Part of our jobs is to prepare students to use mathematics in engineering, physics, chemistry, etc. We don't want to always give problems where the numbers are nice.
Is (3) a problem? How many people can find interpolating polynomials? Very few. Interpolating polynomials are no longer needed by most students of mathematics because computing power is cheap and ubiquitous. The skill set for mastering mathematics has changed given this reality.
Think of it this way. Very few people can start fire without matches or flint. This used to be common knowledge years ago but is no longer needed. Fire making devices are cheap and plentiful.
AI researchers came to this conclusion a long time ago. Reasoning or experience is not enough - you must have codified, accessable knowledge on which to work.
Not knowing basic arithmetic cripples you when you need to work on the next stage. Not knowing that 7*8 is 56 means you can't work out the expansion of (x+1)^8. Working simple cases like that makes it clear what's happening when you differentiate, and gives insight that you can't obtain without doing the work.
The maxim from medicine is "See one, do one, teach one." The middle step is crucial. If you get the computer to do everything, you never gain the experience, and rarely have the insights.
And, personally, I can interpolate polynomials. It's an interesting example, because recently it allowed us to implement a new feature in our system. No one else to work out how to do it, and were amazed when I did.
In your terms, I can make fire in a world where most people can't. It just earned us real money.
The division of labor has rendered many once necessary skills the province of specialists. This is a good thing. Is arithmetic one of those things? I begin to believe so. The expansion of (x+1)^8 does not require the ability to do 7*8. One can do the expansion using Pascal's Triangle.
Besides, the real question is ought one do the expansion of (x+1)^8 by hand? The answer is no. The expansion is a purely mechanical process. A computer does it faster and more accurately. In such a long problem I am likely to make a mistake and this mistake does not mean that I don't know what I am doing.
I do want my doctor to thumb through reference books particularly if my condition is rare. In such a case it is unlikely that the doctor will have encountered my condition and I want her to gain from the shared experience and collective wisdom of experts. That's what a reference book is for and it's a good thing. I want my doctor to be able to read and understand the book. That is what her training is for.
One can get insights into algebra using a computer. It's just that the insights are different than what one could get without a computer. Instead of teaching interpolating polynomials in high schools they are teaching basic statistics and in this present age the latter is much more beneficial than the former.
The point of doing the expansion of (x+1)^8 (to continue the example) is not to get the answer, but to see the patterns building. You say you can do it easily enough by using Pascal's triangle, but finding that insight alone is worth the time.
Just last week I saw someone differentiate x^(1/2) by using a calculator to convert 1/2 to a decimal, putting it at the front, then carefulyl subtracting 1 from it, and getting 0.5 x^(-0.5). Right answer. Then they did the same for x^(1/3) and got 0.334 x^(-0.667) and got full marks. Follow the process by rote, and don't think.
I know I'm fighting a losing battle, because "Get the computer to do it" is too easy. What is lost is hard to see, and once it's gone it's impossible to regain. I wish I'd learned a langauge when I was 15 or younger, instead of now when I'm nearly 50. I'm constantly surprised that I can get answers to questions far, far faster than my colleagues and employees using techniques I learned when I was 12. I'm not using experience or any great gift, I'm using what everyone was taught in school at that time.
It's still useful. Perhaps I should be grateful that these skills are disappearing because I'm certainly never out of work.
It used to be the case that high school students were taught interpolating polynomials but they aren't anymore. It used to be taught to make estimates on what log(1.0356) is based on a table to values. When calculators became common the old timers would say things like,
"Wow, kids these days. They don't know anything. They have to rely on a calculator to find what log(1.0356) is. I can do that using interpolating polynomials and a table of values."
I suspect everyone on this website using a calculator to find log(1.0356).
I've interviewed senior candidates who can't do basic arithmetic in their head. I had one project management candidate, who was doing fairly well up until that point on the interview loop, who literally couldn't multiply 600 by 3 in their head, nor on the whiteboard.
I would tell the students: "The numbers should come out easy. If you get something weird, then one of us made a mistake!"
This simple change made a huge improvement in comprehension in the class. It wasn't about "number jamming" any more, but about learning.
That's painful!
(1) If you're trying to teach mathematical _concepts_ then make your problems come out to whole numbers that are easily done with pencil and paper.
(2) If (1) was true then maybe we'd see some real innovation in the calculator space, especially if the single-purpose electronic device costing $100 from TI had to compete with iphone apps costing $1. Or Google Calculator. Or Wolfram Alpha.
(3) I've anecdotally heard from my math-tutoring friends that there is a growing pandemic of high school kids in America that can't multiply single-digit numbers anymore. O_o