I desagree completely. It's not a trick. This is exactly how multiplication works. The idea behind 2x3=6 is
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In this method the number are decomposed using it's decimal representation, so 23x12 = (2 * 10+3) * (1 * 10+2) = 2 * 1 * 10^2+2 * 2 * 10 + 3 * 1 * 10 + 3 * 2 = 2 * 10^2+ (4+3) * 10 + 6 = 2 * 10^2+ 7 * 10 + 6 = 276
(I'd like to use bigger lines for the dozens.) This is exactly what happens in the method. See: http://imgur.com/S5nOh
If I had to use that in a class I would first use the "all graphical" representation, then the "mixed" representation and finally the "algebraic" representation. The lines are still there, but almost invisible.
I understand that most of the times nobody should use any hand method to multiply two four or five digit numbers. But some properties of the hand method are important, for example:
* Why to use approximated calculation you use the first digit and no the last digit?
* Why is possible to calculate the last digit of the result using only the original last digits of each number to multiply?
* How is this method related to polynomials multiplication?
* How is this related to the casting out nines check?
* Can you imagine the multiplication of a large number by 2 using this method? 3?
Most of these topics are not explored in a usual K-12 math course, and perhaps it's a good idea because some of them are a little tricky. But the proof of how this method works lies in the structure of the decimal representations of the numbers and the algebraic relations between the sum and multiplications. I think that for small children a graphic method like this one can give some insight of these properties, without all the details and formalizations.