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

by Peter Liljedahl

We ran the aforementioned survey after two weeks of implementation, and we saw a definite increase in the number of students who would offer an idea. 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.

Willingness to Collaborate Although many students rail against the groups they find themselves in on Day 1, at the three-week point this resistance is usually completely gone, and they are open to working with anyone they are placed with.

PBIS epistemic

PBIS? Epistemic

Idris It was, but we weren’t so sure. The group next to us had a different answer, and it took a long time working with them before we figured out which one was correct. 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. Or, like it did for Idris’s group, it takes on a combination of these forms.

! Epistemic best line book’s internal definition of knowledge; socially constituted

Reduced Social Stress Despite potential early resistance to visibly random grouping, once it is up and running, many students come to enjoy the elimination of the social stress involved in self-selecting groups. The students who most benefit from this are the students who classify themselves as shy. Researcher How are you liking the random groups? Amanda Love it. I don’t care who I’m with as long as I don’t have to try to get into a group myself. Researcher Why is that? Amanda I’m shy. In social studies the teacher always makes us pick our own groups. I hate that. I hate that feeling of asking if I can join a group and then being told no. I just want to work by myself in that class. But I don’t want to work by myself. It’s just so hard.

use it as a way to keep thinking when they are stuck.

! Inter-disciplinary potential

To enhance the perception of randomness, you can let one student roll a dice and then come up and push the button on a digital randomizer as many times as the dice shows.

Fun!

Transitions are more easily facilitated if your method of randomization doesn’t just tell them what group they are in, but it also tells them where to go to meet their group.

! February instructional emphasis

Although randomizing the groups about once an hour proved to be best, this is not practical if you have your students all day. In those cases, we made use of natural transitions like lunch, recess, and coming back from the library or gym. In essence, you randomize them every time they come back into the classroom.

Elementary

There are several reasons we may wish to keep students apart from each other. Ironically, regardless of the reason, those students often want to be together. And that will happen when you start randomizing your groups—usually on the first day. The best thing to do when this happens is to visit that group first and just say, “Are we going to have a problem here?” More often than not, these students are so thrilled to be together that they do not want to ruin their chances of it happening again. If the situation is one where the students don’t want to be together, then the intervention is different. Treat them being together like it is the most normal thing in the world—just like any other group. The more normal you make it, the more likely they are to be OK with it. Students are highly attuned to the things that you worry about—if you worry about them being together, then they will fixate on that as well. If, despite this, you sense trouble brewing, visit the group, and remind them that it is just for one class and that you expect that they will be respectful to each other.

Buy in Admin support PBIS Emphasis

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. Perhaps this is a by-product of our efforts to stem school and social media bullying and exclusion. In our efforts to stem these social ails we build up an assumption that, without our interventions, all kids would be capable of perpetrating such acts. But this is not true. Children have an unbelievable capacity for empathy for each other. They know which students are strong and weak, which have special designations, and which receive adaptations or accommodations, and yet they still see each other as peers and as friends. Random groups puts this empathy into motion and gives it a venue to play out in.

Lead teacher

Descriptions of Images and Figures Back to image The details are as follows: The left side of the notebook shows the macro-move, with the following checklist: Frequently form visibly random groups. An illustration shows a silhouette of two groups with two and three people in them respectively. The right side of the notebook shows the micro-moves, with the following checklist: In K-2 form groups of two, In Grades 3–12 form groups of 3. Set up your method of randomization such that it tells students where to go. Find a way to randomize such that the students know that you know what group they are in.

New Yorker shouts and murmurs

the notebook has become the catchall for all student work in the mathematics classroom. From taking notes to completing now-you-try-one tasks to doing homework, the notebook is where students do their work.

Interdisciplinary observation

1. how long (in seconds) it took them to start talking about the problem, 2. how long (in seconds) it took them to make their first mathematical notation on whatever workspace they were assigned to, and 3. how long (in minutes) they were willing to keep working without the teacher needing to encourage them to keep going.

