Secondary Mathematics Tuition | How to Catch Up in Maths: Diagnose Gaps, Rebuild Foundations and Rejoin Current Work

Secondary Mathematics Tuition · Catch-up, prerequisite repair and return to current work

Adrian is six chapters behind in Mathematics.

That is how he describes the problem.

It is not yet a useful diagnosis.

Six chapters can contain six different kinds of difficulty: one missing prerequisite that keeps reappearing, several forgotten techniques, slow algebra that makes current work collapse under load, weak question reading, poor checking, or simply unfamiliar material that was never learned securely the first time.

Clara has a different pattern. She understands the lesson while the teacher is explaining it, but the next week’s work depends on fractions, signs and equation structure that never became stable. She appears to be “falling behind” even though the newest topic is not the true starting point.

Ethan reacts by trying to restart everything.

He opens an old textbook at page one and begins again.

After several evenings, he has completed many easy exercises and repaired almost none of the structures that are blocking current schoolwork.

Catching up in Maths is not the same as doing more Maths.

It is a dependency-repair problem.

This guide owns the worldwide Secondary Mathematics learner-facing catch-up layer: how to identify the smallest set of earlier weaknesses that are blocking present work, repair them without abandoning the current syllabus, retest them under changed conditions, and return the learner to ordinary progression as quickly and safely as the evidence permits.

It does not replace the site’s broad diagnostic owner, the worked repair library, a year-specific syllabus page, a local tuition page, or the general study and revision systems.

How Mathematics Diagnosis Works remains the broad diagnosis owner. The Secondary Mathematics Worked Repair Library remains the worked repair directory. How to Study Maths Effectively remains the broad study owner. How to Revise for Maths remains the revision owner. How to Practise Maths Effectively remains the deliberate-practice owner.

This article owns a narrower but common job:

the learner is already behind, current Mathematics keeps moving, and both repair and continuation must happen at the same time.

Adrian, Clara and Ethan are fictional recurring learners. Their scenes are explanatory examples, not reported student cases or fixed ability labels.

The central catch-up architecture is:

LOCATE THE CURRENT FRONTIER → FIND THE EARLIEST BLOCKING DEPENDENCY → TRIAGE BY LEVERAGE → PROTECT CURRENT WORK → REPAIR THE MINIMUM NECESSARY FOUNDATION → RETEST ON CHANGED PROBLEMS → RETURN AFTER DELAY → RECONNECT THE REPAIR TO CURRENT TOPICS → MONITOR WHETHER SUPPORT CAN FADE → EXIT CATCH-UP MODE WHEN ORDINARY PROGRESSION IS STABLE.

That sequence is not a universal calendar or a promise that every learner can close every gap quickly. The duration depends on what is missing, how much current demand continues, how securely earlier knowledge can be rebuilt, and whether the learner is carrying several simultaneous bottlenecks.

Evidence and scope note: the catch-up maps, fictional cases, routing matrices and schedules in this article are original instructional tools, not validated psychometric instruments. The What Works Clearinghouse guide Organizing Instruction and Study to Improve Student Learning supports, with different evidence ratings, principles including spaced study, worked examples, quizzing/re-exposure and explanatory questioning. The WWC Algebra guide gives moderate evidence to intentionally choosing among alternative algebraic strategies and more limited evidence to some other recommendations. NCETM’s Five Big Ideas emphasise coherence, representation and structure, mathematical thinking, fluency and variation. These sources support bounded instructional principles; they do not establish one universal catch-up timetable or one mandatory recovery sequence for every curriculum.

1. “Behind in Maths” usually describes a position; catch-up begins by finding the dependency that produced it

A learner can be behind in three different senses.

Content position.

The class has reached Topic 8 while the learner has not learned Topics 5–7.

Structural readiness.

The learner has technically seen the current topic but lacks an earlier structure it depends on.

Performance readiness.

The learner understands the idea but cannot yet retrieve, execute or check it reliably under ordinary school conditions.

These states require different responses.

Suppose the class is solving linear equations.

Adrian makes sign errors whenever a negative term is moved or an expression is expanded.

Giving him twenty more linear equations may create more exposure.

But if the first unstable dependency is signed-number and bracket control, the repair should begin there.

Clara understands equation balance but is so slow simplifying fractions that every equation with fractional coefficients becomes a working-memory problem.

Her catch-up target is not “learn equations again”.

It is to reduce the cost of fraction operations enough that equation reasoning remains available.

Ethan can solve the equations but repeatedly misreads “at least”, “20% more than” and “without replacement”.

His current scores may look like a broad Mathematics deficit.

The earlier failure is question translation.

This is why catch-up starts with:

What is the first thing that becomes unstable before the visible failure?

A dependency chain might look like:

signed numbers → algebraic simplification → equations → simultaneous equations → coordinate intersections → modelling.

Or:

fractions → ratio → percentage → reverse percentage → compound change.

Or:

angle facts → triangle reasoning → similarity → trigonometry.

When an earlier node is unstable, later chapters can fail repeatedly even after several explanations.

The learner feels as though every new topic is difficult.

In reality, the same weak dependency may be travelling forward.

The catch-up principle is:

repair the earliest high-leverage weakness that is still actively generating later failures.

Not every old weakness deserves immediate attention.

A forgotten topic that does not affect current work may wait.

A small sign-control weakness that appears in algebra, graphs and coordinate geometry deserves priority.

This is triage, not neglect.

Catching up means restoring the dependency system in the order that gives the learner the greatest usable return.

2. Track two frontiers: what the class is doing now and what the learner must repair underneath it

Traditional catch-up plans often imagine one timeline.

The learner goes backward, repairs old work, and then moves forward again.

School rarely waits.

While the learner repairs fractions, the class moves into algebra.

While algebra is repaired, the class begins graphs.

A realistic plan therefore has two frontiers.

The current frontier is the Mathematics the learner must engage with now: this week’s lessons, homework, assignments, tests and immediate prerequisites.

The repair frontier is the earliest unstable prerequisite that is still constraining current performance.

The two frontiers should communicate.

Adrian’s current frontier is simultaneous equations.

His repair frontier is linear equation control and signed algebra.

A strong plan does not spend three weeks only on signed numbers while he misses every simultaneous-equation lesson.

It gives him enough support to participate in the current topic while deliberately repairing the older dependency.

For example:

Current lane:

learn what simultaneous equations mean;

recognise elimination and substitution;

complete carefully selected current homework with support.

Repair lane:

signed-number transitions;

collecting like terms;

single linear equations;

substitution of one expression into another.

Then reconnect:

use the repaired single-equation control inside a simultaneous-equation problem.

This prevents two bad outcomes.

Bad outcome 1 — permanent backward retreat.

The learner spends so long on old content that the distance from current work grows.

Bad outcome 2 — permanent surface survival.

The learner keeps receiving hints and copied methods for current work while the prerequisite weakness remains untouched.

The two-frontier system avoids both.

It asks every week:

What current work must be protected?

What older dependency most limits that work?

What repair can be transferred immediately into the current frontier?

When the repair frontier catches the current frontier and ordinary progress becomes stable, catch-up mode can end.

3. Do not restart the entire curriculum unless the evidence really says the entire curriculum is unstable

“Start from the beginning” sounds thorough.

It can also waste enormous time.

A Secondary learner who is behind may still have many secure structures.

They may understand ratio but have weak percentages.

They may understand geometry but have fragile algebra.

They may have excellent arithmetic and poor question reading.

They may have strong concepts but weak retrieval after long gaps.

Restarting everything treats all old knowledge as equally broken.

It also creates a motivational problem.

The learner spends hours succeeding on work that is far below the true bottleneck.

Success feels easy but produces little movement.

Use a selective audit instead.

For each important prerequisite, ask:

Can the learner explain the relationship?

Can they perform a basic example?

Can they retrieve the method without a label?

Can they handle a small variation?

Can they use it inside current work?

If yes across those layers, do not reteach it from zero.

If no, identify exactly where the control disappears.

Consider fractions.

A learner may add simple fractions correctly but fail:

negative fractions;

algebraic fractions;

fraction coefficients in equations;

ratio-to-fraction translation;

or decimal-fraction-percentage conversion.

“Fractions are weak” is too broad.

The catch-up plan should repair the failure mode that actually propagates.

Likewise algebra.

A learner may expand brackets securely but factorisation is weak.

Or solve equations securely but cannot translate a word problem into one.

Or manipulate symbols but lose restrictions in rational expressions.

Do not turn one gap into a full-course restart.

The principle is:

rebuild from the earliest unstable dependency, not from the earliest page in the book.

4. Build a dependency map before building a timetable

A timetable answers:

When will we study?

A dependency map answers:

What must become stable before the next thing can work?

For catch-up, the second question comes first.

A simple dependency map is not a complete curriculum map.

It is a local route around the learner’s present blockage.

Example:

Current problem:

quadratic graphs.

Observed failures:

cannot substitute negative x-values accurately;

cannot expand (x − 2)²;

cannot recognise intercept meaning;

misreads graph scale.

Potential dependencies:

signed arithmetic;

bracket expansion;

coordinate language;

graph reading.

Now test them.

If signed arithmetic and graph scale are secure but bracket expansion and intercept language are not, the repair map becomes smaller.

Repair:

bracket expansion → substitution into quadratic → table of values → intercept interpretation → current quadratic graph task.

That route is far more efficient than “revise all algebra and graphs”.

Now percentage:

Current failure:

compound change.

Dependencies:

percentage as multiplier;

correct 100% base;

reverse direction;

sequential application.

Test each.

Maybe the learner can multiply by 1.15 but cannot identify which quantity is the base after the first stage.

The repair target is reference-base control, not basic percentage calculation.

A useful dependency map contains four fields:

Current demand.

Observed failure.

Candidate prerequisite.

Small test that can confirm or reject it.

This prevents timetable-first planning.

Once the dependency is known, the timetable can allocate time intelligently.

5. Triage catch-up work by leverage, not by chapter order or emotional urgency

When a learner feels behind, every weak topic can feel urgent.

A catch-up plan needs an order.

Use leverage.

A high-leverage repair does at least one of four things.

It unlocks many current topics.

Signed-number control affects algebra, equations, graphs, coordinate geometry and many later techniques.

It appears frequently.

Fraction and percentage structures often reappear across topics.

It reduces cognitive load.

If algebraic simplification becomes cheaper, more attention remains for unfamiliar reasoning.

It restores independence.

If question reading improves, the learner stops waiting for another person to translate every problem.

Lower-leverage repairs may still matter, but they can wait if they do not currently block progression.

Use a simple triage question:

If this skill became secure tomorrow, how many present problems would become easier?

Adrian has five weak topics:

fractions;

signed algebra;

construction geometry;

probability trees;

statistics terminology.

His current work is simultaneous equations and straight-line graphs.

Signed algebra may have the highest immediate leverage.

Fractions may come next if coefficients and gradient calculations expose them.

Construction geometry can wait if no current or near-term work depends on it.

This is not saying geometry is unimportant.

It is saying catch-up time is scarce.

The sequence should maximise restored access to current Mathematics.

Catch-up is therefore not an equal-opportunity review of every old weakness.

It is targeted repair under moving constraints.

6. Audit evidence from recent work before deciding what to repair

A catch-up plan should begin from evidence, not from the learner’s feeling that “everything is weak”.

That feeling may be understandable.

It is rarely precise enough to guide instruction.

Use recent mathematical artefacts.

Homework.

Classwork.

Marked tests.

Unfinished questions.

Questions solved only after hints.

Questions that took far too long.

Questions where the answer was wrong for a reason unrelated to the current topic.

The aim is not to collect a large archive.

The aim is to find recurring first failures.

Suppose Clara’s last three worksheets contain:

a percentage question where she used the final amount as the base;

a ratio question where she reversed the correspondence;

a probability question where she used the wrong denominator population;

and an algebra word problem where she defined the wrong reference quantity.

These look like four topics.

They may share one underlying weakness:

reference-set control.

That is a higher-leverage repair target than “revise percentage, ratio, probability and algebra separately”.

Adrian’s evidence may show a different cluster.

He reaches the right equation but:

drops negative signs;

expands brackets inconsistently;

copies coefficients incorrectly;

and loses exact fractions.

His issue may be execution accuracy and algebraic state preservation rather than conceptual understanding.

Ethan’s work may show:

correct untimed homework;

poor test completion;

blank later questions;

and rushed arithmetic at the end.

His visible “gap” may partly be fluency and paper pacing rather than missing content.

Evidence should therefore be read by mechanism.

A useful audit asks:

What was the first wrong or missing decision?

Was the relevant Mathematics ever known?

Could the learner explain it without performing it?

Could they perform it on a simple case?

Could they retrieve it without a topic label?

Could they use it inside a mixed or changed problem?

Did the failure appear only under time or cognitive load?

This separates “not learned”, “forgotten”, “slow”, “misread”, “fragile under variation” and “fragile under load”.

Those categories need different repair.

Do not over-interpret one error.

A single sign slip does not prove a signed-number deficit.

A pattern across several tasks is more informative.

Likewise, one correct answer does not prove security.

Try a changed version.

If the learner can succeed only when the question looks nearly identical to the worked example, the structure may not yet transfer.

A good catch-up audit therefore uses:

recent evidence → first failure → small confirming test → changed retest.

That is enough to make the next repair decision more precise.

7. Distinguish missing knowledge, weak fluency, weak reading and weak transfer before prescribing more explanation

Four learners can fail the same quadratic question for four different reasons.

Learner A — missing knowledge.

They do not know what factorisation means or how it connects to solving a quadratic.

Learner B — weak fluency.

They understand factorisation but basic expansion and sign handling are slow and error-prone.

Learner C — weak question reading.

They can solve the quadratic once it is formed but cannot translate the context into the equation.

Learner D — weak transfer.

They can solve textbook quadratics when labelled “solve by factorisation” but do not recognise the same structure inside a geometry or modelling question.

All four may say:

“I don’t know quadratics.”

The repair should not be identical.

For missing knowledge:

teach the concept and method explicitly.

For weak fluency:

stabilise lower-cost components until they stop consuming excessive attention.

For weak reading:

practise target, quantities, conditions and translation.

For weak transfer:

vary surface features, remove method labels and mix near neighbours.

This distinction is important because repeated explanation can be ineffective when the bottleneck is elsewhere.

Clara may understand ratio perfectly in discussion.

Her failure comes when the ratio appears inside a multi-stage word problem.

Another lecture on what ratio means may add little.

She needs practice selecting the ratio relationship inside unfamiliar contexts.

Adrian may correctly explain why both sides of an equation must remain equal.

His errors appear because signed arithmetic is not automatic enough.

Another balance-scale explanation may not fix the execution cost.

Ethan may retrieve methods and execute them accurately at home.

His test failures begin after twenty minutes when he starts skipping condition checks.

The catch-up plan needs timed transfer and control under load.

A simple classification test is:

Explain it.

Can the learner say what the relationship means?

Do it simply.

Can they complete a direct example?

Retrieve it.

Can they choose it without being told the method?

Vary it.

Can they handle a changed surface?

Load it.

Can they still do it inside mixed work or under ordinary time pressure?

The first point of failure identifies the repair layer more accurately than a broad topic label.

8. Protect current schoolwork while repairing old foundations

A catch-up plan that ignores current work can create a moving-target problem.

The learner repairs old Mathematics while accumulating new gaps.

By the time the repair reaches the present, the present has moved.

Current work therefore needs protection.

Protection does not mean pretending the learner is fully ready.

It means giving enough support to keep contact with the current mathematical ideas while prerequisite repair happens underneath.

Suppose the class is studying linear graphs.

Adrian’s weak algebra means he struggles to rearrange equations into a useful form.

Current-work protection may include:

pre-teaching the meaning of gradient and intercept;

using a small number of carefully chosen graph questions;

providing a worked rearrangement when the lesson goal is graph interpretation rather than algebraic manipulation;

and explicitly linking the graph work to the algebra repair lane.

