Category Archives: Problems of Teaching

Teaching Problems: A Model of Teaching Practice

This is the last in a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

I am finishing this book without a great deal of concrete takeaways. Instead, I am a little bit more thoughtful about the different challenges in teaching. If reading Lampert has a positive influence on my teaching, it will be because I am more reflective and better able to learn from my failures in class each day, and more thoughtful when I encounter new problems.

I’d like to finish with Lampert’s visual model of teaching practice:


There are three players involved — the teacher, the student, and the content. My job is to facilitate the student’s practice in learning content, and I do that through my relationship with content, my relationship with the student, and my relationship with the student’s practice of content. Then there is a great deal more complexity when the lens zooms out to consider groups and classes of students.

I like this perspective because, in my view, it simplifies the practice of teaching without stripping away the complexity. It is a relatively straightforward set of relationships, yet each of those arrows involves dozens of decisions each day.

Teaching Problems: Teaching the Whole Class

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

In this chapter Lampert addresses the challenges of teaching a class full of students with different levels of knowledge and different instructional needs. After a quiz assessing addition with fractions, she considers the advantages and disadvantages of separating students who are still just adding numerators and denominators and teaching them without the rest of the class as an audience.

If I divided the class into different groups at this point in the year based simply on how students did on the addition problems on the quiz, I thought I might be able to solve some of the teaching problems that had been gnawing at me. But doing this would mean I would have additional work to do. I would have to figure out how to engage some groups in working independently while I worked with others. I would have to figure out a public explanation for the grouping, even if it was tentative, and explain it in a way that did not cause some students to lower their expectations of themselves in relation to their classmates. I would have to manage new and different kinds of social configurations because of the routines I had already put in place to support work in heterogeneous groups. I knew that different social issues would arise if the groups were to become homogeneous, rendering obsolete the organization of instruction that had been more or less adequate to manage differences in gender, race, ethnicity, and social class among the students (364).

I came out of this chapter with a pretty simple takeaway. Addressing misconceptions is not whack-a-mole. Separating a group of students based on a misconception may solve one problem, but it may surface more. This is not to say that separating the class is the wrong move to take, but a reminder to take a broader view, and only make that decision thoughtfully within the context of students’ larger learning trajectories.

Teaching Problems: Teaching the Nature of Accomplishment

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

Students cannot be said to make progress in learning unless they acquire some knowledge or a degree of skill that they did not already have (329).

This chapter of Lampert’s book focuses on the challenge of ascertaining what students have learned, and communicating that back to students in a way that is meaningful and productive for future learning. in short, assessment and feedback. I’d like to draw a contrast here.

My thinking about the goals of assessment before reading this chapter were informed by a number of sources, in particular this post by Michael. I divided assessment into three goals:

  • Evaluation. Assessment tells various stakeholders what students know, and what they do not know.
  • Feedback. Assessment provides feedback that moves learning forward by creating opportunities for more student thinking.
  • Incentives. Assessment provides incentives for certain behaviors, and disincentives for others.

Lampert uses a different framework. She presents four problems in assessment:

  • Demonstrating knowledge and skill. Students show progress by demonstrating what they know.
  • Multiple dimensions of competence. Learning does not consists of giant leaps, but fits and starts that are multifaceted and complex.
  • Different starting points. Every student does not start in the same place, and these differences should be honored and valued.
  • Public progress. Differences in learning lead to differences in status in the classroom, and this status affects future learning.

It seems like an interesting exercise to evaluate my current assessment system with respect to each set of criteria. I wrote about my current system of standards-based grading here. In short — 75% skills assessments, which isolate individual standards. Students can retake for full credit, with review assessments as the course goes on. 20% synthesis tasks that students have a week to work on and involve multiple standards and sustained reasoning.

My Original Approach
I like the way my system evaluates students, and I think it provides actionable information about what students know and don’t know and what they might do about it. I think it falls short in giving feedback that regularly provides students an opportunity to do more mathematical thinking — retakes are optional, and often pretty far removed (read: right before grades are due). I do like the incentives, as they line up with what I care about, and avoid the negative consequences of trying to incentivize everyday classwork.

