Keep in mind that, even in 1869, nobody is going to calcuate the cube root of 9.3 to five digits by hand, without consulting a table of logarithms. I tried it and it is… not so difficult, yet time-consuming and the chances of making a mistake begin to approach 100%.
So, why is it on the test? The answer is that it is a pure exercise supposed to test that the kids know what they are doing and can come up with and apply an appropriate method. Which brings up the danger that they memorized some algorithm without really understanding what is going on. [Feynman recounts a relevant anecdote.] Even today, it is up to the test writer to devise probing questions even while allowing them to bring along calculators and computers and what not.
Some years back, I was proud of myself for developing a method for approximating square roots that requires only additions and multiplications, aside from a single division at the end. I later learned that the method goes way back, and was possibly even used by Archimedes. Key to the method is that, given two fractions \frac{a}{b} and \frac{c}{d}, the fraction \frac{a+c}{b+d} is always somewhere in between the two.
I kill time by doing multiplication in my head. Up to the 12’s. 11x12 is 132, 7x8 is 56
Gives me something to do at the doctor’s office or waiting in line.
Funny thing. I memorized them in only one order. 6x8 is 48. I can’t tell you 8x6 unless I flip the numbers in my head.
I remember struggling to learn them and flunking a couple 4th grade pop quizzes. I was living with my aunt and uncle. He turned learning the tables into a game. I had them memorized in a couple weeks.
One thing that helped me. Get rid of the verbiage.
My uncle had me say, 3 4’s 12. 5 5’s 25
8 8’s 64. It was a game. I could answer 30 of them in under a minute.
I use arithmetic to turn off my alarm in the morning. I found if I just had to tap my phone to turn off the alarm, it was too easy to fall back asleep with no memory my alarm had even gone off.
Now I’m presented with a problem of the form a \times b+c where a and b are single digit numbers, and c is a two digit number. This is difficulty level 3.
Often my thinking is along the lines of 7 \times 8 + 17
What’s 7 \times 8? Well 6 \times 7 = 42, so +8+8=58+17=75 Blap! Blap! Blap! Try again. 9 \times 5 +12 good, some 5s, 5-1=4, 9-4=5 so 45 then 57. Silence…
I’m not exactly wide awake when I’m done, but it does engage me enough that I don’t fall back asleep and completly forget the alarm ever went off.
I teach elementary math, and the short answer is “yes, of course.”
That said, there’s much more emphasis on being able to solve unfamiliar problems than there was when I was in school. And that’s a good thing. It’s much more common in life to be confronted with a weird numerical situation that must be detangled than it is to need to convert 16/37 to decimal form without a calculator. Knowing what equation will give you the answer you need is a more relevant life skill than solving long division equations.
That doesn’t mean that kids shouldn’t learn facts. I’m a huge believer that by the end of third grade it’s super helpful for all kids to know their addition and multiplication facts with factors and addends up to 10, and to be fluent at deriving subtraction and division facts from those. Fourth Grade math fluency depends on knowing these skills by heart, much as fourth grade reading depends on knowing hundreds of sight words.
Just a quick (possibly unneeded) explainer. What we used to call “times tables” are now called “facts”. No idea when the change was made, or if it is just a regional or pedagogical quirk, but I learned “tables” and my kid learned “facts”.
Thanks–almost exactly right! I prefer “facts” because it can also refer to addition, subtraction, and division, which usually aren’t taught with tables. It’s helpful to know subtraction and division facts, but less crucial than addition and multiplication, since subtraction and division are pretty easily derived from addition and multiplication.
We just finished the IAR for grades 3-8. There are 3 math tests, and two of them have scratch paper that has to be collected and destroyed after the tests. So, yes, they do. The third test is much more conceptual, and calculators are allowed for that test.
I use math every day, and I got as far as, “hmm, they are both a little less than 1/2, so they are pretty close. And I don’t care enough about this random example that doesn’t come up in my regular life to do ny more work”. If I’d cared, I would have used the calculator on my phone. Or Excel, if I had it open. If I really needed to do it, I could have done it without electronics. But why would I really need to?
It is–but in my experience, a lot of kids learn multiplication via a times table, in which factors are the headers in rows and columns. That’s not a method used for learning addition facts. Such tables exist, but they’re rarely used, as far as I can tell.
“Tables” seems like it should refer to the way the facts are organized, rather than the facts themselves.
But “facts” or (“math facts”) is too vague and general. “The derivative of sin x is cos x” and “The millionth digit of pi is 1” and “The Euler characteristic of a dodecahedron is 2” are all math facts. Which facts are we talking about?
I not saying we didn’t learn arithmetic, we just didn’t sit there memorizing a big 10x10 grid of numbers. At least, I don’t remember ever doing it. Which… might be why I’m bad at math?