Though your thesaurus (or the one you use online) may not agree, I believe there is a difference between pushing and shoving. This is especially true when supporting students.
When I think about pushing, I imagine an object with potential that needs some assistance to begin moving. Imagines such as a grocery cart, a rock on a hill, a car that runs out of gas, or a lawnmower come to mind. Each of these objects have potential energy, but need a push to do their work and move forward.
Shoving, to me, is entirely different. When I imagine a shove, I think about the inconsiderate person trying to get ahead in line, Black Friday masses attacking the shelves at midnight, or the response of a school yard bully to a new student.
Pushing is cooperation between an object with potential and a force seeking to help that object do its work.
Shoving is selfish. It is done solely for the benefit of the person shoving.
When you support the potential for students, keep in mind the difference between pushing them and shoving them. Are your requests, activities, lessons, suggestions made to help a student with potential realize that potential or are they made primarily for your comfort. Are you asking them to push themselves or are you simply taking the status quo and making it more difficult (essentially creating the mirage of pushing)?
With all respect to Merriam-Webster, pushing and shoving are not always synonymous.