Writing a book is a big project and Occupy Math is very pleased to announce the first book arising from, well, Occupy Math. This book started at a conference for math educators that Occupy Math was invited to by his former student Andrew McEachern, now a professor at York University. Something that happens when you get math educators together is they tell you their favorite motivating problems, the problems that get students excited and interested. Andrew and I noticed that many of these problems fit into large families of problems. We coined the term Problem Factory for one of these store-houses of problems. Later in the post we have a list of the problem factories that have appeared in Occupy Math posts, to give you the flavor of things and tempt you to purchase the book. The book is Mathematical Problem Factories Almost Endless Problem Generation and it gives a number of problem factories, each illustrated by collections of problems drawn from the factory.
Some readers may find the subtitle “Almost Endless Problem Generation” to sound like what has been happening to all of us during the pandemic. Occupy Math’s editor laughed for several minutes after reading it. Do not worry, in a math book, “problems” mean “exercises” and having lots of exercises is a good thing. At least for math teachers. Never mind.
Andrew and I ended up with eight chapters covering six different types of problems factories. Here are the chapter topics.
- What are problem factories? A short chapter where we explain the idea.
- Sequence extension problems The classic “what comes next?” sort of problem.
- Basic Analytic Geometry Problems Fairly simple geometry problems including finding squares in the Cartesian plane with a given area (e.g. 5) and finding the area of shapes drawn in the Cartesian plane.
- Problems using Whole Numbers This includes pizza counting and a problem that appeared in one of the Die Hard movies.
- Diagrammatic representations of linear systems getting ready to solve simultaneous systems with pictures of flowers and chocolates.
- Polyomino tiling puzzles a combinatorial favorite. An example appears on the cover. Given one or more shapes built out of squares, can you use the shape to fill some area?
- Problems based on graph theory This includes the classic problem “can you draw this without lifting your pencil?” as well as novel problems thought up by Andrew and Occupy Math.
- The road ahead: other problem factories We limited this book to problems that can be done with classical mathematical techniques. In the future we will put out a book on problem factories that need AI to find the problems.
This is Occupy Math’s tenth book and, out of the lot of them, it was the most fun to write. The link to the publisher in the first paragraph will let you purchase a copy if you want to. If you need to learn more about problem factories, then here is a list of Occupy Math posts that give problem factories. Most of the posts that made it into the book are enlarged in the book and the book has lots of example puzzle problems.
- The post can you draw this? is part of Chapter 7, the problem of drawing a picture without lifting your pencil.
- The post on bouquet puzzles is part of Chapter 5. This is the activity that helps students prepare for solving simultaneous equations.
- A very basic programming activity is not from the book, but it’s good for students who will learn to program in the near future.
- Can you add it up? is a problem factory from chapter four about finding ways to express a number as a sum of consecutive numbers.
- Graph puzzles to sharpen your skills is one of the graph-based problems from Chapter 7.
- The geometry of frustration and discovery was the starter set for Chapter 3.
- Number triple puzzles are a problem factory that did not make it into the book.
- Making “guess the next number” puzzles is a starter set for Chapter 2.
- A geometry problem practice factory is also from Chapter 3 and explains some of the methods for constructing problems.
- Can this number of neighbors happen? is a problem factory from Chapter 7, more fun with graphs.
- Number sentence puzzles are a problem factory from Chapter 4.
We are planning another book on problem factories. Once you have the idea of a problem factory, you start seeing them everywhere. The first book is about problem factories that are things you can do with mathematics (and a little help from simple computer algorithms). The planned second book has problems where we need artificial intelligence to help us design the problems. These problems can still be solved by students — and, in fact, that is sort of why we need the AI: to find the problems that a person can solve.
If you were wondering what Occupy Math does with the money from these books, the profits from the books fund graduate students so that they have the summer to work on their own research.
I hope to see you here again,
So remember to get your Covid vaccination!
Daniel Ashlock,
University of Guelph
Department of Mathematics and Statistics
Occupy Math 