The paper presents major challenges to the content and methods of primary and secondary mathematical education presented by the modern society. The new system of priorities of math education should reflect turbulence, uncertainty, probability in nature, society and personal behavior, personal choice, responsibility, and sustainability, human thinking and reasoning, information technologies. Attempts and avenues to implement these priorities are outlined.

School mathematics should provide to students: models of reality, patterns of reasoning, including transfer from real-life to models, and interpretation of results of simulation, experiences of search of solution of a new unusual problem.

The most important and tractable in school, are models for: classical mechanics and optics, nuclear decay, population dynamics, etc., probability, computers and other digital devices automated control and information transmission, information processing and communication by humans, linguistic entities (in different natural and artificial languages including, for example data-base entries), reasoning, formalized behavior, personal choice and game behavior, instability and turbulence, emerging properties

The instruments of reasoning include: classical logic in local context or in a frame of a general theory (like in the case of geometry), formula manipulation, probabilistic and statistical arguments, reasoning about processes, algorithms’ execution, choices and games, dynamical environments, interaction, control.

Problem-solving is still making mathematics an exception among school subjects. It includes: representing a real life (or quasi real-life) situation in a given semi-formal language, executing a given algorithm for a specific input, searching for formal transformations leading to more simple or more complete situation, interpreting a formally obtained result in the original context, experimenting (possibly using computer), interacting with another person or environment (like in a game).

Key factors in the development of “problem-solving” skills:

• maximum of new problems be presented and solved (at least understood and approached, solved partially, etc.).

• discovering and correcting mistakes. The techniques like estimation, checking at the middle point, finding alternative solution, or a counter-example, peer discussion must be a part of math education (not strategies of fast guessing for multiple-choice questions).

• experimenting in the ideal world and its realizations in pen and paper reality as well as in computer microworlds.

Existing and possible implementations are discussed. Visual and palpable objects, computers play important role in them.