- Economic: The important thing is the calculation. In this paradigm, the answer to 4+5 is always 9.
- Philosophic: The important thing is the relationship between numbers. Here, 4+5 could be 9, or 6+3, or 3-squared, or square root of 81, or ...
- Artisanal: The important thing is the units/application. What does the 4 and the 5 mean? If it's 4 feet and 5 inches, adding them up won't make sense.

I accept that these are the historic reasons for teaching and learning math. As civilization developed, people needed to keep accounts, measure fields, and as they had leisure time, pursue math philosophically.

One of the difficulties with teaching math in the present time stems from the fact that we want to include all three of these paradigms. Unfortunately, students are not always clear about what view to use and what we expect of them. For example, many of my students have told me they like math because "there is only one right answer: it's right or wrong." And most of their background has been about performing calculations to get the right answer. When I ask them to think more about the numbers, and get philosophical, some get overwhelmed because 4+5 now has lots of answers, some they know, and some they have never seen. As they try to wrap their minds around that, we throw "word problems" at them where they have to worry about units and whether or not the answer makes sense. (Never mind that I have seen some published problems where the answer "97 watermelons" is supposed to make sense.)

All mathematics is an abstraction, which means the way we view math focuses our work on some aspects and ignores others in order to better understand/use/apply the ideas and procedures. Different methods of abstraction focus on different aspects. So the economic paradigm above is an abstraction of all mathematical ideas, focusing on the calculations. Limiting our understanding of math to this paradigm is good if we want to produce accountants, but bad if we want to produce engineers.* Similarly, limiting math to the philosophical or the artisanal paradigm also gives us an incomplete view of math; each abstraction loses details.

So I will continue to try to teach using all three paradigms, and my apologies to my students who think I'm "doing too much". Perhaps I am, but I promise you it's because I believe in your ability to push past your own limits.

*My apologies if that sounds insulting to accountants, I don't intend that. My father-in-law was an accountant, and brilliant at his job. Unfortunately, I did not understand his work very well despite his trying to teach me some basic accounting ideas late one New Year's Eve. That's a story for another time. The point is that accountants and engineers use math in very different ways, and teaching only one way is limiting.