Research Article

# Preservice Teachers’ Learning to Respond on the Basis of Children’s Mathematical Understanding

## Table 2

Examples of responses coded as most, some, and lack of mathematical details.
 Code Inquiry into Student Thinking Tutoring assignment Most mathematical details He solved it with a direct modeling procedure and drew each individual soccer ball in the designated three bags. Emilio worked on the problem: Dr. E has 4 rolls of candy and 11 loose candies. How many candies does she have altogether? He initially spat out the number 40 and explained on his own that there were candies in each of the four rolls. He had trouble counting up from 40 to 51 for the 11 single candies, but this is an issue he had the first day as well and shows that may be another issue. However, because he knew to count up from 40 by 1 single candies shows that he is able to distinguish groups of 51 from single units which is very significant in using base ten problem-solving strategies. On the JRU problems, Tyler would use a breaking-the-number-apart strategy. He would like to get the numbers into base 10 so that they would be easier to add together. For example, on the first set of numbers (42, 36) for the apples problem, Tyler told me that the answer was 78. When I asked how he knew that he wrote out that 42 + 30 = 72 and then wrote 72 + 6 = 78. I was really excited that he knew a shortcut for how to do the problem. He also used this same strategy for the SRU problems. Some mathematical details Jack throughout the case study counted up by ones to find his answer. From the very first day, Jack miscounted the total number of soccer balls because he had the wrong number of soccer balls in one bag, even though all of the bags had simply 10 balls in each. In his first few sessions, Jack tended to write tall marks to keep track of whatever he was counting, no matter how big the number was. Apple problem: The student started with the original amount of blocks (3) then found the number of picked apples (12). After this the student started counting the blocks starting with 3, counting up to 12 on, starting with 3, 4, 5, 6, 7, 8, 8, 10, … This led me to doing a problem that would include counting since my goal for the lesson was to get my student to be able to count in sequence starting from a given number in the known sequence. Lack of any mathematical details In the beginning, Jack did not recognize ten as a numerical unit. It seemed that, to him 10 was no different than 4 or 9. Because of this, he often counted up to the answer. As the study went on, he began to develop an understanding of ten, first by using a representation of 10 (rather than tally marks or other such one-to-one representations) in session 5 and later by solving number sentences by counting tens rather than counting up by ones (seen in session 9, but also hinted at from session 5 on. Second, any straightforward problem (e.g., 23 + 57 = __) was not difficult for them. It did not seem to matter whether a task was JSU, SCU, SIU or SRU; those sorts of problems were simply too easy for these three students unless the numbers were sufficiently large enough to require them to use paper just to keep track of their carrying …
Note: Join Result Unknown (JRU) Join Change Unknown (JCU) Join Initial Unknown (JIU) Separate Result Unknown (SRU) Separate Change Unknown (SCU) Separate Initial Unknown (SIU).