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

by Peter Liljedahl

What should we do first when making a graph?”, “When do we need to regroup?” It is also an important part of the reifying discourse

Song Chant What should we do first?

then make sure you, likewise, encourage them to turn those pictures into notes to their future forgetful selves.

Focus First, whereas the initial rubrics looked for competencies in student work, this new rubric looks for competencies in student actions. That is, it is an observational rubric (Elrod & Strayer, 2015) to be used by the teacher while students are thinking—not after they are finished thinking. This, it turned out, gave us much more direct access to student competencies and, in return, produced bigger changes in student behaviors. It is an observational rubric to be used by the teacher while students are thinking—not after they are finished thinking.

! Epistemic Wonderfully put

Ironically, we discovered that information in rubrics like the one in Figure 12.1 was ambiguous to teachers as well. In many cases where we secretly scrambled a rubric, teachers were unable to correctly unscramble it—sometimes even scrambling it 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. They just didn’t have a better model to follow. Once we figured this out, we moved to three-column rubrics, and all of those problems went away. The rubrics became easier to create and use, and students found them easier to interpret—they spent more time looking at the feedback, and they were able to take action on that feedback.

Do it! Screenshot

This was true for students in Grades 2–12. 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 (see Figure 12.3).

Epistemic

4. Reduction in Language The fourth visible difference is the absence of language in the middle column. When we were working with three-column rubrics with language in each column, the students treated each column as a discrete level of the competency being evaluated. So, when they were asked to self-evaluate their group’s competency, they highlighted their performance as fitting in one of the three columns (see Figure 12.4). This is how they had been evaluated on four-column rubrics, so it made sense that they would do the same on a three-column rubric. When we removed the language in the middle column, however, their way of self-evaluating completely changed. Students no longer thought of the rubric as a collection of discrete levels, but rather as a continuum with the first and third columns acting as endpoints, and the way they highlighted reflected this (see Figure 12.5). Even students who had previously used three-column rubrics for self-evaluation almost seamlessly switched to seeing this as a continuum. Students no longer thought of the rubric as a collection of discrete levels, but rather as a continuum.

Coconstruction of rubrics allows for the emergence of language and terminology that, although unique and potentially idiosyncratic, is clear to the students who had a hand in creating it.

To do Screenshot !

Teacher Ok. So, what does bad perseverance look like? [Teacher writes BAD at the top of the left column.] Student Giving up right away. Teacher Right away? Student Ok. Just giving up. Teacher When? Student As soon as we get stuck or it gets hard. [Teacher writes this on the board.] … What is interesting about this process is that no matter how lacking a class may be in demonstrating an observable competency, they can always generate a list of indicators of what it would look like to have that competency. In the hundreds of times this has been tried, this has always been true—even if students don’t exhibit the desired behavior, they know what the desired behavior looks like.

the perseverance in the room that day will be through the roof. And not because of the specter of evaluation, but because the coconstructed rubric makes it clear, to every student and every group, what is expected of them that day. The fact that you are evaluating three groups is only necessary to show that you are, truly, evaluating what you value.

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. When they are done, give the three groups that you were evaluating (in Grades 4–12) the rubrics that you highlighted for them. The total time you will have spent on cocreating, presenting, and using this rubric will be less than 15 minutes—but the transformational changes in your students will be monumental. Every time we have done this, the changes in student behavior around whatever competency we choose to focus on, irrespective of how poor it was the day before, is huge. Once students see what behaviors are expected, and that these behaviors are valued, the students begin to see them as valuable as well. And when these competencies improve so does your thinking classroom.

! PBIS

What we learned from the research on rubrics is that students see that we value something if we are willing to spend grades on it—or if we are willing to spend time on it. They know that time is limited and, as such, it is valuable. And if you are willing to give some of it up to focus on their perseverance—or collaboration, or risk taking, or whatever behavior you want to focus on—then it must be valuable.