The support should be temporary and transparent.

Do not allow the scaffold to become permanent.

If the learner always receives the rearranged equation, the algebra dependency never returns to learner control.

Use the principle:

support the non-target weakness just enough to let the learner engage with the current target, then repair the weakness separately and fade the support.

Clara is studying trigonometry but her ratio foundations are unstable.

Current protection might include:

a clear labelled right-triangle diagram;

one reminder of opposite/adjacent/hypotenuse relationships;

and reduced arithmetic complexity on early examples.

Repair lane:

ratio meaning;

equivalent ratios;

fraction interpretation;

then return to trigonometric ratios.

Ethan is studying probability trees but weak fraction simplification slows him down.

Current protection:

allow the tree structure to be practised with simple probabilities.

Repair lane:

fraction multiplication, simplification and complement relationships.

Then reconnect:

full tree questions with ordinary fractions.

This keeps the learner intellectually present in current lessons.

It also gives the repair purpose.

The learner can see exactly where the old foundation is needed now.

9. Run catch-up as two connected lanes: current access and foundation repair

The two-frontier idea becomes operational through two lanes.

Lane A — Current access.

Goal:

keep the learner participating in present school Mathematics.

Typical work:

preview the new idea;

identify the current target;

complete a small number of representative problems;

use temporary supports where an old weakness would otherwise block access;

record which old dependency causes the obstruction.

Lane B — Foundation repair.

Goal:

remove the obstruction permanently.

Typical work:

direct explanation only where knowledge is missing;

worked example;

independent attempt;

feedback;

changed retest;

delayed return;

mixed reintegration.

The lanes should meet.

If Lane B repairs equations, Lane A should soon include a current topic where equations are needed.

If the repair never returns to current work, transfer remains untested.

Adrian’s week might look like:

Current access:

two simultaneous-equation lesson problems with guided algebra.

Foundation repair:

signed-number transformations and one-step/two-step equations.

Reconnection:

one simultaneous-equation problem where the repaired sign transitions appear.

Clara’s week:

Current access:

percentage growth model in class.

Foundation repair:

fraction-decimal-percentage conversion and reference-base identification.

Reconnection:

reverse-percentage and compound-change problem.

The ratio of time between Lane A and Lane B should not be fixed universally.

If current assessments are immediate, Lane A may temporarily need more time.

If the learner cannot access any current work because one prerequisite is severely broken, Lane B may need a stronger short-term emphasis.

The useful rule is:

never allow either lane to disappear for long.

Only current access creates dependence.

Only old repair allows the class frontier to recede.

Connected dual-lane work creates catch-up.

10. Repair the minimum viable prerequisite first, then expand only if the evidence demands it

Catch-up time is limited.

The question is not:

“How can we make this entire topic perfect?”

The first question is:

“What is the minimum prerequisite state needed for the learner to re-enter current work safely?”

Consider simultaneous equations.

Possible prerequisites include:

meaning of equality;

collecting like terms;

signed arithmetic;

single linear equations;

substitution;

multiplying an equation by a constant.

The learner may not need a complete algebra course before beginning simultaneous equations.

They need enough control over the relevant dependencies.

Test them.

If single equations and substitution are secure but sign control is weak, repair sign-sensitive transformations first.

Then return to simultaneous equations.

Do not wait until every imaginable algebra skill is perfect.

Now trigonometry.

Potential prerequisites:

right-triangle recognition;

ratio meaning;

basic equation rearrangement;

calculator use;

angle/length units and diagram reading.

If ratio and diagram reading are secure but calculator inverse-function use is weak, the minimum repair may be tool interpretation rather than broad geometry revision.

Now percentage.

A learner may need:

decimal multiplication;

percentage multiplier;

reference-base control.

They do not necessarily need every fraction topic first.

The minimum viable prerequisite is not a shortcut that ignores understanding.

It is a scope-control device.

Repair what the current dependency chain requires.

Then observe.

If the learner still fails, widen the repair boundary.

This produces a disciplined expansion rule:

start with the smallest plausible repair; expand only when changed retests show that the blockage remains.

That keeps catch-up focused and prevents endless backward drift.

11. Repair number foundations only where they are actively blocking later Mathematics

Number foundations can become invisible once Secondary Mathematics becomes symbolic.

They are still underneath the symbols.

A learner who hesitates over negative numbers, fraction operations, percentage multipliers or order of operations may appear to have an algebra problem because the algebra is where the weakness finally becomes visible.

Catch-up should trace the error backward.

Suppose:

−3(2x − 5) becomes −6x − 15.

The learner may understand distribution but mishandle the sign on −3 × −5.

That is not a reason to reteach all algebra.

It is evidence that signed multiplication must become more reliable inside algebraic work.

Repair with connected examples:

−3 × 5 = −15;

−3 × −5 = 15;

−3(2x) = −6x;

−3(−5) = 15;

−3(2x − 5) = −6x + 15.

Then return immediately to the algebraic context.

Do not create a long isolated signed-number course unless the evidence shows a broad signed-number deficit.

Fractions create another common hidden cost.

Consider:

(3/4)x = 9.

A learner may know equation balance but become uncertain about dividing by 3/4.

Repair the fraction relationship:

x = 9 ÷ 3/4

= 9 × 4/3

= 12.

Then reconnect to equations with fractional coefficients.

Now ratio and percentage.

If the learner cannot move comfortably among:

3/5;

0.6;

60%;

and a ratio representation where appropriate, later probability, data and percentage work become slower.

But again, the repair should match the current need.

A learner struggling with reverse percentage may not need every fraction skill.

They may need reliable multiplier interpretation:

after 20% discount → 80% remains → multiplier 0.8.

After 15% increase → 115% → multiplier 1.15.

A number-foundation catch-up block should therefore be organised around transfers:

signed numbers into algebra;

fractions into equations and ratios;

decimals into calculator interpretation;

percentages into multiplicative models;

order of operations into expression control.

Avoid endless worksheets containing only isolated arithmetic if the learner can already do the arithmetic in isolation.

The success test is:

can the number foundation now survive inside the current Secondary Mathematics task?

12. Algebra catch-up should rebuild relationships, not just symbol-moving habits

Algebra is a major catch-up bottleneck because so many later topics depend on it.

But “weak algebra” can mean many things.

The learner may struggle with:

what a variable represents;

like terms;

bracket expansion;

factorisation;

substitution;

equation balance;

inequalities;

changing subject;

indices;

algebraic fractions;

or translating words into algebra.

Start with the earliest unstable relationship.

For example, if:

3x + 2x becomes 5x correctly;

but 3x + 2 becomes 5x;

the learner may not yet distinguish like and unlike quantities reliably.

Repair with representation:

three x-units plus two x-units = five x-units;

three x-units plus two ordinary units cannot be combined into one x-term.

Then vary:

3a + 2a;

3a + 2;

3a + 2b;

3a² + 2a²;

3a² + 2a.

The goal is not to memorise “same letters combine”.

It is to recognise structural sameness.

Now equation balance.

A catch-up learner may use the language “move it across and change the sign”.

That shortcut can work until the expression becomes more complex.

Repair the underlying equivalence:

if 3x + 5 = 20, subtract 5 from both sides;

3x = 15;

divide both sides by 3;

x = 5.

Once balance is secure, compact notation can return.

Factorisation and expansion should be connected as inverse views of structure.

2(x + 3) = 2x + 6.

Therefore:

2x + 6 = 2(x + 3).

Seeing both directions matters because later Mathematics asks learners to choose the form that exposes the needed relationship.

For quadratics:

x² − 5x + 6

can be expanded form;

(x − 2)(x − 3)

can expose roots;

(x − 2.5)² − 0.25

can expose vertex structure.

A catch-up plan should not teach these as unrelated tricks.

They are transformations that preserve mathematical meaning while changing what is easy to see.

That conceptual link reduces future relearning.

The algebra repair sequence often benefits from:

meaning → direct example → inverse relationship → changed example → mixed use → current-topic reintegration.

If the learner can perform a skill only when the worksheet heading names it, the catch-up is incomplete.

13. Equations and inequalities need separate catch-up attention because boundaries and solution sets change the answer type

Many learners catch up on equations and assume inequalities are “the same but with a different sign”.

That is only partly true.

The algebraic transformations overlap.

The solution object changes.

An equation often asks for one or more values satisfying equality.

An inequality often describes a set or interval.

Consider:

2x + 3 = 11.

x = 4.

Now:

2x + 3 < 11.

x < 4.

The answer is not one value.

It is every value less than 4 within the relevant number system or domain.

Catch-up should explicitly connect wording and symbols:

at least → ≥;

more than → >;

at most → ≤;

less than → <.

Then connect transformations.

For a standard positive multiplier, dividing both sides preserves inequality direction.

For a negative multiplier, the order reverses.

Example:

−2x > 6.

Divide by −2:

x < −3.

This is easy to memorise as “flip the sign”.

It is more durable to connect it to order:

if 4 > 1, then multiplying both by −1 gives −4 < −1.

Negative scaling reverses order.

Now contextual inequalities.

“A budget is no more than 500 dollars” gives:

cost ≤ 500.

If item count must be a non-negative integer, that condition further restricts the solution set.

The catch-up learner should practise:

translate words;

solve inequality;

apply domain;

interpret boundary.

This becomes important in modelling and optimisation later.

Do not treat inequality catch-up as symbolic manipulation only.

The learner must understand what kind of answer has been produced.

14. Ratio and percentage catch-up should centre the reference quantity

Ratio and percentage errors often persist because learners know procedures but lose the reference quantity.

“20%” is incomplete information.

Twenty percent of what?

“3:5” is incomplete until the ratio terms are attached to quantities.

Three parts of what compared with five parts of what?

Catch-up should make the reference explicit.

Example:

Red:blue = 3:5.

There are 30 red counters.

Three parts = 30.

One part = 10.

Blue = 50.

Now:

Red counters are 3/5 of all counters.

If red = 30:

3/5 of total = 30.

Total = 50.

Same fraction.

Different reference.

Different structure.

Percentage works the same way.

Original price 200.

10% discount.

Discount = 10% of original = 20.

Final = 180.

Reverse the problem:

final 180 after 10% discount.

Now 180 is 90% of original.

0.9P = 180.

P = 200.

The arithmetic is not the main difficulty.

The reading state is.

Catch-up practice should therefore mix:

part-to-part ratio;

part-to-total fraction;

forward percentage;

reverse percentage;

percentage increase/decrease;

percentage points;

compound percentage change.

Do not group each type into long predictable blocks forever.

After initial repair, mix near neighbours so the learner must identify the reference.

Use the question:

What quantity represents 100% or the reference whole here?

Once that question becomes automatic, many ratio and percentage gaps shrink dramatically.

15. Graph catch-up should repair representation reading before formula use

Graphs combine visual, numerical and algebraic information.

A learner can know gradient formulas and still misread the graph.

Catch-up begins with representation semantics.

What does the horizontal axis mean?

What does the vertical axis mean?

What are the units?

What is the scale?

What is the domain?

Is the graph continuous, discrete, categorical or time-based?

Only then use formula or interpretation.

Consider two points:

(5,2) and (15,6).

Gradient:

(6 − 2)/(15 − 5)

= 4/10

= 0.4.

If the x-axis is kilometres and y-axis dollars, the gradient might mean 0.4 dollars per kilometre.

If the axes are completely different quantities, the same numerical gradient has a different interpretation.

Now intercept.

For y = 3x + 5, the 5 is the y-value when x = 0.

In a cost model, it may represent a fixed charge.

In another context, it may mean something else.

Catch-up should connect four representations:

words;

table;

graph;

equation.

Example:

“A service costs 5 dollars plus 3 dollars per item.”

Equation:

C = 5 + 3n.

Table:

n = 0 → C = 5;

n = 1 → C = 8;

n = 2 → C = 11.

Graph:

straight line with intercept 5 and gradient 3.

The catch-up target is not “draw a line”.

It is to see the same relationship across forms.

Use changed-context transfer.

Once the learner can read cost graphs, give a temperature or distance graph with the same mathematical structure.

If they still depend on the familiar story, the representation knowledge is not yet portable.

Graph catch-up succeeds when:

labels, scale and relationship control interpretation before visual impression or formula reflex.

16. Geometry catch-up should separate visual recognition from condition-backed reasoning

Geometry often exposes a particular catch-up weakness.

The learner recognises a familiar shape but does not know which conditions make a theorem legal.

A triangle looks right-angled.

Two sides look equal.

Two lines look parallel.

The diagram seems to suggest a midpoint.

If those conditions are not stated, marked or derived, the visual impression is not enough.

Catch-up should rebuild geometry around evidence.

Use three columns:

Given.

Stated or marked.

Derived.

Proved from givens.

Looks like.

Visual only.

Only the first two belong in the mathematical route.

Consider right triangles.

Before Pythagoras:

Is there a right angle?

Which side is the hypotenuse?

What is the target?

If the learner cannot answer those questions, formula practice is premature.

Now similarity.

Many catch-up learners know that similar triangles involve proportional sides.

They still fail because side correspondence is unstable.

Repair correspondence before ratios.

Write:

A ↔ D;

B ↔ E;

C ↔ F.

Then corresponding sides:

AB ↔ DE;

BC ↔ EF;

AC ↔ DF.

Now build ratios.

Rotating or reflecting the second triangle is useful because it prevents position-based matching.

Angles also need relationship control.

Rather than memorising disconnected facts, organise them around structures:

angles on a straight line;

angles around a point;

vertically opposite angles;

parallel-line relationships;

triangle angle sum;

polygon interior/exterior structures where relevant.

Then ask:

what condition makes this relationship available?

Parallel-line angle facts require parallelism.

Isosceles base-angle facts require equal sides or an equivalent established condition.

Geometry catch-up should also reconnect arithmetic and algebra.

If an unknown angle is 3x + 10 and another is x + 30, the geometry creates an equation.

The learner then needs algebra to complete the route.

This is why dependency repair matters.

A geometry “gap” may be partly algebraic.

Or an algebra “gap” may appear only because the learner cannot extract the geometric relationship.

The exit test is a changed diagram.

If the learner succeeds only when the figure has the same orientation and labels as the worked example, catch-up is incomplete.

17. Probability catch-up should rebuild sample space, state change and denominator control

Probability difficulties can look computational.

They are often representational.

The learner must know:

what outcomes are possible;

which event is being requested;

whether stages are independent;

whether the sample space changes;

which outcomes form the denominator;

and whether complement, tree, table or direct counting is the useful representation.

Consider:

5 red and 4 blue counters.

Two draws without replacement.

Probability of two reds:

5/9 × 4/8.

The second fraction changes because one red counter and one total counter have left the bag.

If the learner writes:

5/9 × 5/9,

the error may not be multiplication.

It is state control.

Repair by narrating the state:

before first draw: 5 red out of 9;

after red draw: 4 red out of 8.

Then compress to the tree.

Now conditional probability.

Suppose 18 of 40 Secondary 1 students travel by bus.

“Given Secondary 1, probability of bus” uses 40 as denominator.

“Probability of Secondary 1 and bus” uses the full population denominator.

The same count can sit over different denominators because the reference population changes.

Catch-up should repeatedly ask:

Out of which population?

Now “at least one”.

Learners often mistake it for exactly one.

Repair with event sets:

at least one = one or more;

at most one = zero or one;

exactly one = one only.

Then show complement:

P(at least one success) = 1 − P(no successes)

when that is an efficient route.

The target is not to teach one favourite technique.

It is to stabilise event meaning and state representation.

Probability catch-up succeeds when a learner can explain why the denominator or branch probability changes before multiplying anything.

18. Statistics catch-up should repair what the summary actually says before adding more formulas

Statistics can become a formula list.

Mean.

Median.

Mode.

Range.

Interquartile range.

Frequency.

Cumulative frequency.

