I used to be afraid of mathematics.
I don’t find it strange, therefore, when children tell me that they are afraid of maths. I know the feeling. Mathematics can seem like a subject in which everybody else has been given a set of instructions that somehow never reached you.
There are formulas to remember. Rules about what you can and cannot do. Steps that have to be performed in the correct order. A teacher puts something on the board, moves a few numbers around, and arrives at an answer. You copy it down. Perhaps you can reproduce it in the next problem. Perhaps you can’t.
But somewhere underneath all this there is a more troubling question:
Why am I doing any of this?
For a long time, mathematics for me was something to get right rather than something to understand. Then, gradually, that changed.
I started understanding mathematics. And when I understood it, I started loving it.
That distinction has become increasingly important to me. I don’t think my relationship with mathematics changed because I suddenly acquired some mathematical ability that I hadn’t possessed before. Nor did somebody discover a sufficiently entertaining way of teaching me multiplication.
What changed was that the symbols began to mean something.
And once they meant something, mathematics became a completely different subject.
Mathematics before mathematics
One of the odd things about mathematics education is that we often don’t begin with the most obvious reason for studying mathematics: it is everywhere. A person who says, quite sincerely, “I never use maths”, probably used it several times that day.
Suppose you go shopping and see two packets of something. One costs ₹180 for 400 grams. The other costs ₹240 for 600 grams. Which is cheaper?
That is mathematics.
You have ₹5,00,000 in a bank deposit paying 7 per cent interest and prices are rising by roughly 5 per cent a year. Are you actually becoming 7 per cent richer every year?
Mathematics again.
A shop announces a 30 per cent discount and then offers an additional 20 per cent off the discounted price. Is that a 50 per cent discount?
It isn’t.
If something costs ₹1,000, the first discount brings it to ₹700. Twenty per cent of ₹700 is ₹140, so the final price is ₹560. The effective discount is 44 per cent.
Nothing terribly advanced happened there. But understanding what happened is far more valuable than merely knowing how to calculate 20 per cent of 700.
Or consider a recipe meant for four people. Six people are coming.
Or a room that needs new tiles.
Or a 500-kilometre journey for which you know your car’s approximate fuel efficiency.
Or a medicine that has to be taken at eight-hour intervals.
Or a cricket run rate.
Or the chance of rain tomorrow.
Or an EMI.
Or the storage remaining on your phone.
Or whether the “family pack” in the supermarket is actually cheaper per gram than the smaller packet.
This is mathematics before anyone calls it mathematics.
Children encounter quantity, shape, pattern, comparison, uncertainty, distance and time long before they encounter algebra.
Yet we have somehow managed to create a subject called Mathematics which can feel entirely detached from all of them.
That should bother us.
Why does 10 come after 9?
Here is a question I would like mathematics teachers to ask young children: “Why does 10 come after 9?“
Not what comes after 9. Every child who can count knows that.
Why?
We take this so completely for granted that the question initially sounds silly.
But look at what happens.
1
2
3
4
5
6
7
8
9
So far, every number gets its own symbol.
Then suddenly:
Why?
Why didn’t we invent another symbol for ten?
And why does putting 1 and 0 beside each other mean ten? The symbol 1 means one. The symbol 0 means nothing. Why should putting “one” beside “nothing” produce ten?
There is an enormous mathematical idea hiding inside this apparently childish question.
We use a positional number system.
Imagine putting pebbles on a table. Count them one by one until you have ten. Now tie those ten pebbles into a little bag. Instead of saying you have ten individual pebbles, you can now say you have one bag of ten and no loose pebbles.
One ten. Zero ones.
Add another pebble.
One ten. One one.
Keep going until you have ten bags. Put those ten bags into a box.
One hundred. Zero tens. Zero ones.
Now the notation begins to have a physical meaning. The 1 in 1, the 1 in 10 and the 1 in 100 are the same digit, but they do not represent the same quantity. Position changes value. That idea is astonishingly powerful.
And it becomes even more interesting if we ask children: “What if human beings had decided to bundle the pebbles whenever we reached five instead?”
