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

by Peter Liljedahl

Taken together, these practices signal to students that this class is different: In this class, they’ll be expected to think. Why does it matter? Because most of our students do an awful lot of “studenting,” but not much thinking.

For the health of our students and our societies, we need to challenge institutional norms and build thinking classrooms in which we value students’ thinking and time rather than use legacy practices that encourage students to slack, stall, mimic, and fake their way through the system.

This wonderful patterning, extrapolation, and generalization activity had been reduced to a form of cookbook mathematics that ensured that, within 20 minutes or so, every student had completed it while, at the same time, ensuring that no one would do any thinking.

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

Well, it turns out that some students behave exactly as we expect—but only about 20% of them. The rest do not.

In all instances mimicking was exhibited by more than half of the class, with slacking, stalling, and faking combining to account for about a quarter of the students. Those trying it on their own—which is what the teacher wanted—only accounted for about 20% of the students.

In a typical one-hour lesson, 75%–85% of the students exhibited non-thinking behaviors for 100% of the time. The rest of the students exhibited non-thinking behaviors for all but 8–12 minutes of the time.

Much of how classrooms look and much of what happens in them today is guided by these institutional norms—norms that have not changed since the inception of an industrial-age model of public education.

Everywhere I went I saw students not thinking and, as a result, teachers having to plan their teaching on the assumption that students either can’t or won’t think.

There is no point in researching a practice that teachers are unwilling to implement—irrespective of how positive the results are.

When the students walk into a room that looks very different, however, then they leave their habits and norms at the door and allow themselves to be different—at least to begin with.

Although there are arguments about the exact processes involved and the exact competencies required, there is universal agreement that problem solving is what we do when we don’t know what to do. That is, problem solving is not the precise application of a known procedure. It is not the implementation of a taught algorithm. And it is not the smooth execution of a formula. 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.

Good problem-solving tasks require students to get stuck and then to think, to experiment, to try and to fail, and to apply their knowledge in novel ways in order to get unstuck.

Once routinization happens, students are mimicking rather than thinking—or as Lithner (2008) calls it, being imitative rather than creative.

very few of these tasks require mathematics that map nicely onto a list of outcomes or standards in a specific school curriculum.

Even if there is a rich task that maps nicely to the curriculum you are teaching, it only maps to curricular outcomes if students happen to solve the problem using concepts and skills from their current curriculum.

when students begin to think for themselves, a lot of unpredictable things can happen. If your goal is only to get students to think, then this is not a problem. If your goal is to use a rich task to, for example, get students to think about division of fractions, then this can be a problem.

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.

Asking a high school student to factor x2− 5x − 14 or an elementary student to solve 3.1 + 5.2 after they have been shown how promotes mimicking, not thinking.

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.

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.

mimicking is only an effective strategy when the number of routines to memorize is small.

As the student moves up in grades, the number of routines per topic increases, until this becomes an unmanageable and ineffective strategy.

mimicking tends to create short-term success without the long-term learning that allows students to make connections with other topics in the same and subsequent grades.

Mimicking is bad because it displaces thinking. Mimicking happens not alongside, but instead of, thinking. Likewise, mimicking is not a precursor to thinking. Mimicking requires less energy and less effort than thinking, and once the mimicking has begun, it is difficult to ask students to shift their attention to something that takes more time, more energy, and more effort.

Well selected non-curriculum tasks, with their engaging contexts, propel students to want to begin to think. They create situations where every student gets stuck, which makes stuck an expected, safe, and socially acceptable state to be in. In essence, these tasks make it safe to fail and keep trying. And through these struggles, students begin to build confidence in their teacher’s confidence in them.

In essence, if we just get students thinking about lots of different problems, the curriculum outcomes will eventually be covered, irrespective of which solution paths students follow.

thinking tasks should be asked in the first five minutes from the time you begin the lesson.

For Question 1, over 80% of the students said that it was unlikely or highly unlikely that they would offer an idea, and over 90% said it would be highly unlikely that one of their ideas would contribute to a solution.

To counteract this, some teachers have adopted the practice of assigning a role to each student in a group—leader, recorder, timekeeper, resource-getter, encourager, et cetera. But this assignment of roles only serves to exacerbate the aforementioned problem.

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.

First, we learned that the randomization needed to be frequent—approximately every hour.

We also learned that, from Grade 3 up, the optimal group size was three. Groups of two struggled more than groups of three, and groups of four almost always devolved into a group of three plus one, or two groups of two.

And after six weeks almost 100% of students said that they were either likely or very likely to offer an idea. This, despite the fact that only 50% believed that their idea would lead to a solution. The students were willing to try, irrespective of whether their idea would lead to a solution.

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.

Note taking, as we will see in Chapter 10, is a largely passive activity, which, as we saw in Chapter 1, promotes mimicking. As we saw in the introduction and will see in Chapter 7, both now-you-try-one tasks and homework also rely heavily on mimicking. In short, sitting and working in notebooks promotes and rewards, in continuous and ubiquitous ways, passive mimicking behaviors.

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.

any alternate workspace produced better results than having students work through thinking tasks in their notebooks while sitting at their desks.

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. When students feel anonymous, they are consciously aware that they can shift their focus from the tasks at hand.

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

using vertical whiteboards is enhanced by each group having only one marker. When every member of the group has their own marker, the group quickly devolves into three individuals working in parallel rather than collaborating.

Every time we worked in classrooms that were super organized—desks or tables in perfect rows, in-baskets and out-baskets for all eventualities, everything color coded and in its place—we had more difficulty generating thinking.

If the room was super organized, we had more difficulty generating positive results. But, when we were working in classrooms that were disorderly, but not overly so, we had better results.

a thinking classroom needs to be organized in such a way that says thinking, collaboration, and risk taking are expected. Rows of desks do not achieve this—even if the desks are put together in groups of two or three. Neither do neat rows of tables positioned so students all face the front. These are antithetical to the message we want to send.

giving students a thinking task is pointless if we then proceed to answer all the students’ questions about how to solve it.

in most cases proximity questions consisted of queries about things that students had either already figured out or made decisions or assumptions about. They simply asked the question because it was a habitual studently thing to do when the teacher happened to be standing nearby.

Answering these proximity or stop-thinking questions is antithetical to the building of a thinking classroom.

The only questions that should be answered in a thinking classroom are the small percentage (10%) that are keep-thinking questions.

One of the things that helped with this was the realization that almost all questions asked in the first few minutes of a thinking task are either proximity questions or stop-thinking questions—neither of which needs to be answered.

Working with a team of eight teachers, we came up with a list of 10 things to say in response to a proximity or stop-thinking question. Isn’t that interesting? Can you find something else? Can you show me how you did that? Is that always true? Why do you think that is? Are you sure? Does that make sense? Why don’t you try something else? Why don’t you try another one? Are you asking me or telling me?