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Statistics differs from other mathematical sciences because of “the focus on variability in the data, the importance of context associated with the data, and the questioning of data… [making statistics] particularly relevant for all fields of study (56).” Statistical problem solving involves investigating questions and interrogating data in a non-linear process that includes: formulating questions; collecting or considering the data; analyzing the data; and interpreting the results (as shown in figure 10). Worthwhile investigations depend on question formulation that in addition to having a question that anticipates variability, there must be clear variables and intentions, and the question must be answerable with primary or secondary data. Regardless of the data source, addressing each component of the process is essential, as is constantly returning to the question and potentially identifying new investigative questions.
C.5.a. Collecting, considering, organizing, and representing data
C.5.b. Selecting and using appropriate statistical methods to analyze data
C.5.c. Interpreting data: developing inferences and evaluating predictions
C.5.d. Understanding basic concepts of probability
Probability examines and eventually defines the likelihood of a situation. The key to early probabilistic thinking is understanding that some events are more likely or less likely to occur than others. In later grades, probability provides contexts for making data-based predictions.
Elementary Mathematics Specialists (EMSs) know that children at very young ages have innate notions of variability and probability. EMSs understand that they can capitalize on these notions by exploring data and probability questions that emerge naturally from the classroom and school environment (e.g., fair distribution of snacks or school supplies, likelihood of going to recess after lunch, or that more teeth were lost by older students at the school). They recognize the importance of question formulation and provide opportunities for students to pose questions that they can answer by collecting or considering existing well-defined data sets. EMSs recognize that engagement in these activities builds upon student experiences with counting, sorting, comparing, ordering, and operating on whole numbers, fractions, and decimals as well as provides interesting contexts to apply their mathematical understandings and practice skills. Similar to the modeling with mathematics cycle, statistical investigations may include interrogation of culturally-relevant, real-world contexts (e.g., overcrowded classrooms, water shortages, homelessness, climate change), and these applied experiences not only develop data reasoning skills but also demonstrate mathematics is a useful tool for answering questions and critiquing the world in which they live. When engaging in the statistical problem solving process, EMSs support students and teachers in recognizing the value of curiosity and skepticism as they consider the questions to ask, the data to collect, the foci for analysis, the evidence influencing the interpretation of results, and the best way to report their findings. Finally, EMSs recognize the importance of early work with these data science concepts is what prepares students for the data-based decision-making necessary in their lives both in and out of school.
C.5.a. Collecting, considering, organizing, and representing data
Early work with data focuses on answering statistics-related questions from data that students collect or data that has been collected by others. These data may be collected through direct observation, simple surveys, or measurements from simple experiments. As students engage in collecting, considering, and organizing data, representation opportunities become available. While there are conventions to eventually learn, students benefit from repeated experiences of coming up with their own approaches to collecting, organizing, and representing data, later comparing these invented approaches with standard approaches to understand the benefits and drawbacks of each. Later, students should be allowed to think about and represent multiple variables at a time, and how to represent and analyze data using a variety of representations supported by appropriate technological tools (e.g., Common Online Data Analysis Platform [CODAP], Polypad, IES’s Create a Graph). Statistical problem solving centers on recognizing and understanding variability, and then representing that variability in ways to support analysis and ultimately interpretation.
Initial data investigations include categorical data with sorted physical objects (e.g., shoes, stuffed animals) arranged to form physical bar or circle graphs that can be translated into picture graphs, bar graphs, and circle graphs. Numerical data may be represented using line plots and frequency tables and then in later elementary grades and beyond, expanded to include histograms, stem-and-leaf plots, line graphs, scatterplots, and box plots. Once students have experience creating and analyzing various types of data displays, technology can be a useful tool for creating interactive dynamic representations that support deepened analysis and discussion.
EMSs understand how to design learning progressions from simple, single-variable pre-collected data, to well-defined groups of interest from whom to collect data, to more complex, multi-variable contexts with data that may include errors or missing values. They also recognize there are many opportunities for student voice and choice when it comes to data investigations. EMSs design data contexts that 1) connect to students’ lives, giving them voice as they bring their expertise and interest to statistical problem solving; and 2) empower students by giving them choice about how to collect, organize, and represent data. In either case, EMSs know they can better engage all students when they build from what students know and what makes sense to them.
