In traditional math classrooms—and in classrooms where challenging, unfamiliar work is often assigned, more generally—the progression "I do, we do, you do” often becomes the default approach to learning, according to researcher and professor of mathematics at Simon Fraser University, Peter Liljedahl.
It makes sense in many cases, particularly when difficult concepts need to be addressed in time windows that are compressed by bell schedules, holiday breaks, and summer vacation. But used too frequently, this approach, Liljedahl recently told the Cult of Pedagogy, inhibits higher order thinking and results in students who “mimic” teachers. Students who engage in too much rote work miss out on some of the challenging, sometimes confusing work that builds up their self esteem to face difficult thinking tasks in the future.
“By and large students spend most of their class time not thinking, at least not in ways we know they need to think in order to be successful in mathematics.” Liljedahl explains. “If they’re not thinking, they’re not learning.”
Liljedahl, the author of Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning advocates for “Thinking Classrooms”, which offers a different take on how classroom work is organized, how tasks are assigned, and how students learn and work together. His conclusions are based on more than a decade of research, experimentation, and collaboration with over 400 K-12 teachers.
In a 2017 article for Edutopia, Liljedahl clarified that a “non-thinking” classroom is “predicated on an assumption that the students either could not or would not think.” When presented with difficult problems, students in these classrooms have a hard time pushing through to find their own solutions, he argued, and often wait for teachers to step in to do the heavy lifting for them.
To stimulate independent thinking, Liljedahl says, reorganize some of your classroom approaches: Start with hard puzzles and problems that push kids to their limits; confront the fundamental passivity of classroom seating; and use highly structured group activities to promote discussion, peer review, and iterative thinking.
Starting With ‘Thinking Tasks’
Instead of starting a lesson with direct instruction, give students novel “thinking tasks” they can work on, ideally in groups. Liljedahl describes these tasks as problem solving activities and mental puzzles that, early on in the school year, should be “highly engaging, non-curricular tasks” to motivate students and get them in the mindset of challenging themselves. As the school year progresses and students become more accustomed to this mode of working and thinking, the activities and challenges can be replaced with tasks directly related to the curriculum.
Liljedahl points out that the tasks should be carefully sequenced so they get incrementally more challenging. “The goal of thinking classrooms is not to get students to think about engaging with non-curricular tasks day in and day out—that turns out to be rather easy,” he told Cult of Pedagogy. “Rather, the goal is to get more of your students thinking, and thinking for longer periods of time, within the context of curriculum, which leads to longer and deeper learning.”
Liljedahl has a long list of sample “thinking tasks” to look through. They include challenging dice-related problems, such as: Imagine a typical 6-sided die, and notice that the sum of opposite faces is always seven. The one is across from the six, the two is across from the five, and so on. Now imagine that you were making your own six-sided die that did not have this restriction. How many different dice could be made?
Use Standing, Randomized Group Work
Central to Liljedahl’s approach is student collaboration and group work. Instead of grouping students by ability, or allowing them to choose their own groups to work in, his research has shown him students work more effectively—and are more likely to contribute—in randomized groups. According to Liljedahl, interviews with students show that randomized groups “break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics.”
At Design39Campus, a K-8 school in San Diego, eighth grade math teacher Kyle Asmus puts Liljedahl’s approach into practice by using a random group generator (such as this one from Classtools or this one from Keamk) to ensure that different kids work together across multiple exercises.
To ensure that all students feel included, Asmus assigns group roles: The scribe writes down possible solutions; the speaker communicates the group’s thinking to the broader classroom; the inquirer asks the teacher questions; and the manager makes sure the rest of the group stays on track.
Having students stand while they engage in this collaborative, messy thinking is yet another way to engage them, according to Liljedahl: It makes them much less likely to withdraw from the work, or assume that others will handle it. “It turns out that when students are sitting, they feel anonymous,” Liljedahl told Cult of Pedagogy. “And when students feel anonymous, they disengage.”
Work on Non-Permanent, Vertical Surfaces
In a thinking classroom, students put notebooks away and participate in group work while standing at vertical non-permanent surfaces such as whiteboards, blackboards, or windows—surfaces that Liljedahl believes promote more risk-taking.
According to Liljedahl, his experimentation with students shows that when comparing a group working on a whiteboard versus a group working on flip chart paper, the group working on whiteboards start working within 20 seconds.
“They’ll start making notations on the board. They’ll try anything and everything because they feel like they can just erase it if it’s wrong,” he told Cult of Pedagogy. Meanwhile, students working on chart paper take upwards of three minutes to make a single notation, because, Liljedahl said, they often wait until what they write is perfect—“and that hesitation leads to a lower form of thinking.”
Design39 students also practice Liljedahl’s “vertical surfaces” learning technique, which Asmus says promotes “high quality collaboration” and facilitates “questions and rich conversations” among students. According to Asmus, when students are presented with challenging tasks and work vertically, the time for students to get to a task is not only quicker, but the time they spend on a task is longer.
The large surfaces spread out around the room also allow students to see the work their peers are doing in other groups, and build off of each other’s understanding. “It makes it easy for the whole class to see what everybody else is doing so we can be inspired by each other’s ideas,” said Iniyaa, a student in Asmus’ class.
Answer the Right Questions
As students work vertically in groups, teachers can easily see how they’re progressing and bounce around the room. Questions will undoubtedly arise, but Liljedahl says teachers should avoid answering questions asked for the purpose of reducing student effort and getting to an answer more quickly—such as “is this right?” Instead, they should prioritize addressing student questions that will lead to further independent thinking.
In her elementary school classroom in Brooklyn, teacher Tori Filler says that instead of rushing to provide hints and solutions to the hard parts of a lesson, she often asks them to evaluate what’s tough about it, and encourages them to sort it out on their own before stepping in to help.
If most of the class is struggling, ask more questions and turn the struggle into a productive discussion for everyone. Asking questions like “What makes this hard?” or “What have we tried?” gets students to think metacognitively and develop the skills to push through challenging work on their own.
Evaluate What You Value
To succeed in a thinking classroom, students need to develop skills like perseverance, academic courage, collaboration, and curiosity, among others. But according to Liljedahl, if we want students to develop these competencies, then we should find ways to evaluate them on it.
“What we choose to evaluate tells students what we value, and, in turn, students begin to value it as well,” he writes.
He argues for a mix of both formative and summative assessment in math classrooms that focus less on end products and student ranking, and more on the work leading up to those end products and collaboration between groups to get there.
The educators at Bite Sized Learning, for example, suggest evaluating students on the work they produce, but also on how well they persevere and make an effort in response to challenges, how well they set individual goals and monitor progress toward achieving them, and how well they share information and resources with group members to solve problems and make decisions.
Liljedahl’s own formative assessments focus on informing students “about where they are and where they’re going in their learning.” This can take the form of observations, check-for-understanding questions, or even unmarked quizzes. Summative assessments, meanwhile, “should focus more on the processes of learning than on the products, and should include the evaluation of both group and individual work,” Liljedahl writes.