The math behind elementary math is the Peano Axioms and recursion math. This is not taught in the schools. Nor is it understood by the math ed establishment. Nor is it taught in college to pre-service teachers.
What is taught in college to pre-service teachers and in schools is a garbled version of math. It is a pre-math version of tools and tricks and techniques that humans developed. These were developed as behaviors. They were imitated as behaviors.
The place value notation system was imitated as a behavior. That is how it is taught today. It is taught as stimulus and response.
The emphasis on applications is precisely because the schools and math ed establishment don’t teach the math of elementary math. They teach behaviors. They teach the same behaviors as when these techniques were developed.
In the 19th century, Germans developed the math behind elementary math. This was the work of Grassmann and Dedekind in important ways.
Place value notation is a recursive system that needs to be embedded in the Peano Axiom system to be understood. This was never really done. This is one of the reasons that Peano Axioms didn’t catch on. It is also part of why the math ed establishment doesn’t understand place value notation very well. It is recursive and does have its proper foundation in the Peano Axioms.
The little lemmas in the Dedekind book are what one has to learn to understand the foundations for elementary math. But similar small step lemmas are needed for place value notation as well. Some of these are in my first book on Peano Axioms.
Applications are teaching the behavior, not the concept. The math ed establishment doesn’t know the concepts, so they teach the behaviors. So they teach the applications.
Projects are the same manifestation of their lack of understanding. Projects are another way to teach behavior. They teach group behavior. They don’t teach math.