There is a moment in some Mathematics examinations that I have watched many students mishandle.
They have been working on one question for too long.
The page is full.
There is a diagram.
Two equations.
Something has been crossed out.
The calculator has been used three times.
The student knows she is not making much progress.
Yet she continues.
Another minute.
Then another.
There is a particular emotional force holding her there.
“I have already spent so much time on this.”
Or:
“I should be able to do it.”
Or:
“If I leave it now, I have failed.”
By the time she finally moves on, the question has taken perhaps eight or ten minutes more than it deserved.
Later in the paper, two questions she could have answered comfortably are rushed.
One is left incomplete.
The difficult question did not only cost the marks attached to itself.
It consumed the opportunity to earn other marks.
After many years of teaching Mathematics, I think this is one of the quieter examination skills students need to learn.
Persistence matters.
But persistence is not the same thing as refusing to disengage.
Sometimes the mathematically stronger decision is to leave a problem temporarily, protect the rest of the paper, and return when the search space is smaller and the pressure has changed.
The direct answer
A student should not measure examination strength by how long she is willing to fight one question.
She should measure it partly by whether she can recognise when her present route is no longer producing useful information.
That distinction matters.
A difficult Mathematics question may deserve persistence when each step is reducing uncertainty.
But if the student is repeating the same algebra, trying unrelated formulas, rereading without extracting anything new, or calculating quantities without knowing why, continued effort may have become expensive rather than productive.
At that point, leaving the question is not surrender.
It is resource management.
The student can preserve what she already knows, mark clearly where she stopped, continue collecting marks elsewhere, and return later with fresh attention.
The examination is not asking:
“Can you defeat every question in the order presented?”
It is asking for the strongest total mathematical performance across the paper.
Those are different objectives.
Students often confuse effort with progress
This is understandable.
In school we rightly praise persistence.
Do not give up quickly.
Try again.
Stay with difficult things.
These are good habits.
But they can become blunt if students never learn to distinguish productive struggle from stalled struggle.
Suppose a student is solving:
2x² − 5x − 3 = 0.
She tries factorisation.
She thinks:
Product = −6.
Sum = −5.
After a little testing she sees:
(2x + 1)(x − 3) = 0.
Good.
The time she spent thinking was productive.
Each attempt was constrained by the structure.
Now suppose she is working on a different question and does this:
tries factorisation;
then presses SOLVE on the calculator;
then rewrites the original equation;
then expands something that was already expanded;
then tries the sine rule although there is no triangle;
then rereads the question from the beginning;
then tries factorisation again.
Activity is happening.
Mathematical progress is not.
This is the distinction I want students to notice.
The question is not:
“Am I still working?”
It is:
“Is my current work changing what I know about the problem?”
Productive struggle leaves receipts
When a student is making progress, even slowly, something usually accumulates.
Perhaps she has found a useful substitution.
Perhaps she has eliminated one impossible case.
Perhaps a diagram has revealed similar triangles.
Perhaps she has found one coordinate.
Perhaps she has established the gradient.
Perhaps she knows the equation must have a repeated root.
Perhaps she has reduced a complicated expression to a quadratic.
She may not yet have the answer.
But the mathematical world has become smaller.
That matters.
Take a coordinate geometry problem.
Suppose the student needs the equation of a tangent to a curve at a particular point.
She may not see the entire route immediately.
But she identifies:
find the point on the curve;
differentiate;
evaluate the derivative there;
use the point and gradient to form the tangent equation.
Even if the later algebra becomes messy, she has a route.
Her effort has direction.
Compare that with a student who has spent four minutes differentiating, substituting random coordinates and rearranging equations without knowing what quantity she is trying to obtain.
The second student is not merely slower.
She may be lost.
A lost student needs a different decision from a student who is simply moving slowly along a valid route.
This is why the question “What are you trying to find next?” matters
When I see a student stuck during tuition, I sometimes ask:
“What are you trying to obtain?”
Not:
“What is the final answer?”
The next useful object.
Perhaps:
“The gradient.”
“The value of k.”
“The angle.”
“The second equation.”
“The radius.”
“The stationary point.”
If she can answer clearly, we often have productive difficulty.
If she cannot, I become more interested in whether the current work still has a purpose.
This transfers directly into examinations.
A student should occasionally be able to stop and ask herself:
What quantity or relationship am I trying to create with this line of working?
If there is no good answer, continuing exactly as before is unlikely to become better simply because another two minutes are spent on it.
A-Math makes this more important because questions become chained
Additional Mathematics can create long solution paths.
One question may require a student to:
recognise a relationship,
form an equation,
solve for a parameter,
differentiate,
substitute,
interpret the result,
and then use that result in a final part.
