D1.6 Analyse different sets of data presented in various ways, including in stem-and-leaf plots and multiple-bar graphs, by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions.
Skill: Examining and Analyzing Data Represented in Various Graphs
The skill to interpret results is related to the ability to reason in that it requires a certain amount of thinking and analysis. Teachers need to help students develop this skill by providing a variety of activities that focus on interpreting results and by asking questions that encourage students to look closely at those results. In doing so, they will also be contributing to the development of students' data literacy skills.
Gal (2002) suggests that the interpretation of the results should be done from two perspectives, that of the investigator and that of the reader. From the investigator's perspective, students investigate data that they themselves have collected in a survey and summarized in graphs or tables. They then interpret the data in order to answer the question of interest they posed at the beginning of the survey. From the reader's perspective, students are investigating data that is externally sourced, that is, collected by others. In this context, educators can take the opportunity to introduce them to data that relate to other subjects (for example, science and technology, social studies, physical education, and health).
When educators present a table or graph to students, it is important to allow sufficient time for reflection so that students can investigate the data and form a general idea. Reflecting can be done individually or in small groups. Educators should then ask open-ended questions to help students translate their observations and ideas into their own words and develop ideas from those of their peers.
For example, the teacher might ask them:
- What do you notice about this graph?
- What is interesting about this graph?
- What can you say about this data?
- What can you say about this graph?
These open-ended questions elicit a variety of responses, allowing all students to communicate their observations, descriptions, and conclusions in a general way. Later, educators can ask more specific questions to help students develop the skill of making sense of data. In managing data, this skill involves three levels of comprehension: reading the data, reading between the data, and reading beyond the data.
| Level of Comprehension | Description of the Level |
|---|---|
|
Reading the data |
Identify the data as represented by the table or chart. |
|
Reading between the data |
Compare and combine some data to establish relationships between them. |
|
Reading beyond the data |
Infer or predict implicit or explicit information from a graph and make conclusions. |
According to Friel and collaborators (2001), educators sometimes tend to focus on the first level, but they should be careful to always ask questions related to all three levels, regardless of the representation or organization of the data. This will help students develop independence in interpreting graphs and tables and in using statistical measures.
The following provides a more detailed explanation of each of these three levels, as well as examples of relevant questions that teachers might ask as part of an activity to interpret the results.
Reading the Data
At this first level of comprehension, students are able to determine:
- the components of the representation (for example, the title of the table or graph, the scale or the legend, the labelling of the axes, the choice of categories);
- the value of certain data represented.
Examples of Relevant Questions:
- What is this graph about? (The title)
- How many… are in the category? How do we know?
- How many categories are there?
- What is the scale on the horizontal axis?
- What does the vertical axis represent in this graph?
- What is the largest area?
Reading Between the Data
This level of comprehension requires viewing data less as "[…] an amalgam of personal data each with its own characteristics," and more as "[…] a collective data set with new properties" (Konold & Higgins, 2003). This level is more difficult to achieve because students must analyze individual data by combining or comparing data sets.
At this second level of comprehension, students are able to:
- compare data using expressions such as more than, less than, as much as, greatest, least, a little more than, three times less than, there is a small difference between;
- compare the lengths of bars in a bar graph;
- make connections between different ways of describing a relationship between data;
- combine some data according to certain categories and compare the frequencies of each category;
- describe some advantages and disadvantages of two different representations of the same data;
- determine the value of certain statistical measures (mode, median, mean) of a set of data.
Examples of Relevant Questions:
- What is the most popular category for students in grades 4, 5 and 6?
- Is the number of students with ... greater than the total number of students with ...?
- Of the metals produced in Canada in 2004, did more than half of the production come from Ontario?
- How many more …. are there compared to…?
- Is there one category that outperforms all others in terms of frequency?
- Why were these data presented in two charts rather than one?
Reading Beyond the Data
At this third level of comprehension, students use several critical and analytical thinking skills. They are able to:
- recognize what the diagram or chart does not "tell" directly;
- specify the trend of a data set;
- make inferences and predictions;
- draw conclusions and justify them;
- assess the credibility and logic of predictions and conclusions;
- assess the representativeness of the range, mode, median and mean;
- review the steps in the inquiry process.
Examples of Relevant Questions:
- Do you think that… is the most common in all cities? Why?
- If the survey is repeated with other response choices, do you think the results will be similar? Why?
- How could the data be organized to uncover additional information? (For example, by pooling data from 4th graders and 5th graders, we can analyze student preference in general.)
- What other questions can you answer using this graph?
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 89-97.
Skill: From a Graph, Drawing Conclusions, Formulating Arguments and Making Decisions
As a result of examining and analyzing data (three levels of comprehension: reading the data, reading between the data, and reading beyond the data), students are able to draw conclusions, formulate arguments, and make decisions.
Interpretation of the results allows for relevant conclusions to be drawn in order to answer questions of interest and to make informed decisions.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 89.
Decision-making is very important in the inquiry process because without decision-making, the process becomes meaningless. Why would anyone set up a survey and then collect, organize and analyze data if they have no intention of drawing conclusions? In many cases, decision-making is limited to providing an answer to the original question. In other cases, it is a matter of using that answer to decide whether to act in a particular direction.
Therefore, teachers should engage students in answering the question of interest using all three levels of comprehension, so that they can make decisions based on:
- the relationships established between the data;
- the meaning that students have derived from the data;
- the conclusions they drew from it.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 101.