Lampert’s Approach
I like the way my system allows students to demonstrate knowledge and skill and creates opportunities for students at different starting points. I need to think more about how it frames progress publicly, I’m not sure I’m addressing that area in a positive or negative way. I think I have a great deal of room to grow in multiple dimensions of competence — standards-based grading tends to place a great deal of value on performing individual skills in isolation, which does not value this area.

Both of these frameworks point to overhauling my synthesis tasks, with an emphasis on multiple dimensions of competence and giving useful feedback that moves learning forward. I’m not sure what this looks like, but it provides useful food for thought.

More importantly, I think that the challenges of building an assessment system that works for all students, and works for me, is an ongoing challenge, and I like adding a new tool to evaluate where I can improve and find new areas for growth.

Teaching Problems: Teaching Students to Be People Who Study in School

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

Lampert covers a great deal of territory in this chapter, but for a subset of it she focuses her attention on a single lesson. Students are placing a number of Fraction Bars — shaded rectangles representing different fractions — in order from least to greatest. In the discussion following the task, one student, “Saundra”, asserts that five-sixths and five-twelfths go in the same place on the number line. Lampert’s thought process as she responds is fascinating:

I need Saundra to learn the correct placement of fractions on the number line. I need her to understand why five-sixths is larger than five-twelfths. And at the same time, I need her to learn to think of herself as a person who can study and explain her mathematical reasoning and that she can do it in school, where her peers are watching everything she is doing (305).

Lampert returns several times to these major goals. I identify them as:

  • Students learning content
  • Student learning that math makes sense and they are capable of reasoning logically about it
  • Students learning the courage to take intellectual risks in a public setting

Lampert goes on to describe the tensions inherent in working with these  challenges:

To build Saundra’s mathematical competence, and at the same time maintain her academic self-confidence, I must now work like an air traffic controller, keeping all of the planes that are trying to come in for a landing from crashing into one another (307).

At the same time, she has decisions to make about which goals to prioritize over the course of the discussion. Her first decision is interesting: she helps Saundra to visualize the fractions on a number line at the blackboard, using that simple tool as a support for Saundra to make sense of the problem. Saundra makes a simple mistake — she counts the tick mark at 0 as one-twelfth, and counts up from there, meaning each fraction is off by one-twelfth. Lampert chooses to make that correction herself, rather than opening it up to the class or prompting Saundra to notice it, for the sake of expediency and to focus the class’s attention on the larger questions that she sees more potential for learning in.

As the class begins to discuss the relative placement of five-sixths and five-twelfths, Lampert makes another deliberate decision in who she calls on:

I first called on Charlotte because I had observed a pattern of respectful consideration in the way she responded to other students in the class when she disagreed (308).

Here she prioritizes teaching Saundra to take intellectual risks, by choosing a student to respond who will be likely to be respectful and productive in building up Saundra’s belief in herself as a mathematician.

The class then moves into a broader conversation, with a number of students contributing, and many disagreeing with Saundra’s assertion about five-sixths and five-twelfths. There are a number of teacher moves worth discussing in these exchanges, still working at the tensions between Lampert’s goals for the discussion. Interestingly, in the final exchange, Saundra states, “So I mean, I guess really it could be either way”, stating that either five-sixths and five-twelfths could be equivalent, as she states, or five-sixths could be larger, as several other students articulated. Lampert, in summarizing the discussion, writes: “She performed her understanding publicly and maintained her dignity” — Saundra was willing to take several risks, defended her reasoning, and did not come out of the discussion having been hurt or unwilling to contribute to a future discussion. At the same time, Lampert notes the challenge: “Reasoning mathematically, one would have to conclude that there is a contradiction between these two approaches, but this did not trouble Saundra (324)”. In managing the tensions between her goals, she may have been successful in instilling intellectual risk-taking in Saundra, and provided a new perspective on fractions, but was unsuccessful in teaching Saundra to believe that mathematics makes sense and that her reasoning should be consistent.

This is not a reflection on Lampert’s teaching; rather, it reflects the ongoing challenge of teaching students to be people who study in school, and the fact that this is a challenging, multi-year project, happening in fits and starts, and often without clear victories along the way.

