I will answer you from my own experience: I am a lecturer in a Maths Learning Centre at a University. We provide a drop-in centre, and also revision lectures and other resources.
Students' problems with lecturers skipping the "simple" calculations
Students often talk to me about how frustrating it is when a lecturer skips steps, or at least when the lecturer does not indicate that they have skipped steps. This is not just the struggling students, but the ones who are very clever with most of the rest of the course. They stare at it for 20 minutes trying to figure out what happened, when what really happened was some expanding and simplifying.
I have other students who got through their high school maths, or worse didn't and the university let them do the course anyway (economics students are the classic example), for whom it is precisely the basics that they don't know. In particular, they have a shaky hold on the rules for algebraic manipulation of powers and fractions. Now we can work with them with the general rules, and give them exercises to do, but no matter how much you do this, they still need to see these operations in context with the new stuff. They don't know it well enough to embed it in something else, or to have the instinct of when an algebratic operation might be useful. They need to see full examples with the old stuff fully worked out so they can see where it fits with the new stuff.
Even some very high achievers have this issue -- there is some particular algebraic trick they just have never happened to see, and which only appears as useful while doing something new, such as simplifying the result of a quotient rule. These students will be able to see it as algebra and transfer it to other areas. (Others might need to be told that this is a possibility.)
We are in danger here of thinking that algebra is the only problem, but for many students it's actually the arithmetic that's the problem! Don't forget that many of your students actually have very poor arithmetic skills, having done all their maths on a calculator from a young age. They can often do a lot of high-level maths stuff, but have very few strategies to draw on with the low-level stuff. Students often tell me they really appreciate me doing the most basic whole-number calculations because they learn tricks they had never seen before.
These ideas are not restricted to strugglers with prerequisite knowledge, but at many points throughout the course. Often we teach X in one week, then assume they have it down pat next week and skip all the parts involving X when we teach Y. But of course, they are still trying to assimilate X. The classic example is eigenvalues. Many of my students simply don't have good strategies for calculating determinants, so when they try to find the characteristic polynomial, they struggle. Showing the whole process of actually expanding it out / using row operations cements the previous learning.
Finally, on a purely practical level, the students will have to be able to do all these calculations actually by themselves at some point (ie the exam). There is a danger that if they have never seen an example of doing the whole thing including all the little bits, that they will never realise that it actually is all necessary. They will think they know how the process works, but struggle to actually do it themselves, often failing simply because of the calculations rather than the understanding. To never show the whole thing is to give a false picture of your expectations.
Choosing when to do full calculations
As you have probably guessed, my general rule is to do full calculations whenever you can. Choosing examples where the numbers are easy to work with is one strategy, but you must include harder ones too at some point!
If your purpose is to show the structure of the new solution, I recommend going back through the solution you have done and highlighting the overall structure, noting when you were using theory and when you were doing calculations. I find drawing boxes around the different parts of the solution really helps students to see the parts. So does actually putting headings in that say what you are doing eg: "calculate determinant", "solve equation". Indeed, if they learn to put in helpful signposts like this in their own working then you will find it easier to mark!
For processes that do involve a lot of calculation, these headings can allow you to skip parts of the working without it seeming like magic. The notes say what to do and the students are not staring at it wondering what you did.
Finally, if you have the ability to record your lectures, those who are a bit slower can catch up on the details later.
