D1.6 Analyse different sets of data presented in various ways, including in histograms and broken-line graphs and in misleading graphs, by asking and answering questions about the data, challenging preconceived notions, 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 results should be done from two perspectives: the investigator's and the reader's. From the investigator's perspective, students examine data that they have collected and summarized in graphs or tables. Students then interpret the data in order to answer the question of interest that they posed at the beginning of the inquiry. From the reader's perspective, students examine data that has been collected externally by others. Here, teachers can take advantage of the opportunity to present them with data that relate to other subjects, for example, science and technology, social studies, physical education and health.
When teachers present a table or graph to students, it is important to allow sufficient time for reflection so that students can examine the data and form a general idea. Reflection can be done individually or in small groups. Teachers 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, teachers could 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 graph. |
|
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 or table and draw conclusions. |
Teachers sometimes tend to focus on the first level, but Friel et al. (2001), suggest that 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 key, 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 … 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?
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 and Higgins, 2003). This level of comprehension is more difficult to achieve because students must analyze individual data by combining them 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, the most, the least, a little more than, three times less than, there is a small difference between;
- compare the heights of bars in a bar graph;
- make connections between different ways of describing a relationship indicated by the 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.
Example of relevant questions
- What is the most popular category among 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 Ontario account for more than half of the production?
- How many more of … are there compared to…?
- Is there one category that outperforms all others in terms of frequency?
- Why was this data presented in two graphs 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 graph 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?
- How could the data be organized to uncover additional information? (For example, by pooling the data from primary and junior division students, one 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: Drawing Conclusions, Formulating Arguments, and Making Decisions from a Graph
After examining and analyzing the 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 students to draw relevant conclusions in order to answer questions of interest and 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 an inquiry 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 about using the 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 they have derived from the data;
- the conclusions they drew.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 101.
Knowledge: Misleading Graph
Sometimes graphs present data in an inappropriate way that could influence the conclusions we draw from them, so it is important to always critically interpret the data presented.
Example
The graph below is misleading because the scale for the number of guests does not start at zero. The difference is exaggerated and the intervals on the scale showing the age of the guests are not equal.
image The coloured three-band chart is entitled "Number of Guests at the Tourist Complex by Age". The horizontal axis, named "Age", is graduated from 20 to 80, while the vertical axis, named "Number of Guests", is graduated from zero to 20,000. The tops of the bars are connected to each other by a line. The first bar, which is blue, represents 20 to 40 on the horizontal axis and rises above 15,000 on the vertical axis. The second bar, which is red, represents from 40 to 70 and rises to 20,000. And the third bar, which is yellow, represents 70 to 80 and rises above 10,000.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
It is important to note that throughout the interpretation of results, teachers should encourage students to make critical judgments about the data presented so that they are aware of the potential limitations of the interpretation. The ability to make critical judgments is an important component of statistical thinking. To help students develop this skill, teachers should teach them to:
- question the representativeness and reliability of primary or secondary data by asking questions such as: "Was the data collection done without bias?" "Is the secondary data source reliable?";
- check whether the representation gives an accurate portrait of the data by asking questions such as: “Is the data represented without leaving false impressions? ".
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 104.