I agree with Dan philosophically, but in practice I had to forge somewhat of a middle ground.
The structure of schooling (grades, standardized tests, pacing, prescriptive teaching, etc.) distorts parents and students views of the purpose of mathematics. Thus, teaching from a constructivist epistemology or invention (“discovery”) vantage point can be difficult. By high school many students have bought-in to prescriptive methods of learning, so they are often waiting for steps to manifest on their own. The realities of grading, testing, pacing guides, and students own differing beliefs about mathematics made it difficult to operate entirely from this perspective.
My favorite lessons and many of my students’ were those that they were able to abstract mathematics from a meaningful context. We move from concrete to abstract with different levels of immersion. A given unit might be comprised of 5 phases:
(1) Spend 4-5 days in one context before creating some type of generalize
(2) Then we move to a similar situation for 2-3 days.
(3) 2-3 contexts in a single day
(4) Each context is essentially a 10-minute-ish problems that are set in new or familiar contexts.
(5) typical abstract math that a lot of curriculum starts with.
Autograding helped me with phases (4) and (5). I used a “star” system for feedback and students would then try again, ask a friend, or ask me. Just having simple on-path/off-path feedback was meaningful for students when we got to these stages. Most of my students found this to be incredibly useful and wondered why we didn’t always use it (because I had not taught myself that CL yet).
It was such a success I built quizzes and tests the same way (if interested, see: https://sites.google.com/site/panettamath/desmos/precalculus?authuser=0). Students got feedback on their assessments and had the ability to improve their grade while taking it. The assessments auto graded which also made it easy to re-test for students who did not perform well.
Although the amount of time it took me to build these was immense, it saved on the back-end. The biggest payoff was that students could literally practice the test at home and re-take it at school when they were ready. It saved me time on grading at home and it allowed me dedicate more time to struggling students during phases 4/5.
However, using Desmos in this way was a privilege I have by learning CL. This certainly isn’t something other teachers could do at my school.