I don't understand your comment. The entire article is about why "relying on intuition and physical experience" just wasn't good enough -- how the difficulties

I don't understand your comment. The entire article is about why "relying on intuition and physical experience" just wasn't good enough -- how the difficulties inherent in 19th-century style mathematics forced mathematicians to evolve towards a modern modern style. Algebra and number theory are both heavily reliant on modern mathematical thinking. Abstract algebra is practically the defining example of it. How does one create Mathematica and Maple without modern mathematical thinking? Indeed, formal logic is rather closely involved in computing generally. How does a precollege mathematics system that is based on a 19th century style of mathematics pay more attention to 19th century style mathematics? I find your confident dismissal of modern mathematics to be vexing and oddly misaimed.