“Pure mathematics is, in its way, the poetry of logical ideas.” – Einstein
For many years, I have been fortunate to spend my working life doing what I love. From a very early age, number, and a few years later, mathematics have been a constant joy in my life. It can be hard to explain to others at times that, in our house, we delight in grappling with a mathematical conundrum. Because we are both competitive and creative, we enjoy seeing who can get to a solution first, whether there is more than one solution and whose solution is the most elegant. In truth, I delight in being engaged in mathematical conversation with those I love.
For many, I know that this is at odds with their own experience of mathematics. Too often maths is seen as a right or wrong, rather stilted subject that involves too much rote learning and never manages to excite. Or worse still, that impossible subject that one has to keep studying even though you know you’ll never pass.
Whilst, as with just about every subject I can think of, some rote learning is essential, a sense of creativity can be encouraged from a very early age and there is no end of excitement to be found in even the most basic mathematics.
Quite unexpectedly, due to the rather surreal times we recently found ourselves in, I have been home schooling my 7 year old son. It is perhaps not surprising that, once his standard schoolwork was complete, I dipped into a bit more maths! Initially, we focused on what I would consider core numeracy skills – the number line, the decimal system, times tables and division facts, definition of prime numbers, square numbers and even cube numbers – of course, as a child of a maths teacher, these are ideas that he has met previously! With a little determination, these can be understood and learnt but that is just the beginning. Once done, there is scope to really understand and enjoy numbers and the patterns that can be found within.
Having mastered the tables and primes (definition and identification of primes up to say 50), I posed the question why is it that…
1 × 24 = 24
2 × 12 =24
3 × 8 = 24
4 × 6 = 24
One answer is that they just all happen to make 24. However, if you decide to dig a little deeper you can enter the world of what I like to call “number fingerprints” better known as the product of prime factors. Every whole number is completely unique – just like our own fingerprints, their product of prime factors is unique to them. No two can be the same. Furthermore, every whole number can be formed by multiplying different combinations of prime numbers together. I would argue that is a pretty incredibly, even cool, result and not one indicative of a boring, limited subject but one which encourages creativity, flair and discovery from a very early age. Once children learn to play with numbers, progress accelerates.
Back to the question in hand. The times table learning, and prime number identification makes it possible to re-write each of the sums as follows:
It is then easy to see that each product boils down to the same underlying fingerprint. It is simply because of the commutative nature of multiplication that these prime factors can be multiplied in any order to form the same result. Problem solved.
But why stop there…there are so many natural follow up questions…
What is the number fingerprint of say 30?
How many ways are there of expressing 30 as a product of two or more numbers?
How do you know that you’ve got them all?
This is a story for another day – engendering a systematic approach is equally crucial to develop a strong mathematical mind. For the meantime, I hope the images below can inspire a little mathematical creativity – we had great fun with a little finger painting.