Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (Corwin Mathematics Series)

by Peter Liljedahl

Liljedahl argues convincingly that we need to interrupt the entrenched patterns of school.

Why does it matter? Because most of our students do an awful lot of “studenting,” but not much thinking.

Students from communities that have historically been excluded from mathematics are often denied access to thinking at all.

The goal of thinking classrooms is to build engaged students that are willing to think about any task.”

Thinking is a necessary precursor to learning, and if students are not thinking, they are not learning.

Studenting: is what students do in a learning setting—some of which is learning. … there is much more to studenting than learning how to learn. In the school setting, studenting includes getting along with one’s teachers, coping with one’s peers, dealing with one’s parents about being a student, and handling the non-academic aspects of school life. (1986, p. 39) [as well as] ‘psyching out’ teachers, figuring out how to get certain grades, ‘beating the system,’ dealing with boredom so that it is not obvious to teachers, negotiating the best deals on reading and writing assignments, threading the right line between curricular and extra-curricular activities, and determining what is likely to be on the test and what is not. (1994, p. 1)

In our research into the optimal practices for thinking for each factor, what emerged were a number of what I came to call micro-moves. These are the little things within each of the practices that we found enhanced, streamlined, or made easier to implement the optimal practice.

These are called micro-moves to contrast them against the macro-moves that are the optimal practices for thinking in each chapter. This is not to say they are any less important. In many cases, these micro-moves make the difference between smooth and rough implementation in your classroom.

So, when I talk with teachers about what makes a good task for building thinking classrooms, I don’t talk about what a task is, but rather what a task does.

And what a task needs to do is to get students to think.

Problem solving is a messy, non-linear, and idiosyncratic process. Students will get stuck. They will think. And they will get unstuck. And when they do, they will learn—they will learn about mathematics, they will learn about themselves, and they will learn how to think.

Problem-solving tasks are often called non-routine tasks because they require students to invoke their knowledge in ways that have not been routinized.

Good problem-solving tasks are also rich tasks in that they require students to draw on a rich diversity of mathematical knowledge and to put this knowledge together in different ways in order to solve the problem. They are also called rich because solving these problems leads to engagement with a rich and diverse cross section of mathematics.

Highly Engaging Thinking Tasks are so engaging, so interesting, that people cannot resist thinking. They have broad appeal and can be used across a wide range of grades,

Card Tricks have the same qualities as highly engaging thinking tasks—they are highly engaging situated tasks that draw students in and entice them to think.

Numeracy Tasks are tasks that are based not only on reality, but on the reality that is relative to students’ lives.

Aside from context, all these tasks also have easy entry points (low floor) and evolving complexity (high ceiling), and they drive students to want to talk and to collaborate.

Low-Floor Task: Task with a threshold that allows any and all learners to find a point of entry, or access, and then engage within their level of comfort.

High-Ceiling Task: Tasks that have ambiguity and/or room for extensions such that students can engage with the ...

Open-Middle: A problem structure where a task has a single final correct answer, but in which there are multiple possible correct wa...

In a rich task, once the language has been decoded, the mathematics that is needed to solve it is neither trivial nor procedural. Basically, in rich tasks the problem is in the mathematics, and in word problems the problem is in the words—this is maybe why they are called word problems.

Having said that, it turns out that both of these questions are excellent thinking questions—if they are asked before the students have been shown how to answer them.

These scripts are similar in that they begin by asking a question about prior knowledge, then they ask a question that is an extension of that prior knowledge, and they ask students to do something without telling them how. And, as such, they require students to think, not only in general, but also about particular curriculum. It turns out that almost any curriculum tasks can be turned from a mimicking task to a thinking task by following this same formulation—begin by asking a question that is review of prior knowledge; then ask a question that is an extension of that prior knowledge.

the lesson that was designed around non-curricular tasks (Type 1) got many more students to think than the lesson scripted to get students to think about curriculum (Type 2). Simply turning a standard curricular task into a thinking task was not enough to get all the students thinking.

students can be successful at these types of scripted thinking tasks, even more successful than in lessons designed to promote mimicking, if their willingness to think is first primed with the use of good non-curricular tasks.

