The usefulness of the three approaches described in the article depends on what the introductory course is trying to accomplish.
One professor taught the subject through a steady diet of mathematical models. If the introductory economics course is aimed at those students who are going to major in economics, then that may make some sense. But most students in most introductory economics courses are not going to become economics majors, much less professional economists.
Among those students for whom a one-year introductory course is likely to be their only exposure to economics, mathematical models that they will probably never use in later life, as they try to understand economic activities and policies in the real world, may be of very limited value to them, if any value at all.
If the purpose of the introductory course is to serve as a recruiting source for economics majors, that serves the interest of the economics department, not the students. It may also serve the interests of the professor, because teaching in the fashion familiar in his own research and scholarship is a lot easier than trying to recast economics in terms more accessible to students who are studying the subject for the first time.
Having written two textbooks on introductory economics -- one full of graphs and equations, and the other with neither -- I know from experience that the second way is a lot harder to write, and is more time-consuming. The first book was written in a year; the second took a decade. The first book quickly went out of print. The second book ("Basic Economics") has gone through 4 editions and has been translated into 6 foreign languages.
Both books taught the same principles, but obviously one approach did so more successfully than the other. The same applies in the classroom.
The opposite extreme from teaching economics with mathematical models was described by a professor who uses an approach she characterized as democratizing the classroom, "so that everybody is a co-teacher and co-learner." This has sometimes been called "discovery learning," where the students discover the underlying principles for themselves while groping their way through problems.