What is mathematics?

Ask a grown-up what mathematics is and you might hear "the subject with sums". Ask a mathematician and you will hear something far more exciting: mathematics is the science of patterns, and the search for why those patterns work.

Concept

Patterns are everywhere. The petals of a hibiscus, the ripples on a pond, the tiles on a kitchen floor, the heartbeats on a doctor's chart , all of these have an order hiding inside them. Mathematics is the language we use to describe that order in a precise way.

There are two halves to this work. The first half is noticing the pattern. You spot that every fifth lamp post is taller, or that adding two odd numbers always gives an even number. The second half is explaining why. Anyone can spot that $1 + 3 = 4$, $1 + 3 + 5 = 9$, and $1 + 3 + 5 + 7 = 16$, but a mathematician asks: why do we always get a square? Once we explain it, we can be sure the pattern will keep going forever , even for the first million odd numbers.

Patterns also let us predict. If a bus has come at $8{:}05$, $8{:}15$, $8{:}25$, you can predict the next bus will come at $8{:}35$. The same kind of thinking helps scientists send a rocket to Mars: they spot the pattern in how planets move and use it to plan when to launch.

Mathematics is also a creative subject. To find a new pattern, or to find a new reason why a pattern holds, you need imagination as much as calculation. That is why mathematicians sometimes call their work an art , like music or poetry, but written in symbols.

A small example. Look at this list of dots arranged in a triangle:

*
* *
* * *
* * * *

How many dots are there in all? Counting gives $1 + 2 + 3 + 4 = 10$. Now look at it as two triangles glued together: you would have a $4 \times 5$ rectangle of dots, which has $20$ dots, and our triangle is exactly half. So the count is $\frac{4 \times 5}{2} = 10$. That is a small piece of mathematics , a pattern and an explanation.

Worked examples

Example 1. What is the next number in the list $2, 5, 8, 11, 14, \dots$?

• Look at the gaps: $5 - 2 = 3$, $8 - 5 = 3$, $11 - 8 = 3$, $14 - 11 = 3$.
• The rule is "add $3$ each time".
• Next term: $14 + 3 = 17$.

Example 2. Add the first five odd numbers. What do you notice?

• First five odd numbers are $1, 3, 5, 7, 9$.
• Their sum: $1 + 3 + 5 + 7 + 9 = 25$.
• We notice $25 = 5^2$, a perfect square.
• Why? If you arrange $1$ dot, then add a bent row of $3$ dots, then $5$, then $7$, then $9$, the dots grow into a $5 \times 5$ square. So adding the first $n$ odd numbers always gives $n^2$.

Example 3. A staircase has $1$ step in the first row, $2$ in the next, $3$ in the next, and so on, up to $10$ rows. How many step-blocks in all?

• Total blocks $= 1 + 2 + 3 + \dots + 10$.
• Pair them: $(1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) = 11 + 11 + 11 + 11 + 11 = 55$.
• So $55$ blocks.

Example 4. Look at the sequence $1, 2, 4, 8, 16, \dots$. What is the rule? What comes next?

• Each term is double the one before: $2 = 1 \times 2$, $4 = 2 \times 2$, $8 = 4 \times 2$.
• Next term: $16 \times 2 = 32$.

Try it yourself

1. Find the next two numbers in: $3, 6, 9, 12, \dots$
2. Find the next two numbers in: $1, 4, 9, 16, \dots$
3. Add the first four odd numbers. Is it a square?
4. Add the first six counting numbers. Can you find a quick way?
5. In the sequence $1, 1, 2, 3, 5, 8, \dots$, each term is the sum of the two before it. What comes next?
6. List three patterns you can see in your classroom right now.
7. Look at a calendar month. Pick any $2 \times 2$ square of dates. Add the two diagonals. What do you notice?

Activity

Pattern walk. Take a fifteen-minute walk around your home or school with a notebook. Write down at least five patterns you see , in tiles, leaves, fences, brickwork, wires, anything. For each one, write a single sentence describing the rule. Then ask one friend or family member to try to extend the pattern with you.