Estimated mean.

Learners may remember procedures but not the information each summary preserves.

Catch-up should reconnect calculation to meaning.

Mean:

a balance-type average using all values.

Median:

middle position when ordered.

Range:

max − min, a crude measure of spread.

Interquartile range:

spread of the middle half of data, under the course’s quartile conventions.

Estimated mean from grouped data:

an estimate because the exact values inside intervals are unknown.

Now combined mean.

Two group means cannot generally be averaged equally if the groups have different sizes.

If group A has 18 students with mean 64 and group B has 12 students with mean 76:

combined total = 18×64 + 12×76;

combined count = 30;

combined mean = combined total / combined count.

The essential catch-up relationship is:

mean = total / count.

That relationship lets the learner move backward:

total = mean × count.

It also explains weighting.

Now comparison questions.

If distribution A has higher median and smaller IQR than B, a good response connects both groups:

A has higher central location by the median and less spread by IQR.

Simply calculating the summaries is not enough if the target says “compare”.

Statistics catch-up therefore combines:

representation reading;

formula meaning;

reference population;

estimated versus exact status;

and answer language.

This prevents formula memorisation from hiding a weak statistical model.

19. Question-reading catch-up may produce faster gains than reteaching content the learner already knows

Some learners are not as far behind mathematically as their marks suggest.

They lose the problem during translation.

They misread:

target;

condition;

comparison direction;

percentage base;

unit;

graph scale;

diagram evidence;

or answer contract.

When that happens, the visible wrong answer may trigger unnecessary content reteaching.

Use a reading-first diagnostic.

Take a failed question.

Ask the learner to state:

what must be found;

what information is given;

what conditions matter;

what representation they would build.

Do not let them calculate.

If the mathematical state is wrong before calculation begins, the catch-up repair belongs partly in question reading.

The worldwide owner How to Read Maths Questions handles that system in depth.

For catch-up planning, the key is to route correctly.

Suppose Ethan fails a reverse-percentage problem.

If he can solve 0.8P = 72 immediately once the equation is given, his algebra is probably not the bottleneck.

The missing step is recognising that 72 represents 80% of original.

Suppose Clara fails a geometry question.

Once told “these triangles are similar with A↔D, B↔E, C↔F”, she sets the proportion correctly.

Her issue may be extracting correspondence from the diagram and markings.

Suppose Adrian fails a probability problem.

He knows how to multiply fractions but treats “without replacement” as if nothing changed.

The repair is state interpretation.

Catching up therefore requires one discipline:

do not reteach downstream calculation until you know whether the failure began upstream in reading.

This can save large amounts of unnecessary practice.

20. Use worked examples to restart a broken route, then fade them before they become a substitute for ownership

A catch-up learner often needs a clear model.

When a prerequisite is missing, independent problem solving can become unproductive because the learner does not yet possess a viable route.

A worked example can expose:

what is noticed;

how quantities are represented;

which relationship is selected;

how the method proceeds;

where checking occurs.

But a worked example has a catch-up risk.

The learner may feel recovered because they can follow the solution.

Following is not ownership.

Use a fading sequence.

Complete example.

Read and explain why each decision is valid.

Prediction.

Hide the next line and ask what should happen next.

Completion.

Remove one step and ask the learner to supply it.

Paired problem.

Use the same structure with changed numbers or surface context.

Faded support.

Remove method labels or intermediate prompts.

Changed problem.

Alter the representation or wording while preserving the core relationship.

Delayed return.

Come back later without the model visible.

Consider reverse percentage.

Worked model:

after 15% discount, final is 85%;

0.85P = final.

Next task:

after 20% discount, write only the relationship.

Next:

a salary is increased by 12%; final salary given; find original.

Next:

a population falls to 92% of previous value; previous value unknown.

The learner must extract the invariant:

final = multiplier × original.

The world owner How to Use Worked Examples in Maths handles example fading in depth.

In catch-up mode, use examples as bridges.

The bridge is complete only when the learner can cross without carrying the worked solution beside them.

21. Catch-up practice should be narrow enough to repair the bottleneck and broad enough to prove transfer

Catch-up creates a temptation to assign large amounts of practice.

The learner is behind, so more questions seem reasonable.

Volume is useful only when the practice is pointed at the right mechanism.

If the learner is making one repeated sign error, fifty mixed equations may bury the cause.

If the learner can already solve direct percentage questions, another page of direct percentages may create fluency without repairing reverse direction.

Use a staged practice architecture.

Stage 1 — isolate the failing relationship.

Keep other demands low enough that the learner can focus on the target.

Stage 2 — vary one feature.

Change numbers, signs, orientation, wording or representation while preserving the same structure.

Stage 3 — compare near neighbours.

Mix examples where a small change should change the method.

Stage 4 — remove the topic label.

The learner must recognise when the repaired structure is relevant.

Stage 5 — reconnect to current work.

Use the repaired skill inside the actual topic that originally exposed the gap.

Stage 6 — delay and retest.

Return later.

Stage 7 — mix into ordinary study.

The repaired skill no longer receives special treatment every time.

Example: sign control.

Stage 1:

−3 × −5;

−4 × 6;

7 − (−2).

Stage 2:

−3(2x − 5);

5 − 2(x − 4).

Stage 3:

compare:

−2(x + 3)

and

−2(x − 3).

Stage 4:

mixed algebra with no heading saying “negative signs”.

Stage 5:

use the algebra inside current simultaneous equations.

Stage 6:

return several days later.

Stage 7:

include one or two sign-sensitive questions in ordinary mixed work.

This sequence prevents a common false recovery:

the learner becomes good at the repair worksheet but still fails when the skill is embedded elsewhere.

The world owner How to Practise Maths Effectively covers practice design more broadly.

In catch-up mode, practice has one additional test:

does the repair reduce the gap between the repair frontier and the current frontier?

22. Use retrieval and spaced return to distinguish real recovery from same-day familiarity

Catch-up learners often show rapid improvement during a repair lesson.

That can be encouraging.

It can also be misleading.

A learner sees several examples, receives feedback, completes a few similar questions and feels that the gap is closed.

Same-day success may reflect recent exposure.

Catch-up needs durable access.

Use retrieval before review.

At the next session, ask the learner to reconstruct the relationship before showing notes.

For example:

“After a 20% discount, what multiplier connects final and original?”

Do not begin by displaying 0.8.

Ask the learner to retrieve it.

For algebra:

“What does factorisation reverse?”

“How would you check an equation solution?”

“What condition changes an inequality direction when dividing?”

For geometry:

“What condition must be established before Pythagoras?”

For probability:

“What does without replacement do to the sample state?”

Then return after increasing intervals where performance remains secure.

There is no universal optimal 1-3-7-14 timetable that should be presented as a law.

Use evidence-responsive spacing.

If the learner retrieves easily and transfers correctly, widen the interval.

If retrieval is slow or wrong, shorten the interval and reopen repair.

Spaced return also prevents catch-up from becoming a sequence of forgotten repairs.

Week 1 fixes fractions.

Week 2 fixes equations.

Week 3 fixes graphs.

If fractions never return, the learner may lose them while attention moves on.

Use maintenance samples.

A small number of old repaired structures should appear inside later mixed work.

The worldwide owner How to Remember Maths covers retrieval and spacing in depth.

For catch-up, the rule is simple:

a gap is not closed because it worked immediately after teaching; it is closed when it can be retrieved later and used where it matters.

23. Convert repeated errors into repair decisions instead of repeatedly marking them wrong

Catch-up accelerates when errors change the plan.

A wrong answer by itself is not enough.

Locate the first invalid or missing decision.

Suppose:

2(x − 3) = 10

becomes:

2x − 3 = 10.

The first error is expansion.

Do not spend time analysing later equation steps built on the wrong state.

Repair distribution.

Retest.

Now:

x² = 49

becomes:

x = 7.

If the question asks for all real solutions, the first issue is completeness:

x = ±7.

If x represents a length, the contextual restriction may legitimately remove −7.

The error cannot be classified correctly without the target and domain.

Now reverse percentage.

A learner calculates:

72 × 0.8.

The arithmetic may be flawless.

The first error occurred when final 72 was treated as the original base.

The repair is reference direction, not multiplication.

Use a compact error receipt:

Question type.

First wrong or missing decision.

Repair target.

Changed retest.

Delayed retest.

Return to ordinary work.

Example:

Question type:

linear equation with brackets.

First error:

distributed multiplier to first term only.

Repair target:

whole-bracket distribution.

Changed retest:

−3(2x − 5).

Delayed retest:

5 − 2(x + 4).

Return:

current simultaneous-equation substitution.

The worldwide owner How to Stop Repeating Math Mistakes covers error analysis more fully.

In catch-up mode, error analysis is especially valuable because it prevents the plan from widening unnecessarily.

A repeated narrow error should create a narrow repair.

A broad pattern across several dependencies may justify a broader rebuild.

24. Use a catch-up time budget that protects sleep, current obligations and the quality of attention

Being behind can trigger emergency study behaviour.

Long late-night sessions.

Every topic every day.

No distinction between high-leverage repair and low-value repetition.

This can increase fatigue without closing the right gaps.

A catch-up plan needs a time budget.

The exact number of hours depends on age, school load, curriculum, assessment timing and the size of the gap.

Do not pretend one universal weekly schedule fits everyone.

Instead allocate by function.

Current access time.

Enough to stay engaged with present lessons and immediate assignments.

Foundation repair time.

Focused blocks on the highest-leverage dependency.

Retrieval time.

Short returns to previously repaired structures.

Mixed reintegration time.

Problems that combine current and repaired knowledge.

Assessment or paper time.

Only when the learner has enough stability for broader performance practice to be informative.

A useful catch-up session might have:

brief current-work orientation;

one repair target;

changed retest;

one delayed retrieval item from an older repair;

one current-topic reconnection.

That can be more productive than two hours of unstructured homework.

Quality matters because catch-up work asks the learner to change old habits.

Fatigue makes learners fall back on familiar shortcuts.

If every session ends with exhausted guessing, the plan may be too large.

Use stop rules.

If error rate rises sharply from fatigue, end the high-precision repair and switch task or stop.

If the learner is repeating the same wrong route without new information, change the intervention.

If a topic is secure across retrieval and transfer, reduce its special practice and move it to maintenance.

Catch-up is not a punishment measured in hours.

It is a resource-allocation problem measured by restored access.

25. A tutor’s catch-up job is to control scope, identify the first blocker and return responsibility to the learner

A tutor working with a behind learner can easily become a permanent rescue system.

The tutor pre-teaches every topic.

Translates every question.

Chooses every method.

Corrects every line.

The learner completes more work but independence does not improve.

Catch-up tuition should reduce dependence over time.

The tutor’s first job is scope control.

What is the current frontier?

What is the repair frontier?

Which one or two dependencies have the highest leverage now?

What can safely wait?

The second job is diagnosis.

Find the first failed responsibility.

Do not immediately explain the whole problem.

Ask:

Can the learner state the target?

Can they form the representation?

Can they retrieve the relationship?

Can they execute it?

Can they check it?

The third job is graduated support.

Use the smallest hint that restores movement.

Examples:

“Which quantity is 100%?”

“What condition authorises that theorem?”

“What is the last line you trust?”

“Can you write the equation without solving it?”

“What changed after the first draw?”

The fourth job is transfer.

After repair, change the surface.

Remove labels.

Delay the return.

Mix with neighbours.

The fifth job is exit.

Catch-up mode should not become the learner’s permanent identity.

When current work is stable, repaired dependencies retrieve reliably and support has faded, the learner should return to ordinary progression and maintenance.

A tutor has succeeded when the learner needs less catch-up architecture, not when the tutor becomes indispensable to every new chapter.

26. Parents can support catch-up by protecting the plan from panic, random topic switching and permanent rescue

When a child is behind, families often feel pressure to act immediately.

Buy more worksheets.

Finish every missing chapter.

Add more hours.

Find a video for every topic.

Ask the child to “work harder”.

These responses are understandable.

They can make the repair system noisier.

A parent’s most useful role is often to protect three things:

scope;

consistency;

independence.

Protect scope by asking what the present repair target is.

If this week’s high-leverage gap is signed algebra, do not add five unrelated revision packs because another old weakness suddenly feels urgent.

Record it for later.

Stay with the current repair long enough to produce evidence.

Protect consistency by maintaining a realistic study rhythm.

A short focused repair block repeated over several days can be more useful than a single exhausted marathon.

Protect independence by avoiding full problem translation and method naming too early.

Useful prompts include:

“What is the current target?”

“Which older skill is this plan repairing?”

“Can you show me the last problem you did without help?”

“What will you retest later this week?”

“Which support are you trying to remove?”

A parent does not need to teach the Mathematics to help the plan remain coherent.

Progress can be observed through evidence:

the child starts current homework with less waiting;

the same old error appears less often;

previously repaired skills survive after delay;

fewer questions require full adult translation;

the child can explain what is being repaired and why;

current topics stop creating entirely new crises.

Avoid making catch-up a permanent label.

“You are behind” describes a temporary state.

It should not become the learner’s identity.

The plan should always contain an exit route:

what evidence will show that special catch-up support can shrink?

When the learner can participate in current work, retrieve repaired prerequisites, handle changed problems and recover from ordinary errors, the household should reduce emergency framing.

Normal study should gradually replace catch-up mode.

27. AI can help organise catch-up, generate changed practice and explain gaps, but it should not become a permanent translation-and-solution layer

AI can be useful in catch-up because it can adapt examples quickly.

It can:

generate near-neighbour problems;

simplify wording;

create a direct prerequisite test;

produce a changed retest;

explain a step in several ways;

compare two methods;

or turn an error into a focused practice set.

It can also hide the learner’s true state.

If AI:

reads the question;

chooses the method;

writes the equation;

performs the working;

and checks the answer,

then a behind learner can appear to complete current work without repairing anything.

Use AI inside a graduated catch-up protocol.

1. Learner attempts first.

State target, givens, attempted representation or first move.

2. Ask for the smallest useful help.

Examples:

“Give one hint about what prerequisite I may be missing.”

“Check my equation but do not solve it.”

“Give me a simpler example of the same relationship.”

“Create three near-neighbour questions where the method changes.”

3. Learner repairs.

Complete the work independently.

4. Changed retest.

Ask for a new problem with different surface features.

5. Delayed no-AI return.

Test whether the skill is now learner-owned.

AI is particularly useful for diagnostic contrast.

Suppose the learner keeps confusing percentage increase and percentage points.

Ask for six short statements that alternate the two ideas.

Do not ask for full solutions.

The learner classifies each relationship.

Suppose the learner struggles with geometry conditions.

Ask for short descriptions where Pythagoras is legal in some and illegal in others.

The learner identifies the condition.

Suppose signed algebra is the repair target.

Ask for changed expressions with the same sign-sensitive structure.

AI output still requires checking.

A generated problem can be ambiguous or incorrect.

The original curriculum and official assessment wording remain authoritative where relevant.

The catch-up rule is:

use AI to vary, clarify and critique the repair; do not let it permanently carry the mathematical responsibilities the learner is supposed to regain.

28. A six-week catch-up arc can organise repair, but the sequence should respond to evidence rather than obey a fixed calendar

The following six-week arc is illustrative.

It is not a universal prescription.

Some learners need shorter focused repair.

Some need longer.

Some have several interacting gaps that cannot be stabilised in one cycle.

The value of a six-week arc is that it gives catch-up a beginning, middle and exit review.

Week 1 — Locate and triage

Identify the current frontier.

Collect recent evidence.

Find the earliest high-leverage blocker.

Choose one primary repair target and perhaps one secondary maintenance target.

Do not begin by creating a giant deficit list.

Output:

current lane;

repair lane;

first changed retest.

Week 2 — Build the minimum viable prerequisite

Teach missing knowledge only where needed.

Use worked examples and direct practice.

Keep current work protected.

