Its true, you have to speculate to accumulate! In this blog entry I aim to speculate on how a narrow view of right and wrong that is perpetuated by society is to blame for some misunderstanding of mathematics.

Approaches to trigonometry often brings this to the front of my mind again and I will often address it with questions like the one in the diagram again. I find that students are intent on finding 'the right approach' to solving problems with trigonometry and as such are often frustrated by their inability to do so quickly or at all. When do I use cos, sin or tan? When is it the sine rule or the cosine rule? These and other questions get in the way of simple speculation. The question above aims to encourage students to look at given situation and gather information from it that they could work with if they needed to. Each correctly deduced piece of information becomes a potential problem solver. Once gathered, students can try to discern which is most likely to be of use. For example, if required to find the value of 'c' in the diagram, then one would want an equation linking 'c' with only known quantities. 

How is it that society has developed this widely held view of mathematics as being exclusively about solutions to problems and not about the speculation involved with problem solving? How do we feel about problem solving as teachers? What is it that gives the most pleasure, searching or finding? Is it not true that many people spend days, weeks, years looking for solutions to problems? Is their succes to be judged only by their solutions? I think what I am trying to say is that making progress with a problem can be at least as rewarding as solving it. Like working out how to get one face of a Rubik's cube complete and then two. I have not got any further with that but do enjoy trying. Even if a problem is solved someone might look for an alternative, more elegant solution and this suggests there is much more to consider than just a solution.

Students who are less confident with mathematics are often comforted by slow, careful teaching methods that allow them learn and practise algorithms for numerous 'types' of questions and then reproduce them. It is often said that students experiencing success with this grow in confidence as a result. Of course it is logical that success is likely to breed confidence and so I would suggest that, as educators, we should help to modify the definition of success in mathematics to include the ability to speculate. We should help students believe that following a lead down a dead end is not wrong and not fruitless. Realising and demonstrating that it was a dead end is deeply valuable and if more students had this ability to speculate and reflect then they would be less likely to choose 'the wrong' approach to solving a problem.

Of course we are still bound by assessment tools that place solutions above process and in most cases we could slowly and carefully teach algorithms for all the different problems so that students could achieve well in these exams and get the keys they need for whatever it is they want to do next, but if we only did this I would suggest we are failing them and depriving of them of an opportunity to develop a very important life skill to compliment or even bring to life those often referred to as 'the basics'.

Anyway, having put this into words I am certain that I will think and rethink it a number of different ways and wish I'd said it differently but you have to speculate to accumulate, would you not agree?