Then what we write as 10 in that system would represent what we call five in our usual decimal system.
Suddenly “10” is no longer intrinsically ten. Children can begin to see the distinction between a number and the notation we use to represent it. That is a profound mathematical insight hiding inside primary-school arithmetic.
But it is very easy to turn place value into:
What is the place value of 7 in 7,432?
The child writes “7000”.
Correct. Next question.
Something has been learned. But something much larger may have been missed. And the consequences don’t necessarily remain confined to the chapter on place value.
Consider another familiar instruction from primary-school mathematics:
Carry the one.
Take:
28 + 17.
We add 8 and 7 and get 15. So we write down 5 and “carry 1”. But what exactly have we carried?
Not one. Ten.
We had 15 individual units. We regrouped ten of those units as one ten, leaving five units behind. That “1” written above the tens column represents one group of ten.
The algorithm is simply our positional number system at work. A child who understands this doesn’t merely know how to carry. The child knows why carrying exists. This matters later too.
Children are often taught another convenient rule:
When you multiply by 10, add a zero.
27 × 10 = 270.
It appears to work perfectly.
Until we try:
2.7 × 10.
If multiplying by ten means adding a zero, should the answer be 2.70? Of course not.
The answer is 27. The underlying idea was never “add a zero”. Multiplication by ten changes the place value of the digits. The shortcut happened to imitate that process for whole numbers.
This is where apparently harmless shortcuts can become dangerous.
A child learns a rule that works. The rule gets rewarded because it produces the correct answers. Nobody asks whether the child understands why it works. Then, years later, the rule stops working. And mathematics seems to have betrayed them.
It hasn’t. The shortcut has.
We often teach the answer before the question
This happens repeatedly in mathematics.
Consider area.
A child is told: Area of a rectangle = length × breadth.
Then come twenty rectangles.
Length 7 cm, breadth 4 cm. Find the area.
7 × 4 = 28 cm².
Correct.
But why multiplication?
Give the same child a rectangular floor and some square tiles.
How many tiles will cover it?
Now the formula has somewhere to come from.
If seven tiles fit along one side and four along the other, we could count all 28 tiles individually.
Or we could notice that there are four rows of seven.
7 + 7 + 7 + 7.
Which is 7 × 4.
The formula is not an arbitrary instruction handed down by a textbook. It is a shortcut for a pattern that the child can see. And once that idea is understood, we can ask better questions.
Suppose I give you 40 metres of fencing and ask you to make a rectangular garden. You could make it 1 metre by 19 metres. Its perimeter would be 40 metres and its area 19 square metres. Or 5 metres by 15 metres. Same perimeter. Area 75 square metres. Or 10 metres by 10 metres. Still the same perimeter. Area 100 square metres.
Now something interesting has happened.
We are no longer merely inserting numbers into a formula. We are exploring a relationship. The amount of fencing has not changed, yet the amount of land enclosed has changed enormously.
A student can experiment. Make a table. Draw rectangles. Look for a pattern. Make a conjecture. That feels much more like mathematics to me.
I think we too often give children mathematical solutions to problems they have never actually had. The formula comes first. The need for the formula comes later, if at all.
I would reverse that whenever possible. Create the problem first. Let the formula arrive as a relief.
Fractions are not two numbers with a line between them
Fractions are another good example. Many children become confused by fractions because they meet the notation before they properly understand the relationship.
Ask a child which is larger:
1/3 or 1/4?
A child accustomed to whole numbers may reasonably think 1/4 is larger. Four is larger than three. But put a pizza on the table. Would you rather have one third of it or one quarter?
Now the strange rule starts becoming intuitive.
The larger denominator means we divided the same whole into more pieces, so each piece became smaller.
Then complicate it.
Is one quarter of a pizza always smaller than one third of a pizza? No.
One quarter of a very large pizza could easily be larger than one third of a small one.
Suddenly we have encountered something important: a fraction only makes sense relative to a whole.
And then, a little later, comes one of those wonderful rules of school mathematics:
To divide fractions, invert the second fraction and multiply.
Why?
Consider:
3 ÷ 1/2.