Check whether the repaired component can already re-enter the current topic.

Week 3 — Reduce support and vary

Fade worked examples.

Change numbers, wording, orientation or representation.

Introduce near-neighbour contrasts.

Start delayed retrieval of Week 2 repair.

Week 4 — Reconnect and mix

Use the repaired skill inside current school Mathematics.

Mix it with neighbouring methods.

Observe whether the learner recognises when the repair is relevant without a label.

If not, return briefly to contrast practice.

Week 5 — Add ordinary load

Use mixed homework, short timed sets or multi-step problems where appropriate.

Do not jump to full-paper performance if the learner is still fragile.

Watch whether accuracy collapses only under load.

If so, the issue may now be fluency, pacing or working-memory cost rather than missing concept knowledge.

Week 6 — Transfer and exit review

Retest:

direct skill;

changed surface;

delayed retrieval;

current-topic use;

mixed use;

independent recovery from an error.

Then decide:

move the repair to maintenance;

extend the same repair;

or open the next high-leverage dependency.

The useful sequence is:

LOCATE → REPAIR → FADE → VARY → RECONNECT → MIX → ADD LOAD → RETEST → MAINTAIN OR REOPEN.

The calendar is secondary.

The evidence decides when the learner advances.

29. Twelve common catch-up stalls and what they usually mean

Stall 1 — The learner keeps restarting from easy material

Likely issue: repair scope is too broad or emotionally safe.

Response: identify the earliest unstable dependency near the current frontier and test it directly.

Stall 2 — Current homework consumes all available time

Likely issue: no protected repair lane.

Response: reduce current-work volume to representative tasks where possible and reserve a small non-negotiable repair block.

Stall 3 — Repair work is strong, but current work still collapses

Likely issue: transfer was never tested.

Response: reconnect the repaired skill inside current problems and remove topic labels.

Stall 4 — The learner succeeds only with worked examples visible

Likely issue: support dependence.

Response: fade steps, predict next moves and use paired changed problems.

Stall 5 — A skill disappears a week after repair

Likely issue: no retrieval or spaced return.

Response: reopen briefly, then schedule evidence-responsive delayed returns.

Stall 6 — Every week reveals a completely different “gap”

Likely issue: the plan may be classifying surface topics rather than recurring mechanisms.

Response: look for shared dependencies such as sign control, reference quantities, reading, algebraic equivalence or graph semantics.

Stall 7 — The learner understands explanations but still cannot start

Likely issue: recognition or question-reading gap.

Response: use cue-free first moves, target extraction and mixed method selection rather than more explanation.

Stall 8 — Accuracy is high untimed and poor timed

Likely issue: fluency or control under load.

Response: stabilise accurate routines first, then add pace gradually. Do not treat every timed error as missing knowledge.

Stall 9 — The learner avoids difficult repair tasks

Likely issue: difficulty may be too high, success criteria unclear or the learner may be protecting confidence.

Response: reduce task size while preserving the target decision. Use short successful attempts, then increase responsibility.

Stall 10 — The plan keeps expanding

Likely issue: every discovered weakness is being added immediately.

Response: return to leverage. What most blocks the current frontier? Park lower-priority gaps.

Stall 11 — Parents or tutors give full solutions because deadlines are close

Likely issue: survival pressure has displaced ownership.

Response: use temporary current-work support, but preserve a separate repair lane and retest independently afterward.

Stall 12 — Catch-up never ends

Likely issue: no exit criteria.

Response: define evidence for moving a skill to maintenance and returning the learner to ordinary progression.

Stalls are useful information.

They show where the catch-up system itself needs repair.

30. Catch-up is complete when the learner can participate in current Mathematics without a special rescue system carrying the old dependencies

The purpose of catch-up is not to create a perfect historical record where every old topic has been re-studied.

It is to restore functional mathematical progression.

Use exit evidence.

Current access.

The learner can understand and attempt ordinary current lessons.

Prerequisite retrieval.

Key repaired foundations can be recalled after delay.

Transfer.

The learner can use them in changed problems and representations.

Mixed recognition.

The learner does not need every problem labelled by method.

Execution.

Routine components are accurate enough that they no longer dominate attention.

Recovery.

When an error occurs, the learner can often locate and repair it without full rescue.

Support fading.

Worked examples, prompts and adult translation have reduced.

Maintenance.

Previously repaired skills survive occasional mixed use rather than requiring intensive special practice.

Frequently asked questions

How far back should I go?

Go back to the earliest unstable dependency that is still causing present failures. Do not automatically restart the entire curriculum.

Should I stop current schoolwork while catching up?

Usually not. Protect current access while repairing foundations in parallel. The exact balance depends on how severe the gap is and how immediate current demands are.

How many gaps should I repair at once?

Prefer a small number of high-leverage targets. Too many simultaneous repair goals dilute attention and make progress hard to read.

How do I know whether a gap is concept knowledge or fluency?

Ask the learner to explain the relationship, do a direct example, retrieve it without a label, vary it and use it under ordinary load. The first failure helps locate the layer.

Should I use old worksheets?

Only if they test the dependency you need. Old material is useful when it is diagnostic or targeted, not simply because it is old.

Should catch-up practice be easy?

Early repair tasks may reduce unrelated difficulty so the target is visible. The plan must then vary and reconnect the skill to current work. Easy isolated success is not enough.

How soon should I use mixed questions?

After the repaired relationship is stable enough that mixing tests recognition rather than creating noise. Move from direct repair to variation, then near-neighbour and mixed use.

What if the learner keeps forgetting repaired topics?

Use retrieval before review and delayed returns. If forgetting remains rapid, shorten the interval and check whether the learner ever understood the relationship securely.

What if a test is very close?

Prioritise the highest-leverage current prerequisites and representative exam demands. Temporary support may be necessary, but do not confuse short-term survival with durable catch-up.

Is more tuition always the answer?

No. The relevant question is whether the learner needs explanation, diagnosis, structured practice, feedback, time, language clarification, fluency work or another form of support. More hours without a clear mechanism can add workload without closing the gap.

How do I know when to stop catch-up mode?

When current work is accessible, repaired prerequisites retrieve and transfer reliably, support has faded, and ordinary study can maintain the system without constant emergency repair.

The complete catch-up architecture is:

LOCATE THE CURRENT FRONTIER → AUDIT RECENT EVIDENCE → IDENTIFY THE FIRST BLOCKING DEPENDENCY → TRIAGE BY LEVERAGE → PROTECT CURRENT WORK → BUILD THE MINIMUM VIABLE PREREQUISITE → USE WORKED SUPPORT ONLY AS NEEDED → ATTEMPT INDEPENDENTLY → FEEDBACK → CHANGED RETEST → DELAYED RETRIEVAL → RECONNECT TO CURRENT TOPICS → MIX WITH NEIGHBOURS → ADD ORDINARY LOAD → VERIFY TRANSFER → FADE SUPPORT → MOVE THE REPAIR TO MAINTENANCE → EXIT CATCH-UP MODE WHEN NORMAL PROGRESSION IS STABLE.

This page stops at catch-up orchestration.

For the wider study system, use How to Study Maths Effectively.

For deliberate practice, use How to Practise Maths Effectively.

For retrieval and durable memory, use How to Remember Maths.

For revision toward assessment, use How to Revise for Maths.

For cue-free unfamiliar problems, use How to Solve Unseen Maths Problems Independently.

For reading the question before solving, use How to Read Maths Questions.

The learner has caught up when the repair system can disappear without the Mathematics collapsing again.

Public references: What Works Clearinghouse — Organizing Instruction and Study to Improve Student Learning; What Works Clearinghouse — Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students; NCETM — Five Big Ideas in Teaching for Mastery.

Appendix A — Twenty-four catch-up triage cases: find the first blocker before prescribing the repair

This checkpoint is original instructional material. It is not a standardised assessment and has no validated cut score. Its purpose is routing. For each case, identify the current failure, the likely earlier dependency, the smallest useful test and the evidence that would justify moving on.

Use the sequence:

Visible failure → candidate dependency → confirming test → smallest repair → changed retest → current-work reconnection.

Case 1 — Linear equations keep failing because negative signs are unstable

Current task: solve 5 − 2(x − 4) = 13.

Visible failure: learner writes 5 − 2x − 8 = 13.

Candidate dependency: distribution of a negative multiplier.

Confirming test: expand −2(x − 4), −2(x + 4), 3(x − 4).

Smallest repair: distribute the multiplier to every term and connect sign products explicitly.

Changed retest: 7 − 3(2x − 5).

Reconnection: return to the original equation or a current simultaneous-equation substitution problem containing brackets.

Do not over-repair: if collecting like terms and equation balance are secure, do not restart the whole algebra course.

Case 2 — Fraction coefficients make equations collapse

Current task: (3/4)x + 2 = 11.

Visible failure: learner reaches (3/4)x = 9 and stalls.

Candidate dependency: division by a fraction / multiplicative inverse.

Confirming test: calculate 9 ÷ 3/4 and explain why multiplying by 4/3 reverses multiplication by 3/4.

Smallest repair: fraction division relationship.

Changed retest: (5/6)y = 15.

Reconnection: current equations with fractional coefficients.

Exit evidence: fraction operation no longer consumes enough attention to obscure equation reasoning.

Case 3 — Reverse percentage is wrong although forward percentage is easy

Current task: after a 20% discount, final price is 72; find original.

Visible failure: learner calculates 72 × 0.8.

Candidate dependency: reference-base direction rather than percentage arithmetic.

Confirming test: ask “What quantity is 100%?” and “What percentage of original is 72?”

Smallest repair: final = multiplier × original; 0.8P = 72.

Changed retest: after a 15% increase, final value is 230; find original.

Reconnection: compound-change or growth model in current work.

Do not over-repair: if basic percentage multipliers are secure, avoid weeks of forward-percentage worksheets.

Case 4 — Ratio errors come from correspondence, not arithmetic

Current task: red:blue = 3:5; 45 red; find blue.

Visible failure: learner uses 45 × 3/5.

Candidate dependency: mapping ratio parts to quantities.

Confirming test: ask which ratio term corresponds to the known 45.

Smallest repair: 3 parts = 45, one part = 15, five parts = 75.

Changed retest: boys:girls = 7:4; 28 girls; find boys.

Reconnection: current proportion or probability work.

Case 5 — Learner knows area and perimeter but chooses the wrong one

Current task: find amount of fencing around a rectangular garden.

Visible failure: learner multiplies length × width.

Candidate dependency: target interpretation, not rectangle formulas.

Confirming test: use the same dimensions with three targets: area, fencing, diagonal.

Smallest repair: target sentence before formula choice.

Changed retest: “carpet required” versus “border strip required”.

Reconnection: current geometry questions with shared diagrams and different targets.

Case 6 — Pythagoras is used whenever a triangle looks right-angled

Current task: find a side in a triangle diagram.

Visible failure: learner writes a² + b² = c² without a stated/derived right angle.

Candidate dependency: condition-backed theorem use.

Confirming test: show two visually similar triangles, only one with a right-angle condition.

Smallest repair: classify given, derived and looks-like information.

Changed retest: rotated right triangle with explicit right-angle marking.

Reconnection: current geometry or trigonometry.

Case 7 — Graph gradient formula is known but the scale is misread

Current task: calculate gradient from a plotted line.

Visible failure: arithmetic is correct for the wrong coordinates.

Candidate dependency: axis/scale reading.

Confirming test: read coordinates from two graphs with identical square spacing but different numeric scales.

Smallest repair: axes → units → scale → coordinates before formula.

Changed retest: different graph type with non-unit scale.

Reconnection: current linear-graph work.

Case 8 — Simultaneous equations are understood conceptually but arithmetic errors dominate

Current task: eliminate y from two equations.

Visible failure: learner forms the correct elimination but subtracts coefficients/signs incorrectly.

Candidate dependency: signed arithmetic under algebraic load.

Confirming test: perform the same coefficient subtraction without variables.

Smallest repair: signed addition/subtraction and visible column alignment.

Changed retest: elimination requiring subtraction of a negative coefficient.

Reconnection: simultaneous equations.

Possible second layer: if the direct signed arithmetic is secure but fails only in long equations, working-memory load or layout may be the issue.

Case 9 — Probability without replacement uses unchanged fractions

Current task: two draws from a bag without replacement.

Visible failure: learner uses the same numerator and denominator on the second branch.

Candidate dependency: state change.

Confirming test: ask learner to narrate the bag contents after the first draw.

Smallest repair: state-before/state-after representation.

Changed retest: compare with-replacement and without-replacement versions.

Reconnection: current probability tree or conditional event work.

Case 10 — Combined mean is treated as average of two means

Current task: combine groups of different sizes.

Visible failure: learner computes (mean A + mean B)/2.

Candidate dependency: mean = total/count.

Confirming test: reconstruct total from one group mean and count.

Smallest repair: total = mean × count; combined mean = combined total / combined count.

Changed retest: use very unequal group sizes so simple averaging is visibly implausible.

Reconnection: current statistics or weighted-average work.

Case 11 — Learner can solve a quadratic when told the method but cannot recognise it in context

Current task: geometry or modelling creates x² − 7x + 12 = 0.

Visible failure: learner stalls until told “factorise”.

Candidate dependency: method recognition/transfer, not factorisation execution.

Confirming test: give several unlabeled expressions/equations and ask which transformation would expose the target.

Smallest repair: cue-to-structure recognition; compare almost-same problems.

Changed retest: same quadratic structure in a different context with no method label.

Reconnection: current unseen or mixed-topic work.

Case 12 — Homework is correct, tests are incomplete and error rate rises late

Current task: ordinary assessment.

Visible failure: blank later questions and rushed final work.

Candidate dependency: fluency/pacing/control under load rather than broad missing content.

Confirming test: compare untimed direct accuracy with a short timed mixed set.

Smallest repair: make routine components cheaper while preserving accuracy; practise short timed transfers before full papers.

Changed retest: same skill mix with a modest time constraint.

Reconnection: current assessment preparation.

Do not misclassify: if untimed accuracy is already poor, missing knowledge or execution control still needs repair.

Case 13 — Learner keeps forgetting a repaired topic after a few days

Current task: previously repaired algebra or percentage skill.

Visible failure: same learner succeeds immediately after teaching but cannot reconstruct the method later.

Candidate dependency: retrieval durability rather than initial explanation.

Confirming test: ask for the relationship before showing notes at the next session.

Smallest repair: retrieval before review plus evidence-responsive spaced return.

Changed retest: same relationship with altered numbers or context after delay.

Reconnection: include one maintenance item inside current mixed work.

Exit evidence: the learner can retrieve and use the skill after delay without re-teaching.

Case 14 — Algebraic expansion is correct but factorisation is weak

Current task: solve a factorable quadratic.

Visible failure: learner can expand (x − 2)(x − 3) but cannot reverse x² − 5x + 6.

Candidate dependency: inverse structural relationship between expansion and factorisation.

Confirming test: pair expanded and factorised forms and ask learner to match them.

Smallest repair: connect factorisation explicitly as reverse transformation.

Changed retest: x² + 7x + 12 ↔ (x + 3)(x + 4).

Reconnection: current quadratic solving or graph work.

Case 15 — Algebraic method is secure, but word problems still fail

Current task: form and solve an equation from context.

Visible failure: learner solves the equation instantly once someone else writes it.

Candidate dependency: modelling/question translation.

Confirming test: ask learner to define variables and write only the relationship.

Smallest repair: target → quantities → relationship → equation.

Changed retest: same algebraic structure in a different context.

Reconnection: current modelling or word-problem work.

Case 16 — Inequality boundary is misread

Current task: translate “at least 12” into an inequality.

Visible failure: learner writes x > 12.

Candidate dependency: boundary inclusion.

Confirming test: ask whether exactly 12 satisfies the sentence.

Smallest repair: boundary-value test for at least/at most/more than/less than.

