I teach mathematics at a college. Doing (1) for every type of problem is not feasible. In college algebra courses we now give problems that involve using the best linear, quadratic, or exponential fit. We definitely do not do these problems by hand. The calculator is a black box that spits out the correct approximation and the student is asked to answer questions based on what the calculator spits out for the correct fit.
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.
Do you want your doctor to look at your symptoms and start leafing through reference books, or looking on Google to decide what you have? It doesn't matter how clever you are, or how good at reasoning, you have to have basic knowledge to use as the base.
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.
Interpolating polynomials used to be taught in high schools. I learned it in high school and today almost no high school graduate knows how to do this. Is this a bad thing? I don't think so. With new technology and the advancement of ideas and knowledge the basic skill set that is necessary changes.
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.
I'm sorry I can't convince you, but I've done both, and I know that insights are given in both realms, doing things by hand, and using the computer. To throw away one is to lose an advantage.
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.
I agree with you on the first two points, but interpolating polynomials is not analogous to the ability to multiply single-digit numbers, which is indicative of a certain level of mathematical literacy which one would expect from even a third grader.
My experience is that the third graders know how to multiply without a calculator. But they use the calculator in 6th grade through 12th grade. By the time they get to college the connections in their brain used to multiply haven't been used in a long time and thus they have forgotten it. It isn't a travesty though.
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).
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.