When everything is connected, students can feel compelled to complete the whole chain before moving on.
But examinations often allow something more flexible.
Part (a) may be secure even if part (b) is difficult.
Part (c) may sometimes use a result given in the question.
A student who treats the entire question as one indivisible object may spend too long trying to rescue one broken link.
A stronger student protects what is already available.
She writes the valid working clearly.
She attempts the difficult link.
If no useful route appears after a sensible interval, she moves.
Later, she may return.
This matters because Mathematics papers contain marks of very different costs.
Some marks require one clean observation.
Others require several minutes of reasoning.
The student does not control the paper.
She does control how much time she allows one unresolved question to consume.
There is a simple opportunity-cost calculation hiding underneath this
Suppose a student has ten minutes left.
Question A is difficult.
She has already spent five minutes on it and is still unsure.
Three marks remain available.
Elsewhere, Questions B and C together contain six marks she has not attempted.
If she spends all ten remaining minutes on Question A, she may earn zero, one, two or three marks.
If she moves to B and C, perhaps she has a very high chance of earning four or five marks.
The precise probabilities are unknown.
This is not a formal expected-value calculation.
But the structure is real.
Time spent in one place cannot also be spent elsewhere.
Examinations therefore create a mathematical resource-allocation problem inside the Mathematics examination.
Students who ignore this can lose marks despite knowing enough Mathematics to score them.
This is especially common among conscientious students
Interestingly, the students who struggle with leaving a question are not always careless or poorly prepared.
Often they are responsible students.
They are used to completing what they start.
They dislike blank spaces.
They want closure.
A difficult question feels unfinished in a way that bothers them.
So they remain.
There is also pride.
A strong student sees a question worth five marks and thinks:
“I should know this.”
The question becomes personal.
At that point, the examination has changed.
She is no longer deciding how to maximise her mathematical performance.
She is trying to settle an argument with one question.
That is rarely a good trade.
A paper does not reward indignation.
It rewards marks.
Leaving is easier when the student knows how to leave properly
I do not want students randomly abandoning questions.
There should be a clean exit.
Before moving on, preserve what is useful.
If a diagram has been labelled, leave it.
If an equation is correct, keep it.
If one part has been solved, box or clearly finish it.
If the next required object is known, perhaps make a short note.
For example:
Need second equation in x and y.
Or:
Return: try discriminant condition.
Or:
Have gradient; need point.
This is valuable.
When the student returns twenty minutes later, she does not have to reconstruct the entire problem from zero.
The earlier work becomes a restart point.
The question has been suspended, not erased.
Sometimes another question unlocks the one you left
This happens more often than students expect.
A later question reminds them of an identity.
Another diagram triggers a geometry relationship.
A different algebraic form makes a substitution obvious.
Or simply, the nervous fixation disappears.
The same question that felt completely closed at 10:05 may look surprisingly ordinary at 10:45.
Nothing in the syllabus changed.
The student’s internal state did.
This is one reason returning can be so productive.
Mathematical cognition is not perfectly linear.
Attention can become trapped.
Leaving temporarily can break that trap.
But we should be careful not to romanticise “fresh eyes”
Sometimes the student returns and still cannot solve it.
That is fine.
Leaving a question is not a secret technique that guarantees later insight.
Its value is simpler.
It limits the damage.
The student has already collected the marks she could collect elsewhere.
Now the remaining time can be spent on the hardest unresolved items without sacrificing secure work.
Persistence becomes appropriate again because the opportunity cost has changed.
There are therefore two different kinds of persistence
During learning, I often want deep persistence.
We have time.
The purpose may be precisely to stay with the difficult problem.
We can inspect mistakes.
Try another representation.
Go backwards.
Draw diagrams.
Test examples.
Spend twenty minutes understanding why one method failed.
That is education.
During an examination, persistence has another constraint.
The clock.
The student is no longer optimising only for understanding.
She is also allocating limited time across assessed work.
This is an important boundary.
A strategy that is excellent during tuition can be poor during an examination.
If we always rescue students from difficult problems after two minutes during lessons, they may never build depth.
If we insist they spend twenty minutes on every difficult question during an examination, they may never learn triage.
The environment matters.
This is why examination training should not become ordinary teaching with a timer added
There is a distinct skill here.
A student needs to practise decisions such as:
Should I continue?
Should I change method?
Should I leave and return?
Is there another part I can do independently?
Have I written enough to preserve method marks?
Is the problem conceptual, algebraic or simply time-consuming?
These are examination decisions.
They deserve occasional explicit practice.
Simply handing a student a two-hour paper and saying “work faster” does not necessarily teach them.