Changed retest: “no more than 80”, “fewer than 30”, “at least 5”.

Reconnection: current inequality, probability or optimisation work.

Case 17 — Similar triangles fail when the diagram is rotated

Current task: find a missing corresponding side.

Visible failure: learner matches sides by visual position.

Candidate dependency: correspondence, not ratio arithmetic.

Confirming test: state A↔D, B↔E, C↔F and ask learner to list corresponding sides.

Smallest repair: vertex correspondence before ratios.

Changed retest: reflected or rotated figure.

Reconnection: current similarity or trigonometry work.

Case 18 — Scale-factor questions confuse length, area and volume

Current task: shapes are similar with linear scale factor 3.

Visible failure: learner multiplies area by 3 instead of 9.

Candidate dependency: dimensional interpretation of scale.

Confirming test: use a square of side 2 scaled to side 6 and compare areas directly.

Smallest repair: length factor k → area factor k² → volume factor k³.

Changed retest: scale factor 1/2 or 5/3.

Reconnection: current geometry and similarity.

Case 19 — Learner can calculate gradient but cannot interpret it

Current task: a cost-distance graph.

Visible failure: learner gives gradient 2 but cannot say what it means.

Candidate dependency: unit/representation meaning.

Confirming test: ask for units of gradient and what one additional kilometre changes.

Smallest repair: vertical-unit per horizontal-unit interpretation.

Changed retest: temperature-time or distance-time graph.

Reconnection: current graph modelling.

Case 20 — Learner uses calculator decimals too early

Current task: quadratic formula or trigonometry with a final precision instruction.

Visible failure: intermediate values are rounded aggressively and final answer drifts.

Candidate dependency: exactness/precision control.

Confirming test: compare final results from rounded versus full-precision intermediate values.

Smallest repair: retain exact/full calculator precision until final rounding unless the task requires otherwise.

Changed retest: another multi-stage calculation with 3 s.f. final answer.

Reconnection: current calculator-supported work.

Case 21 — The learner repeatedly copies formulas but cannot choose one

Current task: mixed geometry or algebra.

Visible failure: formula sheet is full, first move is absent.

Candidate dependency: condition-to-method mapping.

Confirming test: remove calculation and ask which relationship is authorised by the givens and target.

Smallest repair: compare candidate methods and the conditions each requires.

Changed retest: near-neighbour problems where one condition changes the legal method.

Reconnection: current mixed-topic work.

Case 22 — Learner needs hints on every unfamiliar question despite strong routine technique

Current task: unseen or mixed problem.

Visible failure: no independent first move without a cue.

Candidate dependency: cue-free first-contact independence.

Confirming test: ask learner to identify target, representation and two candidate first moves without solving.

Smallest repair: low-cost first move, hint ladder and changed-surface transfer.

Reconnection: current unfamiliar problem sets.

Route onward: use the global unseen-problem owner rather than broad catch-up reteaching if the prerequisite Mathematics is already secure.

Case 23 — Learner studies many hours but the same current blockers remain

Current task: general catch-up.

Visible failure: high study volume, low change in current access.

Candidate dependency: practice selection/scope control.

Confirming test: audit what proportion of study time directly addresses the highest-leverage blocker and tests transfer.

Smallest repair: replace random volume with one explicit repair target and current-work reconnection.

Changed retest: compare performance after one focused week.

Exit evidence: study time begins producing visible current-frontier gains.

Case 24 — Several gaps appear genuine and interact

Current task: learner struggles across number, algebra, reading and graphs.

Visible failure: no single narrow repair explains most breakdowns.

Candidate state: multi-layer catch-up genuinely required.

Confirming test: small direct tests show instability across several prerequisite systems, not merely one recurring mechanism.

Smallest useful plan: prioritise one primary repair and one maintenance/current-access lane; sequence the rest by leverage.

Do not: open six equal-priority repair tracks at once.

Reconnection: use current topics to test which repaired foundation now removes the greatest obstruction.

Exit evidence: the number of active repair lanes shrinks over time rather than expanding indefinitely.

How to read the checkpoint as a system

Do not total the twenty-four cases into a score.

A learner may need one high-leverage repair and no others.

Another may need several.

The useful evidence is the mechanism.

Knowledge gap: concept or relationship genuinely missing.

Fluency gap: relationship known but too slow or fragile to support later work.

Reading gap: problem state mistranslated before calculation.

Transfer gap: direct work secure, changed surface fails.

Durability gap: same-day success, delayed retrieval fails.

Condition gap: theorem/method used without its legal condition.

Representation gap: graph/table/diagram meaning misread.

Precision gap: exact/approximate/rounding state lost.

Scope-control gap: catch-up plan itself is too broad, random or disconnected from current work.

The next action should be the smallest repair that can plausibly change the current frontier.

Appendix B — Catch-up routing matrix: symptom → likely blocker → smallest useful repair → evidence for advancement

This matrix is not a diagnostic instrument. It is a planning aid. Its purpose is to stop “behind in Maths” from becoming one undifferentiated problem.

Symptom: the learner can explain a concept but cannot complete routine calculations reliably

Likely blocker: execution fluency or accuracy.

Smallest repair: isolate the unstable component—signs, fractions, substitution, calculator entry, algebraic simplification.

Advance when: direct execution is accurate enough that the concept can remain in attention.

Then: reconnect to the original concept problem.

Symptom: the learner can complete direct questions but fails when the method label disappears

Likely blocker: recognition/selection.

Smallest repair: unlabeled near-neighbour comparisons and “what condition makes this method legal?” prompts.

Advance when: method choice follows problem structure rather than worksheet heading.

Symptom: the learner can solve after someone writes the equation

Likely blocker: translation/modelling.

Smallest repair: target, quantities, conditions, variable definitions and equation formation without solving.

Advance when: the learner can form the model independently on changed contexts.

Symptom: the learner repeatedly forgets methods between sessions

Likely blocker: retrieval durability.

Smallest repair: retrieval before review, delayed return and changed-surface retest.

Advance when: the method can be reconstructed after delay without notes.

Symptom: several later topics fail because basic algebra is slow

Likely blocker: high-leverage algebra fluency.

Smallest repair: identify the precise component—like terms, expansion, equations, substitution, factorisation—not “all algebra”.

Advance when: that component becomes cheap enough inside current topics.

Symptom: every new topic feels impossible, but errors share the same negative-sign pattern

Likely blocker: signed-number control propagating across topics.

Smallest repair: sign relationships inside the actual algebraic forms where they fail.

Advance when: the same sign-sensitive structure survives across equations, graphs and substitutions.

Symptom: percentage, ratio and probability all look weak

Likely blocker: reference quantity / denominator population.

Smallest repair: repeatedly ask “of what?” and map ratio terms or denominator populations explicitly.

Advance when: reference selection is correct across several topics.

Symptom: geometry fails only when diagrams are rotated or not drawn to scale

Likely blocker: condition/correspondence dependence on visual appearance.

Smallest repair: given-derived-looks-like classification and explicit vertex correspondence.

Advance when: orientation changes no longer change the mathematical relationships recognised.

Symptom: graph questions fail even though formulas are memorised

Likely blocker: representation semantics.

Smallest repair: axes, units, scale, domain and interpretation before gradient/intercept formula use.

Advance when: graph meaning is described correctly before calculation.

Symptom: current homework takes all evening, leaving no repair time

Likely blocker: catch-up plan has only a survival lane.

Smallest repair: reduce current work to representative tasks where possible and reserve a small protected foundation block.

Advance when: old blockers begin shrinking instead of recurring every night.

Symptom: old repair consumes all time and the learner falls further behind current lessons

Likely blocker: catch-up plan has only a backward lane.

Smallest repair: protect current access with temporary scaffolds while foundation repair continues.

Advance when: learner remains connected to current topic vocabulary and core relationships.

Symptom: learner completes many easy worksheets but current performance does not change

Likely blocker: repair is below the true bottleneck or transfer is absent.

Smallest repair: move up to the earliest failing changed problem and retest dependency assumptions.

Advance when: repaired work begins affecting current-topic performance.

Symptom: learner becomes overwhelmed by a long list of gaps

Likely blocker: scope overload.

Smallest repair: rank by leverage and activate only one primary repair plus current access.

Advance when: the active plan can be explained in one or two sentences.

Symptom: learner relies on notes for every familiar technique

Likely blocker: reference dependence / weak retrieval.

Smallest repair: attempt before checking notes; use notes only after a precise retrieval attempt.

Advance when: routine relationships are available without external lookup.

Symptom: learner copies worked examples accurately but cannot solve paired questions

Likely blocker: example dependence.

Smallest repair: prediction, missing-step completion and immediate paired problem before more examples are shown.

Advance when: learner can reconstruct the route with support removed.

Symptom: learner recognises the method but cannot recover after one wrong line

Likely blocker: local recovery skill.

Smallest repair: identify last trusted state and first suspect transition rather than restart from the top.

Advance when: the learner can repair local sign/copy/substitution errors independently.

Symptom: test performance is volatile despite decent average knowledge

Likely blocker: several fragile controls under load—fluency, checking, pacing or retrieval.

Smallest repair: compare untimed versus timed mixed evidence and identify where performance first degrades.

Advance when: ordinary load no longer produces sharp accuracy collapse.

Symptom: learner waits for hints on every unfamiliar problem

Likely blocker: first-move independence.

Smallest repair: ask for target, representation and one low-cost information-producing move before any hint.

Advance when: the learner can start unfamiliar problems without route cues.

Symptom: learner gets correct answers but working is too compressed to diagnose failures

Likely blocker: invisible intermediate state.

Smallest repair: make high-risk transitions visible until accuracy stabilises.

Advance when: the learner can safely compress without losing recoverability.

Symptom: learner reads questions correctly but still cannot choose among several known methods

Likely blocker: method discrimination.

Smallest repair: compare methods by conditions, efficiency and target rather than memorising one preferred method.

Advance when: learner can justify route choice on mixed tasks.

Symptom: learner knows current topic but fails because an older prerequisite appears unexpectedly

Likely blocker: hidden dependency not yet integrated.

Smallest repair: isolate the prerequisite and reconnect it immediately to the current problem.

Advance when: learner recognises the dependency when it reappears without warning.

Symptom: learner keeps making different errors after each repair

Likely blocker: either several genuine gaps or one broad state-control weakness.

Smallest repair: compare the first failure mechanisms across questions. Are they all reading, all sign-state, all reference-base, all retrieval?

Advance when: the plan can distinguish one shared cause from several independent causes.

Symptom: learner becomes dependent on AI-generated explanations and solutions

Likely blocker: tool-supported completion has replaced learner-owned retrieval and translation.

Smallest repair: learner-first attempt, hint-only AI, changed retest, delayed no-AI return.

Advance when: the learner can complete the same structure without AI carrying the route.

Symptom: catch-up appears successful in tutoring but not in school

Likely blocker: environment or cue dependence.

Smallest repair: remove tutor prompts, vary notation/wording and use ordinary school-style mixed sets.

Advance when: the repair survives outside the teaching context.

Symptom: the learner is secure on direct repair but still slow

Likely blocker: fluency cost rather than knowledge.

Smallest repair: short accurate fluency work, safe chunking and gradual pace addition.

Advance when: speed increases without sacrificing meaning or accuracy.

Symptom: learner has repaired one dependency but several lower-priority gaps remain

Likely state: normal catch-up progression.

Smallest repair: move the secure skill to maintenance and activate the next high-leverage blocker.

Advance when: the number of active high-priority blockers shrinks and current access improves.

Symptom: current work becomes stable and old repairs survive mixed use

Likely state: catch-up mode may be ready to end.

Smallest action: fade special scaffolds and monitor ordinary progression.

Advance when: the learner continues without emergency repair architecture for a meaningful period of normal work.

The compact routing rule

When a learner is behind, ask in this order:

Is the current question understood?

Is the necessary relationship known?

Can the learner retrieve it?

Can they execute it accurately?

Can they recognise it when unlabeled?

Can they use it after delay?

Can they use it inside current work?

Can they recover when something goes wrong?

The first “no” is usually a better repair target than the final wrong answer.

Appendix C — Eight composite catch-up laboratories: from visible failure to restored current access

These laboratories show the full catch-up job. Each begins with a current failure, traces it backward, repairs the smallest useful dependency, then returns forward. The point is not that every learner follows these exact routes. The point is to make dependency logic concrete.

Laboratory 1 — Simultaneous equations blocked by signed algebra

Current frontier: solving simultaneous equations by elimination.

Question:

2x + 3y = 17

5x − 3y = 4.

Adrian understands that adding the equations eliminates y.

He writes:

7x = 21

x = 3.

Then he substitutes into the second equation:

15 − 3y = 4.

He writes:

−3y = 19.

The simultaneous-equation method is not the main problem.

The subtraction state is.

From:

15 − 3y = 4

subtract 15:

−3y = −11.

Dependency audit.

Test:

15 − z = 4.

Can Adrian find z = 11?

Test:

−3y = −11.

Can he divide by −3?

If those direct forms fail, the repair frontier is signed equation control.

Foundation repair.

Use short direct equations:

15 − 3y = 6;

8 − 2y = 14;

−4y + 9 = 1.

Make the equivalent operations visible.

Changed retest.

3x + 2y = 11

4x − 2y = 10.

Add:

7x = 21;

x = 3.

Substitute:

12 − 2y = 10;

−2y = −2;

y = 1.

Reconnection.

Return to simultaneous equations immediately.

Catch-up receipt: elimination choice secure; sign-state error repaired in substitution. Continue current topic, maintain signed equations in mixed work.

Laboratory 2 — Compound percentage blocked by reference-base control

Current frontier: compound percentage change.

Question:

A population of 50,000 grows by 12% and then falls by 10%. Find the final population.

Clara calculates:

12% − 10% = 2%

50,000 × 1.02 = 51,000.

The arithmetic is fluent.

The dependency failure is that percentage stages use changing bases.

Repair frontier: percentage multiplier + reference base.

Stage 1:

50,000 × 1.12 = 56,000.

Stage 2:

56,000 × 0.90 = 50,400.

Repair contrast.

Compare:

increase 20%, then decrease 20%.

Multiplier:

1.2 × 0.8 = 0.96.

The final is 96% of original, not 100%.

Changed retest.

A price increases by 15% then discounts by 15%.

Ask only:

what multiplier represents the two stages?

1.15 × 0.85.

Current reconnection.

Return to growth/decay questions.

Receipt: no need to reteach basic percentage calculation. Repair is sequential base control.

Laboratory 3 — Trigonometry blocked by ratio and equation rearrangement

Current frontier: right-triangle trigonometry.

Question:

In a right triangle, angle θ = 35°, opposite side = 7 cm, find hypotenuse h.

Ethan correctly identifies:

sin 35° = 7/h.

Then stalls.

The trigonometric relationship is secure.

The blocker is rearranging a fraction equation.

Repair frontier: solving a/b = c style relationships.

Use simpler analogue:

3/x = 0.5.

Multiply both sides by x:

3 = 0.5x.

x = 6.

Then return:

sin35° = 7/h.

h sin35° = 7.

h = 7/sin35°.

Changed retest.

cos42° = 9/h.

Current reconnection.

Mixed trigonometry where sometimes unknown is numerator, denominator or angle.

Receipt: trig ratio selection secure; rearrangement of ratio equation was the prerequisite gap.

Laboratory 4 — Straight-line graphs blocked by coordinate and scale reading

Current frontier: gradient and linear models.

A graph shows x-axis labels 0, 10, 20, 30 and y-axis labels 0, 5, 10, 15. A line passes through labelled points (10,5) and (30,15).

Adrian counts two grid intervals horizontally and two vertically and reports gradient 1.

His formula knowledge is fine.

His representation state is wrong.

Repair frontier: read coordinate values from axis labels, not square counts.

Gradient:

(15 − 5)/(30 − 10)

= 10/20

= 0.5.

Changed retest.

