Anyone tasked with moving Calculus I from a textbook to students sooner or later asks her/himself that question and I am curious as to what people are thinking of when they do. But to make it more explicit, I would like to make a few assumptions.
- Let us assume that your students are the kind to be found in Two-Year Colleges----which, these days, include about 50% of the student population in the US, so that:
- Their "conceptual maturity" is practically inexistent. For example, they decide what calculation to embark on solely on the basis of template examples. While they can compute to an extent, they have no idea on how to verify their result and always want to be told whether the result of their computation is right or wrong.
- While they can usually read the template examples, they cannot read a text to extract conceptual information.
- They are totally unused to the idea that the very text of a problem will often give them an idea on how to attack it.
- They are totally unaware that, in mathematics---as in law, one has to make a case for whatever one asserts, be willing to present one's case to "peers", and be ready to defend one's case.
- Your students are not necessarily declared pre-engineering students, but can also be students who, based on what their experience in calculus 1 will have been, will decide on what to major in or, perhaps, students who have been persuaded to give it a try by a former instructor. In any case, a major issue is that many of the students will likely not continue into Calculus 2.
- You are not bound to use any particular textbook and in fact let us assume that you have facilities to develop the materials you need to help you reach your goal. Also, let us assume that you have the blessing of your department.
At this point, the question of what you would want your students to learn might be, for example and certainly not to limit things, whether you primarily want:
- To remedy some of the shortcomings listed above using as means some or all of the standard topics,
- To improve / consolidate / extend whatever computational skills they already have using as means some or all of the standard topics,
- To let the students acquire a "coherent view" of the calculus even at the cost of not covering all the "standard" topics,
- To let the students acquire the mathematical techniques they will be expected to use in following courses,