PHASE 4 · DIAGNOSTIC PROBE READER GUIDE

Quick Read: what makes a diagnostic Mathematics question useful?

A diagnostic question is useful when different answers or explanations separate plausible causes of the same visible error and lead to a different next teaching action.

The purpose is not to create another test. It is to reduce uncertainty. A student who gets 2(x+3) wrong may misunderstand distribution, misread symbolic notation, lose a sign under load or simply make a one-off slip. A good probe changes one feature at a time so those explanations begin to separate.

One-sentence answer: ask the smallest question that can change what you would teach next.

---

Same error, different cause

Visible error	Possible cause	Discriminating probe
2(x+3)=2x+3	Distribution meaning is weak.	Compare 2(5+3) with 2×5+2×3 before using x.
2(x+3)=2x+3	Arithmetic distribution is fine but symbols disrupt the learner.	Replace only one number by a letter after a correct numerical example.
2(x+3)=2x+3	Concept is understood but execution breaks under load.	Ask the same structure in a short isolated question and compare with a longer mixed problem.
2(x+3)=2x+3	One-off slip.	Ask the learner to inspect and explain the original line without reteaching first.

The visible mistake is not yet a diagnosis. The diagnosis begins when competing explanations are tested.

---

A strong probe changes one thing at a time

1. Hold the mathematical structure steady. Change notation, numbers or representation while preserving the relationship.
2. Observe what survives. Does the learner still know what to do?
3. Change the suspected weak feature. Remove a diagram, simplify the algebra, change the reference angle or alter the wording.
4. Compare performance. The difference between the two responses is often more informative than either score alone.

This is diagnostic contrast. It prevents the tutor from explaining everything at once and then guessing which explanation mattered.

---

Six useful probe families

Probe family	What it helps separate	Example
Representation probe	Concept vs translation weakness.	Ask the same relationship as words, diagram and equation.
Load probe	Knowledge vs working-memory/execution strain.	Compare a short clean version with a long mixed version.
Transfer probe	Memorised template vs portable understanding.	Change the surface while preserving the underlying method.
Reversal probe	Forward routine vs relational understanding.	Expand then factorise; solve then substitute back.
Boundary probe	Rule memorisation vs condition awareness.	Ask when a method no longer applies or give a near-miss example.
Explanation probe	Correct answer vs reasoned understanding.	Ask why the operation preserves the relationship.

---

One wrong answer should not become a learner label

A probe produces evidence, not identity. The learner may be tired, misread one line or use an unusual but valid route. Diagnostic confidence should rise only when evidence converges across more than one observation or when one response is especially discriminating.

• Prefer “the current evidence suggests…” to “this student is weak in…”
• Look for repeated structure across changed questions.
• Allow later performance to overturn the first interpretation.
• Separate temporary state from stable capability where possible.

A diagnosis should remain correctable by the learner’s next piece of work.

---

Three short case examples

Case 1: “My child keeps getting fraction questions wrong.”

First probe magnitude without calculation: place 3/5 and 3/8 on a number line or explain which is larger. If magnitude is unstable, drilling fraction operations is premature. If magnitude is sound but operations fail, the repair moves elsewhere.

Case 2: “My child knows algebra at home but fails tests.”

Compare the same algebraic structure in an isolated untimed question and in a mixed timed sequence. If the concept survives only in the first environment, the active issue may be retrieval, load, pacing or checking rather than missing algebra knowledge.

Case 3: “My child cannot do trigonometry.”

Before reteaching ratios, rotate the triangle or change the reference angle and ask the learner to relabel opposite and adjacent. If the labels fail, geometry/representation is the earlier edge. If they survive, the next probe can test ratio selection.

---

Stop probing when the next useful action is clear

Diagnosis has diminishing returns. Once the tutor can choose a safe, specific teaching response, more probing can delay learning and make the student feel examined rather than helped.

1. Generate a small set of plausible causes.
2. Ask one discriminating question.
3. If ambiguity remains and the next teaching action would differ, ask one more.
4. Once the next useful action is clear, teach.
5. Verify the repair with a changed-looking question.

The verification step matters because a convincing explanation can still be wrong. The learner’s later performance is what determines whether the diagnosis and repair earned confidence.

---

What tutors should record mentally—or formally

• What was directly observed?
• Which causes remain plausible?
• Which probe separated them?
• What teaching action followed?
• Did support reduce afterward?
• Did the improvement survive a changed question?
• Did it remain after delay?

The point is not paperwork. The point is to stop explanations from hardening into certainty without evidence.

---

What parents can ask

• What exactly suggests this is the weak area?
• Could the same mistake come from another cause?
• What small question would distinguish those possibilities?
• What should improve if the diagnosis is correct?
• How will we know the child can do it with less help?
• Will the improvement be checked on a different-looking question?

These questions keep diagnosis practical. The purpose is not a sophisticated label; it is a better next lesson.

---

Frequently asked questions

How many questions are needed for diagnosis?

There is no fixed number. Use enough evidence to separate the plausible causes that would lead to different teaching actions, then stop.

Can one diagnostic question be wrong?

Yes. A learner can guess, slip or respond unusually. That is why interpretations should remain provisional and be checked against later performance.

Is diagnosis the same as testing?

No. Testing often measures performance. Diagnosis tries to distinguish mechanisms so the next teaching decision improves.

What is the strongest evidence that a probe was useful?

It changed the teaching decision in a way that led to more independent, transferable and retained performance.

---

The larger idea: diagnosis should reduce uncertainty, not manufacture certainty

The strongest diagnostic practice is modest. It observes carefully, tests plausible explanations, acts when enough is known and allows later evidence to correct the story.

That discipline protects the learner from unnecessary remediation and helps teaching stay focused on the earliest useful repair rather than the loudest visible mistake.

A good diagnostic question is small because its job is precise: make the next teaching decision better.

COMPATIBILITY
OWNER = DIAGNOSTIC_PROBE_BANK
MATHLAB_ROLE = compose_and_call
BOOT = BTTMathLab/0022
PRESERVE_PROBE_IDENTITY = TRUE
RETURN_EVIDENCE = BTTMathLab/0514