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Drat! Messed up the formatting on the first diagram. The first three circles should be moved over to the right to be underneath Ellen’s set.
Thank you so much for taking the time and effort to craft such a thoughtful, detailed response. I appreciate the insights into some of the decision-making in writing the standards, and I am grateful for the opportunity to deepen my understanding of the comparison situations. This is such a rich topic for learning!
To address your questions:
1) Apologies, I should have been clearer. Children’s Mathematics refers to the first CGI book.
2) My thinking around the difference being “assigned” (maybe not the best word here) to a set is connected to how I expect children would directly model the situation, and also what is linguistically necessary to fully describe the situation. Let’s consider a comparison where Bill has 5 pizzas and Ellen has 2, so the difference is 3. To write a clear problem situation, the statement of difference must be made in relation to a particular set. If Bill has 5 pizzas, (and Ellen’s set is unknown), stating the difference as simply “3 more” is ambiguous. I have to name a set as the subject of the difference, “Bill has 3 more,” before I have enough information to solve the problem.
I see this – the “perspective of the difference” if you will – as influencing how children are likely to try and model the situation. In a comparison problem where one of the sets is unknown, a child might construct the known set and the difference quantity. But what to do with the difference? In “Bill has 5 pizzas, and Ellen has 3 more than Bill,” the difference information is in relation to the unknown set (CQU). In these situations, once the known set and the difference have been constructed, it’s a little easier to see what do to solve the problem – place the “3 more” with the unknown set (as the language in the problem suggests), and then match Bill’s set.
Bill’s set Ellen’s set 0 0 (difference: Ellen has 3 more) 0 0 0 0 } match this 0 0 However, when the difference information is in relation to the known set (CRU), “Bill has 5 pizzas, he has 3 more than Ellen,” it’s not as clear what to do with the difference to solve the problem. A child might try to put together the “3 more” with Bill’s set, rather than seeing that the “3 more” is, in one way of thinking, already contained within Bill’s set, and either putting the 3 more on top of Bill’s set and matching the remaining part of Bill’s, or just setting aside the “3 more,” seeing it within Bill’s set, and matching only the remaining two.
Bill’s set Ellen’s set
0
0
0
0 } match only this part of Bill’s set
0 }Thus, (in my thinking) it is the subject of the difference information in the problem that makes the two comparison situations feel different to a child (though a child might do so, I don’t see that s/he will necessarily need to think about which of the two sets is larger to correctly solve the problem). It is interesting to note, however, that both the CCSS and the CGI classifications would be the same for the two problems described above (if for different reasons). It is only when switching to a “fewer” situation that the results of the two classifications would be different. I need to think more about why this is so and what significance it might hold…
Of course, there are other factors that are going to influence how a child might solve the above problems. These are particularly difficult problems to direct model; I would guess that by the time most children can solve these problems through modeling they are likely to have also developed some number facts and part/whole thinking that they might draw upon in making sense of these situations.
Again, thank you so much for the opportunity to broaden my understanding of the CCSSM problem situations and to share my thinking. This will be of great value to me in my work with teachers.