I want to end this post with a nod to Joe Schwartz. He wrote one of my recent favorites posts over at his blog, and offers an equally fascinating case study of a student who is struggling, both with content knowledge and with his beliefs about mathematics and about himself. These are hard challenges, and I see in Joe’s post another issue — that it is often hard even to notice when a student’s beliefs and dispositions about math are counterproductive unless we ask the right questions and take the time to hear out student answers. That’s a larger challenge for another post, but more food for thought about what is actually happening in the minds of my students while they are in my class every day.

Teaching Problems: Teaching to Cover the Curriculum

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

One of the teaching problems that is particular to teaching “big ideas” like the part-whole relationship is assessing what students already understand and what they still need to learn, and doing this for many different students. Although there is no simple way of averaging their accomplishments, I need to steer the work of the class as a whole. Within that common journey through some mathematical terrain, I need to attend to who needs extra guidance and when they need it. There is no simple metric here as there is in teaching separate topics one after another, where what students have learned can be crossed off a list (238).

Lampert’s view of covering curriculum is in stark contrast with that of certain leading reform efforts today:
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There is a critical difference of perspective here. Lampert is examining a series of lessons about division in different contexts, and focusing specifically on numbers that “don’t work out”. Here is some more of her language:

When I asked Sam where “ninety-six” came from, I was providing the class with an opportunity to study how to connect a problem context and what is done with numbers (235).

Moving back and forth between money and an area model of fractions, I supported talk about the common structure of ratio in both of these problem contexts (240).

I was teaching the topic of “remainders” yet again, but coming at it from quite a different direction than I had in the time-speed-distance unit in November or the cakes and bakeries unit in January (242).

Lampert is not taking a laundry-list view of student learning, where if she just checks off all the boxes students will get what they need from her class. She is intensely humble about the fact that students will not learn overnight, and also intensely focused on the broad, transferable elements of a topic. This isn’t at the expense of the details — she dives deep into several specific contexts, as indicated above, and doesn’t hesitate to probe a student’s thinking, for instance in another exchange where she examines one small piece of the long division algorithm. These details are essential to cover a curriculum and provide students with the facility to apply their thinking in a broad variety of ways. But those details come after the big ideas, in this case the part-whole relationship, and are connected to their broader place in mathematics, rather than existing as procedures that are learned for the sake of procedures.

David Wees has a great, short talk that I think is relevant here, called “From Mistake-Makers to Sense-Makers”. He talks about his shift from looking at student work as a set of mistakes to be avoided, to a set of ideas that have value in moving their thinking forward.

This is a significant shift in thinking, and reading Lampert’s account of her teaching made a connection for me. Looking at students as sense-makers clearly impacts how we respond to student thinking, and honor the ideas that they have rather than trying to pigeonhole students into prescribed modes of thinking. But it also impacts the structure of the curriculum — whether we choose to start with big ideas, and take advantage of students’ attempts to make sense of them, or teach in little pieces that likely send the implicit message that math is a set of disconnected questions that students just need to avoid making mistakes on.

Teaching Problems: Teaching to Deliberately Connect Content Across Lessons

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

This chapter of Lampert’s book has been the most interesting to me so far, but also the most complex. There is no three-step approach to connecting content across lessons. Instead, it’s a great deal of thinking about reoccurring representations, connecting contexts to content, using language purposefully, and formative assessment. I don’t know how I could summarize the chapter as a whole, but Lampert does investigate one domain that I had not thought much about before.

Lampert poses a problem to students:

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a knot is one nautical mile per hour

She spends the class discussing the first problem, which students struggled with, and uses this diagram, which she calls a “journey line”, to relate the context and the mathematics.
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Looking at student work after the class, she notes that only a few students make assertions about the second problem, and only one student, Charlotte, tries to figure out precisely how long it would take the Mimi to go 60 nautical miles at 8 knots. This problem is significantly more complex because the numbers don’t divide evenly. Lampert’s students are still struggling with the their knowledge of division, but Lampert knows she wants to move into teaching about remainders in a future lesson, and uses this as an opportunity to think about how to structure that teaching.

