New style of Mathematics Textbook.

"It may sometimes happen that a truth, an insight, which you have slowly and laboriously puzzled out by thinking for yourself could easily have been found already written in a book; but it is hundred times more valuable if you have arrived at it by thinking for yourself. For only then will it enter your thought -system as an integral part and living member, be perfectly and firmly consistent with it and in accord with all its other consequences and conclusions, bear the hue, colour and stamp of your whole manner of thinking, and have arrived just the moment it was needed; thus it will stay firmly and forever lodged in your mind."

I'm sure many of you agree with this quote. In light of it, I have an idea of a some what new style of mathematics textbook, or rather one that really focuses on exploiting independent inquiry. The vast majority of textbooks has the basic schemeta:

1. Definition
2. Theorem
3. Proof of theorem
4. Problem questions

With different levels of explanation at each stage, but all of which is given without input from the reader. My new style suggests the following schemeta:

1. Definition is given (or a abstract word problem is posed e.g. for calculus what is the area under the curve? Can we chop the area up into infinitely small strips - give a mathematical definition)
2. Ask reader to collect evidence and test definitions and concepts against each other
3. Ask reader to make a conjecture
4. Ask reader to prove conjecture
5. If cannot prove conjecture, provide questions/hints for steps of proof
6. Ask reader to consider possible corollaries
7. Repeat steps 1-6 (in some cases new definitions may need to be created).

So for example consider the implicit function theorem restricted for functions of two variables. The textbook would address this like follows:

1. Definitions of partials, cont, etc given or explored previously
2. Give examples of some levels curves f(x,y)=0 where some can be expressed as y=f(x), x=g(y) and some not, and ask reader to examine what conditions allow for this.
3. Ask reader to conjecture what is necessary for y=f(x), x=g(y)
4. Ask reader for proof
5. If cannot prove reader will then be asked question e.g. why is f_y(x,y)>0 for all (x,y) close to (a,b) etc...
6. Ask reader how theorem can be extended to n variable, ask reader to consider converse etc.

Naturally answers at each of these steps could also be given at back of the book and time limits could be suggested at each stage. Any thoughts? Has anyone come across a similar style of textbook?

submitted by /u/helios1234
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