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

by Peter Liljedahl

hand through the use of hints and extensions

You need to begin by using highly engaging non-curricular thinking tasks to build the culture of thinking in your classroom. Non-curricular tasks are also good to use whenever you introduce a new thinking practice into your classroom—this is why every previous chapter ends with one.

When using flow, our teaching needs only to prepare them to answer the first task they will face and then count on the fact that they will learn something during that task that will help them with the next task.

the list provided a finish line,

Multiple minds thinking together about the same task often provide all of the feedback that is needed as they check their own work.

there are actually two more regions in the flow diagram—perseverance and patience (Liljedahl, 2018). The regions act as buffers between flow and frustration and flow and boredom

What if students pick the tasks out of sequence? Doesn’t that mess up their flow?

In this framework every group starts in the same place—with the same task. What is differentiated, then, becomes more about the timing and pacing that each group moves through the sequence of tasks.

leveling to the top

What this means is that, irrespective of where students are in their thinking or their solution process, the teacher goes over the most advanced and most nuanced aspects of the solution.

Unless a student is close to the answer in their thinking, then a leveling to the top is too big a cognitive jump for them to take.

rather than preparing the students for the next task, they are actually less prepared for it and are even less likely to be able to solve it.

by and large, students can’t learn by being told how to do it.

the consolidation follows the same path as the extensions of increasingly challenging tasks that were used to create and maintain flow while the students worked through the task(s).

three ways in which consolidation from the bottom can take place:

The teacher leads a general discussion about the task(s) and solution(s) but writes nothing down.

The teacher leads a detailed discussion of the task(s) and solution(s) while recording on the bo...

The teacher leads a detailed discussion of the task(s) and solution(s) using student work on the vertical surfaces to work through the different layers of the solution.

The first of these methods is most useful when talking about big ideas and general strategies that have emerged out of the student activity.

The second method is suitable when

more detail is required—such as when consolidating ideas around how to add two-digit numbers or when completing the square.

discussions where the teacher asks very focused questions and the students contribute ideas—ideas

The third method, which turned out to be the most effective method for maintaining

engagement, is used in the same circumstances as the second method, but rather than having the teacher write on the board, the existing work of students is used to demonstrate the details.

having the students standing in a loose cluster around the teacher significantly increased attention and engagement during the consolidation process.

What this does is allow more students to stay with the discussion longer, as the teacher is going over aspects of the tasks that more students were able to do.

This method is often referred to as a gallery walk.

here the students literally walk to

the work to be discussed.

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.

work, this approach changes consolidation from telling to thinking—from passively receiving knowledge to actively thinking about the work at hand. And it positions knowledge as tentative, negotiable, and fallible rather than absolute, definitive, and accurate—a positioning that

offers more space for thinking to happen.

The gallery walk was not a random walk but a focused guided tour through the different levels of the flow sequence.

keep the students moving.

When we evaluate our students, they evaluate us—for what we choose to evaluate tells our students what we value.

For K–1, however, we found that even three columns were initially too many. Students in these grades are still developing their ability to see and sense nuance and subtlety, and they are still experiencing their world through a lens of binary opposites—good-bad, high-low, hot-cold, wet-dry, big-little, and so on (Bettelheim, 1976, Egan 1988, Zazkis & Liljedahl, 2008). Therefore, for these grades the rubrics were only two columns and were constructed, for the most part, using visuals

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.

The purpose of coconstructing a rubric is the ownership that happens when students see that they have a voice in what will be evaluated and how they will be evaluated.

even if students don’t exhibit the desired behavior, they know what the desired behavior looks like.

But, what we do take, we take verbatim from what is in the T-chart. As you see in the script, our chance to massage verbiage occurs as we fill the T-chart, not after.

The students need to see, as much as possible, that it is their ideas and their language that have contributed to the formation of the rubric.

This can be heightened if, during the next class, they can see the new rubric and the original T-chart at the same time. So, if y...

What is important is that the indicators that make it into the final rubric are dichotomous.

At the end of the time spent on the boards, you give every group a highlighter and ask them to self-evaluate how well their group persevered that day.

a tension between how you are teaching your students and how you are assessing them.

real need to assess students collaboratively as well as through day-to-day observations in the thinking classroom.

In the point-gathering paradigm, every point that a student manages to accrue is recorded in your gradebook, and at reporting time you take the number of points a student earned and divide it by the number of points they could have

earned (with some scaling), and out pops a percentage.

I also refer to the point-gathering paradigm as event-based grading because of the way these points are recorded in your gradebooks—with the name of the event (quiz, unit test, project, etc.)

The problem is that, if the goal is to produce a grade that is reflective of what students have actually learned, then events-based grading is neither objective nor accurate.