## Learning Math: Data Analysis, Statistics, and Probability

# Classroom Case Studies, Grades 6-8 Part A: Statistics as a Problem-Solving Process (20 minutes)

A data investigation should begin with a question about a real-world phenomenon that can be answered by collecting data. After the children have gathered and organized their data, they should analyze and interpret the data by relating the data back to the real-world context and the question that motivated the investigation in the first place. Too often, classrooms focus on the techniques of making data displays without engaging children in the process. However, it is important to include children in all aspects of the process for solving statistical problems. The process studied in this course consisted of four components:

Children often talk about numbers out of context and lose the connection between the numbers and the real-world situation. During all steps of the statistical process, it is critical that students not lose sight of the questions they are pursuing, nor of the real-world contexts from which the data were collected.

When viewing the video segment, keep the following questions in mind: See Note 2 below.

• |
Think about each component of the statistical process as it relates to what’s going on in the classroom: What statistical question are the students trying to answer? How were the data collected? How are the data organized, summarized, and represented? What interpretations are students considering? |

• |
What connections among mathematics topics and across subject-area disciplines are apparent in this data investigation? |

• |
Thinking back to the big ideas of this course, what are some statistical ideas that these students are likely to encounter through their investigation of this situation? |

**Video Segment**

In this video segment, the teacher, Paul Sowden, applies the mathematics he learned in the *Data Analysis, Statistics, and Probability* course to his own teaching situation. He starts by asking his students to think about the relative amount of coins with each type of mint mark. The students then sort the coins into four groupings: Philadelphia, Denver, San Francisco, and no mint mark. They will now begin to analyze and interpret their data.

**Problem A1
**Answer the questions you reflected on as you watched the video:

a. |
What statistical question are the students trying to answer? |

b. |
How did the students collect their data? |

c. |
How are the data organized, summarized, and represented? |

d. |
What interpretations are students considering? |

e. |
What connections among mathematics topics and across subject-area disciplines are apparent in this data investigation? |

f. |
What statistical ideas are these students likely to encounter as they investigate this situation? |

**Problem A2
**In this video segment, are the students working with quantitative data or qualitative data?

**Problem A3
**Questions may arise as students examine the nickels in this open-ended investigation. Formulate four statistical questions that students might ask about the nickels that would prompt further investigation.

**Problem A4
**Why is a circle graph an appropriate way to display this data? What characteristics of data are clearly shown through a circle graph?

### Notes

**Note 2
**The purpose in watching the video is not to reflect on the teacher’s methods or teaching style. Instead, look closely at how the teacher brings out statistical ideas while engaging his students in statistical problem-solving.

You might want to review the four-step process for solving statistical problems. What are the four steps? What characterizes each step?

### Solutions

**Problem A1
a. **The question the students are trying to answer is, “What is the relative frequency of each type of mint mark?”

**The teacher brought a collection of nickels to the class so that students could examine the coins’ mint marks.**

b.

b.

**The students organized their data into a circle graph.**

c.

c.

**The students are developing conjectures about the relative frequency of each mint mark.**

d.

d.

**The students are using their knowledge of fractions as they explore this problem. The investigation of mint marks**

e.

e.

involves connections to social studies.

**Some statistical ideas are the nature of data, qualitative variables, variation, relative frequency, sampling, making a circle graph, and interpreting a circle graph.**

f.

f.

**Problem A2
**The students are working with qualitative (categorical) data.

**Problem A3
**Answers will vary. One question might be, “Is this sample of nickels representative of the population of nickels?”

**Problem A4**

A circle graph is an appropriate way to display categorical data. Circle graphs show the fractional relationship of each category or part of data to the whole data set.