The non-curricular tasks (Type 1), in this regard, served as a primer for—and thus made room for—the more curriculum-driven scripted thinking tasks (Type 2).

The key was, however, that in the transition from a non-curricular task (Type 1) to a curriculum thinking task (Type 2), nothing else changed. The teacher posed the task as a challenge—as a problem to solve—without any big declarations that now we are going to start doing curricular tasks in a different way.

these results show that to get students thinking about curriculum tasks, they need to first be primed to do so using non-curricular tasks.

Mimicking is bad because it displaces thinking. Mimicking happens not alongside, but instead of, thinking.

In essence, these tasks make it safe to fail and keep trying.

In essence, no matter how strategic a teacher is in their groupings, when there is a mismatch between their goals and students’ individual goals, it means some students will be unhappy and will disengage. This disengagement is antithetical to a thinking classroom.

While we clearly know the value and importance of student collaboration, the problem is that efforts to actualize collaboration through either strategic groups or assigned roles may be having a negative effect on how our students engage with each other and the task at hand.

At the same time, self-selected groups seem to be of little help, as students group themselves for reasons antithetical to solving mathematical problems.

Although randomizing wrested the control from the teachers, making it visibly random was necessary for the students to both perceive and believe the randomness. We needed visibly random groupings.

from Grade 3 up, the optimal group size was three.

This is because for a group to be generative, it needs to have both redundancy and diversity (Davis & Simmt, 2003). Redundancy, in this context, reflects things that a group of students has in common—language, interests, experiences, knowledge. Without these commonalities they cannot even begin to collaborate. But if all they have is redundancy, they will not achieve anything beyond what they enter the group with. To be generative, they also need diversity; the things that individual members of the group bring that are not shared by the others—different ideas, viewpoints, perspectives, representations, et cetera. Groups of three seem to have the perfect balance of redundancy and diversity.

For Grades K–2, however, the optimal group size was two. Despite the lack of diversity this affords, students at this age are still developmentally in a stage of parallel play, and collaboration consists mostly of polite turn taking.

When the social barriers come down, so too do the barriers to knowledge mobility.

Knowledge mobility takes one of three forms: (1) members of a group going out to other groups to borrow an idea to bring back to their group, (2) members of a group going out to compare their answer to other answers, or (3) two (or more) groups coming together to debate different solutions.

Students tend not to treat knowledge mobility—students call it borrowing an idea—as a way to reduce thinking. Rather, they use it as a way to keep thinking when they are stuck.

aside from mobilizing knowledge, frequent visibly random groups also mobilizes empathy. As a society we give far too little credit for the empathy that children have for each other.

Random groups puts this empathy into motion and gives it a venue to play out in.

in thinking classrooms we start all groups on the same task and then differentiate the hints and extensions we give each group depending on how they are doing.

In the 15 years that I have been engaged in the thinking classroom research, nothing we have tried has had such a positive and profound effect on student thinking as having them work in random groups at vertical whiteboards. Students were thinking longer, discussing more mathematics, and persisting when the tasks were hard.

when they work on whiteboards, they can quickly erase any errors, which, for them, reduces the risk of trying something.

Standing necessitates a better posture, which has been linked to improvements in mood and increases in energy

Standing also afforded an increase in knowledge mobility.

It turns out that when students are sitting, they feel anonymous.

And the further they sit from the teacher and the more things—desks, other students, computers, et cetera—are between them and the teacher, the more anonymous they feel. And when students feel anonymous, they are more likely to disengage—in both conscious and unconscious ways.

Having students standing immediately takes away that sense of anonymity and, with it, the conscious and unconscious pull away from the tasks at hand.