grades 1-5: arithmetic, using the idea in
>>12877689 . Introduce basic algebra. Don't waste time on anything the average person isn't going to use in their normal non-technical job and thus forget. Realistically, most people are going to graduate high school knowing nothing about math except what they learned in elementary school.
Grades 6-8: Introduce kids to the kinds of math that actual math-lovers will enjoy. After a year or two of that, math becomes elective. Only students that care will be required to continue. Don't make them just plug shit into formulae for these years. Show them geometry, set theory, predicate calculus and proofs, and the cool parts of calculus.
Grades 9-12: accelerated precalculus and calculus (maybe statistics is an optional elective too) for students who actually give a shit. Imagine how much progress kids would make in three years of high school calculus if the class is only composed of people who care. Let's be honest, not everyone needs to know 90% of precalulus topics, and they won't remember any of it anyway. Kids that don't take the advanced math track will take something possibly more relevant to their interests and future career/life. In my senior year my teacher turned the class into personal finance class about 75% through the year, and that was pretty great. It's more important to teach that to everyone than it is to teach stuff like trigonometry.
