‘Fluent’ and ‘unencumbered by misconceptions’ are laudable objectives, often lost in excessively progressive approaches that undermine the importance of basic knowledge and skills.

But it shouldn’t be difficult to reach consensus on what the core elements of mathematical thinking are — it would start with some tangible combination of knowledge, skills and problem solving heuristics. And I’d challenge educators to make tangible those profound elements — such as beauty and elegance — because they only embolden your objectives.