Here is Charlotte’s explanation for part two of the problem:
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Charlotte’s interpretation of the remainder is wrong, but that’s not Lamperts focus. She writes:

Using the problem context to give meaning to the remainder is something I would want everyone in the class to be disposed and able to do (200).

This isn’t something I saw when I first read Charlotte’s work. I think it’s an important thing to notice because it is a focus not on the how, the mechanics of a computation or procedure, but instead a why, a habit of making sense of mathematics and relating mathematics to the world. It’s a change in perspective from looking at Charlotte’s reasoning as lacking because her answer is wrong, to looking at her reasoning as useful because it is attempting to make sense.

But more than using this piece of work as an evaluation of Charlotte’s thinking, it provides a “way in” to future mathematics.

Examining the mathematics involved in Charlotte’s assertion led me to think that I should pose problems that would engage students with these ideas. Although Charlotte may have been ahead of everyone else in confronting the problem of giving meaning to the remainder, I could use her work to clue me in to the kinds of things other students might do and what they should be able to do. The challenge for me would be to structure the learning environment so that neither Charlotte nor anyone else would be satisfied with an assertion about the remainder like the one Charlotte ventured without returning to the constraints of the problem situation to see if it fits.

Here, I expected Lampert to launch into a problem the next day to explore this mathematical territory. Instead, the class spends two days working on problems where a remainder is not necessary, so that students can continue to study division and its relationship to a context — dividing 135 by 5, and 180 by 6. Then, on the third day, she poses a problem about a rate. She uses the same distance, 180 nautical miles, as the day before, but changes the speed to 8 knots so that the division will not “come out even”. This problem was not entirely successful. Several students offered interpretations of the remainder, but the class ended before anyone could offer a coherent explanation for why the remainder of 4 represents half an hour.

The next day, Lampert chooses a similar problem, but instead chooses 20 nautical miles and 6 knots, reducing the algorithmic load on students and providing an opportunity to focus more mental effort on the meaning of the remainder. Lampert prompts students to figure out how long it will take for the ship to go one mile, and to use a diagram. This allows her to launch into teaching about what that remainder means, and to relate the hour to parts of the hour, and to do the unit conversion necessary to move between hours and minutes.

This is a bit of an anticlimactic end to a fascinating series of instructional decisions, but I learned several important things from seeing Lampert’s thinking. First, she is enormously patient, and content for students to leave class, perhaps for several days in a row, unable to solve a problem using a certain piece of mathematics. She does this because she has a laser-focus on sense-making, creating situations where students can believe that math makes sense, and they are capable of figuring out problems by making connections with what they already know. Finally, she launches this entire series of problems with an observation from student work, and uses that work to guide her planning for future lessons and frame her thinking about how other students are likely to interpret a new idea.

I’ve never done this level of thinking in my teaching, and it’s a bit intimidating, but also seems like an enormously valuable way to both examine student thinking, and use that thinking to inform future planning. It also highlights the complexity of connecting content across multiple lessons. Fun stuff to get better at.

Teaching Problems: Teaching While Leading a Whole-Class Discussion

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

Lampert has a fascinating classroom routine. Each class, she begins with a problem on the board. Students work to answer it, alone or with other students at their table. If they finish early, Lampert asks them to make more conjectures and try to answer them. At some point during the class, she brings the whole group together, and they discuss the problem and its connections with other mathematics.

In this chapter, Lampert examines the discussion that followed student work on this problem:
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Here are some of my takeaways thinking about discussions in my classes:

Call on students for a reason

The Five Practices advocated this years ago, but Lampert’s writing is a good reminder. Here’s a great quote:

I called on Richard because I wanted to teach him and others in the class that everyone would indeed by asked to explain thier thinking publicly. I also wanted to teach everyone that what they said would be expected to be an effort to make mathematical sense (146).

And later:

As with my choice of Richard to begin talk about problem A, the choice of Anthony here is a deliberate teaching act, meant to get at a particular piece of mathematics and a particular aspect of my relations with students (165).

