**Case 11 – Number of Days in School**

Andrew’s understanding of what day of school it is as fifty-ten instead of 60 makes perfect sense logically speaking. He knows that as you count “up” you move to the next ten. But, he doesn’t quite make the connection of going up in the “ten’s place.” He is understanding what comes next but is missing the connection of 10, 20, 30, 40, etc. Andrew is making the connection of 10’s but is missing the next step.

**Case 12 – Groups and Leftovers**

As I grab a handful of beans and start to count out different groups, I have difficulty keeping track of how many beans there are in total. But, as I count out groups of 10, it is easy to “sum” the beans. This is the idea of using manipulatives to make groupable models and then use the benchmark number of 10 to count to a total. The children came to the same logical conclusion that it was easier to count making groups of 10’s than it was other size groups.

**Case 14 – Who Invented Zero Anyway?**

In this case, the children are trying to figure out the meaning of zero in numbers and trying to find a way to understand and show that it does have meaning and can’t be left out but just what that meaning is is eluding them. They are trying to understand that at the end of the number is not quite the same as a zero all by itself. They are all talking around the usefulness of the zero but not understanding it enough to say that it is holding a place value.

**Case 15 – One Hundred Ninety-Five**

When students were asked to write one hundred ninety-five, they gave many different answers with differing understandings and explanations:

1095 and 10095 – these students knew that 100 meant something and that it included zeros, they just didn’t remember exactly how it worked

195 – was given by one student who just couldn’t explain but knew it to be true. Maybe this child just has more number experience and has a beginning understanding that just can’t be articulated as of yet.

1395 and 1295 are just completely missing the mark but understanding that these are big numbers that are being discussed. There does seem to be some understanding with the “95” portion of the number.

It was very difficult to make a prediction about how these children were understanding and how they weren’t. Also, sometimes I could “see” their thinking but then others I was just not understanding any sort of logic that they were following. Maybe this is when they were just making up their own “new insights” to try very hard to grasp something out of their current understanding.