**I. Maura’s Case 27 – Measures/Features Used to Compare**

I had a difficult time distinguishing the different types of measures or features that the students were specifically identifying. But, here is my best analysis:

1. Students are noticing the mode of the data: “The data tell me that the fourth graders are bigger because the fourth graders have most kids in 53 and 55, and most of the third graders are at 51.” -Aidan

This is developing understanding of mode. Aidan is beginning to understand that “mode” is the number which appears the most in a set of numbers.

2. Students are noticing the range of the data: “I think the fourth graders are taller than the third graders because our [fourth-grade] range is from 52 to 62 and their [third-grade] range is from 51 to 54. 52 is higher than 51, and 62 is higher than 54.” – Zia

This student it using range. Zia understands that “range” is the difference between the highest and lowest values in a set. And further is beginning to understand that the range includes more numbers for fourth grade than it does for third. But, will this information lead to understanding which group is taller?

3. Students are looking at the shape of the data: “I see that with the fourth graders, the X’s are really scattered around, but with the third, it looks like half of a pyramid.” – Beth

Beth is beginning to understand range. She is noting that the differences in heights are along a larger continuum for the fourth graders than for third.

4. Students are beginning to make numerical statements about the data: “The fourth graders have five taller than any third graders.” – Stuart

Stuart is starting to notice that the data on the chart is directly related to students and their height and further noticing that a certain number of 4th graders are taller than all the 3rd graders.

5. Students are noticing that the data is challenging their assumptions: “Some of the third graders are taller than the fourth graders. Weird!” – Anna

Anna is seeing that the data is challenging her assumptions that older students must be taller.

**II. Lydia’s, Phoebe’s, and Maura’s Cases – Beginning Ideas**

1. “If we use the number in the middle, both of Robbie’s numbers [points to the 186 and 152] are inside the middle. There is some of each number in the middle.” – Erin in Lydia’s Case

Erin is beginning to understand and interpret the mean to be a number in the middle that can express more of an average between the two numbers.

2. “We measured all the people in our group, which was four people. We added up all of our heights, which got to a total of 18 feet 11 inches. Then we divided the number by 4, because there were four people in the group, to see how tall the average fourth grader was in terms of our group. We came up with 4 feet 9 inches.” – Trudy in Phoebe’s Case

Trudy is truly understanding that an average can be calculated to estimate the height of the typical 4th grader by using the students in their group as a sample.

**III. Phoebe’s and Nadia’s Cases – Misunderstandings**

1. “What if the people in your group were different sizes from everybody else in the class? Then your wouldn’t actually be correct. It would only work if everybody in the classroom and in your group was about the exact same height, which they’re not.” – Jesse in Phoebe’s Case

Jesses is confused about “average” and mean and is misunderstanding that this number should match a true measure (height) of each person in the room. She is not understanding that an average would just represent more of a middle height between all the students or a central tendency.

2. “…. = 263, and 263/8=32 remainder 7. I got the average. I got 32 remainder of 7.” – Kayla in Nadia’s Case.

Kayla did add up the number of letters in each student’s name accurately. But, then she divided by 8, by the number of different values not by the the number of students in the class. Even though the data numbers were between 10 and 18, Kayla did not reason that a name that had 32 letters could not be an accurate average. Her math computation was correct for the numbers she was using, but she did not question why her answer was so different than the numbers of letters in the students’ names.

I like the examples that you used for the third response as I hadn’t considered either of those as examples of understanding of mean but can definitely see it after reading through the quote and the paragraph (context) again.

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