2013 Conference Proceedings - University of Nevada, Las Vegas

6+7 4+9=6+(6+1) Substitution =4+(10-1) Substitution=(6+6)+1 Associative Property =(4+10)-1 Associative Property=12+1 =14-1=13 =13Instructional DecisionsEven though this was the first time the PSTs had worked with children in this capacity, it wasevident that many instructional decisions were made, both planned and spontaneous. All reportsincluded at least one spontaneous instructional decision. One example was based on what a pairof PSTs observed their child do to confirm their belief that the child was at the INS+ level ofwhole number development. This pair of PSTs wrote the following in their report:Her approach proved to me that she can use the commutative property. She began withthe larger number although it appeared after the smaller number in the problem. Toreinforce this I asked her a similar question (I replaced the numbers with 6 and 9 in thatorder) and she still chose to start with the 9. I asked her why she chose to start with the 9instead of the 6 since it was first in the problem and she told me it was easier to start with9 because it was the bigger number. This supported my belief that she understood thecommutative property and was at least INS+.This excerpt shows the PSTs recognition of the mathematical properties that often were the focusof the content course on Numbers and Operations. It was evident that they were able to maketheir content knowledge usable in their work with children. They were also able to use theframework from class to analyze the child’s mathematics. However, in order for this analysis tooccur they had to use questions to probe the child’s mathematics. It was clear that these PSTswere intentional with their choice of ordering the numbers in the problem. It was clear that thesePSTs were listening for a particular strategy, specifically counting on from largest. They had ahypothesis, and used a specific task to test that hypothesis. We believe that this is an importantpart of the research process. Besides probing questions, which are information seeking, we alsosaw evidence that the PSTs used prompting questions to elicit a particular response.On the project description we asked the PSTs to respond to the question, “if you couldcontinue to work with this child, what concepts or kinds of problems do you think would beproductive work for her or him?” 12 out of 13 pairs were able to thoughtfully respond to thisProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 46