Lampert makes explicit that she calls on students with knowledge of what mathematics they may share with the class. She also makes explicit that her choice of who to call on sends a message to the class about the nature of participation, and the nature of having mathematical ideas. I like this:

Many would no doubt make their own conjectures about why I called on Richard and why I did not call on someone else. They would continue to conduct experiments to learn more about how to get called on or not, depending on their purposes (147).

Lampert gives a searing glimpse into the world inside students’ heads that I too often don’t see in my own classroom, and the consequences my decisions have for students’ beliefs about what it is we do in math class.

Represent mathematics deliberately

Principles to Actions makes an interesting statement about the ideal teacher role in discussions:

Students carry the conversation themselves. Teacher only guides from the periphery of the conversation. Teacher waits for students to clarify thinking of others (32).

I’m pretty skeptical of this assertion. Let’s juxtapose it with a teacher move of Lampert’s. Richard shares an incorrect assertion — that ten times twelve is twenty-two. It becomes clear that he did not misspeak, and has a misconception about multiplication, one that many other students in this fifth grade class are likely to have as well.

One of the things that I came around and did with some people is to draw a picture that would help you to reason about these problems. Twenty-two groups of twelve, you could draw as a twelve, a twelve, a twelve, and so on until you got to twenty-two of them [drawing circles around 12s as I talk] (151).

But let’s look at ten groups of six for a minute [drawing on the board, next to what I had already done] (153).

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Lampert’s representation adds information, clarifies student statements, yet still leaves room for thinking and learning. Those are great criteria for teacher input, and are almost always useful in making a discussion more productive.

Students may at points be able to carry the conversation themselves — but that seems to me better suited to a discussion where students are already confident with the material. Much more typical of my classroom is a discussion where students are still working through their ideas about a piece of mathematics, and a teacher taking an active role to clarify and represent student thinking to create more opportunities for learning seems to me an essential part of discussion.

Give hints to create better opportunities for further discussion

Michael Pershan and I exchanged ideas earlier this year about hints. One goal of a hint that he proposed was to prepare for future learning — if students notice certain problem features, they will be better able to learn from a discussion. Lampert is unafraid to propose a representation or present a mathematical idea if she feels it is an opportunity for learning. At the same time, she is acutely aware of the messages that she sends with these choices — and that when students are generating mathematical ideas, other students learn that it is their role in the classroom to share what they think about a problem. A useful teacher role here is to make the most of independent work time by giving hints when necessary that create a discussion more focused on essential mathematical ideas.

An instance of my teaching practice

I want to zoom in on a moment in my classroom where I could have benefited from this type of thinking. I started an Algebra-II class with this visual pattern as a warmup:

Students were trying to write an expression for the number of pink nubs in the nth step. After some work time, I called on a student with a raised hand to share his expression and reasoning. I did not know what he was going to say, though it turned out to be a useful starting point:
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He shared his reasoning — he saw it as two squares growing, and then subtracted the overlap, which was a square one unit length smaller. His reasoning and explanation were sound, though he was counting Lego pieces rather than pink nubs. I hoped to have another student make this observation, and called on a second student with a raised hand. Instead of commenting on the first student’s expression, she shared a second expression:
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She explained that she saw it as a single square, and then an L-shaped group of squares on the outside, which started at 1 and then increased by 2.

This was a useful alternate way to look at the problem, but still did not get at the point I wanted to make. Returning the focus to the first equation, I changed it to read:
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and asked the class what that might mean. A student volunteered that the expression was now counting the number of pink nubs, rather than the number of Lego pieces.

At this point, the energy in the room seemed to have hit a dead end. I didn’t feel like I had gotten everything I wanted out of the problem, and decided to share one additional interpretation. I wrote this on the board, and asked students what it might mean:
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After a short chance to chat with someone next to them, one group realized that this expression interpreted the pattern as one large square, with two pieces missing in either corner, then multiplying by 4 to account for the transformation from Lego pieces to nubs.

Looking back

I think I made a useful choice in sharing an additional interpretation of the problem. Given the lack of focus in the first part of the discussion, that example was a worthwhile way to cement some of the understanding about the structure of the expressions and its relationship with the pattern — and a number of students in that class have struggled with representing quadratic patterns over the course of the year.

