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.