These days, even the most pure and abstract mathematics is in danger to be applied.
I feel students should be using calculators as little as possible. In my experience this improves number sense.
When I teach a class, below Algebra, I do not allow calculators (except for finding the square root of non-square numbers.) If a student did not have their multiplication mastered after 3rd grade, I will let them use a multiplication table for tests.
In algebra I do not use graphing calculators. Too many students do not fully understand what is happening in a graph, such as the need for even spacing. I like to have calculators that do not use order of operations for them.
When a student reaches Advanced Algebra (Algebra 2) there should be calculator days and non-calculator days.
Research Based News
Peters, Jenny. “The ‘Smart’ Myth and Making Mistakes: Key Elements in Mathematics Learning.” Consider how often well-intentioned teachers encourage math students by praising their smart work when they get the right answers. In “The ‘Smart’ Myth and Making Mistakes,” Jenny Peters argues that such praise, which she often used herself, reinforces the myth that mathematicians […]
Poorly Executed Mnemonics Definitely Addle Students — In this very clever post, Bill Shillito reminds us why most of the mnemonics used to remember the rules order of operations have problems.