Thank you for your response Bob. I understand your approach better now. I do however believe that sooner or later the operations do need to be defined in age relevant ways that can evolve later.
My personal view is that failing to define, results in people not knowing when to multiply by half or divide by half. Maybe that's why 4 out of 3 people don't like fractions. (joke)
Early arithmetic pedagogy had all fraction lessons being taught with fractions. If you wanted to multiply a fraction by two, you had to divide it by 1/2.
Did you use measuring cups to teach dividing by fractions? Division is also a game of finding the missing multiplier.
I enjoy and benefit from your insights with your son. Having been there and done that with three of my own.
One question that was useful to me and may be interesting to you, is "How would you explain that to a child a year younger than you at school?"
This would get my son(s) to translate our conversation into the dialog and mindset of his friends.
Asking "How would you teach this idea to your friends?" is a great way to check if ideas are translating.
Do schools in the USA run a buddy system where every 'senior' child has to 'mentor' a littly?
You might even find your son defining the operations as HE understands them, which gives you the opportunity to be both teacher and student together.
My middle son did math tricks with friends and topped his school in math before moving onto study math/science at Melbourne University. My youngest son organised math study classes at school and also topped his year in math before going onto study science at Melbourne University.
The most moving and meaningful thing for me as a math educator has been my youngest son saying in a speech that he owes his passion for mathematics to me. He now tutors in math as well.
Getting back to the topic about word problems, why not have the students create the word problem and give them an incentive to solve it?
Kids find math boring and irrelevant. They wish we could wave a magic wand so they will like it. Sometimes as you say, it is helpful to provide a variety of examples so the concept is understood. Solving problems in different ways helps build comprehension, yet there still needs to be a central lesson about the operation(s).
I just believe we owe kids a definition that is more helpful than the generic definitions of the operations.
The vocabulary of arithmetic is still important. I am often accused of being pedantic. I prefer to think that I am mathematically or logically correct.
Dictionaries tend to use popular definitions rather than accurate definitions. The reason is not enough people care. So decimate now means the opposite of what it used to mean. It used to mean when an army lost a battle the soldiers would be punished. One in ten would be killed!
Similarly a quantum leap now means the opposite. It should mean very small!
Math professors make educational videos in which they divide a loaf of bread into five. Guess what, they do five slices instead of four!
When I point out examples like this, people say it doesn't matter.
If it doesn't matter to math professors why should it matter to kids?