This semester I'm teaching a course to prospective middle school teachers (grades 5 - 8 in the US), and I've been drawing upon some history of elementary mathematics to design lessons that will enrich the students' understanding of the topics being studied and that they will one day teach.
In connection with introductory algebra, I thought I would draw upon some early developments in algebra, so one topic I looked at was Egyptian algebra. As I was investigating this more closely to teach it to my students, however, I was led to question the standard way of viewing what's going on in the Rhind Mathematical Papyrus (RMP) with respect to algebra. My questions were prompted by something van der Waerden says in his Geometry and Algebra in Ancient Civilizations on page 161. I'll first summarize the standard interpretation, then give my response based on vdW's remarks.
Problem 26 in the RMP says: a quantity and its fourth added together become 15. What is the quantity? The solution goes as follows (calculations omitted): Calculate with 4: 4 + 1 = 5. Calculate with 5 to get 15: 15 divided by 5 is 3. Multiply 3 by 4: the answer is 12.
In explaining this process, most say it is one of false position. You assume the (possibly) false value of 4 and try it out. It only gives 5, not 15, so you determine by division what you need to scale this amount by in order for it to be correct. The scale factor is 3, so multiply your original guess (4) by this amount, giving 12 as the answer.
van der Waerden notes that Tropfke in a 1980 history of elementary mathematics (in German) gives another explanation, which seems borne out by the process itself. Rather than taking 4 as an initial guess, 4 is taken as the part-size (since fractional fourths are involved). Then the sum 4 + 1 = 5 tells you how many fourth-parts you have in total. Since these 5 parts must give 15 in all, each part must be of size 15 divided by 5, or 3. The original quantity was 4 fourth-parts, so it must be of size 4 x 3 = 12.
No false position occurs here at all; it's just quite straightforward calculation with fractional parts and part-size determination. In support of this being the method of part-size (my term) rather than the method of false position, vdW notes that the scribe takes four threes to get 12, not three fours (which would be the natural product to take if you're scaling the original guess of 4 by a factor of 3).
Looking at the resources I have at hand, I'm convinced by what vdW/Tropfke says on this, but then I'm no expert on Egyptian history of mathematics. As I noted above, most historians of mathematics talk about this in terms of false position (e.g., Gillings, Boyer, and Katz all do). So I wonder whether others on this list have thought about this and what they might have to say. Treating this as the method of part-size makes the process more elementary and more transparent to students who are studying it, it seems to me.