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

Table 3

Examples of responses coded as robust, limited, and lack of evidence.

Code

Inquiry into Student Thinking—interpretation

Tutoring assignment—interpretation

Robust evidence

I believe that Jack, by the end of the study, had a much better understanding of how to use his knowledge of base ten in solving problems

Brian had a good understanding of base 10 and basic problems. He was able to count very well and almost never stumbles when switching decades, e.g., 97, 98, 99, 100, 101, … He also demonstrated that he was capable of counting by 10s both forward and backward. I was especially happy to see him easily counting backwards, 204, 195, 184, 174, etc. Brian also understand the use of the equal sign …

One of the problems was this: (JCU) 22 pennies, how many more to have 50. Jack solved this problem by counting up by ones from 22 using tallies to keep track, which solves the problem but shows evidence that he does not fully understand how to use his base ten knowledge to help solve problems. In session ten they did a problem that was as follows: 30 pencils, 29 more. This is a somewhat similar problem from the one in example 1. For both he needed to count up by about 30 to get the answer; but this time to solve the problem he drew a picture that represented groups of tens and then ones. This time he did use his knowledge of base ten to help make this problem easier to solve.

Limited evidence

10 have now become a unit for Jack instead of just the 1’s unit. His thinking for base 10 is fragile though, and he will need more practice. For the last problem, he couldn’t decide between if 45 beads could make 4 or 5 necklaces.

I think both Calvin and Karl seemed to have a good understanding of numbers when counting forward and backward by rote memory. They were able to answer all the questions with ease. I even tried using some of the second-grade questions and they were able to answer them without even thinking.

Lack of any evidence

Jack didn’t have an understanding of base ten at the beginning of the case study but, by the end, he had a concrete understanding of the base ten process. At first Jack got confused with the terminology of loose and thought that he should subtract the balls instead of adding the balls together and he misunderstood the problem type. By the end of the case study, Jack had a better understanding of how to decode problems more properly.

Overall, I was surprised by how much my students know along different strategies to solve the problems. I was also surprised how the students knew how to solve the CGI but, when I presented them with the true or false and open-number questions, they struggles