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You open the door to the classroom and your initial reaction is shock. It seems that chaos reigns as you notice students out of their seats as they talk with one another. Furthermore, the desks are not neatly arranged in rows, but seem to be randomly clustered into small groups as the students use the desk tops to set up laptop computers connected to microphones used for collecting data. Students are recording their work on the white boards that surround the room. Where is the teacher? You are not sure. Concerned with the outward appearance of this classroom, you enter the classroom to investigate. You stop at the first group of students and try to figure out what is going on. Quickly you realize that the topic of their conversations is highly mathematical. They are hypothesizing the result of an experiment where two tuning forks of different frequencies are struck and the combined frequency is measured. Would the frequency be the sum of the two tuning forks? The difference? Or some combination that will be difficult to determine? They follow through with the experiment using a data collection device and a laptop computer and the students begin to analyze the results. They are discussing ideas and concepts foreign to you as they search for the *beat pattern*, calculate the *period* and *frequency, *determine the equation of the *sinusoid* which fits over the *beat pattern, *and ultimately discover the relationship between the frequency of the individual tuning forks and the combined frequency. You ask one of the students, "why are you doing this?" Annoyed by the interruption, she responds that they are trying to solve the problem at hand where, in the situation given, an old World War II mine needs to be deactivated. This can only be accomplished by directing the proper sound frequency at the timing mechanism in the bomb in order to disable it. You find it refreshing to see students motivated to solve a problem and notice that students are simultaneously using technology while communicating their analysis on a nearby whiteboard as they engage in teamwork to find the solution to a problem that has obviously engaged the interest of the students.

You move on to the next group of students. They are equally engaged in what appears to be a different problem. You first notice that the students are huddled around a laptop screen. Suddenly, a shout of celebration is heard as students triumphantly give each other high-fives and fist bumps. You get the attention of one of the students and ask what they are doing. He explains that their team was given data related to the height of waves in the ocean during a particularly dangerous storm. Their job was to create a scatter plot of the data and determine a sinusoidal regression model for this data so that they could calculate the period, frequency, and height of the waves at the moment in time a small touring vessel was capsized. Given the manufacturers specifications of this vessel, the students would be able to determine if it was built to withstand such a storm or not. This would aid in the investigation and inform an eventual lawsuit brought by passengers of the vessel who believed that the vessel's skipper used poor judgment in venturing out into the storm.

Impressed by the level of engagement of the students, the high-level discussions that were taking place between students, and the interesting combination of technology and well-communicated mathematical justifications recorded on the whiteboard that were taking place, you still have one more concern: where is the teacher?

You move on to the next group and find that they are working with tuning forks also. Peering in to see what this group is discussing, you find that one of the members of the group seems to be asking questions. "What would you know about the frequency if the period is 0.123 seconds?" "Do you see a pattern when you consider the different combinations of two tuning forks and their resulting beat pattern?" "Can you explain the difference between period and frequency?" You realize that this is the instructor asking probing questions to the students to help them construct their own understanding of the mathematical concepts being explored.

Standing back to observe, as a whole, the classroom setting that initially brought disgust, you now feel joy as you realize the high engagement of students as they study difficult math (and physics) concepts. But the students seem to be enjoying it. Could this really be a math classroom? What pedagogical framework was employed by the instructor in this classroom to get to this point?

This vignette attempts to portray an example of an active learning environment built on the framework of Peter Liljedahl’s research focused on *Building Thinking Classrooms* (Liljedahl, 2020). Liljedahl’s framework includes 14 practices and for the purpose of this blog, we will examine the first three of these elements. But first, consider that of primary importance is to create a learning environment that allows student thinking to be visible. When student thinking is visible, classroom teachers can react, in the moment, and best support student learning. On the other hand, when the learning environment is focused on the instructor and their presentation of the mathematical ideas and student thinking is ignored, then teaching and learning cannot be responsive to student needs.

The first three practices in the Building Thinking Classrooms framework focus upon:

- The Types of Tasks We Use in a Thinking Classroom;
- How We Form Collaborative Groups in a Thinking Classroom;
- Where Students Work in a Thinking Classroom

In the vignette, the goal was to reveal that students were working on a task focused on solving a problem, using mathematical procedures and calculations in a meaningful way and with appropriate technological tools, and interpreting the results. Students were organized into groups of three using a technique that Liljedahl calls “visible random groups.” Finally, students recorded their work while standing at the whiteboard or a vertical, non-permanent surface (VNPS). For each of these three practices, the following are short excerpts from Liljedahl’s book.

My hope is that this blog post will motivate the reader to explore the idea of active learning in the mathematics classroom. The Building Thinking Classroom framework may serve to provide practical strategies for creating such a learning environment.

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Bret Rickman - Math & Statistics at Portland (Oregon) Community College