Yes, indeed it does. It just provides a little more detail about how it will be used in “later math” (and maybe not that much later, depending on how the curriculum is structured). And naturally leads to questions about the value of the “later math” it leads to. (e.g. “What’s the value of solving quadratic equations?”—part of the answer to which is that it’s used in yet later math.)
When I was a waiter in college, we didn’t have fancy computerized machines to total the bill. We’d write down the items and prices on a table ticket, then we had a couple of adding machines to total them up. I prided myself on my ability to do it all in my head, including sales tax. Of course, it helped that sales tax was exactly 5%, but still.
And in my head, I would not use the old-fashioned method where you carry the one. Most of the prices were $X.95, which lends itself to the “new math” methods. That’s what you can do if you develop a good “numbers sense” in school, and that’s the whole point with the new math methods.
Oh yeah, I agree. I wasn’t slighting your answer 
But that doesn’t require the teaching of Algebra, let alone Trigonometry and the rest of Pre-Calculus. Geometry (the course I specialized in) is very good for that sort of thing, and has the advantage of having a lot of immediate, real-world applications. Probability and Statistics also works well for doing that. But, frankly, so does ANY mathematics course, properly designed and rigorously taught.
There are immediate, real-world applications for the Angle-Angle-Side Theorem?
j/k
Carpentry.
I was only joking. But since you mentioned it, carpenters have to prove that triangles are congruent?
Yup. I’ve said before that, for most students, high school geometry is the first (and usually, last) math course they ever encounter, and a lot are caught off guard by how different it is from the other things called math courses. There ought to be a lot more of that in other math courses. And there are a lot of other subjects in math that could do just as good a job at teaching logical thinking, while possibly holding students’ interest better, such as the “Queen of Mathematics”, number theory.
We’re not likely to be able to get trig replaced by number theory, on the school curriculum. But we can teach trig (and geometry, and arithmetic) more rigorously, and we should do that.
I was thinking of the patterns on doors and constructing molding, considering a square as two triangles.
Seems to me, the method illustrated in the OP is good for indicating conceptually what subtraction is (how far from here to there; what do you add to this to get to that), and it’s basically about as efficient as whatever traditional borrowing algorithm you have in your mind, and there’s no particular point trying to drill children into perfectly optimized rote calculating robots anyway (among other things, actual calculating robots will always outperform them…), so what really matters for children is building conceptual understanding, so…
An IMHO response to a GQ:
On the contrary, IMHO, rote memorization of several dozen math facts is crucial to developing “number sense” and ultimately being good at “mental math” and calculation in general.
Neither my 14-year-old nor my 10-year-old have had to memorize multiplication tables. Accordingly, both struggle with multiplying small numbers (though my daughter turned the corner during middle school). To be good at math, IMHO, you need to be able to look at, say, 9 and 9 … and, depending on the symbol in between them, reflexively and instantaneously come up with 18, 0, 81, or 1.
That lack of rote learning is why I consider circa 1980 math instruction decisively more effective than the 2010s math instruction my kids have received. At least for leading students to master small-number calculation.
Just my opinion, no cites or links.
As a counterpoint, I had to learn the times tables through 12 by rote when I was in third grade and my ability to do mental math is less than stellar.
You’re history’s greatest monster! There is nothing worse than a number line! Its slow, clumsy, and annoying. You should be ashamed for teaching it. 
I prefer pebbles in a circle drawn in the sand. “You have four pebbles. Now, take away two pebbles. How many pebbles do you have? No, not counting the two pebbles in your hand. Yes, I know they’re still your pebbles, but I’m trying to teach you a concept here. Oh, fercryinoutloud, just play along, okay?”
And yes, I hate children.
I was sick for the 8s and 9s and I never caught up.
Quick: How much money is 7 quarters?
I’m guessing it didn’t take you long to say $1.75. I’m also guessing you didn’t determine this by calculating 7 x $0.25 = 7 x $0.2 + 7 x $0.05 = $1.4 + $0.35 = $1.75. Nor did you recall this by having memorized your multiplication tables all the way out to 25s. No, you knew this because working with quarters is something you’ve done often in your life (at first not through memorized multiplication, but through plain addition or other such reasoning). This particular calculation has come up often enough that you didn’t have to work to memorize it; it just happened automatically. Great!
Now what’s 7 times 17? Probably not on the tip of your tongue. It doesn’t come up so often. But you know what? You were still able to figure it out pretty quickly. The fact that you don’t have it memorized is of no consequence to you.
The simplest way to develop number sense, TRUE number sense, is not to worry about memorizing your multiplication tables. There is no task for which this is genuinely useful, other than the contrivance of elementary school tests on memorization of multiplication tables. You can understand and reason about multiplication perfectly well and perfectly efficiently without having memorized the decimal representations of the products of the particular first ten or twenty or however many natural numbers (and, even if you aren’t very efficient at this kind of mental arithmetic, no one at the restaurant will judge you any worse for using your phone to figure out the tip. Really. Do keep in mind, we invented the electronic calculator for reasons other than to be able to forbid its use.).
Those multiplications that it might be genuinely useful to have at your ready disposal because they come up frequently… well, you will eventually memorize them without trying, because they come up frequently. Those multiplications that don’t come up frequently enough for that to happen… well, they don’t come up frequently enough to be worth memorizing, either. Time and effort wasted on memorizing multiplication tables is time and effort which could have been better spent on other skills, whether within mathematics or elsewhere. It’s only historical inertia and misplaced paranoia which keeps us locked into this obsolete curricular paradigm.
