Chapter 21:
Student thinking is the currency of learning.
When students articulately exchange their thinking, the economy of the classroom culture becomes strong and robust. The economy is even stronger when students deliberately capture, produce, and share their ideas.
When students create thinking examples specifically for the purpose of exchanging their thinking, the classroom becomes rich.
Allow me to show you a simple image, along with an example of a story that brings it to life.
Imagine a student who is working with the question, “What is twenty divided by three?” and responds with the loops on this picture.
Now, let’s look more deeply into the intent.
The intent of the graphic is not solely to determine the quotient. A far more important intent is to produce an example that can be used to articulately exchange thinking with other students.
Talking about thinking after the thinking has taken place is vastly different than producing representations of thinking designed with the specific intention of sharing one’s thinking. Knowing you will have the opportunity to explain your thinking to someone else causes you as a learner to anticipate the articulation of your thinking in a way that will personally connect with other learners.
Let’s listen in to some of the discussion between two students who are pointing at the representation of thinking.
“Twenty divided by three is six groups, and there are two left over.” “Two what? Two circles, or two groups?”
“Two circles. These two.” (The student points to the two black circles in the bottom right.)
“You drew an extra circle here. Why did you do that? That changes the number to twenty-one.”
“Do you see how every loop has a group of three inside?” “Yeah.”
“And there are six loops with black circles?” “Seven, if you count this one.”
“True. But there weren’t three circles to draw a loop around. There were only two. So I drew a third circle — but I made it look different than the others so we could tell it apart. The last loop shows that those last circles are not a complete group of three. They would need one more circle to make a whole group of three. So I drew what the group would need to make a whole group of three.”
“Why did you do that?”
“I’ve been waiting for that question! What fraction of the last group is shaded in?”
“Okay … two out of three are black. Two-thirds are black.”
“That’s right. So this example shows that twenty divided by three is equal to six whole groups (pointing to the six whole groups) and two-thirds of another group. My example shows that twenty divided by three is equal to six and two-thirds. Do you want to try it?”
The second student draws twenty dots, loops them in a different way, draws an additional circle, re-examines the representation and writes six and two-thirds.
“Okay,” the second student continues. “I have two more ideas to share.”
Two more pieces of currency, produced by the second student, land on the table.
“The triangle shows how the three numbers are related. So twenty divided by six and two-thirds is equal to three. That means we can circle groups of six and two-thirds. There should be exactly three groups. This is my picture. Let me tell you what it means …”
The representations are built with the specific intent of sharing thinking with others.
The students are leading by example, and they are leading with examples.
“Boys and girls, we are going to learn about a new Big Idea today. ‘Lead by Example.’ I want each of you to lead others toward understanding your thinking. You can do that by creating examples that you can hold up and show to other people, examples that you can use to explain your thinking to other people. When you explain your thinking, you will be leading the learning of others, and your thinking will become even more powerful for you, too.
“My job will be to give you opportunities. Your job will be to lead. To do that, I want you to use examples of your thinking. I want you to lead by using examples.
“Lead by example.”
Representations of student thinking are instantly powerful. Yet I often felt as if something was missing from my classroom practice, something that I, initially, could not detect until I thought back to the old custom of show-and-tell. During weekly show-and-tell times, students brought their prized possessions to school so that they could show them to other students and tell all about them. Sometimes, students went on and on about the item they were so excited to share about.
In classrooms across the nation, teachers repeat the phrase, “Show your thinking.” For example, I may often ask students to show their thinking on the page.
What is missing from that concept? It’s the telling!
If I ask students to show their thinking, I may be communicating that I comprise the entire audience for the thinking. Furthermore, since they are showing their work on the paper, they may also come to believe that I am not necessarily planning to hear from them. After all, why would I need to? I have already asked them to show their thinking. I am not as interested in having students show me their thinking as
I am in having them show and tell others about their thinking. I want students to create thinking that is intended for an audience and will be shared with an audience. I’m not deeply interested in “Show your thinking.” I’m enamored with “Show and tell your thinking.” In the middle of the learning process, as students are making connections and producing meaning, I want them to anticipate that they will be sharing their current thinking with others.
It’s worth noting that no element of perfection hovers near this concept. The point is simply for students to focus on thinking and to anticipate the articulation of that thinking. It is not about the artfulness of the representation. Students are free to tell about their thinking. They also begin to recognize the power of Big Idea 10 (We Learn from Our Successes), which previously seemed to be hiding in the glamorous shadow of Big Idea 9 (We Learn from Our Mistakes). Now the successes are shining through, and as those successes become the currency of the class, several of the other Big Ideas begin to deeply resonate.
In fact, if you look back through the list of Big Ideas, you may recognize that this single Big Idea has the potential to feature the power of nearly every other Big Idea we have discussed so far.
This very Big Idea, which sounds cliché upon first glance, actually has the power to serve as the hub of several other Big Ideas. An example is that when students build thinking representations, those representations can turn into the posterized thinking which I described in chapter 10, “We Learn from Our Successes.”
The more closely I examine this concept, the more connections I discover to other Big Ideas. Making connections, making meaning, writing, valuing multiple pathways and multiple correct answers, constructing learning, leveraging mistakes and successes, building on what we know, encountering the impossible, discovering our potential, holding high expectations for ourselves as learners, growing our ideas, considering our strengths, and designing learning pathways that can be followed by others all intersect in powerful ways within the practice of “Lead by Example.”
Over the years, this idea has evolved significantly from its original intention, which was a generic phrase about leadership. How that well-worn phrase grew from a Big-Idea-in-Training into a powerful, central idea, I am not entirely sure. But the potential at its heart was this: build representations of your thinking that you will share with others. Then use those representations to explain your thinking. By doing so, your ideas will spread to others, and they will also become clearer to you, so you can wrestle with them even more. In addition, as ideas from many other learners multiply throughout the room, we will all become richer thinkers because of one another’s ideas. The very act of sharing our own ideas will prepare us to understand others’ ideas. We will all be leaders of learning. We will lead by sharing examples of our thinking. I clearly remember that when I first placed this Big Idea on the wall, it felt like a risk. But I decided to take that risk, and it has paid off in ways that I never anticipated.
