Mark W. Ellis, Ph.D., NBCT
In the middle grades, students are introduced to four processes of statistical reasoning in Progressions for the Common Core: Grade 6-8 Statistics and Probability:
- Formulating questions that can be answered with data
- Designing and using a plan to collect relevant data
- Analyzing the data with appropriate methods
- Interpreting results and drawing valid conclusions that relate to the questions posed
Data displays such as picture graphs and line plots are not new. For many years, students have learned to look for specific values such as “the greatest value” or “the least value” and make simple comparisons within a sample of univariate data (data with one measurable quality). But now, beginning in sixth grade, students must learn to do the following:
- Examine, represent, and interpret the characteristics of distributions of data such as measures of center, measures of spread, and overall shape.
- Use these characteristics together with the data context to answer statistical questions.
- Apply and extend these understandings to explore and analyze bivariate data representing two categories or measurable qualities—a much more challenging activity.
Statistics students need to understand the concepts of variability and center, since these form the foundation for their later explorations involving bivariate data and probabilistic models. If data never varied (we always had exact measurements and quantities), there would be no use for statistics. We would simply want the exact value! For example, if the height of all sixth graders was 150 cm and you were asked to find the mean, median, mode, and range, you would quickly realize there is no need for these since the data do not vary. So variability is the first big concept of statistics.
Mean, median, mode are all measures of center and provide different perspectives on the center of a data set. Perhaps the most poorly understood is the mean, since students are often given a procedure—add up all values and divide by the number of values—instead of help with understanding the concept. If students learn the mean as an algorithm without first understanding the concept as a measure of center, they often have difficulty making meaningful connections between their calculations and the data context.
As with the other challenging standards, the key to supporting student success in statistics is to give them time to make sense of new concepts and relationships.
To help students make sense of variability, start by discussing the difference between a statistical question and a non-statistical question that lacks variability. Ready® Mathematics Grade 6 Lesson 26 offers a set of activities and prompts designed for this purpose. The goal is to help students recognize questions involving variability. As an extension, students are asked to rank a set of questions in order of greatest variability to least variability, which sets them up well for later work with measures of variability including the mean average deviation (MAD).
When working with data sets—which should be a frequent activity—have students create displays based on individual data points and displays that represent groups of data such as histograms, box plots, and stem and leaf plots. Then ask them to describe the shape of the data, what they see visually and what that means in relation to the data context, including the shape of the data and any characteristics such as typical or common values. Early on, allow students to use whatever language they wish, since the goal is to reinforce the idea that the representations have meaning and to generate curiosity about possible meanings.
To help grade school students understand the mean, use the concepts of fair sharing and balance point. Fair sharing takes a large collection of values, puts them all together, and redistributes them evenly, and the mean indicates how much each person will get.
Acting out a fair share scenario can help students connect the concept to the procedure for finding the mean. To help students think about the mean as the value around which the data are balanced, nicely extending to making sense of the mean average deviation in later grades, you can draw on the article by Robin O’Dell “The Mean as Balance Point.”
Once students understand mean, give them opportunities to compare properties of the mean with those of the median. My webinar for Education Week on March 4, 2015 has an example of an activity you might use, including writing prompts and a real-world check for understanding involving text messaging data.
Students’ early conceptions of variability and measures of center form the foundation for their later explorations of statistics involving bivariate data and probabilistic models. Having a strong foundation in place will better support their later success. This activity involving relationships among Mean, Median, Mode, and Range from the Mathematics Assessment Project provides a way to gauge and reinforce your students’ fluency with these topics.
- This blog by Raymond Johnson and Susan Thomas about middle school statistics topics offers specific activities for Common Core data and statistics standards in grades 6–8.
- This article by Linda Cooper and Felice Shore offers excellent examples you can use with students to discuss the affordances of different types of data and data displays.
- Statistics Education Web has curated lesson plans aligned with GAISE standards and Common Core.
- The Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report offers a framework for developing statistical reasoning across the four steps introduced above across three developmental levels from A to C (see pp. 14–15 of the report). While not directly correlated to grade levels, students in grade 6 will work mostly in Level A activities and transition to Level B activities.
- The Interactivate site offers several free apps to explore statistics including univariate and bivariate data. Both Google Public Data and the UNESCO data site offer free data sets covering a wide range of topics as well as tools for displaying univariate data. The Census at Home site allows you to create displays of data submitted by students from around the globe and invites you to submit data from your own locale.
This blog is the fourth in a series from Curriculum Associates about the most challenging math standards that complements they’re recently released white paper – “Mastering the Most Challenging Math Standards with Rigorous Instruction“.
For more see:
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Measurement
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Modeling
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Fractions
Mark Ellis is a National Board Certified Teacher and professor of education at California State University at Fullerton. He is an author of Curriculum Associates’ Ready® Mathematics program.