Just to stress again what DPRK already noted above, there is actually a really beautiful theory of linear algebra in which all this matrix stuff sits, of natural and ubiquitous use in abstractly analyzing questions which come up either in the world or just for fun or curiosity (i.e., math).
Matrices aren’t intrinsically pointless or anything (though even there, for my tastes, linear algebra is cleanest first viewed without excessive focus on coordinates and matrices and so on; a matrix more directly is an element of the tensor space R^m x R^n than a linear map, and though the former can represent the latter, many concepts are better thought of directly in terms of the latter than the former… Anyway, I digress).
But far more people are introduced to random matrix manipulations unmotivated than ever actually are shown this, making the endeavor pointless. And, it must be admitted, I say, that most people end up living perfectly fine lives without ever bothering with linear algebra. Those who are interested in it can study it and those who aren’t needn’t. (I will grant that abstract linear algebra is one topic that perhaps more people could be exposed to the existence of and discover some interest in than currently happens, but, yeah, nonetheless, still many people will not care about and needn’t be tortured with it)
That’s a possible goal, but even that may mean something which is, as I see it, overly still focused on algorithms as central topic. Why should algorithms be the central topic at all? Certainly, any algorithms which time is spent on should be accompanied by ensuring an understanding of why they correctly accomplish whatever they are intended to accomplish, but it needn’t be that time is spent primarily on familiarizing oneself with various algorithms.
(But, of course, it IS understanding, general conceptual understanding and insight and new perspective, that is the important and useful and interesting thing about math, and that may be all you were getting at…)
If you understand the math, then you can come up with your own algorithms as needed. Indeed, that’s a large part of what STEM professionals do in their work.
Perhaps… but would anyone boldly claim to be able in a few minutes to “come up with” Dijkstra’s Algorithm, the Fast Fourier Transform, a polynomial-time primality test, septimus’s code to compute double-precision logarithms, etc.? I would expect those to be learned in class and/or from technical literature. And a professional who understood such algorithms at the deepest level would indeed be able to come up with their own advanced algorithms.
Back to New/Old/Core Math: so none of those curricula are proven to make any difference as far as facilitating development of mathematical fluency (or STEM fluency, or general development, what have you)? Or is there a statistically significant difference? What about global studies? (New Math was never confined to America.)
In case anyone’s wondering, I believe this is a reference to this other thread.
Of course, all the algorithms you list are useful things to learn in suitable classes. In case anyone takes me as saying “No one should teach any algorithms to anyone ever”, let me make clear that my position is far from it! As I keep saying, people who are interested in things should have opportunity to learn them.
But polynomial-time primality tests are probably not the best thing to focus on in general math classes exposing children to the concepts of primality for the first time (this one I think we can all agree is obvious…), code to compute logarithms is probably not the best thing to focus on in general math classes first introducing the concepts of logarithms (and indeed, the vast majority of people who understand logarithms conceptually and can fluently manipulate them algebraically do not know how to efficiently algorithmically compute them), etc. Calculational algorithms don’t need to be the initial focus, and indeed, all these algorithms don’t need to come up ever except for people particularly interested in learning them. They would be terrible choices to make universal compulsory subjects of instruction, homework, and testing.
And probably very few of the people (in our modern era) who DID have interest in and learnt all these algorithms ever cared to sit down and execute them by hand on examples of any nontrivial complexity (I don’t actually know the details of the AKS algorithm, but I understand the other three examples perfectly well, without having ever done this). That doesn’t turn out to be a particularly vital thing to drill on doing, even for understanding the algorithms!