On Teaching to Deliberately Connect Content Across Lessons (Reflections on Reading Lampert, Part 5)

To me, this chapter was interesting for one key idea: That looking at student work can help us to determine our actions in future classes.

Lampert tells the story of assigning a problem of the day at the beginning of a unit on time, speed, and distance. She has previously anticipated the mathematics she will teach during the unit—some connections between the operations, as well as some ratio and fractions work. During that first problem session, one of the students goes into mathematical territory Lampert didn’t expect to see.

“Because Charlotte ventured into mathematical territory that was new for her (and would be new for everyone else in the class as well) she gave me a resource for preparing to cope with the difficulties students would have in future lessons.”

She uses this information she gleans, in fact, to plan the next few lessons’ worth of problems of the day.

I’m going to admit it: I don’t use student work to my advantage. This reminds me of my readings on formative assessment, and I simply have to put more focus on looking at student work, interpreting it, and USING IT.

As Lampert says, when looking at what students do in their work, we are looking for “mathematical signposts to help… guide the class’s journey.”

She sums up this work quite nicely at the end of the chapter:

“The notes that I made from day to day during the unit illustrate the interplay between assessing students’ performances, reflecting on my own mathematical activities, and designing instructional strategies to get at specific mathematical topics.”


On Teaching While Leading a Whole-Class Discussion (Reflections on Reading Lampert, Part 4)

This Lampert book just keeps getting better and better. She’s so good at breaking down and analyzing her individual “teacher moves,” and in this chapter she talks about the menu of moves that she selects from while she’s teaching. She analyzes a whole group discussion she leads, and in one six-minute section of the lesson she counts 15 teacher moves, as detailed here: 

Teaching and Studying Event #1: Teacher Formulating and Asking a Question to Begin the Discussion

Teaching and Studying Event #2: Teacher Calling on a Particular Student to Answer

Teaching and Studying Event #3: Student Asserting and Teacher Repeating His Assertion

Teaching and Studying Event #4: Teacher Asking a Student to “Explain His Reasoning”

Teaching and Studying Event #5: Student Interpreting “Explaining” and Responding

Teaching and Studying Event #6: Teacher Making Representations of Student Talk

Teaching and Studying Event #7: Teacher Interpreting Symbols in Terms of an Alternative Representation

Teaching and Studying Event #8: Teacher Highlighting Patterns to Give Meaning to Multiplication

Teaching and Studying Event #9: Student Interpreting the Public Representation

Teaching and Studying Event #10: Teacher Relating the Idea of Groups to Practicing the “Times Tables”

Teaching and Studying Event #11: Teacher Again Asking for an Explanation

Teaching and Studying Event #12: Teacher Linking the Explanation to the Public Representation

Teaching and Studying Event #13: Teacher Representing, Student Asserting

Teaching and Studying Event #14: Student Evaluating Earlier Assertion

Teaching and Studying Event #15: Teacher and Student Reason Collaboratively

(She discusses each event in detail… these are merely the titles of each of these sections.)

In this six-minute episode, she gets at how she manages the tension of working with one student while the other students watch, making sure that in the next episode she contrasts this with more involvement from the rest of the class.

She also talks a great deal about her overall goal, which is getting students to “make sense” of the mathematics. She is much more concerned with this than answer getting. (I heard a lot of echoes of this at this year’s NCTM conference in Boston.)

“…Students’ engagement in mathematical sense-making is the foundation on which studying mathematics by working on problems is built,” Lampert writes. “If they learn to expect me or their classmates to step in when they do not make sense, rather than learning to get themselves out of a difficult spot, they will not be likely to do mathematics when completing the assigned problems.”

Finally, here are the unedited notes I made on the menu of teacher moves she presents:


Variety of moves… to address the problems of getting mathematics into the conversation and getting that mathematics to be studied. Elements of the work of teaching within this structure include:

• Creating visual representations of the ideas under discussion as a common record of the class’s journey and referent for our discussion;

• Deciding who to call on from among those who are and are not bidding for attention;

• Simultaneously teaching individual students and engaging the group as a whole in worthwhile mathematical activity;

• Keeping the discussion on track while also allowing students to make spontaneous contributions that they considered to be relevant;

• Monitoring the pace of the discussion with attention to the scheduled end of the class period; and

• Adjusting to the few students who need to leave or enter the room during the period.

The actions I take to address these problems are both managerial and intellectual. They serve both to move the discussion along for everyone and to infuse its content with the mathematics students are to study.

How to begin each new segment of discussion…

• Choosing the question to begin a segment of discussion;

• Choosing who has the floor in response to a question;

• Choosing to give someone who is bidding for the floor an entrée into the discussion.

