My 3 year old can count backwards from 10 - although he thinks that the correct mathematical term for -1 is "blast-off". I'm sure high-level math would benefit from renaming -1 in this fashion. What is the square root of "blast off"? That's irrational!
I have been trying to teach my 5 year old daughter basic addition and subtraction- best done using food. I gave her 6 chocolates, and she 'subtracted' (i.e. ate) each one in turn...
"6 chocolates minus 1 chocolate is?"
"5 ..4 ..3..2..1."
After eating the last one, i thought she'd immediately say "zero" but she said "No chocolates left". Maybe she's right... after all, 1 apple is clearly different to 1 orange but is/are zero apples really the same as zero oranges?* This seems incorrect to me- clearly there is an assumption with a mathematical sequence of apples that you're not 'allowed' to suddenly stick in a watermelon or an elephant. All those places seem to be reserved for apples.
In other words, are there many different 'zeros'?
Maybe we should we be calling the empty space left on the table after the last chocolate has been eaten as "The Empty (Chocolate) Set"... i.e. a virtual space reserved for a number of chocolates- but nothing else. Apparently, however , there is only one "empty set"... so that scotches that idea. This is the problem with D.I.Y mathematics- particularly when you're not that mathematical.
I got to thinking about all the stuff i learned in primary school about numbers.. how we learned how to do (short) division... but in fact only really learned how somebody ELSE did division. It's not as if any of us had any idea why we were "carrying the one" anywhere.
According to wikipedia- here is the REAL explanation: The procedure relies on the division algorithm, which states that given any two integers a and d, with d ≠ 0, there exist unique integers q and r such that a = qd + r and 0 ≤ r < |d |, where |d | denotes the absolute value of d.
You manipulate the symbols in a particular way- hey presto. Ditto for calculus - getting good marks in no way required you to understand what you were doing. In fact, it probably would have been distracting. Leibniz and Newton would roll in their graves.
5th grade geometry: "Parallel Lines Never Intersect"... except in the real world!- and that is why if you walk 400km north, then 400km east, 400km south and 400km west you will very likely end up far far away from where you started. I've always been interested in the idea that, when placed on a map "North & South" seem analogous to "East and West" as if they were equivalent to Cartesian y and x. Walking "South" is not like simply walking East in a different direction.
Luckily, when you get off the surface of the earth and into deep space, the universe really does seem to be "flat" -according to data from the WMAP satellite. There was some suspicion that it was actually a 4-dimensional doughnut shape (no joke) but this apparently has been discredited now much to my disappointment.
* also, why is it correct to use a plural when talking about "zero" somethings???