I came up pretty short in anticipating what students were going to see. Students generated the first three expressions, but I did not sequence them effectively because I did not know what students were going to share. I was reacting to what they said, rather than following a plan I developed while students worked, and too focused on one possible avenue forward. As a result, I missed an opportunity to more effectively connect between expressions, and to create dialogue between students, rather than constantly bouncing off of me.

I also think that the difference between Lego pieces and pink nubs is a less important feature of this task, and I could have used the time more effectively by pointing out that difference to students individually, in their groups, or by being more explicit about it from the beginning.

Lots left to improve here. Back to work.

Problems of Teaching: Teaching While Students Work Independently

This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

Lampert spends this chapter talking about the decisions she makes while students are doing math each day in class. Students are both working on their own and collaborating with others in their group.

I’m fascinated by how many times Lampert uses the word “teach” in this chapter. Some examples:

But I was also teaching Varouna that making a certain kind of picture is a strategy for finding the solution to this kind of problem (125).

I worked on interpreting his responses, both in the interaction and as I thought later about how to teach him to be more mathematically assertive (127).

Perhaps one of them has “taught” the other that it is better to have the boxes filled in with something than to leave them empty, even if there is no clear reason for the numbers chosen, or perhaps they did not interpret the task as having to do with multiplication (130).

His actions presented me with an occasion to teach the task structure of collaboration explicitly (130).

She uses the word “teach” in many contexts that I often don’t think to use the word — in interactions when she redirects student attention, or answers a question students have, or gives a hint to a student who is stuck.

Lampert is incredibly thoughtful with respect to the impact of each action — or inaction — in the classroom, and what they may or may not teach students. I think it’s worth categorizing the types of things that are worth teaching in these situations, based on her examples:

Teaching content. Lampert does this both explicitly, by taking moments to summarize and make points about the problems students are working, and implicitly, through the connections she helps students make in their independent work.

Teaching strategy. Lampert does not do this explicitly, where I think it is less likely to be successful. Instead, she works to reinforce effective strategies that she sees, and model useful strategies that students have not thought of.

Teaching beliefs. Lampert works to teach students that they are capable of making sense of mathematics, that they have ideas worth sharing, and that finding new approaches to problems is more useful than finding a single straightforward solution.

Teaching norms and routines. Lampert pays attention to repeated student actions that are productive or counterproductive for future learning, and works to build norms and routines that will facilitate learning.

Acknowledging that students teach each other. And that this can be both productive and counterproductive, in any of the above realms, and that Lampert’s teaching influences its utility.

An Instance of Teaching Practice

I taught this Illustrative Math task today in a Pre-Calc class:
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It’s an interesting one. There’s a great deal of content here. I taught arithmetic with complex numbers, the complex plane, and geometric interpretation of complex numbers.

But while I think there was a great deal of content learning happening, the way I facilitated the task may have been counterproductive for learning in other areas. I let students try the task independently first. We had worked through this task previously, and most students were comfortable operating with complex numbers, but those operations were an obstacle for a few students and slowed them down significantly.

In addition, no students realized that the geometric operation was a 45 degree rotation with a dilation. The majority of student work was simple computation and placement of complex numbers in the complex plane. I did not provide effective scaffolding to move students individually or in partners toward this realization; instead, we did it on the board, after partner work, and I was doing much of the work. I implicitly taught students that they were not able to make big realizations about the structure of mathematics — that was the role of the teacher in this lesson.

This task also provides strategy for students; they do not choose what operations to use or what representations may be useful. Instead, the problem tells them what to do, they follow directions, and they arrive at an answer. I taught them that, often, problems provide strategy for them, and finding new strategies is  not their role as a student.

Students worked in partners on this task, but the result of that structure was that much of their collaboration was around checking their work and fixing arithmetic errors, rather than doing rich, divergent mathematical thinking and sharing ideas. I taught that partner work is primarily about getting right answers, not sharing new strategies.

Doing Better

I think a simple change might be useful for using this task in the future: remove the direction to graph the powers of z in the complex plane, and have students compute through higher powers of z. With more examples, students are better provided with the resources to make a generalization, and share strategies for how they might extrapolate it forward. Students would be unlikely to draw on the representation of the complex plane in this instance, but it is possible to make valuable conjectures about the pattern without it. I can then provide the representation of the complex plane, teaching students that alternate representations can often make a pattern or conjecture more clear, and that these choices should not be arbitrary or dictated by forces outside of their control.

