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

by Peter Liljedahl

Each of the sequences of these curriculum tasks has another quality that is common among them—the increase in challenge from one task to the next is incrementally small.

You can help make this easier by setting a rule that whatever task they are currently working on needs to be written at the top of whatever vertical surface they are working on—thereby making it easier for others to steal.

the main difference is that hints that decrease challenge are only useful in that moment, whereas a hint that increases ability continues to be useful even as students move on to the next task.

There are two types of hints—hints that decrease challenge and hints that increase ability.

Obviously, hints that increase ability are better in the long run. But frustration is not about the long run. Frustration is an intensely negative emotion that needs a rapid intervention, and sometimes the best way to intervene quickly is to reduce the challenge.

My research showed that this sequence of modes of engagement—doing, justifying, explaining, teaching, creating—not only increases challenge, but does so in a way that can continue to engage, or reengage, even the strongest group. It also explains why students often have a difficult time explaining their thinking—we are asking them to go straight from doing to explaining. We need to first ask them to justify their thinking.

The goal of building thinking classrooms is not to find engaging tasks for students to think about. The goal of thinking classrooms is to build engaged students that are willing to think about any task.

If all students could learn by having us just tell them how to do it, we would not have any of the problems that we have in mathematics education today. For over one hundred years the dominant pedagogy was teaching through telling. If that had worked, then all students would have been in our highest streams, and all students would have gotten the highest marks. But that has not been the result. Student difficulty with mathematics has been a pervasive and systemic problem since the advent of public education—not because students can’t learn mathematics, but because, by and large, students can’t learn by being told how to do it.

students began to mistake being shown how to do it for learning, and they mistook having it in their notes for knowledge.

What we found was that during whole-class discussion, unless there is a punitive structure in place forcing students to listen attentively to other students’ presentations of their solutions, very few students did.

Our fix was that when discussing student work, we ask other members of the class to try to explain what the group was thinking whose work we were discussing.

This selecting and sequencing is both similar to and different from Smith and Stein’s (2011) notion of selecting and sequencing. They are similar in that they both try to achieve a sensible sequence that moves the learners through various ways of thinking about a task or tasks. The difference is that, whereas Smith and Stein do both the selecting and sequencing in the moment, within a thinking classroom, the sequencing has already been determined within the task creation phase—created to invoke and maintain flow. What is left to do is to select the student work that exemplifies the mathematics at the different stages of this sequence. Another difference is that, whereas Smith and Stein have students present their own work, in the thinking classroom the decoding of students’ work is left to the others in the room.

This continual effort to track, write, and keep up with the teachers’ thinking requires a huge amount of cognitive effort, which causes students to fall further behind—to the point where they just stop listening and trying to make sense of what they are writing.

They were simply trying to get the information necessary to fill in the blanks rather than trying to understand what the bigger picture was.

Both I-write-you-write and fill-in-the-blank methods of producing notes were activities that were antithetical to student thinking—and antithetical to building a thinking classroom.

only 18% of students (five or fewer students in most classes) stating that they regularly used their notes in one way or another.

Notes to their future forgetful selves is the essence of mindful (as opposed to mindless) and meaningful (as opposed to meaningless) note taking.

Teachers asked students: “What are you going to write down now so that, in three weeks, you will remember what you learned today?”

What helped the most, however, was circling back three weeks later with questions that required them to use these notes.

Interestingly, the Grade 8 and 9 students struggled the most with this task.

There are three distinct competencies needed for students to be able to produce useful and meaningful (to them) worked examples: creation, annotation, and selection.

Anything you do to force the issue changes, in profoundly negative ways, who they perceive the notes are for.

the same three competencies appear every time: perseverance willingness to take risks ability to collaborate

Is it our job, as teachers, to wait for students to come to us with these competencies in place, or is it our job to develop these competencies within the students that we have in front of us? The answer is always the same—it is our job to develop these competencies. Of course, it is.

Most of the existing rubrics we initially looked at consisted of at least four columns—and in some cases as many as five or six. The problem is that we cannot nuance language well enough to communicate differences among that many columns. The bigger problem is that we think we can. In truth, however, that nuancing garbles the feedback for students.

In our attempts to differentiate student work across four columns, it is easy to create language for the first column. This column generally describes poor work, and that is easy to articulate. Likewise, the good work represented in the fourth column is easy to clearly explain. It is when we try to differentiate between the two columns in the middle that we get into trouble and end up trying to say that mostly right is different from might have some errors, or primarily rather than usually, or sometimes rather than occasionally. Even if we believe it is clear, it is not clear to the students. They can see that they live in different columns, but they don’t know what they need to do to move from one to the next. Put simply, the information is too much and too nuanced to be useful to them. Like anyone, they were more likely to actually read and respond to something that is more direct, more digestible, and more visual.

more. It turns out that that level of nuance was often just as difficult for them to manage and interpret as it was for students.

Interviews with students revealed that many of them were seeing the headings in the initial rubric as descriptions of who they are, rather than where they are.

Students with a growth mindset saw these labels as descriptors of where they were, while students with fixed mindsets saw them as who they were—even when temporal or positional language such as not yet or on the way were being used. This is a problem.

By replacing this language with the arrow, even students with a fixed mindset began to see the feedback as a descriptor of where they were.

As the rubric will, in essence, be a series of indicators each of which is on a continuum, it is also important to select language from the T-chart that is dichotomous in nature.

You cocreate a rubric whenever there is a behavior that you would like to improve within the room. Once you cocreate it, you need to use it right away and for two or three lessons in a row to really show that you value it. Then you can take a break from it for a while and only use the rubric once in a while, or when you see that it is needed.