Persuasive epistemic

You should set rules that students are not allowed to erase someone’s work without their permission.

Thinking is messy. It requires a significant amount of risk taking, trial and error, and non-linear thinking. It turns out that in super organized classrooms, students don’t feel safe to get messy in these ways. The message they are receiving is that learning needs to be orderly, structured, and precise. In these perfectly organized classrooms, the physical spaces in which they are being asked to think is incommensurate with the messiness of thinking. This is a problem. On the other hand, thinking should not be completely unstructured. It needs elements of representation and organization for patterns to begin to emerge. Therefore, overly chaotic spaces are not the answer either. Thinking is messy. It requires a significant amount of risk taking, trial and error, and non-linear thinking. It turns out that in super organized classrooms, students don’t feel safe to get messy in these ways.

relaxed spaces in which students felt safe to take risks, to try, and to fail.

Physical boundaries; minimum disorder

Applying my contrarian experimental methodology, these results

Extremely annoying and reductive posture ! 🤮

A willing class of students can rearrange the desks in a room in less than a minute. Have your students defront the room as soon as they come in. Just prior to turning the classroom over to the other teacher, have your class refront the room. Some teachers place two small maps of the room on the corner of each desk. The first map shows what the desk arrangement is in a fronted classroom, and the second map shows the arrangement in a defronted class. On each map, one desk in the arrangement is shaded in, telling the student who occupies that desk where it belongs in either configuration. Students come into class, grab a desk, and move it to its correct position.

Choose the seating structure based on the learning target. Include a verb(s) next to each of four structures— Explore (structured) Analyze (semi structured) Memorize (lose structured) Socratic (fish bowl) Interviews (fish bowl) Small groups (vertical learning)

You may be familiar with research showing that teachers ask up to 400 questions a day (Vogler, 2008), but a more interesting question as it relates to thinking classrooms is how many questions teachers are answering.

As it turns out, students only ask three types of questions: proximity questions, stop-thinking questions, and keep-thinking questions.

In fact, 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.

What we found was that answering a question with a question (and only a question) was only effective when it was immediately followed by the teacher walking away from the students, with no other statements or suggestions being made. In fact, this was so clear that we decided to try this strategy on its own. We would just walk away. This turned out to be infuriating to students and did cause some negative backlash. But after two weeks we also noticed it caused a sharp decrease in the number of proximity or stop-thinking questions being asked by students—in some cases reducing the number to fewer than 30 questions a day. As students began to realize that their questions weren’t going to be answered, they stopped asking them … except in the primary grades. If a six-year-old asks a question and it is not answered, they ask it again. If it is still not answered, they ask it again. And if it is still not answered they do something that a 16-year-old does not. They reach out and touch the teacher—tap them on the arm or pull on their clothing. And if the teacher walks away, they follow. I have multiple videos of kindergarten and Grade 1 teachers walking around the room with a row of little ducklings following them. As soon as the teacher stops walking, they are immediately surrounded by these little ducklings tapping and tugging at them. These hilarious episodes of primary teachers trying to not answer students’ questions by walking away prompted us to explore some nuances of this...

2 week z pack Walk away and expect two weeks of pushback ! PBIS Admin Mits

I can see how smiling and walking away, although infuriating for students at first, can become a good way to avoid accidentally answering a proximity or stop-thinking question, and thereby a good way to get students to keep thinking. But does it work for all students? Yes and no. It works in that you are not letting them stop thinking. It doesn’t work for all students in the sense that there are students who cannot get past the fact that you have not answered their question. This may be because they are insecure about their own abilities, have learned helplessness, or have a spectrum disorder—such as obsessive compulsive disorder—that does not allow them to move forward without resolution. Alternatively, you may have students who become incensed at your deliberate disregard for their question. After all, they have a lifetime of experiences with teachers answering their questions. Or, it may be a combination of these factors. Regardless, you need to read the situation and know when a nod, a wink, or an encouraging remark—“I have complete confidence that you can figure this out”—is needed. Keep in mind, however, that when you open your mouth, you may be overcome with an almost undeniable desire to answer their question. It is in our nature as teachers. You need to read the situation and know when a nod, a wink, or an encouraging remark is needed.