Are your children not able to sit down and write out multiplication tables for themselves even though they haven’t memorized them? That’d be worrying! If not, then, no biggie. It concerns me about as much as failure to memorize logarithm tables or sine tables or square root tables or…
Geometry serves two conjoined purposes, one of which was hinted at by Chronos in his colloquy with me upthread a bit.
First, by treating Geometry as a theoretical course in mathematics, derived through the establishment of certain basic theses (postulates) and the application of deductive reasoning, we teach the student how to think logically. A good Geometry teacher is one who shows how inductive reasoning can be used to derive basic Geometric principles, and deductive reasoning is used to confirm the induced conclusion.
As an example, we can measure a whole bunch of different rectangles, and reach some inductive conclusions about rectangles as a whole. Examples would be that they all have opposite sides of the same length, opposite sides that appear to be parallel, opposite angles that measure the same, etc. We then apply deductive reasoning to determine that each of these is true about theoretical rectangles. (In the process, we might well make reference to the Angle-Side-Angle Theorem, of which Angle-Angle-Side is but a corollary).
Second, by treating Geometry as a practical course in applied use of shapes, we can put our deductions to good use. In my tests, the final question or two always involved application of some part of what the prior lessons had been discussing to a real-world problem. Thus, the problems addressed things like surveying, mirrors (which are NOT intuitive as to how they work!), buildings, indirect measurement, etc. In applying Geometry, you don’t prove the Angle-Angle-Side Theorem, but you certainly use its properties to accomplish a LOT. For example, determining location through “triangulation” is an application of ASA triangle congruence. When I then describe for my students my summer job reducing filmed weapons tests at China Lake NWC (as it was then known) to numerical data so the base analysts could know what the missile did, and note that this is basic application of ASA triangle congruence, … 
One of the sad things about most high school Geometry classes is that the teachers rarely understand exactly how all this should be put together, because most high school math teachers get into teaching math because they loved Algebra. Understanding applied Logic is not high on their list of interests. 
You are wrong, but in this day and age of electronic calculators, it may be a moot point.
The purpose of knowing math “facts” is to be able to use those facts in an agile way when you are doing other calculations. For example, if you have to calculate 4368 by 235, the old-fashioned laborious pencil-and-paper way, knowing math facts like 3 x 8 and such makes accomplishing the task so much easier. When I was a kid, I did endless calculations by hand, because there WERE NO CALCULATORS to use. For example, I had a “league” that I created of fictitious players of a game I got out of a Pop-Tarts box (as I recall, you used a deck of cards to “bowl” by turning over cards and finding out the result of each roll of the ball). I hand-calculated averages, which involved a LOT of old-style division, and trust me, you want to know your times tables quite thoroughly doing that!!
To give a more “normal” example: when teaching Algebra, as discussed upthread, we have to try and teach the factoring of trinomials (quadratics). One of the most consistent barriers students run into in doing this is that they don’t know their math facts. So if they see a quadratic like x^2 + x - 56, they are dead on arrival because they don’t know that 56 = 7 x 8. God help them if the quadratic has a != 1 .
So you see, math facts aren’t just something we use for the purpose of doing worksheets in elementary school. They actually have a purpose. They make you more numerant literate. And they also help you be more aware of what answers SHOULD look like, alerting you to probable wrong answers. Thus, if you are looking at 8138 ÷ 92, a person who knows their math facts would realize the answer MUST be around 90. You’d be surprised how often modern students have no clue whether or not the answer that pops up on the calculator app is correct.
Honest-to-God true story: When I took Geometry in HS we had a substitute teacher, who was the head of the Math (singular, dammit!) department. He was so used to teaching Calculus (no “the”) that he couldn’t make hears or tails out of our text. Which he had written.
Having learned as little Math as I had, I was at a disadvantage when folks started calling me an engineer. I needed to figure out the area described by an irregular curve, so I redrew it 20x larger, divided it into many rectangular slices and treated the leftover bits at the top as simple triangles. Calculus woulda helped, but I didn’t even know that was why Newton invented it.
I was about to make pretty much this exact reply to Indistinguishable, until I saw that you had already done so. Factoring polynomials is an excellent example of the kind of thing that is much easier to a student who has a good grasp on basic multiplication facts than to one who doesn’t.
Indistinguishable also seemed to be addressing the issue of not just whether but how to commit such facts to memory, and to be favoring the “use the facts until you automatically have them memorized” approach over the “sit down and deliberately memorize them” approach. I love it when things sink into my memory just as a result of using them, and I’m all for that approach when it works. But is it going to work sufficiently well for every student in every circumstance?
Everything I said is taking as a given that we live in a day and age of electronic calculators. It’s true, I would take a different attitude in the past. My position is that this thinking is needlessly stuck in the past.
As I said,
and I will also note that no one will ever be without a readily available calculator again without bigger problems like “How do I get off this desert island?”.
Why is it that we are perfectly well accepting of students getting by using calculators for SOME functions (square roots, sines, whatever), while still developing a sense of what does and doesn’t make conceptual sense, yet find it very very important that they not be allowed to use calculators for OTHER decimal arithmetic calculations (multiplications, whatever), or to do them out by meaningful hand instead of meaningless memory (multiplying 3 by 7 as 7 + 7 + 7 if need be, as they would with exponentiation 7^3)? It’s all arbitrary. It’s just historical inertia.