Comment by trod1234
3 days ago
The classical system was described, but you can find it in various historic works based on what's commonly referred to today as the Trivium and Quadrivium based curricula.
Off the top of my head, the former includes reasoning under dialectical (priori and later posteriori parts under the quadrivium).
Its a bit much to explain it in detail in a post like this but you should be able to find sound resources with what I've provided.
It largely goes back to how philosophy was taught; all the way back to Socrates/Plato/Aristotle, up through Descartes, Locke (barely, though he's more famous for social contract theory), and more modern scientists/scientific method.
The way math is taught today, you basically get to throw out almost everything you were taught at various stages, and relearn it anew on a different foundation, somehow fitting the fractured pieces back together towards learning the true foundations, which would be much easier at the start and building on top of that instead of the constant interference.
You don't really end up understanding math intuitively nor its deep connections to logic (dialectics, trivium), until you hit Abstract Algebra.
You want to teach abstract algebra to middle schoolers?
Up to the first or second chapter, depending on the book being used is more than sufficient to cover the foundational concepts. Sets, and Properties such as closure over given operations, and mathematical relabeling which is a function (f(x), the requirements for it (uniqueness of x, and projection onto) along with the tests for the presence of these, and common mathematic systems properties.
This naturally provides easily understood limitations of math systems which can be tested if there is a question, and allows recognition when they violate the properties that naturally lead to common mistakes, as well as providing a space where they can use numbers/geometry/reasoning at play.