When to Give the Task One of the earliest results from the research was that the same task given either in the middle of the lesson or near the end of the lesson produced much worse results than if it was given right at the beginning of the lesson. There are two main reasons for this—the first of which has to do with the students. As discussed in the previous chapter, students prefer to occupy lower energy states. So, if a lesson begins with the low-energy state of passively receiving knowledge in the form of a lecture or taking notes, it is much harder to then raise their energy level and get them to start thinking. This was obvious in the data through students’ complaints, questions, and slow starts whenever we gave them thinking tasks in the middle or near the end of the lesson. A class given these same tasks at the beginning of the lesson came to it with energy, enthusiasm, determination, and a greater sense of self-reliance. Even when we ran tasks on the same students at different points of the lesson, we saw a huge difference between a task given early versus late in a lesson. It wasn’t about the task, and it wasn’t about the students. It was the timing of the task that made a difference.

In a more simplistic study, we documented how many high school students were looking at their cell phones while sitting versus standing while listening to the teacher. The results were remarkable. In the case of sitting, upwards of 50% of students looked at their cell phones at least once in a five-minute interval. For the students standing, that number dropped to less than 10%. This likely has something to do with the sense of anonymity discussed in Chapter 3, but it also has to do with the degree to which students are engaged. Whereas disengaged students look for distraction, engaged students are not distracted.

!

Nothing came close to being as effective as giving the task verbally. This was a shocking result.

This particular task also highlights a fourth characteristic that is present in some tasks—the constraints of the task emerge out of, and after, actions have been taken.

All students are verbal learners long before they are textual learners. This does not go away when they learn to read. What those students you are thinking about may not be good at is taking verbal instruction in large-group settings. One-on-one, they are likely fine as verbal learners. When they get to their small groups, there will be lots of opportunities for their group members to reexplain the task in a more focused setting. As a teacher, you know for which students such reexplaining may be necessary, and you can observe to make sure this is happening.

Proust and the squid Science of reading

How important is storytelling in giving tasks and instructions verbally? In every case in which we were able to create a story, students’ uptake of the task was better—they had fewer questions, they were able to more quickly begin the task, and they were less likely to misunderstand what they were meant to do. There are loads of research that support these observations (Egan, 1988) as well as research on how to teach mathematics through storytelling (Zazkis & Liljedahl, 2008), but not all tasks lend themselves to being posed as a story, and not all teachers want to be storytellers.

Try to use different locations around the room for presenting tasks.

!

If new knowledge is needed to do the first task, think about what the minimum new knowledge needed is, as well as the minimum things that need to be said and written to pass on that knowledge.

When the students have started working, ask yourself if what is written on the board would make sense to a student who comes in late.

often, homework is displaced by more lofty pursuits like sports, volunteerism, work, family functions, music, and other homework. From the interviews, we learned that a lot of students are legitimately very busy.

Homework, in its current formulation, needs to be upgraded. It needs to be rebranded. It needs to become a thinking activity.

So, you say that we can give worked solutions—but not right away. Should we do this? And if so, when? First, to make sure we are all on the same page, we need to differentiate between answers and worked solutions. x = 7 is an answer. It reveals nothing about how we arrived at that solution. How we arrived at that solution is the worked solution. The research showed that giving worked solutions became more and more important the more complex the questions were. This corresponds loosely with an increase in grade level, but not exactly. The research also showed that worked solutions should not be given out until the students have had a chance to work on the questions, checked the answer they arrived at against the answers provided, and, if needed, retried the questions—sometimes multiple times. As mentioned above, if we give the worked solution at the same time as the question, some students will read the solution and think that, because they understand what is happening, they understand how to do it on their own. However, if the worked solutions are provided the next day, or a few days after, the students now have a chance to compare their thinking to that of the worked solution as well as to get help with any questions they were unable to resolve on their own. Many teachers I worked with gave out worked solutions by posting them on a class website or portal.

