For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor.
If science fair projects or English class writing assignments count as "execution" then certainly solving word problems counts as "execution." None of those things have real-world value, except in the very rare cases of extremely talented students who write a publishable story or produce valuable scientific results. The purpose of English class is to develop skills that will be applied outside English class, just like math.
What I personally saw in high school was that teachers were pedagogically obsessed with application in context, exactly what you call execution, but were at a loss as to how to apply the principle in practice. A project to create "context" for solving math problems could take many hours of class time and result in students using math only a handful of times. Solving a particular math problem is rather like hitting a curveball or playing a piece on the piano. You have relatively few chances to execute in a meaningful context -- a few dozen times per year, if you're lucky. Those natural contexts are just not sufficient for developing skill. So skills are developed in artificial contexts: hundreds of swings in batting practice, hours of practice alone at the piano, and many, many math problems with no real context.
It's easy to provide imaginary context, of course. I'm solving for the side of this triangle because I'm building a bridge and people could die (or I could get fired) unless I can figure out how long this side is. People don't find that very compelling, and that isn't unique to math. Kids taking batting practice or playing etudes aren't immersed in vivid major league or concert fantasies every time they swing or strike a key -- a lot of the time, they're fighting with their minds to fully engage with the task. But they take it for granted that they have to practice to succeed, while in math we have almost reached an attitude that practice is inimical to understanding. It's time we admitted that math is just the same as any other skill: little understanding can exist without competence, practice deepens understanding, and the mind is not freed to combine basic skills fluently until the basic skills become second nature.
It would be nice if it were possible to create a compelling context for every practice problem, but it isn't, not any more than you can create a compelling artistic context for every musical scale or writing exercise.
I experienced one really good example of application in high-school. We had one assignment that was coordinated between Science (in this case Biology) and English where we wrote a paper and the content was given a grade by the Biology teacher and the style/formatting, etc. was graded by the English teacher. It would have been nice if there were more things like this, as it directly addresses a concern pg expressed in one of his essays:
" Certainly schools should teach students how to write. But due to a series of historical accidents the teaching of writing has gotten mixed together with the study of literature. And so all over the country students are writing not about how a baseball team with a small budget might compete with the Yankees, or the role of color in fashion, or what constitutes a good dessert, but about symbolism in Dickens."
My school tried to adopt that approach. It was called "writing across the curriculum." It didn't last long at my school, which I thought was too bad, since it sounded like a pretty cool idea to me.
You may have point there. However, I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me. Execercising it more afterwards does improve my usage, but more in the sense, that I'm quicker to spot situations where I can (or can't!) use that particular feature/theorem. Sometimes I happen to see a new angle which enables new tricks. Learning to play piano on the other hand is a lot about muscle memory, which you train by repeating the same thing again and again. Yes, there is part of this in math too, but thats the handicraft-stuff. Arithmetics. Things a computer can do better.
If science fair projects or English class writing assignments count as "execution" then certainly solving word problems counts as "execution." None of those things have real-world value, except in the very rare cases of extremely talented students who write a publishable story or produce valuable scientific results. The purpose of English class is to develop skills that will be applied outside English class, just like math.
What I personally saw in high school was that teachers were pedagogically obsessed with application in context, exactly what you call execution, but were at a loss as to how to apply the principle in practice. A project to create "context" for solving math problems could take many hours of class time and result in students using math only a handful of times. Solving a particular math problem is rather like hitting a curveball or playing a piece on the piano. You have relatively few chances to execute in a meaningful context -- a few dozen times per year, if you're lucky. Those natural contexts are just not sufficient for developing skill. So skills are developed in artificial contexts: hundreds of swings in batting practice, hours of practice alone at the piano, and many, many math problems with no real context.
It's easy to provide imaginary context, of course. I'm solving for the side of this triangle because I'm building a bridge and people could die (or I could get fired) unless I can figure out how long this side is. People don't find that very compelling, and that isn't unique to math. Kids taking batting practice or playing etudes aren't immersed in vivid major league or concert fantasies every time they swing or strike a key -- a lot of the time, they're fighting with their minds to fully engage with the task. But they take it for granted that they have to practice to succeed, while in math we have almost reached an attitude that practice is inimical to understanding. It's time we admitted that math is just the same as any other skill: little understanding can exist without competence, practice deepens understanding, and the mind is not freed to combine basic skills fluently until the basic skills become second nature.
It would be nice if it were possible to create a compelling context for every practice problem, but it isn't, not any more than you can create a compelling artistic context for every musical scale or writing exercise.