Enjoyed your post. We've been delivering math instruction online for over 15 years, and have enormous documentation that understanding fractions is clearly the single biggest deficiency for students, and one that persists for years. Over that time, I've been struck by the fact that even the best mathematicians don't agree on what a fraction "means". It seems to me the biggest problem is that we persist in defining fractions as "parts of objects." Fractions are not parts of physical objects. We use fractional units to measure the objects and their parts. The applications of fractions are related to measurement, yet we persist in focusing on objects to teach the concept. Until we recognize and deal with this fundamental misconception, I don't think there's much hope in overcoming the problems so common among students.