Same visual line shape on axes with different numeric scales.

Reconnection.

Interpret gradient units in a current cost-distance graph.

Receipt: no need for more gradient-formula drill; scale reading was the blocker.

Laboratory 5 — Quadratics blocked by factorisation recognition rather than factorisation skill

Current frontier: modelling creates x² − 9x + 20 = 0.

Clara can factorise the expression immediately when a worksheet says “factorise”.

Inside the word problem she waits for a hint.

Dependency audit.

Ask:

What is the target?

What equation has been formed?

What forms could make its solutions visible?

If she can factor once prompted but never generates it as a candidate, the gap is method recognition.

Repair.

Compare unlabeled equations:

x² − 7x + 12 = 0;

x² + 4x = 5;

x² = 18;

x² + 6x + 13 = 0.

Ask which representation or method is likely useful and why.

Do not solve all of them immediately.

Changed retest.

Use a rectangle-area context that forms another factorable quadratic.

Reconnection.

Current mixed-topic problems without method labels.

Receipt: factorisation execution is secure. Route to recognition/selection practice rather than reteaching factorisation from zero.

Laboratory 6 — Probability blocked by fraction fluency and changing state at once

Current frontier: without-replacement probability trees.

Ethan understands that probabilities change but struggles to multiply and simplify fractions, so every tree becomes slow.

Current-access protection:

start with a bag where branch fractions simplify easily and focus on changing state.

Repair lane:

fraction multiplication;

common-factor cancellation;

equivalent fractions.

Example direct repair:

4/7 × 3/6

= 12/42

= 2/7.

Then return to the probability tree.

Changed retest.

Use 5 red, 3 blue; two draws without replacement; probability of red then blue.

Reconnection.

Mixed event questions including complement and conditional wording.

Receipt: both state control and fraction execution need maintenance, but they should be diagnosed separately.

Laboratory 7 — Geometry blocked by algebra after the theorem is chosen correctly

Current frontier: angle reasoning.

A pair of parallel lines creates an equation:

3x + 20 = 5x − 10.

Adrian correctly identifies the equal-angle relationship.

He then solves:

3x + 20 = 5x − 10

20 = 2x − 10

10 = 2x

x = 5.

The first incorrect transition is subtracting 3x correctly but mishandling +20 and −10.

Correct:

30 = 2x;

x = 15.

Dependency audit: geometry relation is secure. Algebraic equation movement is not.

Repair: short equations with constants on both sides.

Changed retest: a different geometry diagram producing 4x − 5 = 2x + 31.

Reconnection: current angle reasoning.

Receipt: do not reteach parallel-line geometry; repair the algebra embedded inside it.

Laboratory 8 — A learner with several real gaps: sequencing a multi-layer catch-up plan

Clara’s evidence shows:

signed arithmetic errors;

weak fraction operations;

poor reverse percentage;

graph-scale misreads;

and dependence on worked examples.

This is not one hidden mechanism.

Several gaps are genuine.

A bad plan activates five equal repair tracks.

A better plan sequences by leverage and current demand.

Current class topic:

linear graphs and equations.

Cycle 1 primary repair: signed algebra.

Current access: graph meaning with simplified algebra.

Maintenance: small fraction retrieval.

Once signed algebra stabilises:

Cycle 2 primary repair: fraction operations because gradients/equations expose them.

Current access: ordinary graph/equation work.

Maintenance: signed algebra mixed in.

Later:

reverse percentage becomes active when percentage topics approach.

Graph scale is repaired inside current access immediately because it directly blocks present work.

Worked-example dependence is addressed across every cycle by fading support.

Why this is catch-up rather than ordinary revision:

several missing dependencies are being sequenced while school work continues.

Exit test:

Can Clara participate in current work with only ordinary support?

Can signed algebra and fractions survive delayed mixed use?

Can she start changed problems without the model beside her?

If yes, active catch-up lanes can shrink.

What the laboratories show together

Visible chapter failure is often downstream.

The catch-up system has to move backward only far enough to find the first blocker, then forward again quickly enough to prove the repair matters.

The recurring structure is:

CURRENT FAILURE → EARLIEST PLAUSIBLE DEPENDENCY → SMALL TEST → NARROW REPAIR → CHANGED RETEST → DELAYED RETRIEVAL → CURRENT-TOPIC RECONNECTION.

Appendix D — Compact catch-up planning sheet

This sheet is intentionally short. A catch-up plan that cannot fit on one page is often trying to repair too much at once.

Current frontier

What is the learner expected to understand or do now?

Visible failure

What actually goes wrong in recent work?

First blocker

What earlier dependency plausibly causes that failure?

Confirming test

What small task can confirm or reject that hypothesis?

Primary repair target

Choose one high-leverage skill or relationship.

Current-access support

What temporary scaffold lets the learner stay connected to present work without permanently carrying the missing prerequisite?

Changed retest

How will the same structure be tested with a different surface?

Delayed return

When will retrieval be tested again before review?

Current reconnection

Where will the repaired skill reappear in current Mathematics?

Maintenance

What small future exposure will keep it available?

Exit evidence

What will show that this repair can leave active catch-up mode?

Appendix E — Catch-up receipts: short records that change the next decision

Keep receipts only when they change what happens next.

Signed algebra: “Expansion meaning secure; negative distribution failed in two changed examples. Repair negative multiplier, then return to equations.”

Fractions: “Concept of equation balance secure; stalls only at ÷3/4. Repair fraction inverse, retest with coefficient equations.”

Reverse percentage: “Forward percentage secure. Final amount treated as 100%. Repair reference base, not arithmetic.”

Graph scale: “Gradient formula correct; coordinates read from squares rather than labels. Repair axis/scale scan.”

Similarity: “Ratio arithmetic secure; correspondence fails after rotation. Repair vertex mapping.”

Probability: “Understands without replacement verbally; fraction multiplication too slow. Protect state reasoning while repairing fractions.”

Recognition: “Can factorise when labelled; no candidate method on word problem. Route to unlabeled method selection.”

Durability: “Same-day success high, one-week retrieval fails. Reopen briefly; shorten return interval.”

Transfer: “Direct repair secure; changed context fails. Do not advance to maintenance yet.”

Current access: “Foundation still repairing, but learner can participate in current lesson with one temporary scaffold. Continue dual lane.”

Maintenance: “Three delayed mixed successes. Remove from active repair; sample occasionally.”

Catch-up exit: “Current work stable for several cycles, repaired dependencies retrieve and transfer, support faded. Return to ordinary progression.”

The final receipt is not “all gaps are gone”.

It is:

the learner can now progress normally, and remaining weaknesses can be handled through ordinary study rather than emergency catch-up architecture.

Appendix F — Twelve prerequisite bridge maps: repair the old structure, then prove it works inside the new one

A bridge map is different from a topic list. It starts with a current demand and traces backward only far enough to identify what must be restored. Each bridge below includes a confirming test, a narrow repair, a transfer task and an exit condition.

Bridge 1 — Signed numbers → algebraic expansion → equations

Current demand: manipulate expressions and solve equations containing negative terms.

Typical visible failure: −3(2x − 5) becomes −6x − 15, or 5 − (−2) becomes 3.

Dependency: sign relationships under multiplication and subtraction.

Confirming test: separate the number layer from the algebra. Ask for −3×−5, −3×5, 5−(−2), −4−3. If those fail, the algebra is exposing a number gap.

Narrow repair: connect sign operations to meaning and inverse relationships. Then place them immediately back into expressions:

−2(x − 4);

3 − (2x − 5);

−(x + 7).

Changed transfer: solve 7 − 2(x − 3) = 15.

Current reconnection: simultaneous equations, coordinate substitution or any present algebraic topic with negative transitions.

Exit condition: sign handling remains accurate when it is no longer the labelled focus.

Bridge 2 — Fraction operations → fractional coefficients → algebraic equations

Current demand: solve equations such as (3/5)x = 12 or y = (2/3)x + 4.

Typical visible failure: equation logic is understood, but the learner stalls at division by a fraction or combines fractions incorrectly.

Dependency: fraction equivalence, multiplication/division and common denominators where relevant.

Confirming test: calculate 12 ÷ 3/5 and explain why multiplying by 5/3 reverses multiplication by 3/5.

Narrow repair: use visual or numerical equivalence if necessary, then move quickly to algebraic contexts.

Changed transfer: solve (5/6)y − 2 = 8.

Current reconnection: gradients, ratios, probability, equations or algebraic fractions depending on the present syllabus.

Exit condition: fractions stop interrupting the higher-level reasoning and can be handled without a separate prompt.

Bridge 3 — Percentage multiplier → reverse percentage → compound change

Current demand: model growth, discounts or repeated percentage changes.

Typical visible failure: learner can find 20% of a number but cannot identify the original value after a discount or combines +20% and −20% additively.

Dependency: percentage as a multiplicative relationship around a reference base.

Confirming test: ask the learner to complete:

after 15% increase → final = ___ × original;

after 15% decrease → final = ___ × original.

Narrow repair: 1.15 and 0.85 as multipliers tied to 100% reference.

Changed transfer: final price 170 after 15% discount; find original using 0.85P = 170.

Current reconnection: compound growth/decay or finance-style percentage models.

Exit condition: learner identifies the base and multiplier correctly before arithmetic in mixed forward/reverse questions.

Bridge 4 — Ratio meaning → proportional reasoning → similarity and trigonometric ratios

Current demand: use proportional side relationships.

Typical visible failure: learner knows a formula but reverses corresponding terms or cannot solve a proportional equation.

Dependency: ratio correspondence and scale.

Confirming test: if A:B = 3:5 and A = 18, can the learner identify one part and B? Can they explain why 18 corresponds to 3 parts?

Narrow repair: map quantity ↔ ratio term explicitly, then introduce multiplicative scale factor.

Changed transfer: similar triangles with rotated orientation and stated correspondence.

Current reconnection: similarity, trigonometric ratios or map/scale work.

Exit condition: the learner chooses corresponding quantities from structure rather than page position.

Bridge 5 — Equality → single equations → simultaneous equations

Current demand: solve a system by elimination or substitution.

Typical visible failure: learner memorises elimination steps but loses equivalence when multiplying, subtracting or substituting.

Dependency: equation as a statement of equality preserved by equivalent operations.

Confirming test: solve and check a single equation with variables on both sides.

Narrow repair: make equivalent states visible. Avoid relying only on “move across, change sign” if that language is causing errors.

Changed transfer: multiply one equation by 2 before elimination and explain why the solution set is preserved.

Current reconnection: current simultaneous-equation problems including contextual models.

Exit condition: elimination/substitution choices are understandable as equivalence-preserving moves rather than ritual.

Bridge 6 — Expansion ↔ factorisation → quadratic solving

Current demand: expose roots or rewrite quadratic structure.

Typical visible failure: learner can expand but cannot reverse; or factorises when explicitly labelled but does not recognise why factor form is useful.

Dependency: transformation between equivalent forms.

Confirming test: match x²−5x+6 with (x−2)(x−3), then expand to verify.

Narrow repair: use expansion and factorisation as inverse views of the same structure.

Changed transfer: equation x²−8x+15=0 without the word “factorise”. Ask which form would expose solutions.

Current reconnection: quadratics, graphs, geometry models or algebraic proofs.

Exit condition: learner can select factor form because of the target, not because the topic heading says factorisation.

Bridge 7 — Coordinate meaning → gradient/intercept → linear modelling

Current demand: interpret and construct linear relationships.

Typical visible failure: learner can substitute into y=mx+c but misreads coordinates, axes or what gradient means.

Dependency: ordered pair meaning and graph scale.

Confirming test: identify coordinates from differently scaled axes and explain which coordinate belongs to which variable.

Narrow repair: axes → units → scale → points → gradient.

Changed transfer: compare a cost graph and a distance-time graph with similar visual line shapes.

Current reconnection: straight-line graphs, simultaneous-equation intersections or rate models.

Exit condition: learner interprets gradient as vertical-unit per horizontal-unit and intercept in context.

Bridge 8 — Angle relationships → theorem conditions → geometry reasoning

Current demand: build multi-step angle or triangle arguments.

Typical visible failure: formulas/theorems are known but applied from appearance or the wrong condition.

Dependency: condition-backed angle relationships.

Confirming test: ask which fact authorises an angle equality and whether it is given or derived.

Narrow repair: organise facts by their conditions: straight line, point, parallel lines, triangle, isosceles, right angle.

Changed transfer: redraw or rotate the geometry while preserving markings.

Current reconnection: similarity, trigonometry or proof-style geometry.

Exit condition: method authority comes from stated/derived conditions, not visual impression.

Bridge 9 — Linear scale → area scale → volume scale

Current demand: solve similar-shape scale problems.

Typical visible failure: learner uses the same scale factor for length, area and volume.

Dependency: dimensional effect of scaling.

Confirming test: double the side of a square and cube; compare resulting area and volume directly.

Narrow repair: if linear factor = k, area factor = k², volume factor = k³.

Changed transfer: linear scale 3/2; find area and volume ratios.

Current reconnection: similarity, mensuration, surface area or volume.

Exit condition: learner identifies the dimensional type of the target before choosing the power of k.

Bridge 10 — Sample space → changing state → probability trees

Current demand: multi-stage probability.

Typical visible failure: learner multiplies branch probabilities without updating the sample after a draw.

Dependency: event state and denominator population.

Confirming test: narrate bag contents before and after the first draw.

Narrow repair: state table or simple tree showing numerator and denominator change.

Changed transfer: compare replacement and no-replacement versions of the same bag.

Current reconnection: conditional probability or multi-stage events.

Exit condition: learner can explain branch probabilities from the current state before multiplying them.

Bridge 11 — Mean as total/count → combined mean → weighted data

Current demand: combine groups or reconstruct missing values.

Typical visible failure: learner averages means equally or cannot recover a total.

Dependency: mean = total/count.

Confirming test: given mean 12 for 5 values, can learner recover total 60?

Narrow repair: move flexibly among mean, total and count.

Changed transfer: combine groups of very different sizes and compare with the incorrect simple average.

Current reconnection: statistics, weighted average or data interpretation.

Exit condition: group size automatically influences combined-mean reasoning.

Bridge 12 — Mathematical language → representation → mixed-topic problem solving

Current demand: solve unfamiliar questions without route labels.

Typical visible failure: learner knows techniques but waits for another person to translate “at least”, “per”, “similar”, “estimate”, “hence” or contextual relationships.

Dependency: question parsing and task language.

Confirming test: ask for target, givens, conditions and representation without solving.

Narrow repair: translate relationships rather than keywords. Use near-neighbour wording.

Changed transfer: same structure expressed as prose, table, graph or diagram.

Current reconnection: mixed current problems and unseen questions.

Exit condition: the learner can hand a faithful mathematical state to the solver without full adult translation.

How to use bridge maps without creating another giant checklist

Do not activate all twelve maps.

Choose the one that best explains the current failure.

Test it.

If the confirming test passes, reject that hypothesis and look elsewhere.

If it fails, repair narrowly.

Then reconnect forward immediately.

The bridge is successful only when movement across the current frontier improves.

Appendix G — Twenty-four near-neighbour catch-up drills: train the decision that changes when one feature changes

Near-neighbour practice is useful in catch-up because it reveals whether the learner has rebuilt a relationship or merely memorised one recent procedure. For each pair, do not rush to full calculation. First state what changed, what stayed invariant and whether the route should change.

Drill 1 — Forward percentage versus reverse percentage

A: An item costs 80 dollars and is discounted by 20%. Find the final price.

B: After a 20% discount, an item costs 80 dollars. Find the original price.

In A, 80 is the 100% base and final = 0.8×80.

In B, 80 is the 80% final state and 0.8P = 80.

The percentage and displayed number are the same. The direction of the unknown changes the operation.

Catch-up use: if A succeeds and B fails, do not assign more basic percentage-of-a-number practice. Repair reference direction.

