The willingness to take up questions > to which one does not have the answer takes bravery.
Four years ago, in an attempt to implement this goal, I began using long-term student projects in which the students pick a subject they love and investigate it mathematically. I am learning maths I never dreamed of. So are my students. And I believe they are learning a great deal about mathematics.
In Vol 14 No. 3 1995 of the Journal of Computers in Mathematics and Science Teaching (page 325) there is an article by Sharon Dugdale, et al on Technology and Curriculum Reform: Current Issues, Potential Directions, and Research Questions. in it the authors try to "define" algebra by considering it
"as a way of reasoning involving:
* Variables and functional relationships ... *Generalization and modes of representation. This includes development and use of formulas, translation of ideas to and from various representations, and manipulation of thos representations. ... Prsent applications of technology have enlarged the traditional focus to include new emphasis on graphical representations and possibly others as well.
(my note: Geometric representations are easier to integrate into traditional algebra through technology.)
*Mathematical investigation and argument. ... includes noting patterns, making conjectures and refining reasoning.