My Vision for the Future of Secondary Mathematics Education

I first read Paul Lockhart's "A Mathematician's Lament" when I was earning my undergraduate degree in Mathematics For Teaching. (If you haven't read it yet, you should click on the link and read it right now, at least the first two pages.) It left an impression on me then, but I didn't think until in the middle of my third year of teaching to dig it back up and read it again. Strongly compelled and contemplative after re-reading, I soon discovered that there was a second part to "A Mathematician's Lament" not on the free on-line PDF ("Exultation") and a whole (second?) book written by Lockhart (Measurement). (I think most people would consider "A Mathematician's Lament" to be an essay instead of a book, even though it was published together with "Exultation" as a book, so Measurement would be his first book, not his second book. But please correct me if you find/think otherwise.) I immediately ordered these texts and spent my summer break reading them. (As a result, I was also shamelessly inspired to construct my own proof from scratch that two triangles constructed by taking any triangle and then constructing the line segment from any vertex of that triangle to the midpoint of its opposite side will always have equal area, while on vacation in Germany. I'm quite sure that it wasn't the most efficient proof of this fact that's ever been done, but it was my proof. I owned the proof as my own.)

(This is not the whole proof, just a snapshot of part of it.)

Months later, Lockhart's ideas and passions continue to resonate in my heart. Lockhart has drastically changed my ultimate vision of what I hope mathematics education will one day look like. I don't know if said vision will be attainable in my lifetime (probably not), but I think that that makes it all the more a worthwhile goal to have, a goal that stretches beyond my lifetime, a life-long striving that makes life worth living. Here's my vision...

Imagine a mathematics secondary education in which every student is working on a math problem of his/her own choosing that matters to him/her. Imagine every student getting a weekly one-on-one check-in or mentoring session with his/her instructor, or perhaps a more appropriate name in this world would be a coach. Just as sports coaches mentor their athletes to improve their athletic abilities and reach their potentials, so this mathematics coach would mentor his/her students to improve their problem-solving and critical thinking capacities. At the end of every week, students write a report to explain the progress that they have made in their individual problems. The purpose of this report is at least three-fold: (1) to hold the student accountable to himself/herself as each student innately desires to write a good report that shows significant progress; (2) to hold the student accountable to his/her math coach or whoever might serve as the student's academic evaluator/grader; (3) to create a written record of the student's weekly progress that can act as a portfolio of the student's work and can be shared with other students to spark curiosity and incite opportunities for student collaboration (just as many sports are team sports that require collaboration). These reports will be carefully reviewed by the coaches and discussed with students in their weekly one-on-one check-ins when coaches might provide guiding questions, thoughts, or resources to help students continue to move forward in their inquiries and progress on their problems...

That's just the tip of the iceberg. I originally wrote more (and I am definitely thinking more that I'm still trying to put into words), but I'll stop my description there and save the rest for later (in the hopes of decreasing unnecessary long-windedness and increasing coherency and readership).

Of course this seems fluffy and strange and maybe even impossible or at least impractical to you now. And I'd love to talk about that! I don't have all the answers -- not by a long shot! There are many, many questions this vision brings up. (And as much as I wanted to have all the answers before posting this... that just wasn't going to be possible. I just needed to start somewhere before I could get anywhere.) There are many reasons why this kind of system would not be successful if we suddenly started doing this in our schools tomorrow. But if I believe in this vision, then to make it happen some day, I need to ask:
(1) What conditions would need to exist for this kind of system to be successful?
(2) How can we gradually build bridges to transition from all that is "today" to this vision of math education in the future?

I want to hear your every thought on this and think through every conceivable problem with this system, so please share!