In "The Math Connection Group" there is a discussion on teaching the addition of fractions, and here is my comment on it which I thought might be of interest to the Math History List; the link to that discussion is hereunder:
"I do agree with the opinion that the students should understand the "why" of any Math formula, i.e., the first-principals that feed its notion.
In this regard, in ancient Egyptian mathematics, fractions were observed as parts of the circle, i.e., parts of cake or bread. The so-called Rhind Mathematical Papyrus includes great deal of information on fractions. The subdividing-image(s) is the basic principle behind the concept of the Ancient Egyptian fractions.
Assume that 3/3 is the image of subdividing the circle (bread) into three equal parts. The "denominator" is the number of divisions, and the "numerator" is what is taken out or remained from them. 1/3 is one of the three parts, and 2/3 is 2 of the 3 that are either remained, or taken out (i.e. perhaps was eaten). Similarly, 1/5 is part of the subdividing-image 5/5.
If I would like to add the parts that each belongs to different subdividing-image, e.g., 1/5 to 1/3, I need to combine their images: 3/3 and 5/5, into a unified one that its part divides both, by multiplying the two images [3/3]*[5/5] and get 15/15. That is 1/15 divides 1/3 into 5 parts and divides 1/5 into 3 parts. That means in case of diverse and not the same subdividing-image, the product of addition or subtraction will always be part(s) of a new combined subdividing-image.
If one say adding 1/3 + 1/5 equals (1*5+1*3)/(3*5);
It meant also he multiplied each of the two fractions by their combined image before adding them, e.g., [(15/15)*(1/3)] + [(15/15)*(1/5)].
Of course using a formula is the short way; any simple why-explanation should be narrated as a prelude."