"Five added to itself once is ten", said the bear. yet not if you teach math it seems.
Nearly every mathematician defines mk in the sense of MIRA (math is repeated addition) as m added to itself k times.
I come across this mistake in nearly every math book for children.
mk is actually m added to zero k times, not itself.
For example I quote Hung-Hsi Wu# from the University of California who wrote:
"Sometimes we refer to mk as the product of m and k, and call m and k the factors of mk. We call attention to the fact that mk, the multiplication of k by m, is a shorthand notation for adding k to itself m times, no more and no less. Please be sure to impress this fact on your students."
Yet if we literally take 5 x 3 as described above, the product is 20 as 5 added to ITSELF once gives 10, then added to itself a second time gives 15 and added to itself a third time gives 20. Also 5 x 1 then becomes five added to itself one time to arrive at 10. The correct phrase five added to zero once (not itself) yields the correct product of five.
Nearly all books for children show 5 x 3 as 5 + 5 + 5, yet as any child can see, five has only been added to itself twice, as there are just two addition signs.
I believe mk is m added to zero k times and that children understand this to be right and their teacher (you?) wrong.
So when mk is m added to zero k times 5 x 3 becomes 0 + 5 + 5 + 5 which is correct as their are three addition signs and the answer is of course 15.
Let me give you another example from this forum where Jonathan Groves writes, "And multiplications such as (3/4)*(2/3) = 1/2 are hard to understand with the MIRA mindset because it does not make sense to add 3/4 to itself 2/3 times or to add 2/3 to itself 3/4 times." http://mathforum.org/kb/message.jspa?messageID=7350884&tstart=0
Adding 3/4 to ITSELF 2/3 times means the answer is 5/4. Adding 3/4 to ZERO 2/3 times means the answer is 1/2.
Math is hard enough for little children to understand when they cannot 'see' the same number of addition signs as there are 'times'.
So please from now on, explain that 4 x 3 means 0 + 4 + 4 + 4 and state that mk is m added to 'ZERO k times' not 'ITSELF k times'.