Multiplication tables: up to what number should I memorise?

Question

I never memorised them and suffered immensely through school as a result. Eg. I calculated 7x8 as (7x10=70) minus 7 using my fingers (63) then minus again on my fingers (56).

In Australia children are encouraged to memorise multiplication up to 12 (ie. 12x12, 6x12 etc.)

But why stop there? Would it be helpful to memorise them up to say, 50? Or do good mathematicians find it even easier to calculate an answer like 32x16 rather memorise it?

Answer 1

$32\times 16$ is easily $512$ because I memorised the powers of $2$. That being said, I have never actively memorised the full multiplication table above $10$, and I get along nicely. $11$ comes practically for free, though, and I multiply so often by $12$ that I basically know that one as well.

Above that, it's mainly mental arithmetic and neat tricks for me. For instance, because of $$ a(b\pm c) = ab\pm ac $$ I can calculate basically anything multiplied by, say, $47$ quite easily because I can multiply it by $50$ (multiply by 100 and halve), and I can multiply by $3$ (the hard way, if I must), and then subtract the two.

Using tricks like this, I most likely spend a lot less time doing the calculations that I need when I need them than I would've done memorizing (and retaining) any bigger multiplication table.

Answer 2

Up to $12\times 12$ is good. $12$ has lots of factors. Past that, the next few numbers don't have so many factors. Why do factors matter?

I have a small gig tutoring 6th and 7th grade "at risk" students. Most of them can't add fractions. The first hurdle in adding fractions is that given

$$\frac{5}{42} +\frac{7}{48}$$

they need to find a common denominator. Since they don't have their multiplication facts memorized, the numbers $42$ and $48$ mean nothing special to them. (This is my answer to people who say, "We have calculators now, so why memorize multiplication tables?" Answer:
Because when you see $42$, you need to think $6\times 7$, or you'll play hell trying to add fractions.)

So being able to quickly factor smallish denominators is a useful skill. Working the above addition of fractions is extremely painful if one has to stop and tediously factor $42$ and then $48$ and then figure out what the LCD is.

(Alternatively, the students can multiply the first fraction by $\frac{48}{48}$ and the second fraction by $\frac{42}{42}$. But then we have the same problem. The answer is a fraction in unreduced form, and they still have to factor.)

(Smooth segue into rant:) This is what happens every time someone at one level of math decides that some bit of math isn't useful anymore. Logarithms are the classic. "Now that we have calculators, we no longer need log's, so let's cut them out of the high school curriculum." Now our Calc 1 students don't know log's and ask questions like "I got $\ln 1/2$ but the back of the book says $-\ln 2,$ what did to wrong?" Answer: You went to the wrong high school.