For example, if you see a particular group not persevering, just grab a perseverance rubric and tape it up on their vertical surface. This will signal to that group that you are not impressed with their behavior and that you will be watching them while at the same time communicating exactly what it is that you will be watching for. And once in a while you will see a group member pulling out an old rubric for their group because that individual, or the group as a whole, is aware that they aren’t behaving as they should.

Did not listen, listened well. Did not share, shared. Argued, took turns talking. Worked alone, worked together. Unhappy group, happy group.

“The only solution is to rebuild student-teacher relationships so that the engagement that is essential to learning can be re-established.” — Confronting the Crisis of Engagement: Creating Focus and Resilience for Students, Staff, and Communities by Douglas B. Reeves, Nancy Frey, et al. https://a.co/6szntCS “We do not think it is fair for students to be required to infer what they are learning.” — Confronting the Crisis of Engagement: Creating Focus and Resilience for Students, Staff, and Communities by Douglas B. Reeves, Nancy Frey, et al. https://a.co/1RoCV9b

Bad: Giving up when we get stuck. Good: Not giving up when it gets tough. Looking around for a hint. Asking the teacher for help.

Assess one completely at a time. Use an arrow instead of labels. Keep language at a minimum. Preserve student voices. Have no more than 5 indicators. Use exemplars when co-constructing rubrics of producibles.

you will learn that if we, as teachers, are careful about what and how we communicate with our students, then we will see not only significant improvements in students’ abilities to think about their own learning, but also significant improvements in students’ attainment of content.

Knowledge gap wexler !

The Issue Whether formative or summative, assessment is fundamentally about the communication of information. For much of the 20th century, this information was seen, almost exclusively, as flowing from the student to the teacher for the dual purposes of informing teaching and producing a grade. The methods used to collect this information were either formal (tests, quizzes, assignments, projects, presentations, portfolios, etc.) or informal (observation, conversations, etc.). Whereas grading was typically informed by the more formal means of assessment, teaching was informed by both formal and informal means.

The teacher is interpreting the information against a background of full knowledge and understanding of the concept or concepts in question, a clear picture of what comes next, and a rich history of having taught this same concept many times. The student, on the other hand, has an incomplete—and maybe inaccurate—picture of the concept, has no idea what it is leading to, and is learning it for the first time. How could the same information possibly inform them in the same way? Although the information that is flowing to the student is the same as that flowing to the teacher, the recipients of the information and what they can make of it vary greatly.

Epistemic

information communicated from a teacher to a student who sees the topic as one big unit will only inform that student of what it is that they can do; but because they don’t have a clear picture of the whole unit and all its subtopics, they cannot see what there is still left to learn. The teacher, on the other hand, with their greater sense of the scope and scale of the topic, can use the information that is communicated from the student to determine what that student can do AND what they cannot yet do. In order for assessment to equally inform teaching and learning, we need to find ways to help students see mathematical topics as collections of subtopics, sections, and/or special cases the way teachers do, and to use this knowledge to inform themselves about what it is they can and cannot yet do.

“Relevancy needs to be closer to home. Why are syllables important? Because they help us read big words. Why do we need to know the difference between a simile and a metaphor? So we can use these techniques in our own writing to express ideas without being too obvious.” — The Teacher Clarity Playbook, Grades K-12: A Hands-On Guide to Creating Learning Intentions and Success Criteria for Organized, Effective Instruction (Corwin Literacy) by Douglas Fisher, Nancy Frey, et al. https://a.co/deZhKQP

Frey, Hattie, and Fisher (2018) refer to students who can navigate their learning in these ways as “assessment capable visible learners.”

Another group on whom this navigation instrument was having very little impact were the students who didn’t appear to care about either their learning or their grade. Some of these students were performing at the lower end of their respective classes. For these students, information about where they are and where they are going wasn’t helping them to move forward. They already knew where they were, and they didn’t really have ambitions to go anywhere else. This is not to say that they couldn’t be helped. Just not in this way. This is also not to say that all students performing at the lower end of their class didn’t care. We saw big improvements in many students who were previously performing poorly—many moving from failing to passing a unit test.

