I teach some elementary math’s to adult learners, and I am mandated to teach them how to use a calculator.
When I produce the calculators about ½ way through the course I ask them what a calculator is for. I then explain that it is for when the calculations get to big to do by hand, and we then move on to calculate some really big numbers, like the amount of sand on the local beach, the number of snowflakes that fell in the last big storm, the number of breaths they have taken so far in their lives etc. I use the calculator to introduce the idea of estimating and approximating, and we then try to make an estimate of how good our answers are. The idea is that, now you have the skills, the calculator opens up a whole new vista of doable problems.
That’s for my mainstream students. For some of the students, maybe two in a hundred, I teach more specific calculator use, when I have decided that these are students who don’t have the ability at this time of mastering the skills of hand calculation. This may be an innate deficit, or it may be the product of years of confrontational math education resulting in the student being terrified by the prospect of looking at numbers in any constructive manner whatsoever.
These are the students for whom the calculator will become the mathematical equivalent of dragon dictate, and just as dragon dictate can open up the world of generating texts to individuals who would be unable to write or use a keyboard, I see the calculator as giving access to mathematical results to people who would otherwise be incapable of getting there.
Finally to mention some specific skills. I teach adding fractions by cross multiplying and reducing the result. I explain that there are other methods which they may already know, but that I am avoiding them because they don’t always work, and we don’t have time to go through all the special cases.
I do this, not because I’m short on time, but because often I find students abandoned their attempts to understand fractions at the LCD/LCM stage. Fractions are key, and I explain that these operations are important because if they come to algebra many of the same methods will work.
I spend almost a full class on the concept of dividing by a half.
I teach long multiplication, again stressing that this is a method which will go forward into algebra, and I teach long division in the same way for the same reasons.
The key in all of this is that I am teaching skills which will continue to be of value, both in general life, and in the students further math’s careers, should they care to pursue them.