So what is that old subtraction anyway?

Most people old enough to be parents learned subtraction in the (what I think is poor) method with "borrowing" (or "renaming" as it became known, since it was never "given back"). Their work looked something like this:

Now imagine you asked an adult to explain what is going on. Better yet, ask 10 adults. I think that the phrase "you can't take 8 from 5" and "cross out the 4 and make it a 3", or similar phrases, will be heard. You will probably not hear "50 from 30" or "change 300 to 200". Why? Because people have been trained to think of subtractions such as this in terms of a gathering of individual digits and not 2- and 3-digit numbers.

For years the phrase "number sense" was bandied about, but not enough as far as I am concerned. Generations have been trained in "digit sense" while developing little in the way of number sense.  (I use the word "training" specifically because it connotes the sense of developing a skill or talent without the need to understand why you are doing it. When was the last time you heard of someone being trained to be a PhD?)

What is happening to the numbers in the above subtraction? Here it is written out in some proper mathematical notation. (Keep in mind that once you get (got?) past arithmetic, your math work was predominantly written line-by-line, one step at a time.)

\[\begin{array}{c}345 - 158 = (300 - 100) + (40 - 50) + (5 - 8)\\ = (300 - 100) + (30 - 50) + (15 - 8)\\ = (300 - 100) + (30 - 50) + 7\\ = (200 - 100) + (130 - 50) + 7\\ = (200 - 100) + 80 + 7\\ = 100 + 80 + 7\\ = 187\end{array}\]

If we can accept for the time being this line-by-line approach as having some legitimacy, consider this:

\[\begin{array}{c}345 - 158 = 345 - 100 - 50 - 8\\ = 245 - 50 - 8\\ = 245 - 45 - 5 - 8\\ = 200 - 5 - 8\\ = 195 - 8\\ = 187\end{array}\]

I include this because it is different, perfectly correct, and more easily followed mentally

My method for doing this subtraction is like this:

\[\begin{array}{c}345 - 158 = 245 - 58\\ = 195 - 8\\ = 187\end{array}\]

By having taught myself years ago to work with numbers left-to-right, the same way we say them, I can claim that I haven't "borrowed" (or "renamed") while doing a subtraction in decades.

If all you know is the handwritten version above, odds are you will find few shortcuts and will stumble doing subtraction mentally.  If you find yourself at a checkout counter with a clerk who has trouble making change, especially when you hand over a twenty, two quarters, and a dime for your bill of $19,58, consider that the clerk, who is quite possible younger than you, has only the "borrowing" method to use, which is very very awkward to do mentally.