I think that this is a false dichotomy that is caused by a misunderstanding of what mathematics is and the low quality of many of the questions that students are asked to do in mathematics classes.
A questionHere's a fairly typical question: what is 36 × 9?
Do you think students should do this with or without a calculator?
My answer is (to both questions) - I don't care!
This is an absolutely pointless question. What does it matter what 36 times 9 is? In the absence of any realistic or mathematical context it is meaningless and there is no reason to calculate this product.
The only mathematics worth doing is mathematics that helps us understand the relationship between numbers/shapes or helps us solve a realistic problem.
How much would a worker earning £9 per hour earn for a 36 hour week?
A rectangle has sides length 36 units and 9 units. What is the area of the rectangle and will it be a square number?
Both of these questions require the product of 36 and 9; however, in both cases the important skill here is to be able to identify that product is required, once a student has identified that the answer will require a product I would hope that they had a range of strategies available to them and that they can use the most appropriate one.
Strategy 1: using a calculator.
In the 21st century most people (at least in the UK) have a powerful calculator to hand most of the time (in the guise of a mobile phone). It should be part of students' mathematical education to learn this.
Strategy 2: pencil and paper methods
These are often useful; however, I would argue for developing alternative strategies for processes. I also think it is important to emphasise methods that aid relational understanding over those that are merely efficient. For example, of the two presented here I think that box method gives a clearer representation of the 2-dimensional nature of multiplication.
Strategy 3: in your head
If I wanted to multiply 36 by 9 I would probably do 36 times 10 to get 360 and then subtract 36 to get 324. There are many other strategies that can be used for this but I think it is important to develop a playful nature with numbers through these kind of mental calculations.
Which is the best strategy?
The three different strategies presented here will be useful in different situations - it would depend on the context. Once a student has correctly identified that a product is needed I would be happy with them using any of these three but I think students should be encouraged to develop their skills in all of them.
Developing skills in using these strategies
For all of these strategies it is important that students develop their skills in using them and, with any skill, the best way of doing this is through practice. This practice can either be in the form of drills or play. The problem with much of the mathematical activity that students are asked to do is that the practice of particular skills, such as multiplying a two digit number by a one digit number, becomes an end in itself as opposed to a useful tool to have in one's mathematical toolkit to solve problems. Over-emphasis on practising performing mathematical operations is boring whether they are on a calculator, pencil and paper or in your head.
If the process of multiplying two numbers by hand is seen as an end in itself then obviously using technology to do this could be considered "cheating". I would suggest instead that the goal of the mathematics taught to students should always be explicitly about solving realistic problems or understanding mathematical relationships. If this is the case then where there is a need to develop and practise strategies for performing processes it will be natural to consider both calculator and non-calculator strategies.
In short - if you are asking whether students should be using a calculator or not it's not that you're asking the wrong question it's that you're asking the wrong questions.