Squeaktime.com is a website about solving maths problems, for both teachers and students of maths. It's about doing problems, but also about thinking about doing them.
First and foremost, I am a mathematician who enjoys finding and solving interesting maths problems, so I just wanted to collect particularly interesting problems that I encounter and present them here, and perhaps give suggestions where other interesting problems might be found.
Secondly, as a maths teacher I need to think about the process of solving problems. My job is to try to make my students into successful, inquisitive and confident mathematicians, and a large part of this is teaching them how to solve problems. After 10 years of doing this, I still don't think I'm very good at it - this website is my attempt to explore this process in more detail.
Should we teach students how to solve problems?
Before I go further, please do not think I am one of those teachers who thinks that maths at school should be only taught through solving problems. There are exams to pass and methods and routines that must be learnt. And let's put exams to one side for a moment: even if you only wanted to study maths purely for enjoyment, you must have a grasp of certain mathematical methods before you can access a given problem, or access a given area of mathematics. You can not, and will not, become good at solving maths problems just by learning strategies - you must have a solid foundation in the basic techniques.
However, solely learning techniques will not be enough to make a good mathematician, and will probably deter us from pursuing the subject further. We also need to learn strategies for solving problems - we will not become successful mathematicians without them. Most mathematicians would agree that the real reason, and the real joy, in studying maths comes from that warm fuzzy glow you get when you solve a complex problem, when you discover or understand something new. I hope that the problems I will present here will give those who attempt them that warm glow that I felt when I solved them.
Fields Medal winner Maryam Mirzakhani echoes my feeling about the maths taught at school when she says:
"I believe that many students don't give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers."
What makes someone good at problem solving?
Recently, I was going through some STEP papers with a student, when she asked me: "What makes someone good at problem solving?" I suggested it might be confidence. And you need to enjoy maths, enjoy solving problems, see them as a challenge rather than a test. You need to be open-minded and try things out, don't panic. But you also need to have lots of mathematical techniques at your disposal, so you need to do lots of practice. And... well, lots of things really...
I'm not satisfied with this answer. What exactly is it that makes someone good at solving problems? Is it something that can be taught/learnt? I hope so, but how exactly? What do you do when you get stuck? How do you get unstuck? What strategies and techniques can we teach that will make our students confident, inquisitive and successful mathematicians? What does the research say about how we solve problems? What different types of problem are there, and what skills are required for these? What makes a problem worth solving? What makes a solution elegant and satisfying? What other questions can we ask about a given problem, how can we take it further?
This website is dedicated to trying to answer some or all of these questions; lots of problems to solve... Let's start by trying this one from the 2014 AMC10 test (designed for grade 10 / year 11 students) that I enjoyed doing today:
- How did I start?
- How did I go about understanding what I was being asked?
- Did I get stuck/unstuck? When/how?
- What general strategies did I use?
- What specific maths techniques did I use?
- Did I experience a warm fuzzy glow at any point?
- How could this problem be adapted/extended for use in the classroom?
What other questions and reflections do you have?