Dunno. I don’t recall my kids (or me, for that matter) staring at tables of numbers. We played “schoolhouse rock” to help the kids learn multiplication. I think i just had to do a lot of multiplication problems until it sank in.
My dad had an HP35. It became mine when he got his HP65. We were allowed to use calculators in trig class (circa 1974-5). Everyone else had basic four function calculators, I had my dad’s HP35. The teacher (an elderly priest) was so fascinated with it that he let me use it during tests. Needless to say I ran through the tests in about a tenth of the time as everyone else (no tables!).
This was a very big deal in college, particularly in freshman physics lab. You would get docked pretty severely if you gave too many significant digits in your lab reports, This was in the days before scientific calculators had rounding to a set number of digits built in.
When I was in 4th grade in 1944-45, there was a 12 x 12 multiplication table displayed on the side of the room. We were given problems to do and long before the end of the year, we had all memorized that table without having consciously tried to.
Here is an interesting take on square roots. I am certainly familiar with the method of averaging the trial with the quotient as described above. But once I bought a compiler (for FORTH if you must know) that had a built-in square root algorithm included, based on that averaging method. But it used a lot of divisions and division was painfully slow on the 8088 chip. So I programmed a square root process based on the algorithm I learned in 7th or 8th grade and it was dramatically faster. I sent it back to the programmer and it was incorporated in future iterations. It really uses only doubling and subtraction. Of course you have to guess the next digit, but in binary you always guess the next bit is 1 and then see if it is too large in which case the next bit will be 0. The procedure is much easier in binary.
Oh, yeah. That’s not how times tables were memorized. I graduated in 2003 and we did have a grid, but it was just a tool to help you before you knew them. You could put one finger on the first number, and another on the second, and the answer would be where they met.
The actual memorization was just by doing problems. We didn’t do that weird thing I’ve seen in old movies where they have to recite the table, e.g. “3 times 1 is 1, 3 times 2 is 6, 3 times 3 is 9” and so on.
I would say that, if you can do, say 1435 x 6 on paper without a calculator or other instrument, you have your times table memorized. I suspect most of us could even do it with both factors having multiple digits.
How certain are you of that “we had all”? Unless your classroom did not reflect America as a whole, it seems pretty likely that a lot of kids did not have them all memorized. It’s hard for fourth-graders to have a good handle on what their peers have learned, especially when they’re able to learn something without consciously trying to.
I think I defined it above. Keep in mind that words can have multiple definitions. The “facts” that I’m talking about, again, are those addition and multiplication equations that contain exactly two addends or factors that are whole numbers between zero and ten.
There are a lot of techniques.
One practice tool is filling out a blank “times table.” A lot of kids freaking love this kind of busywork, despite what hippie teachers would have us believe, and it’s a good way to build those neural pathways.
Another technique is the timed drills, a hundred or so facts in random order and you try to complete as many as you can in two minutes. This works for a lot of kids, but some kids have a meltdown under time pressure, and you gotta be ready for that.
Another way is a progress chart, where you work your way through the facts factor by factor, and when you know all the (say) six facts (i.e., 6x0, 6x1, 6x2, 0x6, 1x6, 2x6, etc.), you get a star on a chart, until you have them all memorized. When using this technique, be sure this doesn’t become a source of public humiliation for a kid who’s struggling to proceed at the same rate as other kids.
Another technique is to practice deriving them from “friendly facts”: if you know 10x6 is 60, how can you use that to figure out 9x6? This is really helpful for reasoning, but it’s slower than memorizing 9x6, and you want them to get the fact memorized eventually.
Another is to learn stupid mnemonics for the harder facts: “Five, six, seven, eight! Fifty-six is seven times eight!” When using this technique, make sure kids know how dumb it is–they should be giggling about using these, not thinking they represent some actual mathematical truth.
I am also very familiar with the 12x12 multiplication table. It was a regular part of the math curriculum in the late 70s / early 80s in my school.
I remember misbehaving during lunch once and was put on detention for the period. The punishment was filling out the 12x12 multiplication table multiple times. It is pretty well baked into memory now just don’t ask me to multiply by 13 or higher!
I’ve definitely seen kids do it. When I was subbing in an advanced 5th grade class, though, they were watching a Eureka Math video that made everything seem way more complicated than it needed to be. I didn’t think what the video “teacher” was doing showed anything about the “why”. The students said the same thing, even though I’d kept my mouth shut about what I thought. I’d seen Eureka Math used with younger kids and thought it did a good job, but this was ridiculous.
Yeesh, one and only one student with a calculator with trig functions, on a trig test? That’s not even remotely fair.
Probably more useful in the long term, though, because the techniques you use for those can also be used for more complicated problems.
My mom (a retired teacher) still volunteers in elementary school classrooms, and assessments on addition and multiplication facts are usually a big page of a bunch of problems in random order. She encourages students to go through the page and first do all of the easiest ones, then the ones they’re less sure of, and so on.
If life were fair there wouldn’t be rich people.
Not that we were rich, my dad was just a gadget-happy engineer.
And yes, I was pretty amazed myself - but I would have gotten 100% anyway, just would have taken a little longer. Even without the HP35 I always had time to check my answers on math tests.
I look on it like learning phonics and memorizing sight words. Both are indispensable. Deriving the facts from friendly facts, or from scratch, is really important. But if you’re in high school trying to solve quadratic equations or whatever, and as part of the process you’re still trying to derive 8*6, you’re going to be unbearably slow. It’d be like trying to read Native Son, and having to sound out every word you hit.
Oops.