One useful signal is repetition without new information
Suppose a student has written:
sin²x + cos²x = 1.
Then rewritten:
sin²x = 1 − cos²x.
Then replaced sin²x elsewhere.
Good.
Something changed.
Now suppose she returns to the original identity, rearranges it back, then rearranges it again.
Nothing new is entering the problem.
She is circulating.
The same can happen in algebra.
Expand.
Factorise back.
Expand again.
Move a term left.
Move it right.
Students sometimes experience this as effort because the pencil is moving.
I treat it as a warning.
When the Mathematics begins returning repeatedly to the same state, the student should ask whether another representation or another question deserves attention.
Another warning is collecting quantities without a purpose
This is common in geometry.
A student sees a diagram and starts calculating every angle she can.
52°.
Then 128°.
Then 64°.
Then another 52°.
All correct.
None connected to the required length.
This is mathematically competent wandering.
A student can lose substantial time this way because each small calculation feels successful.
Success itself becomes misleading.
The question to ask is:
How does this quantity help me reach the unknown?
If no relationship is visible, perhaps the calculation should not be done yet.
This is why strong execution cannot substitute for method selection.
A student can accurately calculate the wrong things for a very long time.
Sometimes the correct decision is not to leave but to change representation
Before abandoning a problem, there is often one valuable question:
Have I tried seeing it differently?
An algebra problem may become clearer graphically.
A geometry problem may need an auxiliary line.
A word problem may need a variable.
A repeated expression may need substitution.
A probability problem may be easier through the complement.
A trigonometric equation may hide a quadratic.
This is productive route changing.
But there is a boundary again.
I do not want students performing a frantic tour of every representation they know.
Graph?
No.
Table?
No.
Substitution?
No.
Differentiate?
No.
That is not flexible thinking.
That is mathematical roulette.
Changing representation should be motivated by structure.
A good stop rule cannot be purely based on minutes
Parents sometimes ask:
“How long should my child spend before moving on?”
There is no universal number.
A four-mark question and a ten-mark question deserve different attention.
Students also work at different speeds.
Some questions require a genuine period of thought before anything appears.
An arbitrary rule such as “if you cannot solve it in two minutes, leave” can train shallow behaviour.
The better rule combines time with evidence.
Ask:
Have I identified what is being asked?
Have I found any valid relationship?
Do I have a plausible route?
Is each additional line reducing uncertainty?
Am I making progress, even slowly?
Or am I repeating, guessing and accumulating unrelated work?
Time matters.
But the quality of movement matters too.
This is where marks-per-minute thinking needs restraint
I sometimes hear examination advice reduced to:
“One mark per minute.”
This can be a useful broad pacing guide in some contexts.
It should not become a law.
Mathematics marks are not produced at a constant rate.
A four-mark insight question may require three quiet minutes before the first line and then thirty seconds of writing.
Another four-mark question may be procedural.
A student who panics because nothing has been written in the first sixty seconds may abandon exactly the kind of thinking the question requires.
So pacing needs flexibility.
The real target is not uniform speed.
It is avoiding extreme misallocation.
Ten minutes of genuine reasoning on a hard, valuable question may be entirely sensible.
Ten minutes repeating failed work usually is not.
Parents can notice this during homework
Watch what your child does when stuck.
Does she stop and reconsider the problem?
Does she identify what she knows?
Does she try to explain the obstacle?
Does she change representation for a reason?
Or does she keep performing operations because stopping feels uncomfortable?
You can ask:
“What new thing did the last two lines tell you?”
This is a useful question.
If she answers:
“They gave me the gradient.”
Good.
“They eliminated y.”
Good.
“They showed that k must be positive.”
Good.
If the answer is:
“I don’t know, I’m just trying things,”
then the problem may no longer be effort.
It may be navigation.
This should change how we discuss careless time loss
After an examination, students sometimes say:
“I ran out of time.”
That sentence is incomplete.
Why?
Was the paper genuinely too long for the student’s current speed?
Was algebra execution slow?
Was calculator use inefficient?
Did the student spend too long checking?
Did she hesitate at every question?
Or did one difficult problem absorb twelve minutes beyond its value?
These are different causes.
“Time management” is too broad to diagnose them.
If a student repeatedly finishes ninety per cent of the paper but leaves six easy marks untouched because she fought one difficult part for too long, the repair is not necessarily faster Mathematics.
It may be better disengagement.
That is a much smaller intervention.
We can practise this deliberately
One exercise I like is a short mixed paper with an unusual instruction.
The student is not initially asked to solve everything.
For each question she has to decide:
A — I know a route immediately.
B — I see some structure but need thought.
C — I currently have no useful route.
