Indicator of progress
An important step in early representation of data is to move from a direct one-to-one correspondence of data and pictograph objects, to representing the number of items in each category by the length of a column.
This increasing abstraction depends on previous experience of pictographs or bar graphs where there is a one-to-one correspondence of data and pictograph objects, appreciating the need to use tokens of uniform size. This also depends on students' growing confidence with number.
Illustration 1: Transition from pictographs to bar graphs
Before this transition, students will concentrate on reading individual pieces of information from a pictograph or bar graph. For example in the sample graph 1 below, they will see that Wendy plays in the sandpit on Monday and that William will be able to play with Wendy there. When making their own graphs, they will not fully appreciate the need to make the graph objects of the same size and to align them carefully so that the relative lengths/heights of the columns can be compared.
When they reach this stage, they will be able to make a column graph like sample graph 2, generally with support from teacher or peers. Here they appreciate the need to make the pictograph objects the same size, so that total numbers can be compared.
This graph is a preparation for the construction of the column graph in sample graph 3, where data points are not represented individually.
Illustration 2: Reading before construction of graphs
At every level, it is likely that students will be able to read graphs of a given type before they can construct them, whether with assistance (first) or independently (later).
For example, students able to construct graphs like sample graph 2 are likely to be ready to read graphs like sample graph 3.
The teaching strategy here is to be aware of the stages of students' development and to select data representation activities which support their growth.
In all instances, data representation is best taught when it arises from situations that involve the students, and when it answers questions which have meaning for them. Students should be involved in:
- posing the questions that the graph will help answer,
- collecting the data,
- choosing an appropriate representation and making it,
- interpreting the graph to answer interesting questions,
- evaluating the strengths and weaknesses of the representation.
Activity 1 Constructing a class graph using students' pictures, demonstrates how construction of a simple bar graph from real objects can involve all students.
The chart of stages of development describes characteristics and gives examples of graphs for students at various stages of development. The patterns of posing questions and interpreting graphs outlined above are constant throughout the levels.
Activity 1: Constructing a class graph using students' pictures
Teachers should select a classroom event or routine that is suitable for graphing. It is best to choose a topic that involves every student in the group. For example, on what day each student is allocated a particular activity (e.g. play in the sandpit), or what game each is playing at the school carnival etc.
At this level, the data is categorical e.g. 'what day', 'what game', 'what colour' and not numerical (how many). The words categorical and numerical are not used by students, but in professional teacher discussions. Sample graph 2 gives an example.
- Beforehand, the teacher prepares a large cardboard chart for the graph with the categories (e.g. days of the week) marked on one axis.
- Begin with a discussion about the need to record the data and discuss what benefit a record will be. Pose questions that could be answered and ask for student suggestions.
- Students draw a picture of themselves and place it appropriately on the graph. At an earlier level, the pictures need not be of the same size, nor placed uniformly on the graph, but these are important here so that questions about 'how many' can be answered.
- When the graph is finished, use it to answer questions such as 'how many children …'; 'on what day do the most/least/exactly 3 children …' etc.
- Conclude with students reflecting on the benefit of having pictures of the same size and placed uniformly for answering 'how many' questions.
Transition to next level
In transition to the next level, cover the strips of pictures by columns of plain coloured paper, making a bar graph (or column graph) over the students' pictures, and ask the 'how many' questions again. Students can be asked which questions can be answered using this new graph (the 'how many' questions) and which cannot (questions about individual students).
Stages of development in early graphical representation
||Features of graph
|0.5 - 1.0
Earliest classification of information.
Direct correspondence between 'data' and pictograph objects. Individual information is of most interest, rather than number of items.
Students draw a picture of themselves, and place it appropriately on a chart to show who is allowed to play in the sandpit on each day of the week.
Graph columns may be horizontal or vertical.
Questions to ask: On what day does Wendy play in the sandpit? Who can play there with her on that day?
Sample graph 1
|1.25 - 1.75
Increasing attention to observing the number of items in each category, so the need for tokens of the same size develops. One-to-one correspondence between 'data' and pictograph objects.
Students draw the picture of themselves on squares of uniform size such as sticky notes. Questions to answer: How many students play in the sandpit on Wednesdays? On what day do the fewest number of students play there?
Sample graph 2
|1.75 - 2.25
Transition from pictograph objects representing each item of data to length of bar representing number of items. Can read and interpret graphs with categories on one axis and number (frequency) on the other.
Students record how many students (rather than which students ) are allowed to play in the sandpit on each day of the week and make/draw bars to represent these frequencies.
Sample graph 3
|2.5 - 3.0
Students move from focus on representing individual items of data, to collecting and representing its frequency. Can read and interpret graphs with categories on one axis and number (frequency) on the other OR with numbers on both axes.
Students can draw and interpret bar graphs with numbers of both axes (eg showing how many students have a given number of pets).
Sample graph 4
Sample Graph 1: level 0.5 - 1.0
Children who play in sandpit
This graph may also have been drawn as a vertical column graph with the students' pictures placed on top of one another.
Sample Graph 2: level 1.25 - 1.75
Children who play in sandpit
Likewise this graph may also have been drawn as a vertical column graph with the students' pictures placed on top of one another.
Sample Graph 3: level 1.75 - 2.25
Sample Graph 4: level 2.5 - 3.0