Part of a conversation that I had at the weekend with a friend of mine who is a Maths Teacher was on the subject of kids today not being able to do basic arithmetic.
I work as an accountant; so my entire job is nothing more than applied arithmetic and from an historical perspective, the first and most common use of mathematics in the first place. Arithmetic is the basis for knowing how much stuff there is, how much things cost, how much money you need, and how much tax you need to pay.
But to be honest, arithmetic is really nothing more than applied token moving, where the tokens are mathematical symbols which stand in for small values. Other than the fact that we have 10 digits, there is no inherent reason why we use base 10, when many other bases like 12, 20, or 60 would have been more useful.
One of the useful tools in the arithmetic toolkit is the "Friends of 10". That is, those pairs of numbers which when added together give you 10; such as 1 and 9, 2 and 8, 3 and 7, et cetera et cetera et cetera.
Card players of games like King Tut will notice a similar thing when we look for pairs that add together to give you 13; such as, A and Q, 2 and J, 3 and 10, et cetera et cetera et cetera.
Remember, as an accountant who does nothing other than play with arithmetic and fitting numbers into the most boring Sudoku puzzles (otherwise known as Tax Returns), I have a lot of time where I am putting in hundreds of journal entries but where my mind is off playing with concepts that I can not leave well enough alone. The Friends of 10 is one of them.
IF (big 'if') there are Friends of 10, then there must be Friends of 100, Friends of 1000, Friends of 10000, et cetera et cetera et cetera. If there is a single use case in mathematics, then invariably it follows that there is a general rule and one that applies in all bases.
Armed with the fact that flip digits in sets of accounts give you an off balance which is a multiple of 9, then I reasoned that the general case must be all of the complementary pairs that add together to give 9, plus 1, because that extra 1 is needed to trip the place register and give you a new number with place value.
Consider:
ABCDEF +
abcdef
--------------
Quite literally, pick any number you like.
507,589 will do. Its friend must be:
492,410 + 1.
5 and 4, 0 and 9, 7 and 2, 5 and 4, 8 and 1, 9 and 0. Plus 1 to trip the register.
A plus a, B plus b, C plus c, et cetera et cetera et cetera ad nauseum, will always add to 9, for numbers of any magnitude.
Of course, this is related to the old accounting trick of Casting Out Of 9s, where the Digital Root of natural number in is the single digit value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. You can then do other fun things in the various operations to check that your result is at least sensible.
As there is nothing inherently special about the number 10, then it follows that for all bases X, that everything above should also work for X-1. Remember, 9 is just the X-1 of 10.
Again, Friends of 10 isn't particularly special. 7 plus 3 does equal 10, but the underlying general case suggests that 7 plus 2 is 9 and plus 1 trips the register.
If I can think about this during the idle monotony of entering hundreds of journal entries, then it isn't necessarily because I have a brilliant mathematical mind but rather that the rules of arithmetic are in fact simple enough to discover. Being good "at Maths" at least when it comes to simple operations, is literally just the applied application of those simple rules. What this suggests to me is that primary schools likely need to do a better job at drilling boring maths problems.
Not being able to do basic arithmetic is likely not a function of being good or bad at maths, but as a result of not doing enough boring problems. These are the 'put with', 'take away', 'rows and columns', and 'groups of', tasks which are really nothing more than applied token moving. Make kids play with beads, or little people, or coins, or tokens. Make them play more in the real world, so that they can play in the abstract. That's what I did and I'm nothing special. I am an accountant.
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