Not all of the topics in this section will be in the elementary curriculum; however we believe all of this content is important for prospective teachers to understand thoroughly, both as professionals and as educated citizens. We recognize students may encounter this content in the context of a general education course; we recommend an inquiry-based approach regardless of the audience, believing it gives all students the best opportunity to learn these concepts.
The topics that future teachers will most likely need to teach include the concepts of mean and variability, appropriate graphical representations of data, and drawing inferences from graphs.
Additional recommended resource: The Statistical Education of Teachers
NOTE: This page has five sections, each one aligned to one of the Data outcomes in the Maryland AAT Outcomes document. (Probability is discussed in the next unit.)
D1.a. Understand a statistical question is asked within a context that anticipates variability in data.
D1.b. Understand measuring the same variable (or characteristic) on several entities results in data that vary.
D1.c. Understand that answers to statistical questions should take variability into account.
D1. Teachers providing early experiences in making and interpreting graphs need to help their students select appropriate questions about which to gather data. The Curcio book provides many good examples as a starting point.
As professionals, teachers need to understand how to interpret test results, and the variability that can occur upon re-testing. A student's score is a single data point from which an inference is made.
D1. Activity"Statistical Questions vs Other Questions" Class Activity 15A in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
D1. Book Developing Data-Graph Comprehension in Grades K-8, 2nd Ed., Frances R. Curcio. Includes many examples of suitable data collection tasks for a variety of grade levels.
D2.a. Understand data are classified as either categorical or numerical.
D2.b. Understand a sample is used to predict (or estimate) characteristics of the population from which it was taken (including distinction between population, census, and sample).
D2.c. Understand experiments are conducted to compare and measure the effectiveness of treatments. Random allocation is a fair way to assign treatments to experimental units.
D2. Teachers assisting their students to collect data can lead class discussions about how to ensure the data is representative of the population of interest.
As citizens and consumers, all of us should critically interpret the results of surveys and experiments, and consider possible sources of bias.
D2, D2.b. Activities sequence: "Random Rectangles," "Sampling Rectangles," "Sampling Size," "Sampling Methods. Navigating through Data Analysis in Grades 9 - 12
D2, D2.b Activities from Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed:
15B "Choosing a Sample"
15C "How Can We Use Random Samples to Draw Inferences About a Population?"
15D "Using Random Samples to Estimate Population Size by Marking (Capture-Recapture)"
D2, D2.b Activity "Does Beyonce Write Her Own Lyrics?"
D2, D2.b Activity"How Much Do Fans Love Justin Timberlake?"
D2.b Activity "What is Wrong with These Surveys?"
D3.a. Understand distributions describe key features of data such as variability.
D3.b. Recognize and use tables with counts and percentages as well as appropriate graphs (picture graph, bar graph, pie graph for categorical data, line plots, stem and leaf plots, histograms and boxplots for numerical data).
D3.c. Recognize and use appropriate numerical summaries to describe characteristics of the distribution of quantitative data (mean or median to describe center; range, interquartile range or mean absolute deviation to describe variability).
D3.d. Understand distributions can be used to compare two groups of data with respect to similarities and differences in center, variability (spread) and shape.
D3.b. Teachers need to guide students to recognize the need for area in a graph and frequency counts (or percents) in the data to be in proportion.
Teachers also need to become familiar with common student misconceptions in interpreting graphs.
Teachers and students should become familiar with common ways graphs are constructed to be misleading.
D3.c. Mean and median are frequently taught as procedures. In order to have a robust understanding of mean it can help students to consider models such as "leveling out" a case-value plot or finding the "balance point" of a histogram.
In addition, understanding conceptually how mean and median differ can help identify which is more appropriate for a given situation.
D3.c. Typically students learn to calculate standard deviations for small data sets, and then memorize some facts related to the normal distribution. A more conceptual approach is outlined in the article "Means and MADs," where the mean average deviation is introduced to describe variability. The standard deviation can then be introduced as a similar way of measuring variability with broader applications in statistics.
D3.d. Students generally find it easier to compare data sets represented as histograms. It is essential they also can relate the information in a boxplot to the data set. In addition, they need experience comparing data sets represented as boxplots. These different representations can be compared and contrasted--they each offer advantages and disadvantages.
D3.b. Book. Developing Data-Graph Comprehension in Grades K-8, 2nd Ed., Frances R. Curcio. In particular see "Levels of Graph Comprehension," pp. 7-8 and the examples in Appendix 1 on pp. 99-106. Also includes samples of students' invented representations of data (pp. 12-17), and many examples of suitable data collection tasks for a variety of grade levels.
D3.b Article "Understanding Students' Understanding of Graphs," Friel, et al, Mathematics Teaching in the Middle School, November/December 1997, Vol. 3, p224-227 (NCTM membership needed to access)
D3.b Article "Students' Interpretations of Misleading Graphs," Harper, Mathematics Teaching in the Middle School, Feb 2004, Vol. 9 Issue 6, p340-343 (NCTM membership needed to access)
D3.b Examples of misleading or poorly constructed graphs: pie charts bar, circle, line graphs
variety of graph types
D3.b book Navigating Through Data Analysis in Grades 6 - 8: In particular, note the contrast of a case-value plot and a frequency bar graph on pp 21-22 (frequent student error).