I’m looking forward to teaching this again, and I’m bummed I won’t have the chance for a while. But I hope this perspective on teaching through independent work — teaching content, strategy, beliefs, norms and routines, and cooperative learning — will help me better analyze and reflect on more of my teaching in the future.

Problems of Teaching: Teaching to Establish a Classroom Culture

This is one in a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines a challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.

Lampert teaches fifth grade in this book, yet I am constantly struck by how much of what she writes applies to my high school classes. I intended to read her chapter, “Teaching to Establish a Classroom Culture”, on how she works to establish a culture conducive to learning mathematics in the first weeks of school, and find one big takeaway to write about. Instead, I have a whole bunch, but here’s my attempt at paring that down to one big idea.

Lampert spends much of her time talking about three activities that students would engage in over the course of the year, and that she teaches deliberately from the first week.

In the context of these lessons, I taught my students three new activities and named them as such for public identification:

  • finding and articulating the “conditions” or assumptions in problem situations that must be taken into account in making a judgment about whether a solution strategy is appropriate;
  • producing “conjectures” about elements of the problem situation including the solution, which would then be subject to reasoned argument; and
  • revising conjectures based on mathematical evidence and the identification of conditions (66)

These are well illustrated by a series of problems Lampert gives her students. Here is one:
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Lampert allows students to work on the problem, and challenges those who finish early to try to find all of the possible additions. She finds that, as students try to find more solutions, some use unusual approaches. Some use unit fractions, others use negative numbers, and others put multiple digits in a single box. Creating “conditions” for a problem here is making explicit a practice of mathematics to encourage precise thinking and provide opportunities to interrogate the assumptions of a problem. 2015-11-30 21.01.10
In this problem, Lampert begins to make conditions explicit. She’s very thoughtful about this choice. There is clearly value to thinking creatively about different ways to answer a problem, but she points out that

These multiple interpretations pose a problem for teaching because if all these ways of making combinations are allowed, students will not be able to evaluate the assertions their classmates are making about the total number of possibilities (74).

She then takes this a step further and poses a question with conditions that seems to me to be very difficult for the first weeks of a fifth grade math class.
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She makes students’ conjectures transparent to the class by putting them all on the board
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Students discuss the conjectures, and in the course of doing so, Lampert provides multiple opportunities for students to revise their thinking based on the reasoning and conjectures of their classmates.

This seems to me like remarkable teaching. She introduces significant mathematical ideas — conditions, conjectures, and revisions — and uses these ideas to teach students their role in the math classroom. Students learn that everyone’s ideas have value, that mistakes are normal and are part of learning, that communicating about mathematical ideas is important, and that it is the student’s job to make sense of mathematics.

She also has an interesting way of framing student work. She refers to this as independent work, though students are working in groups and have group norms:

  • You are responsible for your own behavior
  • You must be willing to help anyone in your group who asks
  • You may not ask the teacher for help unless everyone in your group has the same question (82)

Lampert’s choice to call this “independent work” emphasizes individual accountability, and her norms both reinforce this accountability, and provide built-in support for students without requiring the involvement of the teacher at every stumble.

I found it fascinating that Lampert launches right into norming the way her class does mathematics. She spent the first class talking about revision and the importance of revision in learning math. Even more, she makes explicit her priorities in setting routines:

The work of ‘routine setting’ is more than telling students what the rules are going to be and enforcing them. It is largely a matter of guiding student talk and action in such a way as to establish shared understandings among everyone present about what it means to teach and to study and how it is to be done, here, with this class and this teacher (93).

This obviously ignores the many challenges of maintaining these routines, and what to do when students challenge them. I have never been particularly good at classroom management, and I look forward to her insights over the coming chapters. But framing expectations in class around the mathematical norms that facilitate learning, and then using those norms as a framework for teaching students what it looks like to do school, strikes me as a more purposeful way of managing a classroom. I have looked at behavior norms and mathematical norms as separate entities in the past; I’m considering now what those norms could look like, synthesized together, in my class.

