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9:30 in the morning. The 3rd grade students at Riudellots de la Selva School enter the classroom. It’s time for math, and they will be working on Challenge 8 of Adventures, which is about statistics. During this activity, they will learn to:
- Differentiate between quantitative variables and qualitative ones.
- Collect data and analyze results.
- Represent the results with bar graphs.
- Work with concepts such as median and mode using real data.
You can refer to the activity here.
Introduction to the activity
Several vertical boards, identified with geometric shapes, catch students’ attention in different classroom corners. The students, curious, look at each other. What will they do today?
To introduce the activity, the teacher projects a video where the Bmaths receive a call from the Ministry of Statistics, which proposes a fictional survey:
- What is the length of the names in the class?
- What color is your hair?
The students discuss the difference between qualitative and quantitative variables with these and other examples, such as: “How tall are you?” (quantitative variable) or “Where do you live?” (qualitative variable).
The teacher distributes numbered cards from 1 to 7 to form random groups of 3 students. Each group stands in front of a vertical board with a single marker.
“Why only one marker?” asks a student.
“Because this way, you’ll discuss the answer before writing it,” responds the teacher.
This limitation transforms the dynamics of teamwork: any decision requires consensus, enhancing dialogue, and collective reflection.
Peter Liljedahl’s Practices: Learning by Thinking
Peter Liljedahl, an expert in mathematics education, presents in his book Building Thinking Classrooms in Mathematics 14 practices for fostering critical thinking and active participation in the classroom. In this article, we highlight random groups and vertical non-permanent surfaces.
Random Groups
BENEFITS
- Equitable collaboration: Randomness breaks the power dynamics or labels associated with “strong” and “weak” students. This encourages active participation from everyone.
- Diversity of ideas: varied perspectives enrich work in Problem Solving.
- Reduction of interpersonal conflicts: avoids fixed subgroups and promotes inclusion.
- Strengthening the classroom community: students interact with more classmates, which helps create an inclusive environment.
CHALLENGES
- Initial resistance: some students may feel uncomfortable or insecure when working with new people.
- Pacing issues: different skill levels can generate frustration if not managed properly.
- Dependence on teacher dynamics: Requires a skilled teacher to guide and supervise interactions.
Vertical Non-Permanent Surfaces
Liljedahl recommends using vertical, non-permanent surfaces instead of notebooks or horizontal boards to encourage active and visual interaction among students.
BENEFITS
- More physical and mental engagement: working standing up and as a team at a vertical board makes students more engaged.
- Visual accessibility: all group members can see the work at the same time, facilitating collaboration.
- Visibility of the teaching process: the teacher can easily observe how groups progress and offer real-time feedback.
- Experimentation and quick erasing: boards allow for correcting mistakes without fear, which encourages exploration and active learning.
CHALLENGES
- Limited space: having enough boards for all groups can be difficult in large classrooms with many students.
- Infrastructure dependence: not all schools have portable boards or enough vertical surfaces.
- Potential physical exclusion: for students with physical disabilities or limited mobility, vertical boards may not be accessible without adaptations.
Challenge Development
In this challenge, we work on designing data collection to answer a question. We aim to generate conflict when collecting data, so students realize that in the case of the length of class names, everyone should get the same results, but they can use more or less efficient strategies to collect them. In the case of hair color, we will see an element of subjectivity, so they will need to emphasize establishing coherent classification criteria since otherwise, each group, analyzing the same sample, could obtain different solutions.
Phase 1: Name Length
Let the letter counting begin! The boards fill with names as students look for ways to classify them. Gradually, they’ll realize they must be thorough so they don’t leave anyone out. Those who follow a logical order complete the list of names before those who do it from memory.
Decision-making is done collectively, which generates small debates. Some strategies observed:
- Initial annotation and classification in columns: they write all the names in a list and, next to each name, indicate the number of letters it contains. They organize the information by creating columns with the number of letters at the top. Below each number, they write the people’s initials with names that are that length. Meanwhile, they cross off the names from the list to ensure they don’t miss any.
- Grouping names by length: they arrange the names according to the number of letters they contain to simplify counting.
- Organizing names by groups: they classify the students’ names according to the random groups created. Next to each name, they indicate the total number of letters it contains. Later, they make a tally indicating the number of names for each letter count. Example: 9 → 2 (With nine letters, there are two names).
Students debate, correct errors, and help each other, creating a count they record in their logbooks.
Phase 2: Hair Colors
Similarly, they collect data about hair color, and each group uses different strategies to organize the information:
- Verification: They write a color and the total number of students who have that hair color, and, next to it, make a list with the names to verify that no one is missing.
- Organization by columns: They write the colors in columns and, under each color, list the names of the people.
- Counting by groups: They write each hair color and, next to it, the number of people from each group who have that color. For example, “2 students from group X have black hair” and “1 more student from group X also has black hair.” Finally, they add all the values to get the total people count for each color.
- List: They rewrite the list of names and, next to each name, note the corresponding hair color.
During the class discussion, they reflect on the differences between the variables. The length of the name is objective, while hair color is subjective and can generate debate. This, as we had planned in the session objective, helps them understand the importance of agreeing on criteria when collecting data.
Phase 3: Representing the results
Finally, the groups represent the data in bar graphs. This allows them to visualize statistical concepts such as the mode and the median. At a glance, students can see which number of letters appears most frequently (the mode) and which name length falls in the middle of all the data (the median). Looking at the bars, a student exclaims:
“Oh, the mode is like what’s most in fashion, right? It’s what’s most popular!”.
Conclusion: Fostering a collaborative learning environment
The two teachers emphasize that preparing to implement Peter Liljedahl’s strategies has been straightforward, especially when working together. Setting up vertical boards and organizing the space has been key to ensuring the activity’s success.
Regarding random groups, they highlight that this approach avoids conflict, reveals unexpected roles, and encourages collaboration among diverse profiles:
“Working with different profiles, some students have stepped forward and taken on more responsibility.”
The vertical boards have had a positive impact on concentration and participation:
“We’ve been surprised by certain students who normally get distracted by their pencil case, backpack, or chair when seated. However, when facing the board with a marker, they have fewer distractions and can focus their attention better, participating actively like never before.”
They also highlight how using a single marker has enhanced dialogue and joint reflection:
“At first, everyone wants to be the protagonist and grab the marker. But this forces them to ask: ‘Wait, what will you write?’ So, they must explain the entire process before writing it down. With just one marker, they’ve learned to reach consensus on decisions.”
The teachers also emphasize the role of Innovamat resources in facilitating group work:
“Group work is facilitated throughout Innovamat’s curriculum, especially because students can write down their thinking and thus consolidate their learning.”
This experience demonstrates how certain dynamics can transform the classroom into a space where students not only learn mathematics but also develop critical thinking, collaboration, and active participation.
