Student Reasoning About Chance

 

Developing Critical Numeracy Across the Curriculum

Student Reasoning About Graphing

Over the primary and middle years, it is likely that students will be exposed to various conventional graphical forms of data representation. It is also important that students are given the opportunity and encouraged to create their own graphical forms to tell the stories that are inherent in data sets. This can be done by presenting students with raw data, with using data that students have created themselves, or by providing a verbal description that describes a relationship or trend. It is also possible for students to use data analysis software, such as TinkerPlots, to create plots for specific data sets. Three potential sequences of development are suggested here: (1) for conventional graphical forms, (2) for graphs created from a verbal description, and (3) for a constructivist software package such as TinkerPlots.

Conventional graph forms

Examples of the graphical forms discussed here can be found in many text books or by searching the web. Generally speaking the complexity of the demands of these representations involve the type of data (continuous, discrete, or categorical), the number of variables involved, and whether values (of the data) or frequencies of the values are being plotted. Various combinations increase the demands on students. A likely sequence of increasing complexity is the following.

  • Pictographs, where a single icon represents one data value; counting icons gives frequency.

  • Pictographs, where a single icon represents more than one data value; frequency requires counting icons and multiplying.

  • Bar graphs, where icons are merged into rectangular bars and a frequency is recorded against a scale.

  • Value graphs, where horizontal bars from the vertical axis represent data values, scaled along the horizontal axis.

  • Dot plots, where value bars disappear with a dot remaining above the appropriate value on the horizontal scale.

  • Stacked dot plots, where the value dots are stacked along the horizontal scale to reflect frequency.

  • Stem-and-leaf plots, vertical representations showing frequency in relation to place value representation of data, with frequency determined by count within groups.

  • Histograms, where measurement data are represented in continuous intervals along a horizontal axis, with a vertical scale for frequency.

  • Box-and-whisker plots, derived from stacked dot plots (or equivalent) where quartiles and extreme values determine the middle edges of the box and ends of the whiskers.

  • Side-by-side representations of the above types, in considering a single variable across another categorical variable such as gender or state.

  • Scatterplots of two measurement (continuous) variables, where a single point represents the relative position of the object with respect to each variable simultaneously.

  • Graphs based on verbal descriptions

Graphs based on verbal descriptions

An example is provided here of a context where students were asked to draw a graph showing the "almost perfect relationship between automobile usage and heart deaths." The expected progression in understanding is shown in the following sequence

Context only picture
Graph outline only
Graph with no context or trend
Graph with variables but no relationship
Graph with time and one variable
Graph showing two variables graphed against time

Graphs created in TinkerPlots

Examples of graphs from TinkerPlots show student work for arm span data collected by a middle school group. This sequence shows an increasingly sophisticated representation of the data.

Value plot



Below:
Stacked dot plot - two different scales
Stacked plot with hat plot
Stacked data plot with box plot
Stacked plot separated by category (grade)

Many more representations are possible for graphing and although it is important to provide students with models, it is also necessary to give them freedom to develop sophisticated ways of telling stories on their own.