Conclusion
You’ve reached the end of the book. That’s an achievement in itself.
Now is a good time to think back over what you’ve learnt and reflect on what it means for you.
Throughout the book we’ve asked you to have a go at two fundamentally different types of problems.
The first type we’ve called ‘routine problems.’ These are the problems you’d usually find in a textbook: they have an established methodology for solving and a very clear answer at the end of it. For many of us, that’s what we think of when we hear the word ‘maths.’ It’s important not to dismiss this kind of learning lightly, because having a grasp of some of these methods and techniques gives us tools for modelling and understanding the world. As you will no doubt have noticed, knowing how to apply these methods can make solving problems a whole lot easier. Algebra is probably the most obvious example of this, because it is such a powerful tool to help us figure out the relationship between things.
The next type we’ve called ‘non-routine problems.’ These are the problems for which there’s no clear cut method, and it’s not always straightforward to know if we’ve arrived at the precise answer. Sometimes there is no precise answer. For these problems, we need to figure out the method for ourselves. This can be a powerful way to learn mathematics, not only because it links what we’re learning to real world applications, but because it forces us to really understand and be able to justify the method we’ve chosen.
This, in turn, promotes mathematical reasoning. This centrally important, critical thinking skill teaches us how to make reasoned conjectures, test them and then come to reasoned conclusions. The process of doing so will mean that most of us need to challenge our assumptions, keep a mind that is open and strategic at the same time, justify our own thought process and think about what it means to solve problems in the first place.
If there’s one thing that both routine and non-routine mathematical problems can teach us, it’s the skill of patiently working through problems creatively, systematically and methodically.
Your final problem – create a new one!
Pose a problem
All throughout the book, you’ve been asked to develop an understanding of mathematics using problems. We hope that, by creating your own methods to solve non-routine problems, you understand the principles underpinning your solutions at a different level to problems you’ve learnt to solve from a routine textbook. We also hope it’s helped you to practice that creative process of problem solving and mathematical reasoning.
Now it’s your turn to pose a problem. Think for a minute about what makes something a “problem” and not just a routine skills question is not always straight forward and sometimes it depends on the level of the student. For example, asking a late high-school student to tell you the volume of a sphere is more-or-less a routine problem, but for a student in year 7 or 8 it could be quite perplexing. As a rough guide, a problem should
- not have a straight-forward solution;
- require the solver to define terms or set the scope of the solution (i.e., it can be ill-defined);
- have a solution that could be verified or approached in a number of ways;
- be generalisable to other contexts or situations;
- require the solver to use some form of abstract thinking.
After attempting a number or problems yourself, you might have some other opinions that apply – e.g. that the problem should be “interesting” and motivate the solver to want to solve it.
