It always feels good when you work through a maths problem and you get to the right answer, doesn’t it? How about when you work through a problem and you get to an answer that’s wrong? It doesn’t feel so good then. If it’s a homework sheet that you need to hand in and you can’t see where you went wrong then you might be tempted to copy someone else’s work; at least you’ll get some nice ticks all over your work telling you what a great job you’ve done instead of those horrible crosses. If it’s a past exam paper that you’re working through you might just look at the mark scheme to see how to solve a problem instead of bothering to spend time figuring it out and before you know it you feel like you can solve anything – the solution is always obvious when you know how it’s done.

But is the answer what you really need right now? I agree it feels nice to get a right answer but a wrong answer is telling you something; like it or not you still have some learning to do.

When I set homeworks for my students, or if they’re working through past papers (shockingly, some schools seem to try and teach the whole GCSE or A Level maths courses through past papers) it’s all too tempting for them to just look in the back of the textbook at the answers or just siphon off a model answer from the mark scheme and expect that I’ll be happy with that. The thing is – I know the ability levels of each of my students and I can recognise what is their work and what isn’t fairly well. If I see something that I know isn’t their work or their answer I will challenge them about it; I will ask them how they arrived at their answer or how they came up with their solution. The silence can be deafening.

So I have before me a page of beautifully correct answers and solutions – yet my student has learned nothing. Is that a good result for the student? As far as I’m concerned, every one of these answers is incorrect because if there is no evidence to support the answer, if the student can’t even begin to explain the solution then how can it be believed to be correct? Obviously if I let this continue then my student will get the feeling that all they need to do is write the correct answer and their job is done. It doesn’t seem to dawn on some, though, that lifting the answers from the back of the book or from the mark scheme is not something that they will be able to do in an exam – but more importantly, what if there is currently no answer to the question that they’ve been asked and it is up to them to provide an answer. After all, at some point in their lives they will probably have a job where they have sole responsibility for certain decisions and the solutions to certain problems – the answers can only come from them; they won’t be able to consult the back of the book and nor will they be able to just give up because they didn’t know where to start otherwise their competency will be seriously questioned.

There is a reason that maths problems are called problems and that’s because you have to look for solutions. It takes practice to find solutions; it takes practice to understand concepts and piece bits of knowledge together and it may take some time before you get the hang of things and consistently produce correct answers. But not going through this process, and indeed denying yourself the opportunity to go through this learning process, means that your problem solving skills will remain shrivelled and weak. You won’t be able to be decisive about what to do; you will not have any degree of confidence in you solutions or results and you will be entirely dependent on external sources to validate your solutions and answers, you might not even be able to make your own mind up about where to make a start.

I put much more emphasis on the solution to a problem than I do on the answer and I make this clear to my students so most of them (if not all of them) learn quite quickly that whatever antics they might get up to with their school homeworks won’t wash with me. I only tutor for ages 14 and up so I feel that I can treat my students as young adults; so I make it as clear as I possibly can to them that if they continue to copy their answers then they only damage their own chances and that they just give the appearance of understanding. They now have to take some responsibility for their own learning; I would rather see several pages of unsuccessful attempts at a problem then either a page of correct answers without solutions (big red flag that one) or simply no attempt at all because they “couldn’t think where to start”. These unsuccessful attempts are the starting point for understanding and there is no way to bypass this stage. Of course you would prefer to get the right answer straight away, who wouldn’t? And sometimes that will happen. But you have to be prepared to think through problems and make a lot of false starts – it is the solution that matters at this stage and not necessarily the final answer.

 

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