate judgments S06, S08, S09, S10 and the scope limit S13; H⇒G by conjunction. H is derived forwards from explicit fields and rules; G is never used as a premise.
• S13 · premise_audit · G3, G5, S12, mko_diag_record — This step fixes the scope of the conclusions: it derives no new positive claim and only excludes inferences beyond the record. It is not claimed that many records would suffice; that would need a separately built valid measurement model.
• S12 · premise_audit · G6, S11, contrapositive, numeric_evaluation — The argument needs one pair with the same record and different procedures; G6 states this pair explicitly and both outputs are computed exactly (N11). It is not claimed that either Gen_A or Gen_B is 'the real one'.
• S11 · premise_audit · S01, G1, function_def — This follows directly from the definition of a function (each input has exactly one output); it is not claimed that different records receive different labels.
• S10 · premise_audit · G1, S08, S09 — Judging D3 needs only the missing-field clause of G1; S08 and S09 only show that guessing a rule would change the outcome and are not premises of the judgment.
• S09 · premise_audit · S08, gcd_lowest, numeric_evaluation — R_low is a conjunction of conditions, and failure of the lowest-terms condition suffices for 'no match'. It is not claimed that R_low is more correct than R_val; they are simply different explicit rules.
• S08 · premise_audit · S04, numeric_evaluation — This uses the '⇐' direction of the equivalence S04: 12a=11b gives a/b=11/12. Completeness is checked as in S05.
• S07 · premise_audit · numeric_evaluation, G5 — Only arithmetic is needed; no converse is claimed, that is, no generating procedure is inferred from this equation.
• S06 · premise_audit · S04, S05, contrapositive, numeric_evaluation — It needs D1 complete (S05) and the contrapositive of S04: 12a≠11b gives a/b≠11/12. The label claims nothing about who produced the response or why.
• S05 · premise_audit · G1, G2, G3, S03 — Completeness checks only that the fields exist and have the right form; this step does not judge the response.
• S04.2 · inverse_equivalence · frac_rules, mul_assoc, mul_comm — These are identities valid for all integers when b≠0 and d≠0, so the step holds both ways.
• S04.1 · inverse_equivalence · eq_congruence, zero_product, mul_inverse_nonzero — Multiplying by bd and by 1/(bd) undo each other given bd≠0, which follows from the lemma's hypotheses b≠0 and d≠0 by the zero-product property. If b=0, a/b is not even defined.
• S04 · inverse_equivalence · L_cross, G2 — Both directions of the lemma are proved in S04.1 and S04.2; the condition b≠0 is guaranteed by the answer format in G2, and d=12≠0.
• S03 · inverse_equivalence · frac_rules, numeric_evaluation — Backward, 11/12 splits back into 9/12+2/12 (since 11=9+2); addition over a common denominator is a rewriting of equals and holds both ways. The gcd is one exact check.
• S02 · inverse_equivalence · frac_rules — The backward move is reduction: 9/12=(3·3)/(4·3)=3/4 and 2/12=(1·2)/(6·2)=1/6; expanding and reducing use the same rule a/b=(ak)/(bk), so both expressions have the same value.
• S01 · premise_audit · G1, mko_diag_record — This step only states the definition and derives no new fact; the range of the label is fixed by the definition.

Error controls and negative controls

One mismatch does not establish a learning disability

The common mistake named by the MKO is to declare a learning disability, a missing concept or failed teaching from one wrong item. D1 here is synthetic, and its label says only 4/10≠11/12 (S06); S12 shows further that two different generating procedures produce exactly the same record, so the record cannot determine a cause. A conclusion about a person drawn from one record has neither a subject nor a basis.

The label depends on the rule

The same response 22/24 is a match under R_val and not under R_low (S08, S09). If the record does not state its rule, a reader cannot replay the judgment or tell whether 'no match' refers to the value or to the form. This is also why D3 can only be labelled 'incomplete'.

Compare by cross-multiplication, not decimals

The decimal expansion of 11/12 starts with 0.91666, with the digit 6 continuing forever, and comparing truncations can go wrong; S04 turns equality of values into the integer equation 12a=11b, which is decided exactly. L_cross needs b≠0 and d≠0, and the answer format G2 already guarantees nonzero denominators.

• D1 is labelled 'no match', so the student who answered does not understand common denominators → reject — The response is synthetic and there is no real student (a1); moreover Gen_A and Gen_B produce the same record, so the record cannot determine a cause (S12).
• 22/24 is written differently from 11/12, so D2 is 'no match' under every rule → reject — 12·22=264=11·24, so the values are equal and R_val gives 'match' (S08).
• D3 lacks a rule, so R_val may be assumed and D3 labelled 'match' → reject — A missing field allows only 'incomplete' (G1); assuming R_low instead would give the opposite label (S09).
• 4/10=(3+1)/(4+6), so the response was certainly produced by adding numerators and denominators separately → reject — Gen_B outputs the same 4/10 from a preset list and the record is identical (S12); the equation is only a relation between numbers.

Result and limitations

The expected result of item Q-FRAC-01 is e=11/12. Under R_val, D1 (response 4/10) is 'no match' because 12·4=48≠110=11·10, and D2 (response 22/24) is 'match' because 12·22=264=11·24; under R_low, D2 is 'no match' because gcd(22,24)=2; D3 lacks its rule field and can only be labelled 'incomplete'. These labels describe only agreement with explicit rules and support no conclusion about a real student's ability, error cause or learning outcome.

• Multi-item scales, validity, reliability, human review and privacy governance belong to the MKO's formal next step; this guide builds none of them and does not claim that any collection of records supports an educational inference.
• R_val and R_low are two rules written for this demonstration; which rule an actual item bank uses must be stated by the item's designer.

Verification and source binding

12 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
• adopted source SHA-256: 68c57629a9fbd59bc53c99b3d573ddd78386c61f4559822c6aa9a0484a453820
• original source SHA-256: c887249d698314e2f1a1000c157d36ba15ec9b3910807e5b370612fbfe6b57ee
• Lean: NOT_RUN · independent AI: NOT_MEASURED

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