D1.2 Collect continuous data to answer questions of interest involving two variables, and organize the data sets as appropriate in a table of values.
Skill: Formulating and Answering Questions of Interest
In planning their inquiry, students should first ensure that they have a clear understanding of the situation. This can be done by clarifying the problem and formulating one or more questions that can be answered with data. Students should also consider, to the extent possible, some of the variability factors that may affect the results of the inquiry or experiment.
Clarify the Problem
When presenting a problem-solving situation, teachers should ensure that the problem is understood by all students. They can check this understanding and, if necessary, help them clarify the problem by facilitating an exchange of ideas about the situation.
Formulate Questions
Once students have a clear understanding of what they are looking for or would like to know, they can begin to plan the inquiry. The students must first clearly formulate the question of interest to clarify the intent of the inquiry. A question of interest is a question that can only be answered from variable data. So, "How many children are there in your family?" is not a question of interest, since the answer is fixed (for example, three children) and does not depend on variable data. However, "How many children per family are there among the students in the class group?" is a question of interest, since to answer it, it is first necessary to collect data relating to the number of children in each of the students' families. In the light of the answers obtained, it will then be possible to conclude, for example, that in the majority of the students’ families, there are two children.
Konold and Higgins (2003) argue that in the inquiry process, students' first challenge is to turn a general inquiry into a question of interest. Teachers need to help students understand the importance of framing the question of interest correctly and ensuring that it accurately reflects what is being sought.
Students should also learn to recognize that the choice of the question of interest affects the type of inquiry that should then be conducted. The following are some examples of questions of interest that students might address. Each question is accompanied by the type of inquiry it suggests.
| Question of Interest | Type of Inquiry |
|---|---|
| What is your favourite food in the school cafeteria? | Survey |
| How many students eat lunch outside of school? | Observations |
| How far can you travel in less than 20 minutes? | Measurements |
|
When simulating an earthquake using the robotics equipment, how strong are the buildings you have constructed? |
Scientific experiment |
| If two names are chosen at random from the class list, are they more likely to be two names beginning with a letter from A to M or from N to Z? | Probability experiment |
| What were the results of the last Ontario election? | Secondary data collection |
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 61.
Skill: Collecting Data
Planning a data collection and carrying it out provides authentic and meaningful data.
The inquiry process is a comprehensive one that involves four steps: identifying the situation, collecting the data, organizing the data, and interpreting the results.
Once students have clarified the problem and formulated one or more questions of interest, they should plan and conduct a data collection. It is important to consider the different types of inquiry, the different kinds of data, and the difference between population and sample when planning data collection. By actively involving students in planning data collection, it encourages them to make thoughtful choices and look critically at the entire inquiry process.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 44.
Questions to Ask
It is important to provide students with a variety of opportunities to plan a data collection. By asking students questions throughout this stage, teachers help them to understand the importance of choosing the right type of inquiry and data for the question of interest, and then to identify the population and, if necessary, the sample to be surveyed. In doing so, teachers help students to develop critical thinking skills, which will be very useful in the fourth stage of the inquiry process.
Here are some ideas for questions that teachers can use to guide students through the planning of data collection.
Type of Inquiry :
- What type of inquiry is most appropriate for your question of interest? Why?
Type of Data :
- What type of data will you collect?
- Does this kind of data lend itself well to your question of interest? Why or why not?
- If you are going to use secondary data, where will it come from? Is this source reliable?
Target Population :
- What is your target population?
- Is this the group that your inquiry is targeting?
- Will your inquiry be conducted with the entire population or only a portion of the population?
Sample Size :
- What will your sample size be? How did you determine it?
- With a sample of this size, will the results be representative of the target population? Why or why not?
- Do you think the results would be similar if the sample size were smaller? larger? why?
Sample Composition :
- Is the sample composition free of bias? Convince me.
- How will you go about selecting your sample at random?
- Does your sample need to be stratified? Why or why not?
- What strata will you use in the composition of your sample? What will be the size of each stratum?
Modalities (Where, When, How) :
- Where will you conduct your inquiry?
- When will you conduct your inquiry? Why is this a good time? If it were conducted at another time, would the results be the same?
- How will you go about getting the data you need?
- How will you record the results of your inquiry?
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 57-58.
Skill: Organizing Sets of Data
Organizing data and representing it in tables and graphs helps communicate information for interpretation. Once students have identified the situation and collected data, they must organize it.
Why Organize Data
Gal (2002) indicates that we organize the data to better analyze it or to communicate information. Since the purpose of the inquiry is to find an answer to one or more questions of interest, it is very difficult to base that answer on data that is presented in a disorganized fashion. By organizing the data collected, it can be presented in a way that summarizes it, highlights certain information, communicates its main characteristics, and facilitates its interpretation.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 62.
Knowledge: A Two-Variable Question of Interest
The type and amount of data to be collected is based on the questions of interest. Questions of interest involving two variables require the collection of two data sets from the same sample or population.
Many science experiments involve the relationship between two variables. The independent variable is what the researcher gets to change, and the dependent variable is what the researcher gets to observe or measure during the experiment.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020
Knowledge: Types of Inquiry
Data Collection by Observations
In a data collection by observations, we record what we see or do.
Examples
- We count the number of birds we see in the schoolyard at specific times.
- We count the number of cars that pass through an intersection during a given time interval.
- We count the number of times we go to the sports center in a month.
- Every day, for a week, we note the time we go to bed and the time we get up.
In planning data collection by observations, it is necessary to consider where, when, what, and sometimes how to observe; for example, how to distinguish between a car that makes an incomplete stop at an intersection and a car that makes no stop at all. It is also important to determine whether all observations will be made by one person or by several people at the same time to ensure greater reliability.