When a student responds… There are several subsequent moves that can be made to turn that response into a resource productive of teaching and studying:

• When an assertion is made, choosing to stay with the student who made it and requesting an explanation;

• When an assertion is made, choosing to stay with the student who made it and suggesting my interpretation;

• When an assertion is made, moving to other students and requesting a counterspeculation; or

• When an assertion is made, moving to other students and requesting an explanation.

Whichever of these actions is chosen, the teacher can continue by

• Asking additional students to comment on another student’s thinking;

• Rephrasing a student’s explanation in more precise mathematical terms and asking him or her to comment; or

• Creating a representation of the students’ talk on the chalkboard.

To address the problem of infusing mathematics into the discussion in conjunction with attending to social issues, a teacher can:

• Alternate between persistent engagement with one student and quick moves around the class, or

• Alternate between single student answers and chorus-style participation.

At any point in the discussion the teacher can step out and

• Comment on what kind of problem or what kind of work this is, or

• Name a process or a kind of number with either a contextually invented term or a term from the public domain of mathematics.

Each move is designed and enacted in a particular moment to bring a particular piece of mathematics to students and particular students to mathematics. The work of teaching is not only deciding what to do at each of these levels, but also doing it, and keeping track of the studies it enables for students.

The teacher consistently works at teaching students both mathematics and how to study mathematics by asking students to reason, to explain, to attend to and interpret the assertions of others, and by reasoning, explaining, attending, and interpreting the mathematics herself in concert with their responses.


I left this chapter with a much better sense of the types of teacher moves that exist, which I had wondered about since reading Elizabeth Green’s Building a Better Teacher last summer. It would be interesting to videotape my own lessons and try to break them down and analyze them in this way. I can only imagine the amount of growth that would help me make going forward.

On Teaching While Students Work Independently (Reflections on Reading Lampert, Part 3)

In Chapter 6, Lampert describes what she does during the first thirty minutes of a sixty minute class period. She describes this work as follows: 
I need to watch and listen…

I need to enable relationships among the students, and between the students and the subject matter…

I also need to learn more about how they interact with one another, how disposed they are to talking about mathematics, and what their communications skills are like when I am not around to help or guide them.

I need to acquire information I can use later in the lesson and in future lessons.

I need to watch and see who gets along with whom, who would prefer to work alone, and where the social trouble spots may be.

In thinking about my own class and the way I have run it up to this point, I do not spent nearly enough time doing these things.

And what I like about Lampert is that she admits that this is quite difficult to manage. She notes that she needs to direct students’ work “in a way that was not a simple ‘telling’ of the answer.” She notes that she needs to work to “locate each of [the students] in the mathematics I had anticipated that the lesson would be about.” And she needs to make decisions about what direction she needs to take them in the second half of the lesson.

She also notes that some student-student relationships might be unproductive, and that this is a risk in this work (and thus, something to which she must pay close attention).

I still wonder how these structures could work in my classroom, though she has said that a classroom does not necessarily need to be problem driven or problem based to adhere to her philosophy of teaching.

If nothing else changes, I at least need to figure out how to monitor students while they are doing independent work, and use that information I glean in future lessons.

On Preparing for Lessons (Reflections on Reading Lampert, Part 2)

Chapter 5 is a much shorter chapter in the book, but it has a few key points I’d like to highlight.

First, a couple of quotes from the text:

“The problem I am working on is how to engage this class, with its particular variation of skills and understanding, in the study of the ideas surrounding this piece of mathematics.”

“Teaching a lesson begins with figuring out where to set the particular students one is teaching down in the terrain of the subject to be taught and studied.”

Second, two main ideas struck me as important here:

  1. The importance of anticipating what students will do with the problem, and thinking about “teacher moves.” This reminded me of Phil Daro’s talk at NCTM this year, in which he compared teaching to a chess game. As a teacher, you must think about how things may proceed in the classroom, but teaching is responsive—you never know quite what you’re going to get back from the students, and this may alter the direction of the discussion that follows. It also reminded me of the lesson plans from the CIMT Mathematics Enhancement Programme. (See here for an example.)
  1. The importance of getting to know the students as learners. Perhaps by knowing students as learners, you can anticipate what will happen better. I was fascinated by the depth of notes she had in her teaching journal about students. I wonder how I can organize these kinds of notes. (Let me know if you have any suggestions!)

Third, this quote gave me a big sigh of relief:

“It would be unusual for a teacher always to invent activities for students ‘from scratch.’”

At the same time, I am still concerned about a percieved overreliance on textbook materials to guide my lesson design. I feel I am often just doing examples from the book in front of students, and I’d like to figure out how to change this.

Thoughts? Please leave a comment!

On Creating a Classroom Culture (Reflections on Reading Lampert, Part 1)

I’ve just finished reading the first “meaty” chapter of Teaching Problems and the Problems of Teaching by Magdalene Lampert, titled “Teaching to Establish a Classroom Culture.”

