So often, I hear children excitedly telling me that they know how to perform a particular calculation, but then go on to recite some line of poetry, a riddle or a seemingly random word. Some of the more common ones I hear include:
- Keep, Flip and Change for dividing fractions
- Ignoring the decimals when multiplying or divining – then putting them back in afterwards
- A positive and a negative make a negative
- Write 1-9 down, then up right next to it to calculate the 9 times table
The frustrating thing from my perspective is that all the above hacks WILL GET YOU THE CORRECT ANSWER!
Now, surely if the goal is to help children succeed, these ‘time saving’ and ‘fun’ methods should be celebrated. I must confess that I have indeed used these methods myself on numerous occasions, so why exactly do I have such an issue with these, and other, ‘tricks’?
Well, at the risk of getting particularly ‘meta’, I think my whole argument is summed up on a number of levels in this 26 second clip…
As Mr Holomisa tried to explain, if you give someone a trick, answer, or quick fix solution, you will help them out for that specific situation only. However, if you guide them into an understanding of key concepts in problem solving, you can equip them to use those concepts with a level of understanding that allows them to be applied to different situations, with a full and technical understanding how ‘how’ and ‘why’, rather than simply presenting the finished answer.
However, in a cruel twist of fate, what he also managed to do was prove this to be correct, as, it later transpired that he was reading the autocue machine and it malfunctioned. The TV company provided him with a fish, but omitted giving him that understanding of the proverb and the significance of its meaning to the issues of the day.
As mathematicians, it is essential that we can understand the how’s and why’s, not simply be able to present the answer. At GCSE level, there are three clear ways to tackle a ‘multi mark’ question (that doesn’t explicitly ask for working out)… Put the correct answer and get all the marks, put the wrong answer (possibly due to a basic calculation error) and get no marks, or provide a clear and complete method for finding the answer and get most marks, even if your final answer is wrong. The gamble you take by using a trick is that you have no mathematical backing to support your answer, so if it is wrong, you score zero.
A short reflection from my own maths education… when doing my A-Levels for the first time, I found myself really struggling with all subjects. In maths, we had done some basic calculus work and had moved on to logarithms. Desperate as I was to succeed, I tried to use what I knew to get the right answers. Unfortunately, that resulted in my trying to evaluate a logarithm by differentiation, which, if you know these topics, you will understand are not connected pieces of maths at all, other than they both can involve indices in some way.
You see, I had no mathematical understanding of these topics in order to use them wisely. Sure, I knew that to differentiate an equation I could multiply the index by the coefficient, subsequently reducing the index value by 1, but I didn’t know or understand what this related to, other than letters and numbers on the page. I appreciate this is a reference to a movie most of you haven’t seen, but, I felt like the NASA scientists trying to get a square plug to fit into a round hole (or maybe it was the other way round?)… they used what they’d got, whether it was useful or not.
Had I a better concept that differentiation provided the gradient of a curve at a specified point, or that integration allowed me to find the area under the curve, then I would have never used these methods to evaluate a logarithm.
This is my biggest concern with these so called ‘tricks’ in maths. Go with me… to divide fractions, we use KFC (keep, flip, change). Imagine this is the only way you’ve been taught to divide fractions, with no deeper understanding offered, other than ‘it works’. Later on, I am presented with a different question that happens to involve division. Excitedly, I recall the greasy flavour of the colonel’s secret recipe, and proudly go forward into solve, for example, 4 ÷ 2. Right then… here we go… Keep the first one… so, 4. The divide turns into a multiply, so that’s gives me 4 x 2 = 8. Now I know what you’re thinking… ‘you’ve forgotten to flip’… don’t worry, that is the last bit. My answer needs to be flipped, so my final answer will be 1/8. Perfect – full marks to me – move on! Wait, what do you mean, that’s not right? I’ve seen that method used with fractions, so I did the same thing here..?
Let’s look at this same question from a different perspective… to divide fractions, it’s important to recognise that, for example, 6 ÷ 2 is the same as 6 x ½. Looking at this, we can see that if we divide 6 by 2, we get the answer of 3. In the second calculation, we also find that 6 x ½ (which means; what is the value of 6 lots of ½) also gives the answer 3. The whole calculation relates to what we’ve known since an early age, that if a x b = c, then c ÷ a = b.
How do 2 and ½ interact with each other? Well, first of all, if you multiply them together, you get the answer 1. Secondly, one is the reciprocal of the other (meaning, the value you get when you divide 1 by the number).
Of course, in practise, you can just keep the first, flip the second, and change the symbol, but, that’s not going to help you with a question that requires you to perform an abstract calculation with. Take this exam style question:
James is thinking of a number.
He writes down the reciprocal of the number.
It is the same as his starting number.
What is the number James is thinking of?
KFC isn’t going to help you with that, not matter how hard you try, but, with a little understanding of what we actually do with KFC, and the effect of the reciprocal, then suddenly this question becomes quite a bit more accessible.
So… by all means use the tricks you’ve been taught, but please don’t end up like Mr Holomisa – up the creek without a fish, erm, rod, erm, paddle. Take the time to understand the maths behind the trick, it will absolutely take your maths further. Just remember, though, that while the trick may at first appear to work, it’s not always obvious what the damage you’ve done is until it’s too late. I’ll leave you with this rather visual demonstration…