We had a whole course on numerical methods. My guess is they have become even more important over the decades since.
Fair enough. When I was in the physics curriculum we didn’t have any requirement to take any kind of numerical methods but I took it as a math/comp sci elective because it seemed like it would be useful. When I went over to mechanical engineering, some of the upper division classes would introduce simple approximation methods (e.g. Newton-Raphson and 4th order Runge-Kutta) but no theory behind it such as how to estimate error and determine the correct step size for a problem.
For the most part, working engineers are using turnkey tools or calculation systems that have numerical simulation tools (Matlab, Python/NumPy/SciPy) so they don’t actually have to actually code up a numerical method by scratch (I haven’t cracked Numerical Methods in C in years) but it would certainly be good for engineers and scientists working with numerical approximation methods to understand the fundamentals, especially if they are working on optimization problems where using the wrong method or parameters can give totally wrong answers or an unstable model. I do recall briefly covering approximation methods in Introductory Differential Equations but obviously you don’t spend much time on it.
Stranger
That does sound pretty nutballs, especially when finding all the integer roots of, let’s say, Ax^2+Bx+C , when A, B, and C are integers, can be done by factoring C. And “completing the square” is just
\begin{align}
x^2 + 2bx + c &= (x+b)^2-(b^2-c) \\
&= \bigl(x+b+\sqrt{b^2-c}\bigr)\bigl(x+b-\sqrt{b^2-c}\bigr)
\end{align}
and does not involve any integer roots.
I’m pretty sure she was not talking about van Hoeij’s algorithm either
but who knows…
And coding up Gaussian elimination into spreadsheets and pulling them up is even quicker yet. And probably unnecessary, because your spreadsheet probably already has built-in functions for Gaussian elimination.
I actually think that it should be the other way around: Teaching arithmetic based on polynomials. For that matter, algebra should be taught before arithmetic. That nebulous “number sense” that standards say arithmetic students should develop? That’s just algebra, and it’s no wonder that most students never learn it, because it’s never taught.
But current textbooks do include polynomial multiplication using the same algorithm as is usually used for multiplication.
The main value of high school geometry isn’t the geometry itself; it’s that it’s the first class most students encounter that’s focused on proofs. Which is important, because that’s what math is. Now, why it’s geometry that’s presented that way, rather than other high school math subjects, that’s just an accident of history.
@Stranger_On_A_Train , I have no idea what procedure your teacher was calling FOIL, if it was a technique for factoring. In any event, whatever it was, it’s clear that your teacher just sucked at teaching it.
I mean, if you really have to do matrix algebra in Excel:
I’d rather threat bamboo skewers through my sinuses than attempt to do numerical analysis in Excel if I had literally anything else available to me, but some people really love their spreadsheets for every problem. I still shudder at an engineer who was trying to do FFTs in an iterative spreadsheet and just refused to use Matlab even after I set it up to import her data and re-export the result to a .csv file.
Stranger
Probably. But it probably doesn’t have functions for Gaussian elimination… over a Galois field. Which is something I did once, and therefore appreciated knowing how to do it in the more ordinary case. And anyway, I had to do it in C for speed…
That looks a lot more complicated than what we used it for.
In Algebra I, that’s how the teacher told students they could do the multiplication—so you could multiply (a + b)(c + d) and be sure to get all four products needed. I don’t think they required anybody to do it that way, but if the student missed the question that’s what the teacher would reference.
I thought multiplying them was pretty easy, but I inferred that 9th graders must often screw it up, so the teachers gave them a visual.
Indeed. Possibly a course was available, but it wasn’t required.
Really, a course on numerical methods with a focus on modern computing techniques would be even better. A clear understanding of floating point math and when it goes awry would be a good start. And modern graphics and deep learning techniques depend on lower-precision math than standard CPU types: 16-bit down to 10-bit floats, fixed-point math, and so on. It all comes down to power efficiency, and wasting femtojoules on unneeded precision is not good for power.
As a math major and PhD statistician I think that over the years I’ve probably used pretty much every math concetpt that I was forced to learn. But there were a few elective courses that I took that were pretty much useless to me. Number theory in particular was fun, but always felt a bit like circus tricks. Yes perfect numbers are kinda cool and Fermat’s last theorem makes a nice historical anecdote, but other than the fact that an random obscure theorem happens to produce a function that’s helpful in cryptography, it seems mostly recreational.
I concede your point. Proofs are important and geometry emphasizes these.
Was this supposedly a technique for factoring a trinomial, ax2+bx+c ?
About the time I was 40 or so, a math teacher told me about a method he called “the AC method” that I had never heard of before, that is rather easier because there is less trial-and-error involved.