! If only the will was governed by clear-eyed approaches to Cognitive achievement. “As is the case in many school systems, achievement has taken priority over personhood.” — Belonging Through a Culture of Dignity: The Keys to Successful Equity Implementation by Floyd Cobb, John Krownapple https://a.co/eDvDuoY ““The biggest problem with tracking by a million miles is the equity issue,” says John Hattie (Corwin, 2015), lead researcher of the largest-ever study of educational research. “For instance . . . there is a high probability of African Americans and [Latinx students] being in the lower tracked groups. How can you possibly defend that apartheid in our schools?”” — Belonging Through a Culture of Dignity: The Keys to Successful Equity Implementation by Floyd Cobb, John Krownapple https://a.co/8cCOJ6a “The fact is that Maslow’s insight about belonging is an inconvenient truth when it comes to how schools currently function. School policies, practices, and behaviors all assume that achievement is all-important and is the determiner and entrance to belonging to that culture. The obsession with achievement has damaged relationships and the health and well-being of our students.” — Belonging Through a Culture of Dignity: The Keys to Successful Equity Implementation by Floyd Cobb, John Krownapple https://a.co/7rkk8X3 “This is the area illuminated by the streetlight. In addition, people often fear or ignore whatever is outside the radius of light, that is, whatever is beyond their experience and comfort zone. We believe we need to look beyond what is already known and practiced. The key to equity is still in the shadows. It’s time to bring it into the light.” — Belonging Through a Culture of Dignity: The Keys to Successful Equity Implementation by Floyd Cobb, John Krownapple https://a.co/1RDAiLi “They were persuaded, that, by some ridiculous arts of divination, they could discover the will of the superior beings, and that human sacrifices were the most precious and acceptable offering to their altars.” — The History of the Decline and Fall of the Roman Empire by Edward Gibbon https://a.co/eojcdOW “If we contemplate a savage nation in any part of the globe, a supine indolence and a carelessness of futurity will be found to constitute their general character. In a civilized state every faculty of man is expanded and exercised; and the great chain of mutual dependence connects and embraces the several members of society. The most numerous portion of it is employed in constant and useful labor. The select few, placed by fortune above that necessity, can, however, fill up their time by the pursuits of interest or glory, by the improvement of their estate or of their understanding, by the duties, the pleasures, and even the follies of social life. The Germans were not possessed of these varied resources. The care of the house and family, the management of the land and cattle, were delegated to the old and the infirm, to women and slaves. The lazy warrior, destitute of every art that might employ his leisure hours, consumed his days and nights in the animal gratifications of sleep and food.” — The History of the Decline and Fall of the Roman Empire by Edward Gibbon https://a.co/0OFRq2p Carelessness of futurity being characterized by the slaveowning, complicit, colonial hegemon “If polarization is not the natural condition of most of the country, then defining our national life by the views of those who are political obsessives is a misrepresentation of American national life and also affects that life, creating a profound weariness that cripples the will to participate in public affairs. Those of us in the press have an obligation to be clear that America is not defined by its most partisan citizens.” — The Hardest Job in the World: The American Presidency by John Dickerson https://a.co/dKlbXGk

For the students who saw units of study as one big topic, there were no connections. For them it was a bunch of disconnected and discrete routines to memorize. Once they could clearly see the different complexity levels, they could begin to see how, for example, multiplying proper fractions is a special (and basic) case of multiplying mixed fractions (advanced). They needed to see the distinction to see the connections.

Epistemic

Self-assessment, as used in classrooms for the last few decades, has largely been based on students’ opinions of their abilities. The navigation instrument, on the other hand, is a form of self-assessment that is based on data about students’ abilities. In truth, it could be argued that opinion-based self-assessment is not a form of feedback at all. One student in our research referred to these types of self-assessments as “feedforward,” because students tell it what they can do, not the other way around.

Important epistemic consideration

no matter how much freedom we had to break the institutional norms within the classroom, three to four times a year we still had to dock with the mother ship and take all of the thinking and learning that was happening day-to-day in our thinking classrooms and report out a grade.