Drill 2 — Percentage of versus percentage more than

A: A is 30% of B.

B: A is 30% more than B.

A gives A = 0.3B.

B gives A = 1.3B.

The phrase “more than” creates an additional whole B before the 30% increase.

Catch-up use: ask the learner to test B=100. A should become 30 in the first statement and 130 in the second. A hypothetical value exposes whether the translation is structurally correct.

Drill 3 — Percentage increase versus percentage points

A: A rate rises from 40% to 50%. Find the increase in percentage points.

B: Find the relative percentage increase.

A gives 10 percentage points.

B uses original 40% as reference: 10/40 = 25% relative increase.

Catch-up use: this pair checks whether the learner can preserve the type of percentage comparison rather than treating every difference between percentages as a “percent increase”.

Drill 4 — At least versus more than

A: x is at least 7.

B: x is more than 7.

A includes 7: x ≥ 7.

B excludes 7: x > 7.

Catch-up use: test the boundary. If exactly 7 satisfies the sentence, the inequality is inclusive. This is a reading repair, not an algebra repair.

Drill 5 — At most versus less than

A: A container holds at most 20 litres.

B: A container holds less than 20 litres.

A allows exactly 20.

B does not.

Catch-up use: connect wording to feasible sets. Later optimisation and probability problems depend on preserving the boundary correctly.

Drill 6 — Fixed fee plus rate versus direct proportion

A: Cost = 5 dollars plus 2 dollars per item.

B: Cost = 2 dollars per item with no fixed fee.

A: C = 5 + 2n.

B: C = 2n.

B is directly proportional to n in the standard school sense; A is linear but has a nonzero intercept.

Catch-up use: if both are called “direct proportion” because both are straight lines, repair the meaning of proportionality rather than graph drawing.

Drill 7 — Average of two speeds versus average speed over a journey

A: Find the arithmetic mean of 20 km/h and 40 km/h.

B: Travel 30 km at 20 km/h and 30 km at 40 km/h. Find average speed.

A = 30 km/h.

B requires total distance / total time:

60 / (30/20 + 30/40) = 60 / 2.25 ≈ 26.67 km/h.

Catch-up use: the displayed speed numbers tempt a shortcut. Repair the definition of average speed, not arithmetic averaging.

Drill 8 — Consecutive integers versus consecutive even integers

A: two consecutive integers.

B: two consecutive even integers.

A can be n and n+1.

B can be 2n and 2n+2, or x and x+2 with x restricted even.

Catch-up use: this checks whether the learner translates the condition into the variable structure instead of writing x and y with no relationship.

Drill 9 — Sum of twice x and 3 versus twice the sum of x and 3

A: the sum of twice x and 3.

B: twice the sum of x and 3.

A = 2x + 3.

B = 2(x + 3).

Catch-up use: if bracket structure is wrong, the gap is language grouping. Practise clause parsing before more algebra expansion.

Drill 10 — With replacement versus without replacement

A bag contains 4 red and 3 blue counters.

A: two draws with replacement; probability of two reds.

B: two draws without replacement; probability of two reds.

A = 4/7 × 4/7.

B = 4/7 × 3/6.

Catch-up use: the phrase changes the second state. If the learner keeps the same second fraction, repair state tracking.

Drill 11 — At least one versus exactly one

Three independent trials.

A: probability of at least one success.

B: probability of exactly one success.

A includes one, two or three successes and may be efficiently found by 1 − P(no successes).

B includes only one success and usually requires the appropriate cases.

Catch-up use: event-language repair should precede tree arithmetic.

Drill 12 — Linear scale versus area scale

Two similar figures have linear scale factor 3.

A: corresponding length changes by factor?

B: corresponding area changes by factor?

A: 3.

B: 9.

Catch-up use: if both use 3, repair dimensional scaling. Test with a 2-by-2 square scaled to 6-by-6 so the area change is visible.

Drill 13 — Area scale versus volume scale

A solid is enlarged with linear factor 2.

A: surface area factor?

B: volume factor?

A: 4.

B: 8.

Catch-up use: target dimension decides the power. The repair should connect geometry to dimensional reasoning, not memorise three unrelated rules.

Drill 14 — Similar-looking triangles versus stated similar triangles

A: two triangles look similar in a sketch, no conditions given.

B: the problem states the triangles are similar.

In A, visual appearance alone does not authorise proportional corresponding sides.

In B, similarity is a given condition.

Catch-up use: train evidence authority. A diagram is a representation, not a licence to invent conditions.

Drill 15 — Right-looking triangle versus marked right triangle

A: the angle looks 90°.

B: a right-angle mark is present.

Only B provides direct diagram evidence for right-angle methods unless A has another derivation.

Catch-up use: if the learner chooses Pythagoras in A, repair given/derived/looks-like classification.

Drill 16 — Same graph shape, different axes

A: horizontal segment on a distance-time graph.

B: horizontal segment at 12 m/s on a speed-time graph.

A means stationary.

B means constant nonzero speed.

Catch-up use: graph semantics come from axes. If the learner says both mean “stopped”, repair axis interpretation.

Drill 17 — Same grid, different scale

Two graphs use identical square spacing.

A: x-axis increments by 1, y-axis by 1.

B: x-axis increments by 10, y-axis by 2.

A visual slope can correspond to different numerical gradients.

Catch-up use: require coordinate values from labels before applying the gradient formula.

Drill 18 — Exact answer versus decimal approximation

A: give √18 exactly.

B: give √18 to 3 significant figures.

A = 3√2.

B ≈ 4.24.

Catch-up use: if the learner gives the same form to both, repair answer-contract reading rather than the underlying calculation.

Drill 19 — Use the graph versus solve algebraically

Two models intersect.

A: “Using the graph, estimate the intersection.”

B: “Solve the equations algebraically.”

The target can be identical while the requested representation changes.

Catch-up use: teach the learner to preserve method constraints in the answer contract.

Drill 20 — Mean of equal groups versus mean of unequal groups

A: two groups each contain 20 students, means 60 and 70.

B: one group contains 10 students, mean 60; another contains 30, mean 70.

In A, the combined mean equals the simple average 65 because group sizes are equal.

In B, group means must be weighted by counts:

(10×60 + 30×70)/40 = 67.5.

Catch-up use: if the learner always averages means, repair total/count structure.

Drill 21 — Equation solution versus contextual solution

A: solve x² = 49 over the real numbers.

B: a length x satisfies x² = 49; find the length.

A gives x = ±7.

B gives x = 7 because negative length is inadmissible in the context.

Catch-up use: algebraic solution set and contextual domain are different layers. Preserve both.

Drill 22 — Same equation, different target

Given y = 3x + 5.

A: find y when x=4.

B: find x when y=20.

A substitutes x.

B solves 20 = 3x + 5.

Catch-up use: the relationship is unchanged; the unknown direction changes the route. This trains target control inside familiar algebra.

Drill 23 — Same rectangle, different mathematical object

A rectangle is 12 by 8.

A: floor covering required.

B: border strip required.

C: diagonal length.

A invokes area, B perimeter, C right-triangle reasoning.

Catch-up use: one diagram, three targets. If the learner always performs the same rectangle routine, repair target recognition.

Drill 24 — Same repaired skill, direct versus embedded use

A: simplify −3(2x − 5).

B: solve 7 − 3(2x − 5) = 10.

C: use that algebra inside a simultaneous-equation substitution.

The sign-distribution skill is the same, but the surrounding cognitive load grows.

Catch-up use: if A succeeds and B/C fail, the learner may need reintegration practice or more fluency rather than another explanation of distribution.

How to use the drills

Choose pairs connected to the active repair.

Ask first:

What changed?

What stayed the same?

Which decision changes?

What condition or target causes the change?

Then solve only enough examples to verify the discrimination.

The purpose is to prevent catch-up from producing one more memorised procedure that works only in a narrow block.

Appendix H — Six full catch-up programme examples: how the architecture changes when the bottleneck changes

These programmes are illustrative rather than prescriptions. Their purpose is to show how the same catch-up architecture produces different plans when the evidence is different. Timings are examples, not universal schedules. A learner should move on when the relevant evidence is secure, not because a calendar says the week is over.

Programme 1 — The algebra dependency chain: signed numbers → brackets → equations → simultaneous equations

Starting state.

A Secondary learner is currently studying simultaneous equations. They understand the idea of two equations describing the same unknowns. They can explain elimination when coefficients already match. Their recent work shows repeated failures after substitution and whenever negative coefficients appear.

Three recent questions reveal:

correct method choice;

incorrect negative distribution;

incorrect subtraction of a positive term from both sides;

and a tendency to restart the whole question after a sign error.

The learner says:

“I am bad at simultaneous equations.”

The evidence suggests a narrower diagnosis.

Current frontier: simultaneous equations.

Repair frontier: signed algebra inside equivalent equation transformations.

First confirming tests.

1. Expand −3(x−4).

2. Solve 8−2y=14.

3. Solve −4x=12.

4. Substitute x=−2 into 3x−5.

If these fail, the repair target is confirmed.

Current-access lane.

Do not stop simultaneous equations completely.

Use one or two current problems where the conceptual job remains visible. Permit slightly cleaner coefficients initially so the learner can practise:

what elimination does;

what substitution does;

why both equations must be satisfied.

The goal is to preserve contact with the current mathematical idea.

Repair lane, first cycle.

Day/session A:

signed multiplication and subtraction inside short expressions.

Do not use a long arithmetic worksheet. Use expressions that look like the algebra where failure occurs:

−2(x−5);

4−(−3);

7−2(x+1).

Day/session B:

equations with one sign-sensitive transition:

9−2x=15;

−3x+4=19;

5−3(x−2)=11.

Day/session C:

return to simultaneous equations.

Choose one elimination and one substitution route.

Require the learner to mark the line where the repaired sign control is used.

Changed retest.

Change coefficient signs and method:

2x−3y=11

4x+3y=7.

Then a substitution case:

y=2x−5

3x+y=11.

The learner must preserve the negative 5 through substitution.

Delayed retrieval.

Several days later, before reviewing notes, ask for:

−3(2x−5);

solve 10−4x=18;

then one simultaneous-equation problem.

Recovery training.

Deliberately present a worked solution with one sign error.

Ask:

What is the last trusted line?

Where is the first invalid transition?

Repair locally.

This prevents the learner from treating every error as a reason to restart the entire problem.

Evidence for moving the repair to maintenance.

The learner should be able to:

expand negative multipliers accurately;

solve sign-sensitive single equations;

use the same control inside simultaneous equations;

retrieve the skill after delay;

and locate a local sign error without full tutor rescue.

If those conditions hold, signed algebra no longer needs a major special block. Include it occasionally in mixed work while current simultaneous-equation learning continues.

What this programme does not do.

It does not reteach all algebra.

It does not stop current work for several weeks.

It does not interpret one sign error as evidence of a global number deficit.

The repair stays narrow until evidence requires expansion.

Programme 2 — The percentage and ratio chain: reference quantity → reverse percentage → compound change → data/probability transfer

Starting state.

A learner performs percentage arithmetic quickly. They can find 15% of 240 and convert 0.35 to 35%. Yet they repeatedly fail reverse percentage, compound change and conditional probability questions.

Recent evidence shows:

after a discount, the final price is treated as 100%;

ratio terms are applied to the wrong quantity;

in a two-way table, the denominator is always the total population even when the question says “given that”.

These failures appear across different chapters.

The shared mechanism may be:

reference-set control.

Current frontier: compound percentage change and data probability.

Repair frontier: identify the quantity that acts as the reference whole or denominator population.

Confirming tests.

1. “A is 30% of B.” What is the reference quantity?

2. “A is 30% more than B.” What is the reference quantity and relationship?

3. Red:blue=3:5, 45 red. Which ratio term corresponds to 45?

4. Given Secondary 1, 18 of 40 travel by bus. What denominator should a conditional probability use?

If the learner cannot state the reference reliably, the cross-topic diagnosis is supported.

Repair lane, first stage.

Use translation-only tasks.

No calculator.

No full solutions.

For each statement write:

reference = ?

relationship = ?

unknown = ?

Examples:

after 20% discount, final = 80% of original;

red counters are 3/5 of total;

boys:girls=7:4;

given bus travellers, probability of Secondary 1.

Repair lane, second stage.

Convert relationships into symbolic forms:

0.8P=72;

R=(3/5)T;

7 parts boys ↔ stated boys quantity;

P(S1|bus)=number(S1 and bus)/number(bus).

The notation can be adapted to curriculum level.

The important job is reference preservation.

Current-access lane.

Continue current compound-change work, but require a stage map:

original → after first change → after second change.

Each arrow carries a multiplier.

Example:

×1.12

then

×0.90.

Do not allow +12−10 as a substitute for the changing-base model.

Cross-topic changed retest.

Move from money to population growth.

Then from percentage to ratio.

Then to a two-way-table denominator question.

The surface changes.

The common decision remains:

what is the reference set?

Delayed retrieval.

Several days later present six short prompts:

two forward percentages;

two reverse percentages;

one ratio correspondence;

one conditional denominator.

Ask for the relationship only.

Evidence for advancement.

The learner should correctly identify:

100% base;

ratio correspondence;

conditional denominator;

and changing percentage base

across mixed contexts without being told which category each question belongs to.

What this programme teaches about catch-up.

Several chapter labels can conceal one mechanism.

When the mechanism is repaired, gains can appear in several topics at once. This is why catch-up should search for dependencies rather than count weak chapters.

Programme 3 — Geometry to trigonometry: conditions → correspondence → ratio → equation control

Starting state.

A learner is beginning right-triangle trigonometry. They remember sine, cosine and tangent mnemonics. They still struggle to solve unfamiliar triangle questions.

Recent evidence shows four different breakdowns:

sometimes a non-right triangle is treated as right-angled because it looks that way;

sometimes opposite and adjacent are labelled before the reference angle is identified;

sometimes the correct ratio is written but cannot be rearranged;

sometimes a rotated diagram causes side relationships to be lost.

This is not one trig-formula problem.

It is a small dependency chain.

Current frontier: use trigonometric ratios to find sides/angles.

Repair frontiers: theorem condition, reference-angle labelling, ratio meaning and simple equation rearrangement.

Confirming tests.

1. Given several triangle sketches, identify which are known to be right-angled from evidence.

2. In a right triangle, change the reference angle and relabel opposite/adjacent.

3. Solve 3/x=0.6.

4. Match corresponding sides in rotated similar triangles.

The results show which layers are genuinely weak.

Current-access lane.

Keep trigonometry present.

Use clean right triangles with explicit markings while repairing the upstream layers.

Ask the learner to say:

right-angle evidence;

reference angle;

known side;

target side;

candidate ratio.

Only then calculate.

Repair lane A — condition authority.

Use diagram pairs:

one looks right-angled;

one has a right-angle mark.

Ask which permits right-triangle methods and why.

Repair lane B — ratio correspondence.

Without trigonometry, use simple ratios such as 3/5=6/10 and solve missing terms.

Connect this to:

sinθ = opposite/hypotenuse.

The trig ratio is a relationship among quantities, not a magic calculator command.

Repair lane C — equation rearrangement.

Use:

sin35°=7/h.

Multiply by h:

h sin35°=7.

Then:

h=7/sin35°.

Compare with:

sin35°=x/10

so x=10sin35°.

The location of the unknown changes the algebraic route.

Changed retest.

Rotate the triangle.

Change the reference angle.

Make the unknown an angle rather than a side.

Use a contextual ladder or height problem.

Delayed retrieval.

Ask the learner to reconstruct:

what makes right-triangle trig legal;

how opposite/adjacent depend on the reference angle;

how to choose among sin/cos/tan;

and how to rearrange one ratio equation.

Evidence for advancement.

The learner handles changed orientation without losing side roles, checks the right-angle condition, selects a ratio from the target/known sides, rearranges appropriately and interprets the answer with units/angle status.

