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As this course comes to a close and you reflect on ways to bring your new understanding of data analysis, statistics, and probability into your teaching, you have both a challenge and an opportunity: to enrich the mathematical conversations you have with your students around data. As you are well aware, some students will readily grasp the statistical ideas being studied, and others will struggle.

The problems in Part D describe scenarios from three different teachers’ classrooms, involving young children’s developing statistical ideas. Some student comments are given for each scenario. For each student, comment on the following:

**• **Understanding: What does the statement reveal about the student’s understanding or misunderstanding of statistical ideas? Which statistical ideas are embedded in the student’s observations?

**• **Next Instructional Moves: If you were the teacher, how would you respond to each student? What questions might you ask so that students would ground their comments in the context? What further tasks and situations might you present for the child to investigate? See Note 6 below.

**Problem D1
**Mr. Mitchell’s second-grade class investigated the number of raisins in a box. The line plot below displays the raisin data collected by the students:

After plotting their data, here is what the students had to say:

**a. **Janet: “That’s not fair. I only got 26 raisins. I think we should tell them to put the same number of raisins in each box so that it would be fair.”

**b. **Sahar: “I think there will be between 32 and 38 raisins in that [unopened] box because most of our boxes had between 32 and 38.”**
c. **Jermaine: “Forty must be an outlier because there’s a gap at 39.”

**Problem D2**

Ms. Jackson asked her kindergarten students to formulate questions. She said, “Each of you will conduct a survey today to find out something about our class. I want you to think of a survey question you can ask your friends that they can answer by saying either yes or no.” The students formulated their questions, wrote them on their papers using words or pictures, and then collected the data. The following conversation took place in a whole-group discussion of the children’s results:

Teacher: | Bryce, what was your question? |

Bryce: | “Do you like dinosaurs?” |

Teacher: | How many people said yes? |

Bryce: | Eight. |

Teacher: | Eight people in your survey said yes. Now what do you think I want you to count? |

Bryce: | The nos. There were seven nos. |

Teacher: | Eight people said yes, and seven people said no. What does that tell us about dinosaurs? |

Here is how the students responded:

Bryce: | “More people like dinosaurs.” |

Nicole: | “Lots of people like dinosaurs, but lots of people don’t like them, too.” |

Wallace: | “It’s pretty close — one more, and we would have a tie.” |

Maggie: | “I think the boys said yes and the girls said no.” |

Zulay: | “I don’t think we can tell if more people like dinosaurs, because two people aren’t here today.” |

**Problem D3
**Mr. Kettering teaches first grade. Almost every Friday since the beginning of the year, his class has collected pocket data. The children count the number of pockets on their clothes and then record the data on a class line plot. Early in the year, they decided to count only the pockets on the clothes they were wearing at that moment, such as pants, shirts, vests, and sweaters; they did not count pockets on their coats or their backpacks. One day, Mr. Kettering displayed the two line plots below, showing pocket data from two different days, and asked the children to compare the line plots and describe what they noticed about the number of pockets:

Here’s how the children responded:

Ben: | “The blue dots are a lot more spread out than the green dots.” |

Damon: | “I think the green dots are from the beginning of the school year, when everyone was wearing shorts, and the blue dots are probably from December, when it was cold out.” |

Emma: | “I think that blue dot on zero is someone who forgot we were counting pockets that day.” |

Tarra: | “With the green dots, most people only had zero or two pockets, and with the blue dots, most people had six.” |

Isaiah: | “Look at that gap with the blue dots — and there aren’t any gaps on the green dots.” |

For more information about statistics problems for children like the problems in this session:

Burns, Marilyn (1996). *50 Problem-Solving Lessons.* Math Solutions Publications.

Economopoulos, Karen and Murray, Megan (1998). *Mathematical Thinking in Kindergarten.* Dale Seymour Publications.

Economopoulos, Karen; Akers, Joan; Clements, Douglas; Goodrow, Anne; Moffet, Jerrie; and Sarama, Julie (1998). *Mathematical Thinking at Grade 2.* Dale Seymour Publications.

Russell, Susan Jo; Corwin, Rebecca B.; Rubin, Andee; and Akers, Joan (1998). *The Shape of the Data.* Dale Seymour Publications.

**Note 6**

If you’re working in a group, make a two-column chart with the labels “Understanding” and “Next Instructional Moves” for recording the group’s responses to Problems D1-D3.

**Problem D1
a. **Janet is not willing to accept that there is variation in the raisin data. The teacher might point out to Janet that the raisins are packaged by weight, then have her examine the various sizes of the raisins more closely, and finally ask her to think about why the raisins are packaged by weight and not by number.

d.

e.

f.

**Problem D2
a. **Bryce is noticing the mode of the data. The teacher could ask Bryce to think about the closeness of the numbers seven and eight and acknowledge that eight is more.

b.

c.

d.

e.

**Problem D3
a. **Ben is most likely reasoning by looking at the range of each data set. The teacher could ask the students to pretend that the data value at 0 is not there and to then think about Ben’s statement.