Epistemic “epistemic goods such as information or education.” — Epistemic Injustice: Power and the Ethics of Knowing by Miranda Fricker https://a.co/275NK3r

Even if you work in a jurisdiction where it is either suggested or required that you triangulate each student’s performance using observation, conversation, and product, these suggestions or mandates are often not accompanied by practical suggestions for how to implement them. Tensions beget tensions.

! Ethical epistemic

Romagnano (2001) further argues his point by drawing on both first- and second-hand data to show how inconsistent grading is on everything from a quiz all the way up to the SAT-I mathematics test. This inconsistency creates what is called in the sciences a measurement error. For example, the measurement error on the SAT-I mathematics test is 30 points. This means that if a student scores 470, we can say with 95% certainty that their score is somewhere between 410 and 530 points. This is a huge measurement error. And it is on the SAT-I, one of the most tightly controlled assessments in the world.

Objectivity is a myth. Yet, the myth endures. And it leads to what I have come to call the tyranny of objectivity, which is the harm that we do with the points we have recorded for each student across a number of events. Believing that the points we record in our gradebooks are objective, we then further believe that the sum of these points convey truth—truth about what our students have learned. But this, too, is a myth.

“Rev. Martin Luther King, Jr., tied the dignity of sanitation workers to their contribution to the common good. One day our society will come to respect the sanitation workers if it is to survive, for the person who picks up our garbage is in the final analysis as significant as the physician, for if he doesn’t do his job, diseases are rampant. All labor has dignity. 43” — The Tyranny of Merit: What's Become of the Common Good? by Michael J. Sandel https://a.co/4yUyQI9 Epistemic

If we look at Ben and Charlie through a data-gathering lens as well, we quickly discern that something tragic has happened to Ben part way through the unit. His parents got divorced, or his grandmother died, or he got caught up with the wrong crowd. The same can be said about Charlie. Her inconstancy is cyclic. What is happening? I taught a Charlie once—her parents were divorced, and every two weeks she alternated who she lived with. In the weeks when she lived with one parent, she performed much better than the weeks when she lived with the other parent. The tyranny of objectivity that comes from point gathering ignores the very human elements of grading that are clearly evident in Ben and Charlie’s cases.

Ethical and developmental epistemic Emotional intelligence

Description Figure 14.2 Instrument for recording student data on a repeating patterns unit. Description Figure 14.3 Instrument for recording student data on a fractions unit. These elegantly simple instruments allow us to record all of our data in one place. Whether those data come from an observation, conversation, or a test, they are just data points within this table.

Ethical grading

a test is a collection of discrete opportunities for students to demonstrate learning, and these demonstrations are recorded on the table in a disaggregated format.

For anecdotal reporting, the instrument easily and clearly helps you to structure your comments.

Epistemic Learning target success criteria I will be able to demonstrate an understanding of summary and main idea with at least 2 conversational “busted”s or “getting caught” in academic discourse !

Description Figure 14.5 Benjamin’s performance on the repeating patterns unit. For example, if Benjamin demonstrated attainment at the advanced level only once (see Figure 14.5), that corresponded with the teacher’s subjective assessment of that student’s attainment at that level less than 60% of the time. Conversely, if performance at the intermediate level was demonstrated five times (see Figure 14.5) it corresponded with the teacher’s subjective assessment 100% of the time. But it was felt by both the researcher and the teachers that this was too much evidence—there was too much redundancy. It turned out that tipping point is two consecutive demonstrations of attainment. That is, two positive data points were sufficient to match with teachers’ subjective assessments of a student provided that the two positive data points were consecutive. So, whereas ✓ ✓ was enough to show attainment, ✓ ✗ ✓ was not—more data may be needed.

In my jurisdiction we are mandated to triangulate our data and gather grades of how students are performing through the framework of conversations, observations, and products (COP). Although what is in this chapter helps me in this regard, it appears that some students’ grades may still end up being the result of only products (tests). The COP framework you speak of is a huge step forward in the way teachers are beginning to think about grading and the reporting out of grades. But, in the jurisdictions in which it is mandated, it is largely misunderstood.