Data Collection by Measurements
In a data collection by measurements, we make simple measurements in situations that do not require special attention to various variables, as is the case in an experiment.
Examples
- We measure the height of people and the length of their feet to determine if there is a relationship between the two variables.
- We measure the amount of time required for Grade 8 students to read a given text.
- We measure the amount of rain (in mm) that falls each day in May.
In planning data collection by measurements, it is important to consider where, when and how to collect the measurements, and whether all measurements will be collected by one person or by several people at the same time to ensure greater reliability.
Data Collection by Means of an Experiment
In data collection by means of an experiment, the data are derived from scientific activity that requires adherence to certain preset parameters and, often, the use of precise measurement techniques and tools.
Examples
- At specific intervals, we measure the growth of plants, some of which have received a small amount of nutrients, some of which have received a larger amount, and some of which have not received any, to see if nutrients contribute significantly to plant growth.
- Every 30 seconds, the temperature of any liquid that has been heated to 100°C and left to cool. The experiment is repeated with various liquids in order to compare the rate at which they cool.
In planning data collection by means of an experiment, the scientific approach must be used and the reliability of the data collection method must be ensured. Variables that may render the results invalid must be controlled and neutralized.
Survey
In a survey, data is collected by asking a number of individuals about a particular topic. The questions often take the form of a questionnaire that can be answered in writing or orally.
Examples
- Students in the class are asked how many hours they spend watching television each week.
- Grade 8 students are asked what kind of music they prefer.
In planning a survey, it is important to carefully formulate the survey questions to ensure that they are clear and objective. It is also important to determine the responses that can be given and sometimes group them into categories.
Secondary Data Collection
When gathering existing data, the data is usually found in an electronic database such as a website, or in a printed document such as a book, magazine, or encyclopedia.
Examples
- We want to compare the population of Canada's provinces and territories.
- We want to compare the subject preferences of junior students to intermediate students across Canada.
When planning secondary data collection, it is important to check whether the data are available, where and how to obtain them, and to ensure that the source is reliable.
Source: adapted and translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 46-48.
Knowledge: Sampling Techniques
Selection Process
Simple Random Sampling
Students should understand that one of the best ways to have a good bias-free sample is to choose it randomly, so that all members of the population have an equal chance of being included.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 55.
Systematic Random Sampling
Systematic random sampling is used when the subjects from a population are selected through a systematic approach that has been randomly determined. For example, a sample could be determined from an alphabetized list of names, using a starting name and count (for example, every fourth name) that are randomly selected.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020
Stratification Process
In some surveys, one might want to ensure that certain subgroups of the population are well represented in the sample (for example, the junior and intermediate subgroups). In this case, the population is said to be stratified (divided into mutually exclusive groups), and one wants each stratum (group) to be represented in the sample.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 56.
Stratified Random Sampling
Stratified random sampling involves partitioning the population into strata and then taking a random sample from each. For example, a school population could be divided into two strata: one with students who take a bus to school and the other with those who don’t take a bus. Then a survey could be given to 10% of the population randomly selected from each of these strata.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020
Knowledge: Primary Data
Primary data is data collected by the person conducting the inquiry. It is well-suited to questions that involve objects and people in the students' immediate environment, and is ideal for introducing students to data management, since they are usually more interested in the data they have collected themselves. Once students are familiar with the possible answers, they can facilitate the recording of data by using a frequency table.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 50-51.
Knowledge: Secondary Data
Secondary data are data that have been collected by an individual or organization (for example, researcher, company, association) other than the person conducting the inquiry. Secondary data can be found in books, encyclopedias, journals, newspapers, and on the Internet. They are particularly useful for answering questions of interest for which it is difficult or impossible to collect primary data (for example, over the years, what was the size of the Francophone population in major Canadian cities?). They can also be used to interpret other data with which they are related to.
Students should be able to choose, depending on the type of inquiry and the nature of the question of interest, whether they will use primary or secondary data. Teachers should help students develop the ability to judge the relevance of secondary data to which they are exposed on a daily basis. To do this, teachers need to continually educate students about the importance of checking the reliability of various sources of information, as well as the importance of making good use of the data presented. Graphs and data in newspapers provide an authentic and meaningful context for dealing with the data.
Survey Process and the Internet
Access to the internet gives students the opportunity to participate in national and even international projects that place them in authentic data collection and exchange situations, thus fostering collaboration among students from different countries.
For example, the Census at School project "is an international online project that introduces students in Grades 4 to 12 to the world of surveys and statistics. The project began in the United Kingdom in 2000, and schools in Australia, Canada, New Zealand, and South Africa are now taking part. Students in each participating country anonymously fill in an online questionnaire in class. They answer non-confidential questions about topics such as their height, travel time to and from school, and favourite subject. The responses become part of a national database, which is later added to an international database that is maintained in the United Kingdom.
Since students are expected to recognize the difference between primary and secondary data by Grade 4, such projects are valuable tools for investigating a topic of particular interest and concern to them.
*From Statistics Canada, Census at School - Canada! (Accessed June 23, 2022).
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 51-52.
Knowledge: Continuous Data
A form of quantitative data. Continuous data is data that can be measured but not counted, such as time, height, and mass.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020
Knowledge: Table of Values
A table used to record the coordinates of points in a relation.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020
Note: When coordinates are linked by a relationship, the value corresponding to the x is the independent variable and the value corresponding to the y is the dependent variable.
Source: The Ontario Curriculum. Mathematics, Grade 1-8 Ministry of Education, 2020