The trouble with graphs

It’s tempting to assume that constructing graphs of data is a relatively straightforward task, requiring students to learn a small set of rules for each type of graph, to be used with a multitude of data sets - in some senses an algorithmic approach. However, maths teachers’ experiences suggest otherwise. Students seem to come up with a plethora of problems and misconceptions when faced with any task involving graph construction; for example unusually inconsistent axes and best fit lines that are in fact quite the opposite.

Much of this difficulty can be traced back to two specific and related problems. Firstly, teaching the mechanics of graph construction encourages students to attend to the ‘how’ without considering the ‘what’, the ‘where’ or the ‘why’ of the different features. What does each axis represent? Where are the individual data values represented? Why is this representation appropriate for this data?

Secondly, because each graph rule seems simple, it is easy to ignore the fairly deep mathematical ideas that underpin each one. Students whose broader mathematical knowledge is insecure may be forced to learn graphing rules by rote, rather than understanding their meaning and relationship to the wider mathematical ecosystem.

Let’s start by focusing on what is generally considered to be one of the most straightforward charts: a bar chart for a small categorical data set such as ‘favourite colour of students in my class’. What concepts are necessary to understand the various non-arbitrary features of such an entity?

Correspondence – at the most basic level, students must understand that their individual responses to the question “what is your favourite colour” are represented by an icon (e.g. a symbol, shape, or mark) on the graph. More often than not this icon is an imagined square (or rectangle) whose horizontal position signifies their answer to a question.
Equivalence – any two identical responses are equivalent and hence interchangeable. The value of each answer can be represented by the horizontal position, but each response is functionally identical to any student who gave the same response and therefore their own value cannot be distinguished within this set of responses.
Exclusion – once a data value is placed on the graph, an equivalent value cannot be placed directly on top of it. This results in the formation of bars.
The number line – it’s essential for students to understand this model of number in order to be able to read information from the y axis of a graph. Evidence that students either have a weak model of the number line, or do not associate the axis of a graph with the (linear, non-logarithmic) number line can be seen when the spacing between consecutive values are of inconsistent size.
Representing a count by length – students often initially judge the frequency of a data value by counting icons (e.g. in a dot plot, or by colouring squares on a grid), but must transition to the idea that the length of the bar itself corresponds to the value of the count.
Visualisation of an orthogonal grid – students equate the length of each bar to a position on the y axis. In their early work on measure, students usually position the ruler closely to the object being measured. When drawing or reading from a graph, a different situation is likely to happen with different sized gaps between the scale and the bar, meaning students have to visually scan across in some way. So students need to either visualise a grid on the plane of the graph, or use a straight edge at 90º to the frequency bars to relate the bar length to the scale.

Other types of chart will share some of these conceptual underpinnings, but many will differ. Students who are taught the mechanics of drawing graphs without attending to these underlying concepts find the rules more confusing and inconsistent the more charts they are shown. The exclusion principle, for example, does not apply in a scatter graph. What reason might a student give for this?

There is an increasing volume of evidence that students’ graphical literacy is enhanced by creating and refining their own representations rather than by being taught specific chart types, but this is not a panacea. Research shows that time must be spent critiquing and analysing students’ own representations, and the representations of other students, in order to reach a consensus that resolves toward the more standard graphs. The process of designing and critiquing graphical representation encourages students to attend to the ‘why’ and the ‘how’ of data representation, and may support graphical literacy when interpreting unfamiliar graphs both in school and in the wider world beyond. A further point may be considered: if we would like students to be able to work creatively with data presentation, they need to be able to traverse design decisions orthogonally too, in both the aesthetic and the mathematical directions.

CORRESPONDENCE:

"this is a representation of me amongst others"