Then she starts with the A questions.
After those, she returns to B.
C comes later.
The letters themselves are not important.
The habit is.
The student learns that question order and solution order do not have to be identical.
This is surprisingly liberating for some adolescents.
They have treated the examination booklet almost like a story.
Page one must be completed before page two is morally available.
It is not.
The paper is a field of opportunities.
Then I make the classification less comfortable
Some questions that look difficult should actually be quick.
Some that look familiar should contain a trap.
One early part may be difficult while a later part is independent.
Now the student has to read rather than simply rank by appearance.
This matters.
I do not want triage becoming another superficial rule:
“Do the short-looking questions first.”
The goal is better judgement.
A long question may contain three easy marks at the beginning.
A compact question may hide a substantial proof.
The student should learn to inspect before deciding.
Returning should also be practised
Students often practise leaving.
They do not practise returning.
When they return, they restart from the first sentence and lose time.
I prefer a cleaner routine.
Read the earlier working.
Identify the last secure statement.
Ask:
“What remains unknown?”
Then continue from there.
For example:
Earlier work established:
dy/dx = 3x² − 12x + 9.
The question asks for stationary points.
On returning, do not re-differentiate.
The next step is already visible:
3x² − 12x + 9 = 0.
Factorise.
Continue.
The secure work should function as stored progress.
Measurement is straightforward
I would not measure improvement only by whether the student finishes more papers.
Track something simpler.
How many questions consume excessive time without producing marks?
How many easy later questions remain unattempted?
How often does the student return successfully to a skipped question?
How much valid working is preserved before moving?
Does she recognise dead-end repetition earlier?
Across several papers, these patterns become visible.
The aim is not to create a student who abandons difficulty quickly.
Quite the opposite.
The aim is to create one who can persist intelligently because persistence has become a choice rather than a reflex.
There is an emotional maturity inside this too
Adolescents often experience a difficult Mathematics question as a judgement.
A blank space feels public.
An unsolved problem feels personal.
The stronger the student’s academic identity, the more uncomfortable leaving can become.
But examinations are not improved by turning each question into a test of character.
Sometimes the mature response is:
“I do not have this yet.”
Then:
“I will protect the rest of the paper and come back.”
That sentence is calm.
It contains neither panic nor denial.
It respects the difficulty without giving it unlimited authority over the student’s time.
I think that is worth teaching.
The useful next route
If your child regularly runs out of time despite knowing much of the Mathematics, do not begin by demanding that she “work faster”.
Take one recent paper and reconstruct the time loss.
Find the question where the paper stopped flowing.
Then ask:
What did she know when she reached it?
What valid progress did she make?
At what point did new information stop appearing?
What did she do after that?
What unattempted marks remained elsewhere?
Then repeat one similar paper with a deliberate rule:
When progress stalls, preserve the last secure line, mark the question, continue elsewhere, and return later.
Compare the two papers.
Did more accessible marks get collected?
Did returning help?
Was the difficult question actually easier later?
Did the student leave too quickly?
That last question matters too.
We are calibrating judgement, not teaching avoidance.
With a few repetitions, students begin recognising their own patterns.
One student leaves too late.
Another too early.
One panics whenever she cannot see the first step.
Another spends ten minutes trying to force every question to completion.
The repair should match the student.
What long teaching has made me notice
We often describe a good Mathematics student as persistent.
I still believe that.
Mathematics contains ideas that do not surrender immediately.
Students need the patience to sit with uncertainty.
They need to try, fail, re-represent and try again.
But I now think persistence becomes stronger when it acquires a boundary.
A student should know why she is still working.
She should be able to feel the difference between a difficult path that is slowly opening and a closed loop she has been walking for five minutes.
She should be able to preserve what she has earned.
Move elsewhere.
Return.
Begin again from the last secure place.
That is not less determined.
It is more controlled.
And perhaps this is one of the useful things examinations can teach when we handle them properly.
Time is finite.
Attention is finite.
Not every problem deserves every remaining resource merely because it appeared first or because we have already invested heavily in it.
Mathematical adulthood includes knowing how to commit.
It also includes knowing when the evidence says:
not yet.
A strong student does not need every question to yield immediately.
She does not need to defeat the page in sequence.
She can hold an unresolved problem without allowing it to consume the rest of the examination.
She can say:
“I have taken this as far as I can responsibly take it for now.”
Then turn the page.
And later, when the rest of the marks are safer and the mind is quieter, she can return.
Sometimes she solves it.
Sometimes she does not.
But either way, the question has not been allowed to decide the fate of the entire paper.
That is not giving up.
It is judgement.
And in difficult Mathematics, judgement is part of the subject.