Comment by trod1234

Mathematics pedagogy today is in a pretty sorrowful state due to bad actors and willful blindness at all levels that require public trust.

A dominant majority in public schools starting late 1970s seems to follow the "Lying to Children" approach which is often mistakenly recognized as by-rote teaching but are based in Paulo Freire's works that are in turn based on Mao's torture discoveries from the 1950s.

This approach contrary to classical approaches leverages torturous process which seems to be purposefully built to fracture and weed out the intelligent individual from useful fields, imposing sufficient thresholds of stress to impose PTSD or psychosis, selecting for and filtering in favor of those who can flexibly/willfully blind/corrupt themselves.

Such sequences include Algebra->Geometry->Trigonometry where gimmicks in undisclosed changes to grading cause circular trauma loops with the abandonment of Math-dependent careers thereafter, similar structures are also found in Uni, for Economics, Business, and Physics which utilize similar fail-scenarios burning bridges where you can't go back when the failure lagged from the first sequence, and you passed the second unrelated sequence. No help occurs, inducing confusion and frustration to PTSD levels, before the teacher offers the Alice in Wonderland Technique, "If you aren't able to do these things, perhaps you shouldn't go into a field that uses it". (ref Kubark Report, Declassified CIA Manual)

Have you been able to discern whether these "patterns" as you've called them aren't just the practical reversion to the classical approach (Trivium/Quadrivium)? Also known as the first-principles approach after all the filtering has been done.

To compare: Classical approaches start with nothing but a useful real system and observations which don't entrench false assumptions as truth, which are then reduced to components and relationships to form a model. The model is then checked for accuracy against current data to separate truth from false in those relationships/assertions in an iterative process with the end goal being to predict future events in similar systems accurately. The approach uses both a priori and a posteriori components to reasoning.

Lying to Children reverses and bastardizes this process. It starts with a single useless system which contains equal parts true and false principles (as misleading assumptions) which are tested and must be learned to competency (growing those neurons close together). Upon the next iteration one must unlearn the false parts while relearning the true parts (but we can't really unlearn, we can only strengthen or weaken) which in turn creates inconsistent mental states imposing stress (torture). This is repeated in an ongoing basis often circular in nature (structuring), and leveraging psychological blindspots (clustering), with several purposefully structured failings (elements) to gatekeep math through torturous process which is the basis for science and other risky subject matter. As the student progresses towards mastery (gnosis), the systems become increasingly more useful. One must repeatedly struggle in their sessions to learn, with the basis being if you aren't struggling you aren't learning. This mostly uses a faux a priori reasoning without properties of metaphysical objectivity (tied to objective measure, at least not until the very end).

If you don't recognize this, an example would be the electrical water pipe pressure analogy. Diffusion of charge in-like materials, with Intensity (Current) towards the outermost layer was the first-principled approach pre-1978 (I=V/R). The Water Analogy fails when the naive student tries to relate the behavior to pressure equations that ends up being contradictory at points in the system in a number of places introducing stumbling blocks that must be unlearned.

Torture being the purposefully directed imposition of psychological stress beyond a individuals capacity to cope towards physiological stages of heightened suggestability and mental breakdown (where rational thought is reduced or non-existent in the intelligent).

It is often recognized by its characteristic subgroups of Elements (cognitive dissonance, a lack of agency to remove oneself and coercion/compulsion with real or perceived loss or the threat thereof), Structuring (circular patterns of strictness followed by leniency in a loop, fractionation), and Clustering (psychological blindspots).