The most important mathematical concept adopted in Tonna's procedure is that by multiplying or dividing the same number in both numerator and denominator, the value of the fraction is unchanged. This is conceptually simple and student should know very well already ( at least by dividing the same whole number from both numerator and denominator when they do the reducing of fraction ).
This concept and its extended version: operate the same operation and number on both side of equal sign, the equal sign is still valid. are also the most valuable ones in developing the algebraic manipulation algorithm. The beauty is that both these basic concepts can be very easily demonstrated and proven.
This is may be the reason why it "is" (or "was"?) the "standard" algorithm been taught in the "old" classroom world wide.