Forget the rule.
What does division mean here? How many halves are there in 3?
There are six.
So:
3 ÷ 1/2 = 6.
Now try:
3 ÷ 1/4.
How many quarters are there in three wholes?
Twelve.
So:
3 ÷ 1/4 = 12.
The strange result—that dividing by a smaller number can produce a larger answer—begins to make sense once division is understood as asking how many of one quantity fit into another.
Eventually, yes, teach the general algorithm. Teach reciprocals. Teach students how to manipulate fractions efficiently.
But let the algorithm come after the idea. “Invert and multiply” is easy to remember. Understanding why it works is mathematics.
Children can do algebra before we tell them it is algebra
Imagine I tell a child:
I’m thinking of a number.
I double it.
Then I add 6.
The answer is 20.
What number was I thinking of?
A child may reason backwards.
Take 6 away from 20.
Half of 14 is 7.
The number was 7.
Nothing terribly frightening happened.
Now write:
2x + 6 = 20.
Suddenly it is algebra. The reasoning hasn’t changed. Only the notation has.
This is worth thinking about because many children who are perfectly capable of algebraic reasoning become intimidated when letters appear.
Perhaps algebra is sometimes made difficult not by the idea itself but because we introduce the symbolic language before allowing children to discover that they already understand the problem.
The equals sign is not an instruction to calculate
Even the humble equals sign is more interesting than it appears.
Ask a young student to fill this in:
8 + 4 = ___ + 5
Some children will find it surprisingly uncomfortable.
Why?
Because through years of arithmetic they can acquire the idea that “=” means now write the answer.
3 + 4 = 7.
6 × 5 = 30.
18 − 9 = 9.
The left side contains the problem. The right side contains the answer. But that isn’t what “=” means. It means the thing on this side has the same value as the thing on that side.
So:
8 + 4 = 7 + 5
is not an unfinished calculation. It is a statement that two quantities are equal. That small conceptual shift becomes enormously important when algebra arrives.
If a child understands an equation as a balance, then:
x + 7 = 12
isn’t a mysterious instruction to “take 7 to the other side and change its sign”. It is a balance.
If I remove 7 from one side, I must remove 7 from the other.
x + 7 − 7 = 12 − 7
so x = 5.
Nothing actually jumped across the equals sign. Nothing magically changed its sign. We performed the same operation on both sides and preserved equality. The familiar shortcut comes later.
Again, the shortcut isn’t the problem. The shortcut without the understanding is.
Mathematics should not be a collection of incantations
There are a surprising number of sentences in school mathematics that can become almost like incantations.
“Carry the one.”
“Add a zero.”
“Invert and multiply.”
“Take it to the other side and change the sign.”
“Minus into minus is plus.”
Say them often enough and they begin to sound like explanations. They aren’t. Take that last one.
Why should multiplying two negative numbers produce a positive number?
We can tell a child:
minus × minus = plus
and require them to remember it. Or we can let them notice a pattern.
3 × (−2) = −6
2 × (−2) = −4
1 × (−2) = −2
0 × (−2) = 0
Look at the answers.
−6, −4, −2, 0.
Each time the first number falls by one, the answer increases by two.
What should come next?
(−1) × (−2) = 2.
Then:
(−2) × (−2) = 4.
The rule begins to look less arbitrary.
We can later show why it follows from the structure of arithmetic itself. But already the student has seen something important.
The rule wasn’t invented merely to make life difficult. It has to fit with everything else. This, to me, is one of the most beautiful things about mathematics. Ideas constrain one another.
You can’t simply decide that negative times negative should remain negative without breaking other relationships that you already rely upon.
And sometimes mathematics goes further and tells us that something our intuition insists must be true simply isn’t.
Consider:
0.999999…
with the 9s continuing forever.
Surely that must be a tiny bit less than 1?
But how much less?
Let:
x = 0.999…
Then:
10x = 9.999…
Subtract the first equation from the second:
9x = 9.
So:
x = 1.
Or notice that:
1/3 = 0.333…
Multiply both sides by 3:
1 = 0.999…
There isn’t a microscopic gap hiding after infinitely many 9s. They are two representations of the same number.
