OP: What seems to be missing as taught is an explicit indication of the plan.
Even though this is far afield from your concentration on geometry, it
illustrates your point. Michael Sipser's text,
Introduction to the Theory of Computation,
includes many "proof idea" sections prior to launching into each formal proof.
I've taught from this text and ...
I think maybe the correct answer to this question is something simple like
Sure, do something else if you like.
I also recommend reading this question because the good answer there may give you some insights into how imposing a structure on proofs can help and how it can hurt.
It sounds to me like you need exposure to mathematical topics beyond what is covered
in an undergraduate math major. Here's one recommendation:
Fuks, Dmitrij Borisovič, and Serge Tabachnikov. Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Soc., 2007.
There are similar collections, but this one is both broad and ...
(I had tried to add a clarifying comment to @AlexanderWoo's good answer, but it was mysteriously deleted.)
My point was, and is, that it is not constructive to think of "proofs" as a thing separate from normal human discussion of things. Rather, proofs are really explanations, or discussions that persuade. There is no magic formula, and "proof&...
I have that book. It's actually a very easy gentle book. I think easier than the medium difficulty calculus books or ODE books (not Spivak) that I have. I don't think you'd have any issues self studying it. Note, that it only has answers for 50% of the exercises, but the amount of exercises is vast (and like I said, the content is easy). So I think you ...
This is a text for an "introduction to proofs" course.
It might not be well-known outside mathematical circles, because mathematics educators don't like to advertise this fact, but, outside of fairly selective universities, most students taking such courses fail to learn the material despite earning a passing grade. The majority of students never ...
The text says it is designed for a 14-week semester. With an instructor, we could guess that means 3 hours a week of class and 6 hours a week of additional work ( = 126 hours?). I cannot say whether someone without an instructor could do it in that time, or could do it at all. That will vary greatly depending on the student.
When doing a proof by contradiction of "if A then B", you get to assume A and you also get to assume NOT B. Proving it directly, you only get to assume A. So proof by contradiction gives you more to work with from the start. I think that's why students like it.
Several professors I've had over the years, when asked why they were showing us a proof well beyond our fabrication, justified it as follows.
Mathematics essentially contains a finite number of "tricks". As you develop your skill in mathematics, you accrue more of these tricks that you have seen employed in various contexts. It is rare that a proof ...
I assign this reading early on in the kind of undergraduate classes where I expect students to understand proofs:
This handout is based on the following research:
Once students understand the ...
Your quest for measurable learning outcomes suggests a framework rooted in Bloom’s Taxonomy and the higher-education assessment industry that has been built around it. As I think about your question, I think about inquiry-based learning [IBL], especially when implemented in upper-division proof-based mathematics classes. In this Notice of the American ...