It’s funny. I worried a lot more about this when I first started teaching, but it still remains crucial to establish some routines at the beginning of the year. I’ve always done it in one way or another, and the truth is I probably do it subconsciously at this point, but I felt after reading this chapter like I could do it even better.

What I really liked from this section was how she framed setting routines to create an environment in which students could study and learn mathematics

There are some key things that Lampert talks about in this chapter that I’d like to take into next school year.

  1. Structuring how students write in their notebooks. This goes so far as getting students to organize their work into four sections: Date, Problem of the Day, Experiments, and Reasoning. I like this idea of getting students to think consciously both about their experimentation when trying to solve a problem as well as the thinking they used to approach the problem. Students in Lampert’s class used black felt-tip markers to write in their notebooks (which were grid-ruled), which helped Lampert see student work easier while walking around the classroom (genius!) as well as copy work to present to colleagues (Professional Learning Community, anyone?).
  1. Teaching students how to speak about and listen to math (“linguistic routines”). Including using the words condition, conjecture, and revision; putting conjectures on the board and then discussing them and figuring out which “made sense.” Also reflecting on a problem’s possible solutions.

Finally, these are the things that Lampert says teachers need to take into consideration in terms of devising and choosing structures for studying:

  1. Devising/choosing a method for ending the previous class and starting the new class
  2. Devising/choosing a method for students to record their work
  3. Devising/choosing a seating arrangement
  4. Devising/choosing strategies to teach students how the class will operate on a daily basis
  5. Anticipating what students will do in response to these methods and strategies
  6. Devising/choosing methods of handling their responses

I still need to do a bit of thinking on this, especially about seating arrangements and figuring out how to construct groups. I have done this quite haphazardly in recent years, and I know I need to do a better job of this–basing it on data.

Summer Planning, So Far

Here’s a quick update of what I’ve been up to the past three days:

Monday: Nailed down my units/scheme of work for all of my courses this coming year, as well as the individual topics I’ll be teaching in each unit.

Tuesday: Created UbD documents for each unit of each course, listing all of the standards and/or topics to be covered in each unit.

Wednesday: Finally (!) made my way through the nrich maths curriculum mapping document, placing links to appropriate tasks/activities in the appropriate unit document. Did the same for Illustrative Mathematics site. Amazing resources…

I still have a lot of work to do looking at possible activities for these courses, but I’m well on my way!

Building a Better Teacher, Year 2

Since I finished teaching June 23rd, I’ve taken a couple of weeks off of thinking overly seriously about teaching. Inside my brain, though, things have been germinating for quite some time.

It’s been a little over a year since I joined Twitter, and that’s led me to all sorts of reading over the past year, including Embedded Formative Assessment by Dylan Wiliam (thanks @cheesemonkeysf for the recommendation), Building a Better Teacher by Elizabeth Green (thanks @nytimes for running an excerpt), and a few chapters of Teaching Problems and the Problems of Teaching by Magdalene Lampert (thanks @elizwgreen for spending so much time talking about Ball and Lampert in your book).

Last year I was also fortunate enough to be able to attend the annual NCTM conference in Boston. Just before that conference I found out that in addition to IB Mathematical Studies SL (Standard Level) and two sections of the 10th grade Standard Level Advanced Algebra course we offer at my school, I would be teaching one section of IB Math HL (Higher Level) next year. That led me to a second workshop in Florida in mid-June to prepare me to teach the HL course.

In addition to all of this, I started a book study at my school last semester with about 10 of my colleagues; we read (and enjoyed) Make It Stick: The Science of Successful Learning by Peter Brown, Henry Roediger, and Mark McDaniel.

And so here I am, at what I consider to be the beginning of year two of me “building a better teacher.” Last year I tried to execute more of a problem-based learning approach and also made inroads into adopting a standards-based grading approach, but I still don’t feel I’ve changed enough.

Here are a few things that are on my list of improvements I’d like to make:

  • From Embedded Formative Assessment, more assessment in the classroom to guide instruction
  • Tied to this, true implementation of standards-based grading (a school goal for this year).
  • From Make It Stick, the idea of interleaving and spacing practice
  • Make classes more varied in terms of activities we do

Just yesterday I looked back at some of the resources I found last year from TeachingWorks and the Boston Teacher Residency, and these few improvements seem to coincide well with their instructional goals:

Instructional Goal #1: Build a productive learning environment where every student matters and participates.

Instructional Goal #2: Teach lessons with high cognitive demand, maintaining a consistent focus on student reasoning and enabling students to understand big ideas in academic content areas.

Instructional Goal #3: Assess students’ understanding everyday to inform instructional decisions and plan cognitively demanding lessons and units of instruction.

Instructional Goal #4: Ensure Students Read and Write in All Content Areas, in meaningful ways, and on a regular basis.

That’s a tall order, but I like how they break down the instructional goals in their document (see link) into nice chunks.

So that’s where I begin this year’s journey. Can’t wait to continue thinking about this and making progress.