This is where mathematics becomes much more interesting than calculation. It can force us to examine what we mean by numbers, infinity and equality.
A student may disagree initially. Good.
That disagreement is an opportunity.
“Why can’t that be right?” is a far more interesting beginning to a mathematics lesson than “copy this formula from the board.”
Percentages should make you suspicious
Percentages are particularly important because adults encounter them constantly. Suppose someone’s salary rises by 10 per cent one year and falls by 10 per cent the next.
Are they back where they started?
Start at ₹100.
A 10 per cent rise gives ₹110.
A 10 per cent fall from ₹110 is ₹11.
You end at ₹99.
Or suppose a company’s sales increase from 100 units to 200 units.
That’s a 100 per cent increase.
Then sales fall from 200 to 100.
That’s a 50 per cent decrease.
Same absolute change: 100 units. Different percentage change.
Why?
Because percentages have denominators. They are always percentages of something. That “something” is often more important than the percentage itself.
“Risk increased by 50 per cent” sounds frightening.
But suppose a risk went from 2 in 10,000 to 3 in 10,000. That is indeed a 50 per cent relative increase. It is also an absolute increase of 1 in 10,000. Both descriptions can be mathematically correct. They do not communicate the same impression.
A mathematically educated person should instinctively ask: Fifty per cent of what?
That question is not merely useful for an examination. It is useful for life.
Averages can lie without anyone lying
Or take averages. Suppose five people earn:
₹20,000
₹22,000
₹25,000
₹28,000
₹5,00,000
Their mean income is ₹1,19,000.
Would you say that ₹1,19,000 describes the income of a typical person in that group particularly well?
Probably not. The median is ₹25,000. Neither calculation is wrong. They answer different questions.
There is another, simpler example that I like.
Suppose four families have 0, 1, 2 and 3 children.
The mean is 1.5 children. Obviously there is no family in our group with one and a half children.
So what exactly is an average?
This is a much more interesting question than simply learning how to calculate one. An average is a mathematical summary. It need not describe any actual member of the group.
That distinction becomes extremely important when we encounter claims about salaries, house prices, wealth, examination scores and almost every other kind of social data.
I would much rather a student leave school understanding why mean and median can tell very different stories than merely remembering:
mean = sum of observations / number of observations.
The formula takes ten seconds to look up. Understanding what the resulting number means—and when it might give a misleading impression—is the mathematics.
Probability is where our intuition gets into trouble
Probability deserves much more attention for similar reasons.
Suppose I toss a fair coin five times and get:
H H H H H.
What is more likely on the sixth toss?
Heads?
Tails?
Neither.
If the coin is genuinely fair and each toss independent, the probability remains one half.
But tails feels due.
That feeling is so strong that adults gamble money on it.
Or consider another question.
Which sequence looks more likely from six fair coin tosses?
H T H T T H
or
H H H H H H
The first looks random. The second looks suspicious.
But each exact sequence has the same probability: 1/64.
And here is another question.
How many people need to be in a room before there is a better-than-even chance that at least two of them share a birthday?
A hundred? Perhaps fifty? There are 365 days in a year, after all.
The surprising answer, under the usual simplifying assumptions, is just 23. The important part isn’t memorising 23. It is understanding why our intuition was wrong.
If I ask whether someone in a room shares my birthday, I am making one set of comparisons. But that isn’t the birthday problem. We are asking whether any two people share a birthday.
With 23 people there are 253 different pairs of people. Suddenly the result becomes less mysterious.
Our minds are pattern-seeking machines, but they are not naturally good probability calculators. That is usually fine. But there are situations in which our intuition can mislead us badly.
That is why probability should not be taught merely as coloured balls being removed from imaginary bags. Those exercises have their place.
But the bigger lesson is extraordinary: human intuition about uncertainty is often unreliable.
Mathematics gives us a way of checking it.
Estimation deserves far more respect
I would make estimation a major part of mathematics education. Ask students questions for which they cannot possibly know the exact answer.
How many litres of water does your school use in a day?
