Statistical diagrams inquiry
Mathematical inquiry processes: Speculate and verify; create more examples; make connections. Conceptual field of inquiry: Bar and pie charts; bi-variate data and scatter graphs; other representations of data.
This prompt is suitable for students between the ages of 10 and 14, although it could be made appropriate for younger children by presenting a pictogram and frequency polygon with the bar chart or for older children by adding a histogram. When the bar and pie charts have been presented as a pair, students have made comments related to the following:
Missing information - there should be labels on the axes of the bar chart (for example, 'frequency' on the vertical axis) and titles.
Reading the charts - the bigger the piece of the pie chart (or the higher the bar in the bar chart), the greater the amount represented; the charts tell you how many things there are in something; most of the time, pie charts are to do with percentages; the bar on the left is a quarter of the highest bar.
Construction of the charts - the pie chart has degrees; why is a pie chart harder to draw than a bar chart?
Interpretation - what do the charts represent?; the pieces of the pie chart are split into 5%, 20%, 25%, and 50%; the charts represent the same information; they show favourite school dinners.
An initial discussion might focus on the data set represented by the charts. How was the data collected? What sampling technique was involved? What was the size of the sample?
To test if the charts come from the same data, students have decided to draw a scale on the bar chart (for example, 1mm represents two people), worked out a set of frequencies and drawn the pie chart accurately. The pie chart they end up with will be slightly different to the one in the prompt. However, each pie chart will be the same as the others in the class (or very similar), regardless of the scales that students have used. An important lesson to draw out at this point is that pie charts show the proportion of quantities, not the quantities themselves.
The scatter graph introduces the concept of bivariate data and challenges students to compare its meaning with the two charts. Often, a class will argue that the graph is linked because of its similar shape to the bar chart. Students might interpret it as a time series with the line of best fit as a trend line (see box). The inquiry teacher could use the regulatory cards at this point to decide how to proceed. Students might ask the teacher to explain or provide resources so they themselves can inquire into the meaning and purpose of scatter diagrams.
Insight and understanding
Our student teacher, Jonathan Li, from the Faculty of Education at the University of Windsor, took a risk and used the prompt to create an open and engaging learning task for students using a Google Jamboard. The inquiry developed out of the need for students to be able to read and interpret data summarized in the form of graphs. Jonathan saw this need and felt that providing the students something that would be open to different interpretations would be an effective way of doing this. It was his first time conducting an inquiry in this way and he felt somewhat uncomfortable turning so much over to the students. As the students made observations, raised ideas and questions, and created key learnings, Jonathan quickly realized that when the students were left to read the graphs on their own they were able to demonstrate true insight and understanding.
Ju Garcia, a grade 5 teacher at the United World College in Phuket (Thailand), orchestrated a data handling inquiry with her class. She posted these pictures on twitter. They show the pupils' questions and observation about the prompt and their plans and lines of inquiry. Ju reports on how the inquiry developed:
We launched our inquiry into data handling with the rich statistical diagrams prompt. It was a great lens into what students already knew and what they were curious about. At the start of the inquiry, I could already see some interesting inquiries taking shape. Students formulated plans, investigated, made conclusions and considered next steps as they initiated their own personal inquiries in graphs for the next few day.
We spent five or six 50-minute blocks from exploring the prompt to reaching a conclusion. Engaging with partners and collating initial ideas took about one hour. The rest of the time students planned, conducted and finalised the inquiry. Students finished at different times, which gave them open opportunities for the next steps.
We moved from the prompt to collecting, organising and representing data connected to our migration unit. The students engaged with data sets from surveys they created, school enrolment data (provided by the enrolment department) and 'The Refugee Project' data available for Thailand.
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), posted these pictures on twitter. They show the students' questions about the prompt. He explains that Shaleen Cuffe, a trainee teacher working with the class, conducted the inquiry: "It was her first attempt at running something so open and that had such little direction. The inquiry developed out of the students already working with graphs, and we wanted to see what they would do with the information. They had not been able to identify that certain graphs are used under specific circumstances. In taking the risk to run the inquiry, Shaleen successfully got students to think about and read graphs and the inquiry moved their thinking along."
Making thinking visible through inquiry
The reports below come from grade 5 pupils at Castle Oaks Public School in Brampton (Ontario, Canada). As their teacher says, the pupils have made their thinking visible by asking and answering their own questions:
Do all the graphs represent the same thing but in different ways?
How can we figure out what they are trying to graph?
If the line is not connected to the X's is it still called a line graph?
What is the scale?
The spirit of inquiry in the classroom is summarised in the teacher's encouragement to others: "Keep engaging in open problems and critical thinking fellow math inquirer!"
Reasoning through inquiry
The picture shows the questions and observations from grade 5 students at the Western Academy of Beijing in China. Nathaniel Atherton, their teacher, reports on the inquiry that developed:
The students quickly began to identify the graphs, first by naming them and then by listing the missing information. They were able to name the first two easily but really struggled with the scatter graph, which they argued was like a line graph, but not a real line graph. The students then began analysing the values represented and drew connections between the growing patterns shown in each graph. The class debated whether all graphs were representing the same data.
Students then divided into sub groups to either prove or disprove the correlation between the graphs. Some groups focused on all three graphs while others looked at just the first two. The students quickly identified tools which would possibly aid in their inquiry including rulers and protractors. They specifically worked on the bar and pie chart, trying to develop a standard measurement unit in which they could compare the two sets of data.
The inquiry proved challenging yet engaging for the students and they enjoyed comparing their results.
These are the comments and questions of Ann Macdonald's year 8 class. Ann is a teacher of mathematics in Brighton (UK).The class used the regulatory cards to decide to inquire collaboratively. Ann describes how the inquiry progressed:
"All of the students were able to make up some data to fit the bar chart and the vast majority measured the vertical axis to create a scale. Very few had actually considered that the bar and the pie chart represented the same data, but a couple did. When this was brought up, the students wanted to draw their own pie charts, with those that knew how to helping the ones that didn't. Some figured it out once percentages had been mentioned. In the first lesson, no one thought to check whether their completed pie chart looked like the one on the prompt.
"The majority of the class thought the scatter graph was a 'line graph' so I decided to focus on the bar and pie charts. At the end of the first lesson, one girl volunteered to present how to draw a pie chart to the rest of the class as a re-cap. The rest of the class decided that this was a good idea."
During the first lesson, one student interpreted the scatter graph as a time series with the line of best fit viewed as a trend line. Ann distinguished the two by adding a time series to the prompt and, in the discussion that followed, drew out the knowledge that existed in the class about scatter graphs.
At the end of the second lesson, Ann used the learning journey to encourage students to reflect on how their inquiry skills had developed.
Emma Rouse, lead practitioner at Brittons Academy (Rainham, east London), posted the picture above on twitter, explaining that she had enlarged the prompt to cover more of the department's scheme of learning. The prompt below features a cumulative frequency graph, which follows the 'pattern' of the bar chart and scatter graph.