What this programme avoids.

It avoids assigning hundreds of trig questions when the true blockers are geometry evidence and algebraic rearrangement. It also avoids retreating into a full geometry restart when only a few dependencies are unstable.

Programme 4 — Graphs and functions: scale reading → coordinate meaning → algebraic form → model interpretation

Starting state.

A learner is studying linear graphs and beginning simple functions. They can substitute numbers into y = mx + c and can draw a line from a table. Yet they perform poorly on graph questions.

Recent evidence shows:

coordinates copied from the grid rather than the axis labels;

gradient calculated from square counts;

y-intercept identified numerically but not interpreted in context;

straight lines automatically called direct proportion;

and intersections described as “where the lines cross” without connecting them to equal output for the same input.

The broad label “graphs are weak” is too vague.

The failure chain involves representation semantics.

Current frontier: linear models and graph interpretation.

Repair frontier: axes, units, scale, ordered pairs, gradient meaning and intercept meaning.

Confirming tests.

Use two graphs with identical visual line shapes but different scales.

Ask for the same labelled point from each.

Use a distance-time graph and a cost-distance graph with a horizontal or straight segment.

Ask what the shape means in each context.

Give y=2x+5 and ask what 2 and 5 could mean if x is items and y is dollars.

If those interpretation tasks fail while substitution arithmetic succeeds, the diagnosis is supported.

Current-access lane.

Keep the learner in current linear-model work.

Before every calculation use a four-line state:

x means ___;

y means ___;

x-unit ___;

y-unit ___.

Then mark the scale.

This is temporary scaffolding.

The aim is for it to become a silent scan.

Repair lane A — coordinates and scale.

Use graph snippets where the task is only to read coordinates.

Vary scales:

1 per square;

2 per square;

5 per square;

unequal x/y scales.

Do not calculate gradient yet.

The learner should learn that graph-paper distance is not itself mathematical coordinate difference.

Repair lane B — gradient as rate.

Once coordinates are stable:

gradient = change in vertical quantity / change in horizontal quantity.

Attach units.

For cost against distance:

dollars/km.

For distance against time:

km/h.

The units expose meaning and often detect inversion.

Repair lane C — intercept.

Ask:

what does y equal when x = 0?

In a taxi model, that may be fixed charge.

In a temperature model, it may be initial temperature.

Do not memorise “c is fixed fee”.

The interpretation depends on context.

Repair lane D — direct proportion versus general linear.

Compare:

y = 3x;

y = 3x + 5.

Both are straight lines.

Only the first passes through the origin and is directly proportional in the standard school sense.

Changed retest.

Use a storage-cost model, then a distance-time model, then a function with no story.

Ask the learner to interpret or identify gradient/intercept appropriately in each.

Current reconnection.

Move to intersections of two linear models.

Ask:

what does the intersection mean?

For two cost plans:

same cost at same usage.

For two functions:

same y-value at same x.

Evidence for advancement.

The learner reads axes/scale before shape, extracts correct coordinates, calculates gradient with units, interprets intercept by context, distinguishes direct proportion from nonzero-intercept linear models and explains intersections in terms of the represented quantities.

Catch-up principle illustrated: formula competence can hide representation weakness. The repair must begin where the graph acquires meaning, not where the formula is written.

Programme 5 — Near an assessment: prioritising survival, durable repair and honest limits

Starting state.

A learner has an important school assessment in two weeks. They are behind in several areas:

fractions;

linear equations;

graphs;

percentage;

geometry.

The family wants to “cover everything”.

There is not enough time to rebuild every weak topic deeply before the assessment.

The plan must distinguish:

what can improve the upcoming assessment;

what must be repaired for long-term progression;

and what cannot honestly be promised in two weeks.

Step 1 — map assessment demand.

Identify the major topic families and the prerequisite structures that appear repeatedly.

Do not assume every syllabus chapter has equal weighting or equal leverage.

No invented weightings should be used; rely on the actual teacher guidance, syllabus or available papers where relevant.

Suppose recent class materials show that equations, graphs and percentage relationships recur heavily, while one geometry subtopic is smaller.

Step 2 — choose a high-leverage emergency repair.

Linear equation control may unlock:

algebra questions;

graph questions;

formula rearrangement;

some geometry calculations.

That can justify priority.

Step 3 — protect current test familiarity.

The learner still needs to recognise the formats likely to appear.

Use representative tasks, not an enormous paper stack.

For each task, record:

target;

first failure;

repair needed;

whether it is realistic before the assessment.

Two-week illustrative structure.

Days 1–3:

diagnose and stabilise the highest-leverage equation/fraction controls.

Use short direct practice and changed retests.

Days 4–6:

reconnect to graph and percentage questions that depend on those controls.

Introduce short mixed sets.

Days 7–9:

review the next most frequent blocker from evidence.

Begin delayed retrieval of the first repair.

Days 10–12:

short assessment-style sets.

Focus on question reading, route choice, working and checking.

Days 13–14:

reduce new repair work.

Use retrieval, representative questions, sleep-protective workload and clear error receipts.

What not to do.

Do not promise that every historical gap will close.

Do not spend the final nights cramming low-leverage old topics merely to say they were “covered”.

Do not hide dependence behind full tutor solutions.

Do not let exam survival erase the post-assessment repair plan.

After the assessment.

Read the script as evidence.

Which errors were:

knowledge;

reading;

fluency;

selection;

accuracy;

pacing?

Return to the two-frontier catch-up architecture without the short-term deadline dominating every decision.

Evidence of useful short-term progress.

The learner completes more high-leverage familiar tasks;

makes fewer repeated prerequisite errors;

starts questions more independently;

and retains the core repair after the assessment.

Catch-up principle illustrated: urgent performance and durable learning are related but not identical. A responsible plan serves the deadline without pretending that emergency preparation has repaired the whole dependency system.

Programme 6 — Strong understanding, low fluency: catching up by reducing the cost of routine Mathematics

Starting state.

A learner understands lessons well and can explain methods in conversation. Untimed, they solve many questions correctly. Under ordinary school pace, they are too slow. Later parts of tests are blank. Current lessons feel overwhelming because each algebraic step consumes too much attention.

The learner describes this as:

“I am behind because everyone else is faster.”

The evidence suggests that the main catch-up task is not conceptual reteaching.

It is to make routine components cheaper while preserving accuracy and meaning.

Current frontier: mixed Secondary Mathematics under normal classroom and assessment pace.

Repair frontier: fluency in a small set of recurring routines.

Possible high-cost routines:

fraction simplification;

basic algebraic manipulation;

substitution;

calculator entry;

factor recognition;

graph coordinate reading;

common exact values where curriculum-relevant;

or checking every low-risk step too slowly.

Confirming test.

Separate concept and execution.

Ask the learner to explain the method without calculating.

Then time or observe a small routine set—not to rank the learner against others, but to locate where time is actually spent.

For example:

five substitutions;

five fraction simplifications;

five bracket expansions.

Look for:

hesitation;

repeated checking;

calculator navigation;

rewriting unnecessary lines;

or genuine calculation difficulty.

Repair lane.

Choose one routine.

Stabilise accuracy first.

Then use short focused repetitions with immediate feedback.

Compress only safe steps.

For example, if the learner can solve:

3x + 5 = 20

with full balance steps accurately, later fluency may allow:

3x = 15

x = 5

without writing every verbal explanation.

The Mathematics remains valid.

The visible load shrinks.

Current-access lane.

Continue mixed questions, but initially reduce unnecessary complexity so the learner can practise decision-making without routine execution consuming the entire session.

Then reintroduce ordinary load.

Fluency ladder.

1. Accurate direct routine.

2. Same routine with small variation.

3. Routine embedded inside a larger problem.

4. Mixed routine selection.

5. Short timed set.

6. Ordinary current work.

7. Assessment-style load.

Checking control.

Some slow learners verify every line fully because they do not trust themselves.

Teach risk-weighted checking.

Check expensive transitions:

sign changes;

calculator entry;

model formation;

final interpretation.

Do not spend equal time checking every low-risk arithmetic step once it is stable.

Changed retest.

Use the same routine inside a different topic.

For example, algebraic rearrangement first in pure equations, then in trigonometry or formulas.

Delayed retest.

Return after several days.

Fluency that disappears immediately was temporary.

Evidence for advancement.

Routine work becomes faster while accuracy remains stable;

the learner completes more of mixed tasks;

attention is available for unfamiliar reasoning;

and speed survives changed contexts.

Catch-up principle illustrated: being behind is not always missing content. Sometimes the learner knows the Mathematics but routine execution is too expensive for the pace of the current frontier. The repair should reduce cost, not reteach understanding the learner already owns.

What the six programmes show

The phrase “catch up in Maths” can describe very different systems.

Programme 1 needed a narrow signed-algebra repair.

Programme 2 found one reference-set mechanism across several topics.

Programme 3 needed a short geometry-ratio-algebra dependency chain.

Programme 4 repaired graph semantics rather than formulas.

Programme 5 balanced an immediate assessment with honest long-term repair.

Programme 6 treated fluency cost as the main gap.

The common architecture remains:

current frontier → first blocker → narrow repair → changed retest → delayed return → current reconnection → fade support → ordinary progression.

The content of the repair changes because the evidence changes.

Appendix I — The catch-up handover: how a repaired skill leaves emergency mode and becomes ordinary Mathematics again

Catch-up is incomplete if every repaired skill remains on a special list forever.

The final job is handover.

A repaired dependency should move through four states:

active repair → transfer watch → maintenance → ordinary use.

Each state has a different workload.

State 1 — Active repair

The learner still needs deliberate attention on the weak relationship.

Examples are selected because they expose the target.

Feedback is immediate.

Worked support may still be visible.

Changed retests happen frequently.

A skill belongs here when direct performance is not yet reliable or the learner cannot explain the relationship.

Examples:

negative distribution still fails;

reverse percentage base is still reversed;

graph scale is still read incorrectly;

ratio correspondence still changes with diagram orientation.

Do not hurry a skill out of active repair just because one familiar worksheet went well.

State 2 — Transfer watch

Direct performance is secure.

The question becomes:

does the repair survive when the surface changes?

Use:

different wording;

different numbers;

different diagram orientation;

different representation;

no topic label;

or a neighbouring method that must be discriminated from it.

Example:

a learner can solve direct reverse-percentage questions.

Transfer watch asks whether they still identify the correct base inside a population, profit or growth context.

Another learner can expand negative brackets in isolation.

Transfer watch places that structure inside an equation or substitution problem.

If the learner fails, return to active repair briefly.

Do not interpret transfer failure as proof that nothing was learned.

It means the repair has not yet become portable.

State 3 — Maintenance

The learner succeeds on changed problems and can retrieve after delay.

The skill no longer needs a dedicated block every session.

Sample it occasionally.

Maintenance should be small enough that it does not crowd out current work.

For example:

one sign-sensitive algebra item inside a mixed set;

one reverse-percentage relationship during later percentage review;

one rotated similarity problem inside geometry;

one graph-scale question in a future function set.

If maintenance samples remain secure, frequency can decrease.

If the skill deteriorates, increase retrieval frequency or reopen repair.

Maintenance is evidence-responsive, not calendar-driven.

State 4 — Ordinary use

The learner no longer experiences the skill as a special repaired topic.

It is simply part of Mathematics.

Signed-number control appears when algebra needs it.

Ratio meaning appears in probability or geometry.

Equation balance appears inside formulas.

Graph reading appears in modelling.

The learner does not receive a warning that the old weakness is about to appear.

This is the strongest evidence that catch-up has worked.

Handover example — fractions

Active repair:

simplify, multiply and divide selected fractions accurately.

Transfer watch:

fraction coefficients inside equations and gradients.

Maintenance:

occasional fraction-rich mixed problems.

Ordinary use:

fractions appear naturally without a special reminder or separate support sheet.

Handover example — question reading

Active repair:

state target, givens, conditions and answer contract explicitly.

Transfer watch:

different topics and representations with no prompt telling the learner what to underline.

Maintenance:

brief target/condition scan only on dense questions.

Ordinary use:

the learner reads mathematical wording accurately as part of normal solving.

Handover example — algebraic factorisation

Active repair:

connect expansion and factorisation and practise direct examples.

Transfer watch:

unlabelled quadratic equations and geometry models.

Maintenance:

factorable forms appear inside mixed algebra.

Ordinary use:

the learner chooses factor form when it exposes the target, without being told “factorise”.

Handover example — graph scale

Active repair:

read axes and coordinates from varied scales.

Transfer watch:

different graph types and contexts.

Maintenance:

occasional non-unit scales in mixed graph work.

Ordinary use:

scale checking becomes automatic before gradient, area or interpretation.

What should trigger reopening?

Reopen active repair if:

the same mechanism fails repeatedly across changed problems;

delayed retrieval collapses more than once;

the learner needs the old scaffold again to participate in current work;

or the skill becomes a new bottleneck as later Mathematics increases demand.

One isolated error is not automatically a reopening trigger.

Use a pattern.

What should not trigger reopening?

Do not reopen merely because:

one difficult unfamiliar problem was wrong;

a later topic requires genuinely new knowledge;

the learner makes an unrelated arithmetic slip;

or performance varies slightly from day to day.

Catch-up should not make the learner or teacher hypervigilant about every mistake.

The purpose is to restore ordinary learning, not create permanent deficit surveillance.

The final handover question

Ask:

If we remove the special catch-up support, does the learner continue to access current Mathematics, retrieve the repaired prerequisite, transfer it to changed problems and recover from ordinary errors?

If yes, move the skill out of active repair.

That is the point of the whole system.

Catch-up succeeds when the learner’s Mathematics can carry itself forward again.

Appendix J — Catch-up exit gate: nine questions before special repair support is withdrawn

This final gate is intentionally practical. It does not certify mastery in a formal assessment sense. It asks whether the learner can now function without the special catch-up system carrying the old dependency.

1. Can the learner enter current work?

They should be able to understand the current lesson target, attempt representative questions and identify where they are stuck without needing every problem rewritten for them.

2. Can the repaired prerequisite be retrieved before review?

A skill that appears only after notes, a worked example or a reminder is not fully handed back to the learner. Ask for the relationship first, then allow review only if retrieval fails.

3. Can the learner use the repair on a changed surface?

Change numbers, wording, orientation, representation or context. A sign-control repair should survive a different equation. A percentage-base repair should survive money, population and data contexts. A similarity repair should survive rotation.

4. Can the learner recognise when the skill is needed without a topic label?

Ordinary Mathematics rarely announces every prerequisite. Mixed work should not require headings such as “use reverse percentage” or “factorise here” for the repaired structure to become available.

5. Can the learner execute with ordinary accuracy and reasonable cost?

Perfect speed is not required. The routine should be accurate and cheap enough that attention remains available for the new reasoning in the current topic.

6. Can the learner check a risky transition?

They should know where an error is expensive: sign changes, calculator entry, diagram conditions, percentage base, unit conversion, final interpretation. Checking does not need to become a full second solution every time.

7. Can the learner recover locally?

When a check fails, can they find the last trusted state and inspect the first suspect transition? If every error still requires a tutor to restart the problem, support has not fully faded.

8. Does the skill survive delay and mixed use?

Same-day success is not enough. The repair should remain available after a gap and when mixed with neighbouring Mathematics. If it repeatedly disappears, return it to transfer watch or active repair.

9. Is the current frontier still moving forward?

The strongest evidence is practical: current Mathematics no longer produces the same emergency because of the old gap. New topics may still be difficult. Ordinary mistakes may still happen. The learner is nevertheless progressing rather than repeatedly falling back through the same prerequisite hole.

If most of these questions can be answered with evidence, special catch-up support can shrink. If one or two fail, target those failures rather than reopening the entire historical curriculum.

The exit gate protects both sides of the process: it prevents premature withdrawal, and it prevents a repaired learner from being kept in catch-up mode longer than necessary.