D3.b Activities from Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
Class Activity 15E Critique Data Displays or their Interpretation
Class Activity 15F Display and Ask Questions About Graphs of Random Samples
Measures of Center
D3.c. Activities from Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
Class Activity 15K The Mean as "Making Even" or "Leveling Out") (Note: this interpretation only works if the graph is a case-value plot)
Class Activity 15M The Mean as "Balance Point" (Interpretation applicable to frequency graphs)
Class Activity 15L Solving Problems About the Mean
Class Activity 15N Same Median, Different Mean
Class Activity 15O Can More Than Half Be Above Average?
Class Activity 15P Critique Reasoning About the Mean and the Median
D3.c. Article: "Making the Mean Meaningful" (Research study with 6th graders using the "leveling" approach with case-value plots):
D3.c. Video comparing mean and median:
D3.c. Article: "How Much Does the "Average" Wedding Cost?"
Measures of Variability
D3.c. Article describing activity "Means and MADs," Kader, G., Mathematics Teaching in the Middle School, March 1999, pp 398-403. (NCTM membership needed to access)
D3.c. Activity: "Representing Variability with Mean, Median, Mode, Range"
D3.c. Activity: Boxplots (Open Middle)
D3.c. Activity: Absolute Deviation (Open Middle)
D3.c. Activity: Mean Absolute Deviation (Open Middle)
D3.d. Activities from Mathematics for Elementary Teachers with Activities, Beckmann, 5th Ed
Class Activity 15Q What Does the Shape of a Data Distribution Tell About the Data?
Class Activity 15S Comparing Distributions: Mercury in Fish
Class Activity 15T Using Medians and Interquartile Ranges to Compare Data
Class Activity 15U Using Box Plots to Compare Data
Class Activity 15V Percentiles vs Percent Correct
Class Activity 15W Comparing Paper Airplanes
D3.d. Activity "Representing Data with Grouped Frequency Graphs and Box Plots" (Note: this compares the information available in histograms vs box plots)
D3.d. Activity "Representing Data with Frequency Graphs" (Note: this generalizes histograms to curves)
D3.d. Activities from Navigating Through Data Analysis in Grades 6 - 8:
Batteries (comparing two brands) pp 39 - 42; 89
Migraines (construct and compare box plots) pp 63-65; 97-99.
D3.d. Article describing activity: "Sticks to the Roof of Your Mouth?" Friel, S and W O'Connor, Mathematics Teaching in the Middle School, March 1999, Vol. 4 Issue 6, p404-411 (NCTM membership needed to access)
D3.d. Activity "Comparing Data Using Statistical Measures"
D3.d. Article "So Many Graphs, So Little Time," Wall, J and C Benson, Mathematics Teaching in the Middle School, September 2009, Vol. 15 Issue 2, p82-91 (Note: this article summarizes how graphs indicate attributes such as mean and variability with area, colors, shading, etc. Includes additional types of graphs.)
Recommended software and apps:
Excel or Google sheets
Tinkerplots (small fee required)
D4.a. Recognize the difference between a parameter (numerical summary from the population) and a statistic (numerical summary from a sample).
D4.b. Recognize that a simple random sample is a ‘fair’ or unbiased way to select a sample for describing the population and is the basis for inference from a sample to a population.
D4.c. Recognize the limitations of scope of inference to a population depending on how samples are obtained.
D4.d. Recognize sample statistics will vary from one sample to the next for samples drawn from a population.
Probability provides the foundation for statistical inference; some earlier experience with probability will provide a deeper understanding of this indicator. Experience with the normal distribution, particularly the Empirical Rule (68% - 95% - 99.7% rule) should precede this unit.
It is important that experiences with simulation be used as opportunities to build conceptual understanding of a sampling distribution. Can re-do or refer to prior activities with repeated random sampling carried out when discussing variability (D3.c) Online apps can be used to create distributions with sufficient samples so the normal distribution becomes visible.
Prior experiences with probability based on the normal distribution then can be connected to statements of statistical inference such as confidence intervals or margin of error statements or p values.
D4. Activities from chapter "Prediction and the Law of Large Numbers (impact of sample size)", Navigating Through Probability in Grades 6 - 8, pp 51-71
D4. Activity. What's the Proportion of Orange Reese's Pieces?
D4. Activity. How Tall Are We?
D4. Activity. Activity 15C "How Can We Use Random Samples to Draw Inferences About a Population?" from Mathematics for Elementary Teachers with Activities, 5th Ed.
D4. Activity. Activity 15R "Distributions of Random Samples" from Mathematics for Elementary Teachers with Activities, 5th Ed.
D4. Activity. "Does Beyonce Write Her Own Lyrics?"
D4. Activity sequence. "Discrimination or Not?", "Simulating the Case," "Simulation Results," "What Would You Expect?" from Navigating Through Data Analysis in Grades 9 - 12
D4. Activity. "Flint Water Crisis," (scroll down page)
D4. Article with description of activity. "Cents and the Central Limit Theorem," in "An Activity-Based Statistics Course," Gnanadesikan, et al., Journal of Statistics Education v.5, n.2 (1997)
D4. Applet to illustrate Central Limit Theorem
D4. chapter in book with activities. "Connecting Probability and Statistics, "Navigating Through Probability in Grades 6 - 8, pp 73-85 (Designing and carrying out simulations)
D6.a. Interpret student test results (percentiles, normal distribution, standard deviations, stanine).
D6.b. Understand data and analysis presented in journal articles.
As professionals, teachers will need to use weighted averages, interpret box plots, and interpret numerical data from standardized testing (often related to a normal distribution).
D6. Activity. Activity 15V "Percentiles vs Percent Correct" in Beckmann, Mathematics for Elementary Teachers, 5th Ed