Mits

He noticed that their actions became a seamless and efficient extension of their will.

only one element should be varied at a time.

We have used this exact script and a similar sequence to move through all of solving one- and two-step equations in a single lesson. In fact, I have personally taken a thinking Grade 5 class through this script and sequence in 35 minutes, by the end of which every group was solving tasks as complex as ☐ ÷ 15.3 – 8.27 = 3.01.

Figure 9.6 Thin slicing sequence of tasks. As mentioned, the benefit of thin slicing sequences of thinking tasks is that it helps avoid frustration. The drawback is that groups can move through the tasks very quickly, putting an extreme pressure on you to get to groups that are done before they get bored

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 benefit of thin slicing sequences of thinking tasks is that it helps avoid frustration.

Micro moves: Make groups responsible for the learning of every member of the group. Have groups write the task they are working on at the top of their vertical surface. Start with low challenge tasks to ensure the groups start in flow. Create sequences of tasks that get incrementally more challenging by varying one thing at a time. Create a parallel sequence of tasks.

Parallel sequence of tasks.

To reify: is to make concrete and real something that is abstract or intangible.

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

Epistemic “I know your notebook knows.”

This came through in classroom observations and student interviews over and over again. If you have been implementing the thinking classroom in concert with reading this book, it has likely come through in your experiences as well. When students ask, “When are you just going to go back to teaching us math?” what they are really asking is when are you going back to telling (showing) us how to do it so we can just write it down in our notes.

The gallery walk needed to be a gallery walk. Given the effectiveness of the guided gallery walk in maintaining student engagement and necessitating student thinking, this should be the most frequently used method of consolidation. This is not to say that the other two methods are ineffective, but they should be used only in situations where the guided gallery walk is not suitable. FAQ If the guided gallery walk is to be effective, we need to have student work from every task, or every solution strategy, present on the boards at the time of consolidation. How do we ensure this when students have the option of erasing whenever they want?

! Closure walks w/ student work

Taking pictures is a great idea. The only caution is that pictures create a temptation to use them as a slide show during the consolidation—with students sitting in their seats. Regardless of what method you choose to consolidate, or how you choose to preserve the students’ work, putting students back in their seats drastically diminishes the thinking and engagement during the consolidation. Keep them standing.

you can add notation to their work during the gallery walk as you are facilitating a whole class discussion. It is perfectly OK to add notation with your red marker to any student work during the gallery walk. However, it is not OK to erase students’ work. Erasing devalues their work and should only be done when absolutely needed and only with permission.

What if I don’t know exactly what a group was thinking? Should I still include it in the gallery walk? The purpose of the discussion of student work during the gallery walk is not to figure out, with absolute certainty, what a group was thinking. The goal is to use their work, the traces of their thinking, to get the whole class to think and explain. So, not knowing exactly what a group was thinking is OK. But if you do want to know, you can always ask them prior to the start of the consolidation.

Important and practical epistemic; humility good shit

Should I try to select something from every vertical surface for the guided gallery walk? No. This would create too much redundancy and would cause the consolidation to take too long. Over time, every student will have their work honored. It doesn’t have to happen every day. In this chapter you talk about consolidating and reifying at the level where students got to. And you also talked about leveling to the top as something that doesn’t work. Does that mean we can never lift students’ understanding above the level they got to? We can. But there is a limit to how far we can lift their understanding at a single point in time. What we saw in the research is that when we consolidated from the bottom, moving up through the levels that they had already worked through, we were able to lift their understanding up beyond where they got to. Leveling to the top doesn’t work because it starts the consolidation above (often too far above) where the students reached, rather than collecting them at the bottom, reifying ideas and terminology, and then moving upward.

Double Epistemic