In the usual method, you factor a and factor c and try every combination of one factor-pair of a and one factor-pair of c to see which gives you the right middle term. PITA amount of trial-and-error.
In the AC method, you multiply the a and c coefficients, and then factor that. Go down that list of factor-pairs to see which add up to the middle coefficient. Split the x term into two x terms with these coefficients, and replace the original x term with these two. Then factor by grouping.
Confusing? It’s actually simple and has less trial-and-error. For the actual detailed step-by-step process with an example, see:
https://gato-docs.its.txstate.edu/slac/Subject/Math/Algebra/The-AC-Factoring-Method/The%20AC%20Factoring%20Method.pdf
You can also google for: ac method of factoring trinomials
and find many other pages explaining it, with more examples.
Instead of something useless that I learned in algebra class, here was something useful that I didn’t learn in algebra class.
ETA: The AC method has another interesting thing about it: As you go through the steps, you can actually see the multiplication of the two binomial factors via the FOIL method happening, step-by-step, in reverse (almost).
But but but… Are they still focusing on proofs in High School Geometry in these modern days? Maybe not. Too old-school, I guess.
I stumbled across a High School Geometry book, dated circa 2001, recently. Needless to say, I “acquired” it, because of course I did. (I have a habit of collecting old math textbooks when I come across them. I have a trig textbook from 1918, and I learned some things from it that I never saw when I took trig in school.) But I digress…
This geometry book doesn’t seem to say much about formal proofs. It might talk about proofs in some informal way, maybe. I haven’t actually looked too closely at it. Out of 16 chapters, it only has one chapter of about 40 pages on formal proofs, of which only about 10 pages are about those old two-column proofs.
The rest of the book seems to cover a lot of territory, but focused in a way to make geometry “fun and interesting” I think. Not that there’s anything wrong with that. The cover has a collage picture of a cantilevered bridge, some kind of building with lots of domes, and a starfish. That sort of tells you what you need to know about this book.
ETA: When I got to college, I took a symbolic logic class. We spent the entire quarter doing two-column proofs of logical propositions.
Surely no trial-and-error is required to factor 2x^2-11x+5, over the integers or otherwise. With a higher-degree polynomial like x^{4} - 2 x^{3} + 19 x - 28 it might be a different story.
As for geometry, perhaps people are looking to cash in with crappy textbooks that solve the problem of Euclid being in the public domain (though I always thought that digital books would make it much less easy to coerce students into paying inflated prices for those).
I’ve taught (and used a textbook that teaches) both the “usual method” and the “AC method”; I figure students can decide which they like better. The AC method is, indeed, more systematic (involves less trial-and-error); but at least for some trinomials, like @DPRK’s example of 2x² – 11x + 5, I think the usual method is quicker and easier.
Now, since you’ve opened the door to anti-examples of things you weren’t forced to learn in school that would have been useful, I’ll offer another example of this: checking calculations by “casting out nines.” I never saw this in school, but I ran across it in a math textbook I happened to find in the library and found it to be a very useful way of checking paper-and-pencil calculations with big numbers (maybe not so useful today now that calculators are ubiquitous, but doing those calculations was a big part of the curriculum when I was in elementary school).
Here are some links that explain how “casting out nines” works:
https://www.chronicle.com/blognetwork/castingoutnines/math-monday-what-is-casting-out-nines
All of them. But luckily, I forgot them almost instantly.
I have dyscalculia, something not even heard of in my school days. I was told I just didn’t try hard enough and that anyone could do basic math. Well, not if you have dyscalculia you can’t. The part of my brain that deals with all numerical and spatial relationships is essentially a white emptiness with a faint hum.
Not only that, but since all I have is a kind of feeble memorization of skills, which degenerates over time, I am gradually but noticeable getting worse. I now need a calculator to add a couple of two-digit numbers without getting it wrong. I suppose eventually I’ll stop being able to tell time on an analog clock (which I mastered at age eleven).
People with dyscalculia develop strange little workarounds, like giving numbers colors or emotions, counting on their fingers under the table, and memorizing landmarks instead of directions.
At some point they explained how to calculate roots (realistically, square roots) of decimal numbers via “long division”. This might be of theoretical interest or good for some computer algorithm somewhere but otherwise not all that useful (may as well stick to your slide rules etc.) It was not really “required” though.
Yes, I vaguely remember learning that, and yes it is genuinely useless. Even if you didn’t have a calculator, there are more efficient algorithms for calculating square roots by hand.
Critical Path Analysis. Have never had to use it since.
I think we learned (proofs using) induction before geometry?
Calculating square roots with the long division method has kept me awake during boring meetings though - so it’s very useful