Ideally, how would math be taught?

Rather than "memorize formula, solve problems, tommorow add more to formula, solve more problems, repeat," how should math ideally by taught?  What kind of teacher would you need?  What kind of student interaction would there be?  And are there any close examples from other countries or societies, past or present?

There is a general issue in that the goals of education are not what they ostensibly should be, and they will likely never be such. "There is no power but power over other people." True mass education is undesirable to those who would have to arrange for it.

The manifest goals are several. The first appears to be training children to obey orders, in particular to remain sedentary. The second appears to be indoctrination. The third is to confine persons of the relevant age group to remove them from potential criminal activity. The fourth is to remove them from competition with older persons in the workforce in order to artificially inflate wages.

There are probably other ways of phrasing all this to make it more palatable or perhaps less controversial; however, the root cause of the math (and indeed, general) education issue is that the actual "motive forces" propelling it are contrary to the goals of actually educating people.

If the goals were such, different overall behavior would be observed. For example, progressively higher standards would be put into place, attempts to significantly advance the amount of content delivered in a given period of education, and the like would be observed. Instead, the focus is on capturing larger numbers of students, dealing with "problem students", and the like, and the standards degrade over time to reduce the rates of withdrawal from the compulsory education program. Those are all advancements toward more complete control of the society, not higher-quality education.

I'm largely unconvinced there is a "solution" as such. While I could review proposed solutions and try to anticipate their effects, my overall assessment is that this cause is far beyond futile and will remain so in the absence of extremely powerful external influences on global society. One of the only ones I suspect would suffice is an impending natural disaster for which the entire population of the planet must be recruited to devise the technology to avert. Another would be if economics became so heavily reliant on technology that countries engaged in "technological arms races" to recruit as many creators of technology for economic purposes as possible, which is far from the case now. It does not appear plausible that either of these situations will occur in the near future.