Private: Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades K-2 Part D: Examining Children’s Reasoning (30 minutes)
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.
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.”
d. Ramel: “The range is from 25 to 43.”
e. Ava: “The raisins must all be kind of the same size since most of the data are bunched together in the middle.”
f. Paul: “I think there will be 34 raisins in that [unopened] box, because 34 had the most.”
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:
|Bryce, what was your question?
|“Do you like dinosaurs?”
|How many people said yes?
|Eight people in your survey said yes. Now what do you think I want you to count?
|The nos. There were seven nos.
|Eight people said yes, and seven people said no. What does that tell us about dinosaurs?
Here is how the students responded:
|“More people like dinosaurs.”
|“Lots of people like dinosaurs, but lots of people don’t like them, too.”
|“It’s pretty close — one more, and we would have a tie.”
|“I think the boys said yes and the girls said no.”
|“I don’t think we can tell if more people like dinosaurs, because two people aren’t here today.”
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:
|“The blue dots are a lot more spread out than the green dots.”
| “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.”
|“I think that blue dot on zero is someone who forgot we were counting pockets that day.”
| “With the green dots, most people only had zero or two pockets, and with the blue dots, most people
|“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.
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.
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.
b. Sahar is able to consider an interval of the data that is most representative of the data. The teacher could ask the class to speculate on the likelihood of her prediction.
c. Jermaine reasons incorrectly that 40 is an outlier because it is the largest number separated from the others by a gap. The gap, however, is only one raisin, so based on this data, 40 is not unusual enough to be an outlier. The teacher might ask the rest of the class to discuss further the meaning of an outlier.
d. Ramel is looking at the range of numbers written on the horizontal axis rather than at the span of the data points. The teacher could use the following questioning to get Ramel to focus on the data points: “What is the smallest number of raisins that we found in a box? Point to it with your left hand. What is the largest number of raisins that we found in the box? Point to it with your right hand. This distance from the lowest to the highest number is what we call the range.”
e. Ava reasons about an interval of the data that contains the most data points and provides an interpretation of raisin size based on this observation. This would be an opportunity to write out Ava’s conjecture and ask the other children to evaluate it. Do they agree or disagree? Would they want to modify the conjecture in any way?
f. Paul is reasoning with the mode as a summary statistic that describes a measure of center. The teacher might ask the rest of the class to react to Paul’s prediction. How many of them agree, and why? How many disagree, and why? You could also ask Paul and the other children to comment on the likelihood that an unopened box contains 34 raisins.
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. Nicole is considering the variance of the data and the whole data set. The teacher might ask the rest of the class to react to the accuracy of Nicole’s statement.
c. Wallace is noticing that there is not much variation in the data. The teacher could ask Wallace to further explain what he means by “one more” and by a “tie.”
d. Maggie is formulating a conjecture or hypothesis about the data. This would be an opportunity for the class to plan a follow-up data investigation to test Maggie’s conjecture.
e. Zulay is thinking about the sample of students in attendance today and the population of the class. The teacher might ask the rest of the class to consider Zulay’s comment and to then predict how they think the two children who are absent would respond to the question about dinosaurs.
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.
b. Damon is providing an interpretation of the data. The class could be asked to evaluate Damon’s conjecture and to propose other reasons for the differences in the data.
c. Emma is developing a hypothesis about the outlier 0. The teacher could ask the class to formulate other conjectures about the outlier 0.
d. Tarra is reasoning about the data by looking at the modes of each data set. The teacher could introduce or reinforce the meaning of the term mode.
e. Isaiah is interested in examining areas with no data as well as areas with data. He also notices that the green dots are more concentrated with little variation. The teacher might use this as an opportunity to focus further attention on the importance of examining gaps and considering how the data are spread out or bunched together.
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.