MATHEMATICS DIAGNOSIS · FIND THE EARLIEST WEAK LINK

Do not diagnose the chapter. Diagnose where the route first becomes unreliable.

A low mark, a difficult chapter or a loss of confidence is a signal. It tells us that something is wrong. It does not yet tell us what is wrong.

At Bukit Timah Tutor, Mathematics diagnosis means tracing the learner’s work backwards until we find the earliest important concept, representation, decision or prerequisite that is no longer carrying the later work reliably. We then repair that point, reconnect it to the current syllabus and test whether the repair survives when the question changes, support reduces and time passes.

Current failure → gather evidence → narrow the possibilities → probe → locate the earliest unstable connection → repair → reconnect → verify.

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Diagnosis is a process. A test is one possible instrument.

There is no rule that every student must begin with the same standard paper. Sometimes a recent school script already exposes the problem. Sometimes the tutor needs to watch the student attempt a question. Sometimes a conversation reveals a useful history but not the mechanism. Sometimes a short set of deliberately chosen questions is needed to distinguish between two competing explanations.

These are not four competing diagnostic systems. They are four sensors feeding the same question: where does the route first become unreliable?

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The visible mistake may have several different causes.

Two students can produce the same wrong answer and require different teaching. That is why the intervention should follow the cause rather than the label on the chapter.

Possible weak link	What it can look like	What the tutor needs to distinguish
Concept	The idea itself is not understood.	Can the learner explain what the operation or relationship means?
Representation	The student cannot move reliably between words, diagrams, symbols or equations.	Does the difficulty disappear when the same idea is represented differently?
Prerequisite	The current chapter exposes an earlier dependency.	Which earlier skill must be stable before the current method can work?
Retrieval	The student once knew the method but cannot bring it back when needed.	Is the knowledge absent, or merely unavailable without a cue?
Recognition	The student can solve a familiar form but cannot identify when the same idea applies elsewhere.	Can the learner recognise the structure when the surface changes?
Method	The intended route is known but the procedure is incomplete or incorrectly sequenced.	Where does the method first depart from a valid route?
Execution	The mathematics is known but signs, substitutions, algebra or working break during performance.	Is this conceptual failure or performance leakage?
Load	The student becomes slow, overloaded or loses track during multi-step work.	Does accuracy return when the task is decomposed or time pressure is removed?
Transfer	Single-topic practice works; mixed or unfamiliar questions fail.	Can the learner select and adapt the idea without the topic label?
Examination control	The student knows the Mathematics but loses marks under paper conditions.	Is the limiter retrieval, timing, reading, decision-making, checking or recovery?

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A diagnostic probe should discriminate, not merely measure.

Suppose a student sees 2(x + 3) = 14 and writes 2x + 3 = 14. Calling this “an algebra weakness” is too broad. A better diagnosis asks which explanation predicts the error.

• Can the student explain what 2(x + 3) represents?
• Can the student expand it correctly when explicitly asked to expand?
• Does the same error occur with 3(x + 5)?
• Can the student identify the mistake in somebody else’s working?
• Does the problem disappear when variables are replaced with numbers?
• Can the learner distribute correctly in isolation but fail only while solving an equation?

Each question changes the evidence. The aim is to eliminate explanations until the smallest useful repair becomes clear.

The smallest useful repair is not necessarily the smallest error. It is the earliest unstable connection whose repair restores the largest amount of downstream work.

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A diagnosis is provisional until reality answers back.

The tutor’s first interpretation can be wrong. So diagnosis should not end when a plausible explanation is found. It should make a prediction: if this is the important weak link, repairing it should improve these later behaviours.

If the predicted improvement does not appear, we revise the diagnosis. That is an important part of the philosophy: the learner’s later performance is allowed to contradict our first explanation.

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The complete loop

1. Signal — notice the visible problem.
2. Gather evidence — conversation, work, papers and history.
3. Narrow — identify competing explanations.
4. Probe — ask only the questions needed to discriminate between them.
5. Locate — identify the earliest important unstable connection.
6. Repair — intervene at that point.
7. Reconnect — return the repair to current Mathematics.
8. Verify — change the question, reduce support and revisit later.
9. Re-diagnose when necessary — if reality does not behave as predicted, change the explanation rather than defending it.

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What this is — and what it is not

This is a structured teaching and mathematical error-analysis process. It uses evidence to decide where teaching should begin and whether a repair has worked.

It is not presented as a standardised psychometric assessment, clinical diagnosis or independently validated diagnostic test. A short diagnostic worksheet can be useful, but its role is to provide evidence for teaching decisions—not to turn one score into a complete description of a learner.

Diagnosis should make the next piece of work smaller, clearer and more justified. If it does not change what we teach next, it has not yet done enough.