Keith Devlin, long-time mathematician, gave a talk at the Fondation Helvetica Educatio in 2018. I've embedded the YouTube video below if you would like to watch and added my thoughts
It's nice to see others wanting to pull away from purely procedural mathematics to the bigger picture. This is a lot of what Jo Boaler talks about with Mathematical Mindsets. It's not about getting the final answer, it's about the process.
I am, however, a bit lost with the application of the Archimedes style mathematics that Devlin talks about. At the moment we do, as he says, teach individual procedures. We split the year into 5-7 topics and at the end we test the students' application of the rules. During their early years, the students get very little exposure to real critical thinking and decision making. Sure, we ask them what tools to use and what steps to take, but this scaffolding is to get them to complete the procedural or computational side of mathematics.
It would be good to open it up to more abstract pictures where the students need to determine their own path to solve a problem, but at this stage I struggle to see how we can assess their abilities. Am keen to read other ideas on how this might be achieved or if others think it's the right direction to head in.