How many sheets of paper are in all the classrooms?
How many grains of rice are in a one-kilogram bag?
How many people could stand on a football field?
How many piano tuners might there be in Kolkata?
The last sort of question is sometimes called a Fermi problem.
The answer isn’t really the point. The reasoning is.
Suppose you want to estimate how many cups of tea are drunk in Kolkata every day.
You don’t know. Fine.
How many people live in the city? Roughly how many drink tea? How many cups might each drink? What assumptions are reasonable?
Perhaps your estimate is badly wrong. Excellent. Now ask why.
Which assumption caused the error?
That is mathematics behaving like thought rather than calculation. And it develops something that I consider more important than the ability to calculate quickly: number sense.
We use words like thousand, million, billion and trillion very casually. But do we actually have a feeling for how large those numbers are?
Consider seconds.
A million seconds is about 11.6 days.
A billion seconds is about 31.7 years.
A trillion seconds is about 31,700 years.
The difference between a million and a billion suddenly feels rather larger than it does when we merely add three zeroes.
A child who can correctly write 1,000,000,000 may know how a billion is written without having the faintest feeling for how large a billion is.
Those are not the same thing.
I want a student to be able to see that 19 × 51 is somewhere around 1,000 before calculating it exactly.
I want them to know that a restaurant bill of ₹2,340 cannot become ₹2,925 after adding a 5 per cent charge.
I want them to look at a calculated answer of 0.003 metres for the height of a door and know, before checking anything else, that something has gone wrong. The ability to recognise an absurd answer is enormously valuable.
Mathematics is also the art of asking whether a number means what it appears to mean
There is another kind of mathematics that schools need to take more seriously: learning to distrust numbers intelligently. Not reject them. Interrogate them.
Suppose a newspaper headline says:
“City crime doubles.”
That sounds alarming. Then you discover that a particular category of crime went from two reported cases to four. The statement may be perfectly true.
But you understand it differently now.
Or imagine a graph showing sales increasing from 95 to 100. If the vertical axis begins at zero, the increase looks modest. If the graph begins at 94, the line can appear to shoot upwards dramatically.
Same numbers. Different visual impression. A graph can be numerically accurate and rhetorically misleading.
Children should know this.
They should learn to ask:
What is being measured?
What isn’t being measured?
What is the sample size?
What is the baseline?
Is this an absolute change or a relative change?
Is this correlation or causation?
Where did the data come from?
These are mathematical habits, but they are also habits of intellectual self-defence.
We live in a world full of numbers used by advertisers, governments, businesses, journalists, campaigners and social-media accounts to persuade us of things.
Knowing long division is useful.
Knowing when a statistic is being used to manipulate your intuition may be even more useful.
And then there is compound growth
There are some mathematical ideas that I think everyone should encounter because our intuition simply isn’t built for them.
Exponential growth is one.
Imagine a sheet of paper about 0.1 millimetres thick.
Now imagine that every time we fold it, its thickness doubles.
Real paper obviously cannot be folded indefinitely. This is a mathematical thought experiment.
After ten doublings, its theoretical thickness is roughly 10 centimetres. After twenty, roughly 100 metres. After thirty, roughly 100 kilometres.
At around 42 doublings, the theoretical thickness has passed the average distance from the Earth to the Moon.
That sounds absurd. And physically, of course, it is. No actual sheet of paper behaves this way indefinitely.
But the mathematics is telling us something important about repeated multiplication. Our intuition is reasonably comfortable with linear growth. Add 10 centimetres forty times and we can imagine what happens. Doubling forty times is another matter entirely.
The same mathematics appears in less spectacular forms all around us.
Compound interest is repeated multiplication wearing much duller clothes.
Suppose you have ₹1 lakh earning 10 per cent annually.
After one year you have ₹1.1 lakh.
After two years you don’t earn 10 per cent on the original ₹1 lakh alone. You earn it on ₹1.1 lakh.
Then on the new amount. Then on that new amount.
The same mechanism can work against you with debt and for you with investments. It appears in inflation, population growth, epidemics and many natural processes.