I think the other important takeaway here is that Lampert is explicit in teaching what these actions look like, and why they are important for the class. Too often, I just put students in groups, or ask them to make a prediction, or give them feedback on their work, without teaching them why and how these things are important for their learning, and what student actions will contribute to their success. My meta-curriculum project broached several of these ideas, but I think I have a ton of room to grow in thinking through a progression of how these ideas develop and how students become normed to being productive and doing math in my class.

Complexity and the Problems of Teaching

A recent project of mine has been to work on developing tools to evaluate my teaching and help focus my energy as I try to get better. There are plenty of resources to build off of. Deborah Ball’s Teaching Works project names nineteen high-leverage practices that span instruction, relationship building and planning, and could be used with multiple content ares.The Danielson Framework provides another lens, this time breaking teaching into four different parts and attempting to describe what goes into each part. But the best resource for me has been Principles to Actions, and its eight teaching practices.

I spent a bunch of time over the last few months developing these into a rubric that adds more detail than the teaching practices alone, but is more compact than the book. My current draft of the rubric is linked here — it consists of the eight practices with 3-5 bullets for each practice, sourced from Principles to Actions and other resources.

I hoped to use this as both a reflection tool for myself, and a tool for soliciting feedback from others, in particular when my Professional Learning Community at my school observed me. While I’ve found it useful for big-picture reflection, I’ve tweaked the wording and structure of each practice more times than I can count. I’ve found the ideas here difficult to communicate to teachers who aren’t already familiar with the Principles to Actions practices, and not particularly useful for non-math teachers — it’s just too hard to communicate a vision of great teaching in so few words.

This last week I’ve been digging into Magdalene Lampert’s great book, Teaching Problems and the Problems of Teaching
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Lampert uses the word “complex” to describe teaching. While her book precedes much of the recent writing on the difference between “complicated” and “complex”, her reasoning fits into neatly into that narrative.

One reason teaching is a complex practice is that many of the problems a teacher must address to get students to learn occur simultaneously, not one after another. Because of this simultaneity, several different problems must be addressed by a single action. And a teacher’s actions are not taken independently; they are inter-actions with students, individually and as a group. A teacher acts in different social arrangements in the same time frame. A teacher also acts in different time frames and at different levels of ideas with individuals, groups, and the class to make each lesson coherent, to link one lesson to another, and to cover a curriculum over the course of a year (2).

I had been treating the evaluation of teaching as a complicated problem. If I could just break it down into the right parts, of the right size, I could build a perfect tool to analyze my teaching. But it’s not complicated. It is a complex problem with innumerable parts, small and large, that interact with each other continuously during the teaching day.

Lampert frames her approach differently.

Different teachers, in different kinds of communities, with students of different ages, teaching different subjects, work on the same kinds of problems, although the problems themselves, and certainly their solutions, will be different (6).

Instead of trying to establish the outputs of teaching, she considers the inputs. Teachers have a set of similar challenges in deciding what actions to take each day, and examining these inputs from a critical perspective, and developing greater skill and wisdom in meeting these challenges, presents a very different approach to improvement in teaching.

Lampert’s book examines ten problems of teaching, and does so through an in-depth look at a year in her teaching practice. The book uses extensive transcripts of her lessons and a detailed look into her thought processes to examine how she worked to solve these problems, what went well, and what could be improved. The problems she identifies, though she acknowledges this is not a complete list, are:

Teaching to Establish a Classroom Culture
Teaching While Preparing for a Lesson
Teaching While Students Work Independently
Teaching While Leading a Whole-Class Discussion
Teaching to Deliberately Connect Content Across Lessons
Teaching to Cover the Curriculum
Teaching Students to Be People Who Study in School
Teaching the Nature of Accomplishment
Teaching the Whole Class
Teaching Closure

Lampert offers no quick-and-dirty solutions, no simple tips or tricks, and no bulleted lists. Instead, with constant humility, she examines the challenges she faces, and her best attempts to meet those challenges.

This sounds like a fun project. Take one teacher’s examination of the challenges of her teaching experience, and see what I can learn from it.