C.5.b. Selecting and using appropriate statistical methods to analyze data
Data analysis can begin with a completed data representation or once students have formulated questions and collected and organized data. When given the opportunity to notice and wonder, students very naturally observe larger and smaller amounts, how the data seem to clump together, and other stories the data may be telling. From there, they can make informal decisions related to range, frequencies, and “what’s typical” within a data set. As students more formally analyze data, the most common descriptive statistical methods used relate to measures of center (i.e., mean, median, and mode) and variability (i.e., range, standard deviation) of the data.
Students will often recognize how data collection and representation of data are both similar and different. They should informally discuss range as they note both the largest and smallest amounts (e.g., monthly rainfall accumulation), or most and least often chosen selections within a set of categorical data (e.g., weather pattern choices from among sunny, cloudy, rainy, snowy, or other). Consider the following investigation of typical household sizes for students in Mrs. Lopez’s class as a way to build intuition about measures of center and range without actually applying a statistical method or computational procedure. The following question was posed: “How many people, including yourself, live in the household you are living in now (58)?” First, nine students represented their family size using connecting cubes. Then they arranged the stacks of snap cubes in the increasing order of their family sizes (see figure 11a). As the connecting cube data were represented, students could recognize that family sizes varied and there are some family sizes that show up more often. Given the chance to notice and wonder about these data provides opportunities to discuss the range of family size; the most frequent or modal number of family members; and good estimates of both the median and mean number of family members. To develop conceptual understanding of mean as “equal sharing,” students can explore different approaches to equalizing the cube column lengths to find the average or mean family size (see figure 11b) and consider how the results would change if two of the nine students had three more family members. These observations about the distribution of family sizes can provide the opportunity for teachers to formally name and define terms as well as consider which measure of center is the most useful in describing the typical family size in Mrs. Lopez’s class. They can also engage students in thinking about and discussing how changes in a data set impact the distribution. Middle grades students will build upon these understandings as they learn how to determine the mean, median, range, and interquartile range and use these measures to compare data sets.
Noticing and Using Mathematical Structure.
Long before students formally explore measures of center and variability they notice attributes of data sets as they collect and organize data. For example, when examining the “typical” number of goals scored in soccer games, students noticed that in most games either 1 or 3 goals were scored, but in one game a non-typical 8 goals were scored. As they progress through the grades, students formalize their observations using standard data displays and statistical methods or computations and recognize the connections between these structures and earlier intuitions.
C.5.c. Interpreting data: developing inferences and evaluating predictions
Data analysis includes representing and describing key features of data distributions to support data-based decision-making that may involve making inferences or predictions. For example, an analysis of student attendance data collected over a week could support predictions about attendance for the following month while taking into consideration trends related to illness or upcoming events that may affect the data.
From the time that younger students begin categorizing and organizing data, they should be encouraged to both acknowledge and discuss differences that might explain the variability between groups. With more experience, for example, students can make statements about the results from a survey of third graders about their field trip preferences even when results vary from one class to the next.
Data are all around us. EMSs know that students need to regularly consider, frame, and respond to questions involving data, be able to understand and critically think about the use of data, and finally develop inferences or evaluate predictions based on data. They recognize the importance of statistical literacy in navigating today’s world and work with teachers to consider the multitude of mathematical and statistical learning opportunities that data analysis provides.
Contextualizing and Decontextualizing. Moving between the real world and the mathematical world is central to the statistical problem solving process– with formulating questions, collecting, representing, and considering the data and what is happening in context, decontextualizing when creating models and analyzing the data, and recontextualizing when interpreting the results.
C.5.d. Understanding basic concepts of probability
Early and informal understandings related to probability are related to whether a particular event is likely to occur. Prior knowledge and related understandings of numbers and in particular fractions, will assist students in understanding that the probability of an event occurring is the fraction of the time that event will occur theoretically (e.g., if you need to spin a 4 on an equally divided 1 through 4 spinner, the probability of the spinner landing on 4 would be 14, one chance in 4 trials). Students engage in probabilistic contexts by conducting and analyzing the results of experiments.
EMSs support teachers as they plan and facilitate instruction on probability at the elementary school level, which would typically focus on understandings related to likely and unlikely events as well as representing and understanding the results of simple experiments (e.g., games, surveys) and perhaps making and testing predictions. Such probability-related opportunities extend work with data analysis and fractions.
(56) Bargagliotti et al., 2020, p. 8.(57) Bargagliotti et al., 2020, p. 13.(58) Adapted from: Bargagliotti et al., 2020, p. 28-29.
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