This is exactly why mathematics is useful. It allows us to think beyond the limits of intuition.
So what should a mathematics classroom look like?
I don’t have a grand curriculum to offer. And I am suspicious of grand educational revolutions that promise to solve everything. But I do think some principles follow naturally from all of this.
Start, wherever possible, with the problem rather than the formula.
Before teaching area = length × breadth, give students something that needs covering.
Before teaching percentages, give them competing discounts.
Before teaching ratios, give them recipes to scale.
Before teaching averages, give them data where mean and median tell noticeably different stories.
Before teaching equations, give them balances and unknown quantities.
Before teaching probability formally, ask them to predict outcomes and discover where their intuitions fail.
Before teaching graphs, show them two graphs of exactly the same data with different scales and ask why they feel so different.
And perhaps sometimes start with a question that sounds almost stupid.
Why does 10 come after 9?
What exactly are we carrying when we “carry the one”?
Why does dividing by one half make a number larger?
Why does minus times minus become plus?
Why can’t we divide by zero?
How can 0.999… possibly be 1?
Why does changing the shape of a rectangle change its area even when its perimeter stays the same?
Why can an average describe nobody in the group being averaged?
A teacher doesn’t necessarily need to answer every question immediately.
In fact, perhaps the temptation to answer immediately is itself part of the problem.
Let children argue.
Let them make predictions.
Let them be wrong.
Let one child convince another.
Let them find an example that breaks somebody’s proposed rule.
Let them discover that the answer they were certain about cannot be right.
Sometimes not knowing yet is the beginning of mathematics.
Misunderstandings accumulate
This is another reason I think conceptual understanding matters so much. Mathematics is unusually cumulative. A misunderstanding does not necessarily remain where it began.
A child who doesn’t really understand place value may nevertheless learn the algorithms for addition and subtraction.
Then decimals arrive.
A child who doesn’t understand fractions may learn to manipulate them well enough to pass an examination.
Then ratios, percentages, probability and algebra arrive.
A child who thinks the equals sign means “the answer comes next” may do perfectly well with arithmetic.
Then equations arrive.
At every stage, another compensating rule can be memorised.
Carry this. Borrow that. Move this. Flip that. Change the sign. Add the zero.
And a surprisingly capable child can continue for years this way. Until eventually there are too many rules.
Then we say: “They’ve reached the stage where maths became too difficult for them.” Perhaps. But perhaps the difficulty began years earlier.
Perhaps what finally collapsed was not mathematical ability but a tower of procedures built on ideas that had never become secure. That possibility should matter enormously to mathematics teachers.
When a student repeatedly makes a mistake, the interesting question isn’t only: What are they doing wrong?
It is: What do they think they are doing?
Those are very different questions. The second one can reveal the mathematics that needs to be taught.
Stop confusing speed with intelligence
There is another change I would make quite strongly. We should stop treating mathematical speed as a proxy for mathematical intelligence.
Some children calculate quickly. Good.
Some don’t. That does not tell us nearly as much as we think it does.
A child may take a long time over a problem because they are confused. But they may also take a long time because they are thinking deeply.
They may draw it. Try one approach. Abandon it. Notice something. Go backwards. Ask whether there is another way. That process is not a failure to do mathematics. It is mathematics.
Timed arithmetic has a place if the purpose is to build fluency. Fluency matters. Having to laboriously reconstruct every multiplication fact makes more complicated mathematics unnecessarily difficult.
But fluency is a tool. It is not mathematical intelligence. When speed becomes the culture of mathematics, slower students begin to believe that they simply don’t possess a mathematical mind.
I know how powerful that belief can become. And it is very difficult to learn a subject while simultaneously believing that the subject belongs to other people.
Calculators are not the enemy
For the same reason, I am not particularly frightened of calculators. Or computers. Or, now, AI.
A calculator can multiply 7,438 by 926 much faster than I can. So what?
That isn’t the most interesting part of mathematics. The interesting questions are:
Why are we multiplying these numbers?
What do they represent?
Approximately what should the answer be?
Does the answer the machine produced make sense?
What can we infer from it?