Figure 14.9 The COP framework as triangulation of data.

Triangulating data means that we are gathering data from multiple sources in order to seek correspondence within the data. Correspondence indicates that you are getting close to the truth about what a student has learned. So, if a student’s performance on a test corresponds with the evidence gathered in an observation, you have correspondence and you are close to having a true measure of what that student knows. Once you have correspondence between two forms of data, then you can stop. Additional data from the third data form will either be redundant and unnecessary or an outlier you can disregard. Only if the first two data forms do not produce correspondence will you need to go to the third. Triangulating data means that we are gathering data from multiple sources in order to seek correspondence within the data. Correspondence indicates that you are getting close to the truth about what a student has learned. So, if the correspondence is between a test and a conversation, the final grade will be equivalent to that of the grade the student received on the test. This is not to say that all the data come only from the test—it only means that equivalence, as a product of correspondence, exists.

Love it

looming tyranny of an external standardized test.

Are there still measurement errors? Yes. Measurement errors also occur through our own inconsistencies. We know, for example, that test papers graded early on are often graded differently than ones that are graded near the end. For some teachers the grades improve over time; for others they get worse. We also know that if we take a break in the middle of our grading, the papers that are graded prior to the break are graded differently than the ones that are graded after the break. If that break is extended and involves sleep, a meal, and some positive or negative social events, this may exaggerate the difference. And this doesn’t even take into account the variance introduced by the students and the complexity of their lives. We know, for example, that some students perform better in the morning than in the afternoon, or on Mondays better than Fridays, et cetera. Taken together, it is safe to say that there can be a measurement error on any given student’s test paper at any time. Let’s conservatively say that this error is only 1%. What this means is that the grade they received on the test is an accurate reflection of what that student knows to within ±1%. This is a pretty good margin of error—much less than reality. But this is on a single test. The thing about measurement errors is that, over multiple measurements, it compounds. So, if over the year, this student will take 10 tests for you, at the end of the year her grade will be accurate to within ±10%. Depending on y...

To change the norms, then, means that you need to also change the pattern of student behaviors and habits. From system theory we know that when we try to change a stable system, the system will defend itself. “In a system, all the features reinforce each other. If one feature is changed, the system will rush to repair the damage” (Stigler & Hiebert, 1999). When that system is a classroom, these defenses look like resisting, complaining, and apathy.

The three practices in the first toolkit, however, are not like journaling. Putting students in random groups to solve engaging thinking tasks on vertical non-permanent surfaces is enough of a departure from classroom norms that the students will notice that a change has happened. At the same time, the changes to the classroom routines are radical enough that they overwhelm the system’s ability to defend itself and, as a result, the students allow themselves to change, to be different, to deviate from their normative mimicking behaviors, and to begin to really think. And so the system changes.

asynchronous use of hints and extensions to create and maintain flow before starting to think about simultaneously planning for consolidation.

you are able to decrypt the mysteries around content and clearly communicate to students not only the demarcations between outcomes, but also to what level

The learning we were seeing in the groups was real. But it was synergistic, temporal, and contextual—existing only in that moment, in that context, and within that collective.

Gross

Lesson planning in the thinking classroom

Lesson planning thinking classroom

In essence, the only difference between homework and check-your-understanding questions is a shift of responsibility. This is why this practice appears first of the three responsibility practices, and it is the only one of the three to be included within the second toolkit.

you will realize the full benefits of a thinking classroom—greater student engagement, more active learning, greater student enjoyment, increased student responsibility, higher performances, easier and faster movement through content, greater satisfaction as a teacher, et cetera.

Macro lesson 1

Macro lesson 2

the original framework is governed by the acclimatization periods for both students and the teacher, the rebuilding framework is governed only by the acclimatization rate of students.

Acclimatization epistemic PBIS

building classroom norms as opposed to trying to change classroom norms—a much easier task.