My calculus textbook is organised as follows.
Each topic is introduced by means of a “lecture worksheet.” The lecture text introduces the topic and explains the key points. But it also has many problems embedded in it, from short conceptual queries to substantial applications. The aim is to strike a balance between teaching by instruction and learning by self-driven inquiry. There is enough explanation to give a solid structure and a platform on which to build, yet enough integrated problems to make the student an active participant in the development.
Building a mathematical narrative by means of embedded problems is in my opinion the best form of mathematical exposition. It is the clearest way of explaining any mathematical argument, even to readers who will not actually solve the exercises. Phrasing an argument as a multi-step exercise requires the author to set out all innovative key ideas that the reader could not be expected to think of, while outsourcing to the reader precisely those parts of the argument that can easily be glossed on the spot. Spelling out all the details adds nothing but confusion: you can no longer see the forrest for the trees. When reading a conventional exposition, with all details included, most of the reader’s effort must be devoted to extracting the key ideas from all the routine connecting of the dots that goes on in between. With the exercise-based form of mathematical exposition the key ideas and the line of thought is laid bare at once.
At the end of each chapter I include a “reference summary” containing key results and step-by-step guides for typical problem types. In my opinion much material is better presented in such bullet-point-style reference tabulation than in running text. Placing such matters in the reference summary allows the lecture text to be clean and streamlined and focussed on key ideas and stimulating problems rather than being cluttered with incidental remarks about terminology, notation, or technical details.
My step-by-step problem guides give to the students what, in a conventional course, they usually spend most their time trying to figure out. This is “what they really need to know,” so to speak, for the narrow purposes of passing tests. This kind of information is usually conveyed mostly by means of examples in standard books, but there are in my view a number of problems with such an approach. First of all, examples are cluttered with particulars, from which the students need to extract the general process. My step-by-step guides cut straight to the general process. Another downside of examples is that they are very inefficient at conveying any process that splits into various cases. My step-by-step guides simply give the general flowchart. Additionally, some portion, maybe a third or so, of the “examples” in any given textbook are not in fact the kind of thing that would ever show up on an exam. With my form of presentation, such “enrichment” material is clearly separated from staple prototype-problems.
But the primary advantage, as I see it, with my step-by-step problem guides is that it “takes out of the equation” students’ obsession with the kind of rote learning that they have been led to believe is the essence of mathematics. The reference summary section contains everything you need for those kinds of standard problems. Thereby it liberates the main text and the class meetings from wasting time on the inefficient and irrational transmission of procedural and reference matters by lecture, leaving instead more time for genuine mathematical thought.
A calculus class should involve assigned sets of practice problems of the types that my “Reference summary” sections are intended to help you solve. In my opinion such assignments are best handled through a system such as WeBWorK, which has numerous advantages over the inefficient and outdated model of cluttering textbooks with tons of problems of this type.
The problems at the end of my sections are aimed at intellectual enlightenment, not routine practice. It would take a very ambitious student to tackle all of these problems. A more realistic use of these problems in a class setting is to use a few of them for written homework assignments. This complements the “answer-only” focus of routine exercises by demanding carefully crafted mathematical prose, including many higher-order, explanation-type tasks.