A student who knows which buttons to press but understands none of those things has gained very little. But so has a student who can perform the multiplication beautifully on paper and has no idea why they are doing it. Calculation is a tool. Mathematics is the thinking around the calculation.
In fact, the availability of calculators and computers makes understanding more important, not less.
If a machine gives me an answer, I need enough mathematical sense to recognise when the answer is nonsense. A calculator will quite happily give me a precise answer to a question I should never have asked. Precision is not the same thing as understanding.
Don’t make mathematics “fun”. Make it interesting.
I also think we should be slightly careful with the constant demand to make mathematics fun. I understand the intention. But mathematics does not need to be disguised as entertainment in order to justify itself.
Sometimes mathematics is fun.
Sometimes it is beautiful.
Sometimes it is frustrating.
Sometimes it is tedious.
Sometimes you can spend an hour getting nowhere.
That is all right. The thing I would insist upon is not that mathematics should always be fun. It should mean something.
And interesting is not the same as easy.
“Why is 0.999… equal to 1?” is interesting precisely because the answer initially feels wrong.
“How many piano tuners are there in Kolkata?” is interesting because nobody in the classroom knows the answer.
“Can you make two rectangles with the same perimeter but very different areas?” is interesting because it gives the child something to discover.
A good mathematical problem creates a little irritation in the mind.
Something doesn’t quite fit. You want to know.
There is a particular pleasure that comes from struggling with an idea for a while and then suddenly seeing it. Anyone who has genuinely understood a piece of mathematics knows this feeling.
One moment there is confusion. Then something shifts. You see the structure. Of course. That’s why.
That little moment is difficult to describe to someone who has never experienced it. But it is one of the reasons people fall in love with mathematics. I did.
Understanding first
Looking back, I don’t think my fear of mathematics was irrational. I was afraid of something I didn’t understand.
Once mathematics became a collection of ideas rather than a collection of instructions, my relationship with it changed completely. That is why I am uncomfortable when people casually say, “I’m just bad at maths.”
Perhaps some people really do have much greater mathematical aptitude than others. Of course abilities differ. But I wonder how many people who say they are bad at mathematics were actually defeated much earlier by mathematics education.
Perhaps they learned procedures without meanings.
Perhaps they fell behind at one important conceptual point and everything subsequently built on that gap.
Perhaps they were slow and were made to feel stupid.
Perhaps they became frightened of giving the wrong answer.
Perhaps nobody ever showed them why any of it was interesting.
Perhaps they became quite good at producing correct answers without understanding them, until one day that stopped being enough.
And eventually they concluded: Mathematics isn’t for me.
That is an enormous loss. Not because everybody needs to become a mathematician. They don’t. But because mathematics gives us a particular way of looking at the world.
It teaches us to notice structure.
To quantify.
To compare.
To estimate.
To recognise scale.
To reason about uncertainty.
To distinguish intuition from evidence.
To ask what follows from what.
To recognise when something cannot possibly be right.
To become comfortable saying, “I don’t know the answer yet, but perhaps I can work it out.”
That last habit may be the most valuable one of all.
I don’t want mathematics education merely to produce students who can solve the kinds of problems that appear in mathematics examinations.
I want someone standing in a supermarket comparing two packets to recognise a ratio problem without needing to call it one.
I want someone reading a frightening percentage in a newspaper to ask what the original number was.
I want someone looking at a graph to inspect its axes.
I want someone taking a loan to understand what compound interest can do.
I want someone hearing an average to wonder which average.
I want someone hearing “one billion” to have some feeling for how enormous a billion actually is.
I want someone presented with a probability to understand that their intuition may not be enough.
And I want a child looking at 10 to realise that even something as ordinary as the way we write numbers contains an idea worth thinking about.
Most of all, I want children to experience that wonderful transition from:
I don’t understand this.
to:
Wait. I think I see it.
That was the transition that changed mathematics for me.
I used to be afraid of mathematics. Then I began to understand it. And once I understood it, I discovered something I wish I had discovered much earlier. Mathematics was never really about remembering what to do with numbers.
It was about understanding why things make sense.