Asking for methods to produce the sum of natural numbers from the disjoint union of sets, it seems that the obvious way is to use the general definition, as coproduct, of the sum of sets. The accepted answer in that question suggested to use pairing with a "label", but the coproduct just needs a pair of "disjoint" injections, for example here in "Sets for Mathematicians", chapter 2:

Lawvere Sets

Amusingly I have found a textbook in Spain, around 1974, that show the required injections.


(I found another one https://i.stack.imgur.com/j0QmF.jpg EDIT but in this case the author confirms me that the intention was just to show the idea of "increasing by one").

As you can see, it is very like the examples from Lawvere category of sets.

And now I am puzzled if the Spanish textbooks could have originated from some foreign, probably USA, material for primary school. Is this kind of representation, or drawing, used in New Math textbooks? Do you know any?

  • $\begingroup$ I recall that my first grade (ages 6-7; this was 1965-66) mathematics book --- which I still have (and saw a couple of years ago while moving to a "new" house, but I don't know where it is at the moment) because it was a workbook that one writes in (rather than the school-loaned books I used after the first grade) --- has lots of arrow diagrams somewhat like this (mostly for 1-1 correspondence of objects for how to count things), although nothing this complicated. For your question, I think this was all very much "in the air" in most places and thus things like this were (continued) $\endgroup$ – Dave L Renfro May 15 '18 at 9:39
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    $\begingroup$ probably independently developed/created many times. That said, this all strikes me as super-silly for young children, and a diagram like this for adding 4 and 5 seems like something that should only be in a teacher's manual, and not something a student should see. Indeed, it seems to me that one has to have a very good understanding of what addition is just to understand the relevance of this diagram, which is putting the cart before the horse. $\endgroup$ – Dave L Renfro May 15 '18 at 9:45
  • $\begingroup$ I'm not entirely clear on what is being asked; there are mentions of coproduct, Lawvere, Spanish textbooks, and SMSG... But, it seems the base question is whether there are texts where addition is represented as in the picture? In this case, I agree with @DaveLRenfro 's guess that it would've been "probably independently developed/created many times." Anyway: I'm not sure what a satisfactory answer would look like. $\endgroup$ – Benjamin Dickman May 15 '18 at 21:24
  • $\begingroup$ @BenjaminDickman Well, a satisfactory answer would be any pre-1970 textbook, or teacher material, for primary school, using this same kind of diagram to represent addition with sets. It would still be better an authoritative source -say, Papy?- telling that it is a valid construction. $\endgroup$ – arivero May 16 '18 at 0:02
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    $\begingroup$ This all seems too abstract for such young children. The X's should be recognizable and distinguishable objects from their daily lives. The "sets" should be something like objects in boxes or other kinds of enclosures or sub-categories IMHO. $\endgroup$ – Dan Christensen May 16 '18 at 2:44

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