Teach-Primary-Issue-19.7

Definitions and properties A precise definition matters. The schoolroom formulation – that a prime number is divisible only by one and itself – is familiar but flawed. By that logic, one would be prime, since it meets the condition – but it is not. The definition lacks the clarity necessary to be useful. Instead, we should teach that a prime number has exactly two factors. This excludes one, includes two, and reveals two as the only even prime. Every other even number has more than two factors, since it is divisible by two. This clearer definition supports deeper mathematical reasoning, and encourages pupils to connect prime numbers with multiplicative relationships, divisibility, and fractions. Understanding primes helps children compare fractions, simplify expressions, and identify common denominators. Prime numbers also often indicate a point of stability; the number is as simple as it can get, or the solution is nearly complete. In this way, primes shape learners’ choices and reasoning. Representation and reasoning Prime numbers should not be treated in isolation. Like Roman numerals, their power lies in the connections they illuminate across mathematics. We should define them, explore their properties, and revisit them whenever they appear. Arrays are a powerful way to visualise prime and composite numbers. Take eight as an example. It can be arranged as two rows of four, four rows of two, one row of eight, or eight rows of one. These are multiple arrays, each showing a factor pair. By contrast, three can only be shown as three rows of one or one row of three. This reinforces the definition: exactly two factors. The sieve of Eratosthenes complements this structural view with a procedural one. Using a hundred square, children first remove one (not prime), then circle two and eliminate its multiples. The same is done with three, then five, then seven. What remains are the primes. These are the numbers with exactly two distinct factors. This process is not simply a visual activity. It reinforces knowledge of multiples, factors, and the cumulative nature of elimination. When teachers accompany it with questions such as "Why isn’t one prime?", "Are all odd numbers prime?", or "What’s the largest prime under 100?", pupils are pushed to reason, reflect, and generalise. Arrays and sieves do different things. One shows structure. The other shows process. Both are essential in conjunction. Other representations, including factor trees, Venn diagrams, and factor sorting, offer even more ways in. Each reinforces the same truth from a different angle. The wider story Ultimately, the question is not just what prime numbers are, or how we teach them, but why they matter. Where do they sit in the broader mathematical narrative? What do they unlock in pupils' understanding? Mathematics is a story. It began at the dawn of the universe and continues with every child. When we reduce mathematics to disjointed procedures, we exclude students from that story. When we connect ideas meaningfully, we offer them a chance to enter it. Prime numbers are not simply facts to memorise. They are threads in the great tapestry of mathematics. If we help children follow those threads, they will not only understand primes. They will begin to see the deeper logic and wonder of the subject, and they will, as is their right, have full access to the story of mathematics. TP Kieran Mackle is the creator of The Story of Mathematics, the author of Tackling Misconceptions in Primary Mathematics and Thinking Deeply About Primary Mathematics , and the host of the Thinking Deeply About Primary Education podcast. “Prime numbers are threads in the great tapestry of maths” Prime numbers at a glance What is a prime number? A number with exactly two distinct factors. This excludes one, and makes two the only even prime. What do they do? They underpin key mathematical concepts such as factorisation, fractions, and multiplicative reasoning. Beyond school, they lie at the heart of modern cryptography and data security. More than a list Prime numbers offer a perfect example of how apparently simple facts connect to deep mathematical structures. They form part of a narrative stretching from ancient civilisations to the digital world. alta-education.com thinkingdeeply.info 62 | www.teachwire.net