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What constitutes good quality mathematical reasoning?
If there’s one big question we’ll be looking at throughout the book, it’s this.
During the first section, we’ll begin to answer that huge question by starting with a slightly more manageable one: what does a good approach to problem solving look like?
We begin with the topic of mathematical reasoning, where we will give an overview of the three-stage framework of problem solving: entry, attack and review. In the next section, we’ll refresh some of the fundamentals of maths and algebra. You may have already come across algebra during your schooling, so here you can put your focus on deep understanding – so that even if you know how to do something, you could comfortably explain it to someone else and understand how to approach it from different angles.
The sub-topics of problem solving and algebra will lay the foundation for how we approach future topics. In general, problem solving has the potential to draw on all of your mathematical learning. However, it should also bring your understanding together so that mathematics progresses from being a series of unrelated tasks to a much more cohesive framework for understanding the world.
Problem solving can make you a bit of a hero
As the video discusses, there appears to be a deep structure to many of the stories we tell ourselves about what makes someone a hero. The main character (who in this analogy would be you) leaves their comfortable home and sets off on an odyssey in order to solve a problem that troubles a group of people. Along the way, they consult with their forebears and overcome a series of challenges on the path to their ultimate victory. They then return home as a hero, having solved the problem at its source, once and for all. For our purposes, the ultimate victory might look something like a method for solving not only this particular problem, but any similar problem.
Let’s keep that in mind as we try our first problem.
The handshake problem
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You and 19 others are at a networking event. As the event closes, everyone shakes hands with everyone else. How many handshakes take place altogether? |
Before you go any further, try to solve the handshake problem. You can use any tools at your disposal to do so.
If you’re able to find an answer, can you generalise your solution? In this case, generalising might mean coming up with a way of quickly solving the problem, no matter how many people there are at the networking event.
What did you do?
Did you:
- Skip the process of trying and use google to find an answer?
- Work out quickly in your head what would roughly be required and then assumed we’ll show you the answer later in the book?
- Remember that you’ve solved this problem before and so you already knew how to solve it?
- Jump straight into scribbling diagrams and writing sums to try and find the answer?
- Re-read the question to work out what it was actually asking and why it might be considered a problem in the first place?
For the moment, any of these is fine – a problem has to grab you, and depending on your approach to this topic, you’ll attempt some of the problems and skip over the details of others. It’s just important to bear in mind that what we’re really interested in when it comes to problem solving is the journey and the opportunity to improve.
What makes a good problem solver?
Stop and think
Do you agree with Simon’s description of what a good problem solver looks like?
Do you think that you are a good problem solver? What could you work on to improve your problem solving skills?
In the context of this book, to generalise is to come up with a method for solving any problem that is similar to the one you're working on.
