Mathematical Problem Factories: Almost Endless Problem Generation (Synthesis Lectures on Mathematics & Statistics)

A problem factory consists of a traditional mathematical analysis of a type of problem that describes many, ideally all, ways that the problems of that type can be cast in a fashion that allows teachers or parents to generate problems for enrichment exercises, tests, and classwork. Some problem factories are easier than others for a teacher or parent to apply, so we also include banks of example problems for users. This text goes through the definition of a problem factory in detail and works through many examples of problem factories. It gives banks of questions generated using each of the examples of problem factories, both the easy ones and the hard ones. This text looks at sequence extension problems (what number comes next?), basic analytic geometry, problems on whole numbers, diagrammatic representations of systems of equations, domino tiling puzzles, and puzzles based on combinatorial graphs. The final chapter previews other possible problem factories.

Andrew McEachern is an Assistant Teaching Professor in the Mathematics and Statistics Department at York University in Ontario, Canada. His research has been in the fields of DNA analysis, evolutionary computation, game theory, and education in mathematics. He has over a decade of experience in outreach mathematics at all levels. He is currently working on the problem of getting students of all ages to better understand fractions, as well as engaging with mathematics without dread. Part of his mission is to demonstrate that popular mathematics is all around us, in the form of puzzles and games, and it is his opinion that mathematics should mostly be fun, and at least a little useful. He is a huge fan of tabletop roleplaying games, which are a combination of mathematics and storytelling, his two favorite things.Daniel Ashlock was awarded a Ph.D. in Mathematics from the California Institute of Technology. He has taught more than 50 different classes from abstract algebra to bioinformatics but enjoys set theory because it is the first time most students meet abstract mathematics. Dr. Ashlock is a Professor of Mathematics at the University of Guelph where he uses mathematics to help biologists with their research as well as continuing his own work in how to represent information to make it easier to understand and easier to work with on a computer. Dr. Ashlock is a Senior Member of the IEEE and has chaired technical committees in both bioinformatics and games, demonstrating the broad usefulness of an education in mathematics.