The traditional approach to teaching rigorous, proof-based mathematics is to provide students with models of excellent mathematical exposition and let students learn by emulation. Typically students will first absorb by reading the textbook and listening to lectures, and then they work through similar examples and exercises until they have mastered the techniques and thought processes.

This model has been challenged in recent years. An increasingly favored approach emphasizes learning through independent discovery, with variations like inquiry-based learning and the inverted classroom. In these scenarios students learn by doing. Starting with minimal pre-guidance, students are given problems to think about individually and in collaboration with others. Instructor involvement is thus pushed toward the end of each lesson. Though this idea turns the traditional methods of teaching on its head, the number of proponents is growing quickly, and studies suggest that the trend will continue.

How do textbooks fit in with this new approach to teaching? Since class time focuses on group projects and exercises, lectures must be more flexible and adaptable than before. From this point of view, textbooks may seem inappropriately rigid and only useful for their exercises. Instead, one could post or hand out lists of notes that can be easily changed on the fly. There are certain crucial downsides to this approach, especially when accommodating large numbers of students, such as losses in conformity across sections and in continuity within curricula. Also, in the long run, one of the great values of textbooks is that they help perpetuate a universal language and culture within mathematics globally. In the old approach standard course textbooks balanced the effects of stylistic differences among instructors.

So what would the ideal textbook for a modern, active-learning oriented classroom look like? A great book has a sense of narrative — a compelling story that makes you keep turning the page, and a sense of charm and wit — you can follow “the voice” with confidence knowing that the journey and destination will be full of delights. These can indeed be incorporated into a problems-based book, as the highlighted example below shows.

Please send more examples in comments!

**Featured Book of the Day**

A Problems Based Course in Advanced Calculus by John M. Erdman

This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. The tone of the book reflects the author’s years of experience balancing the need to give students helpful guidance while maintaining the principle that less teaching leads to more learning. All this culminates in prose that is conversational and inviting, yet efficient and economical, allowing plenty of room for the reader to discover for themselves.