Author: David Wees (page 8 of 97)

In 1965 a pair of researchers, Robert Rosenthal and Lenore Jacobson, set out to study the Pygmalion Effect, which hypothesizes that if we hold high expectations for people’s performance, their performance will be better than if we hold low expectations.

Rosenthal & Jacobson, 1968

What they found was startling, especially for younger kids. Students who had been randomly selected as being likely to show potential based on an imaginary test and then this information communicated to their teachers, showed tremendous gains in the intellectual ability compared to the control group.

Rosenthal and Jacobson, 1968

The reverse effect, called the Golem Effect, has also found to have an impact through randomized-control-sample experimental design. This means that one factor into student performance is the expectations teachers have, either high or low, on that student performance.

Every time you refer to your students as “low kids” you are re-enforcing expectations of future achievement that limit children. #NCSM16

— David Wees (@davidwees) April 12, 2016

This is why I get incredibly frustrated when teachers use labels like “high” or “low” to describe their students. Invariably these labels are like self-fulfilling prophesies that lead to strengthening or weakening student beliefs about their own ability to be successful and lead to teachers, subconsciously, taking actions that lead to either higher or lower student performance, and as performance is related to student learning, this can be devastating.

Of course, this pernicious labeling of students is built right into our systems of education so we can hardly consider individual teachers to be at fault. We have the ELL students, the SpEd students, the “lower third”, the students who only achieved a 1 or a 2 on the NY state exam, the students who failed algebra last semester, and other unhelpful labels for students ad nauseam.

So what do we do? I propose a good starting place is to eliminate unhelpful labels from our own vocabulary. Another step is to talk to our colleagues about our shared use of labels for students. Find ways to look for and talk about strengths in students rather than perceived deficits.

CLIME (The Council for Technology in Math Education) is an affiliate of NCTM with the mission to:

Empower math communities to improve the teaching and learning of math through the use of dynamic tools in a Web 2.0 world

Last night members of CLIME and other interested people attended a meeting of CLIME to discuss the its future.

In order to understand the role of CLIME in promoting the use of technology in math education, one has to understand a bit of the history, so Ihor Charischak (the long-time President of CLIME) started us off with a brief recap.

We then discussed some ideas for how we could better support the meaningful and productive use of technology through the NCTM annual meeting. Note that for this meeting our focus was on improving the NCTM conferences rather than all of the other ways we can support technology use. We brainstormed the following list of ideas.

1. We could find people doing interesting work with technology and invite them to submit proposals on that use.

2. We could set up an area in the exhibit hall and run mini-technology based sessions where educators could come to learn about how to use dynamic geometry software, learn how to get started with blogging, how to set up a Twitter account, etc… One benefit of this arrangement is that we could offer to help people install software (or find and bookmark websites) so that people who wanted to run workshops on the same technology later would be more likely to have a group of attendees with the software already ready to go.

3. We could suggest the labeling of sessions on technology as beginner versus advanced so that people who need help installing software, finding the menus in that software, and getting started with their initial exploration of the technology can have support and that people who are already experts in the use of technology can share ideas back and forth.

4. We offered that the program NCTM has started where presenters add additional information about their sessions and invite participants to comment on and ask questions about sessions could be extended. This way the 50 words or so presenters have to describe their work could be increased without dramatically changing the experience of conference organizers (who have to read all of those descriptions and make decisions about who gets to present at the conference).

5. We could continue to review the existing program after it is published and offer feedback to the NCTM program organizers to use with the next conference.

6. We could run our own technology in math education conference. We noted the importance of a face to face conference for encouraging networking between math educators but we still considered a hybrid or entirely online conference as well.

7. We wondered about ways we could encourage the younger generation of math teachers to participate in NCTM’s conference.

8. We could form a technology study group with the aim of cataloging and reviewing different technologies in use in math education and then potentially presenting our findings at an NCTM conference.

If you were tasked with promoting the meaningful use of technology in math education through a conference experience, what else would you do?

If you or your students are going to talk about mathematical ideas in your class, it is critical that everyone understands the idea being discussed otherwise they are less likely to either remember it or be able to participate in the discussion.

Every time you or your students make logical leaps when explaining mathematical ideas, your students must fill those logical leaps with what they understand about mathematics or invent their own logic to fill the gaps.

Consider this task:

Which line segment is steepest? (source)

Make the task explicit

This task as currently written is actually ambiguous. There are a lot of vertical and horizontal line segments in this picture; are they meant to be included, or not? Are we only supposed to focus on the bold line segments?

We could change the prompt to something like “Which of the bold line segments is steepest?” This of course assumes that students understand what a line segment is and interpret bold to mean the same thing their teacher means. (It’s fine if students don’t completely understand what steepest means though so since the goal of a task like this is to come to a common definition of steepest.)

Use Gestures

Another approach is when asking students to solve the task, having it projected on a screen, and using one’s hand to trace and emphasize the line segments in the image, while asking the question.

Push for clarity

Now consider these (simulated) student strategies for solving this task and imagine students are describing their strategies out loud to share with the class.

Student 1

Student 2

Student 3

If you have taught students how to interpret lines or line segments on a graph (or remember the mathematics associated with the task from when you learned it), you can probably figure out what these strategies mean. But there are gaps or missing steps in each explanation and since the explanations are out loud, there are ambiguities in each explanations as well.

With respect to the first strategy presented, what does it look like to extend all of these line segments so that they are the same length? How does the first student know that just because the line segments are now the same length, that one of them is steeper than another? And which line segment did they actually find to be steeper anyway?

In the second strategy, where are those little triangles drawn? Are they connected to the line segments in some way? And even if they are, one of the line segments is horizontal; how do I draw a triangle under that? Why does the largest rise over run correspond to the steepest line? Is that always true?

In the third strategy, where exactly are the angles between the line segments and the x-line? And what is an x-line? And why does the largest angle correspond to the steepest line segment?

Use questioning

One strategy is to ask clarifying questions about the strategy or to prompt students to ask clarifying questions of each other. In order to be able to ask critical questions “on the fly” it is extremely helpful to have anticipated the approaches students will use and at least some of the possible leaps in reasoning they may make, so that you can prepare questions in advance.

Use annotation

If you or students talk about mathematical ideas with no public written record of what was discussed, chances are high some students will either not be able to follow the argument being made or will quickly forget the argument. You can, and should, keep this record for students during discussions and use color and symbols to make the connections between mathematical ideas clear.

Here are some examples of annotations related to the student strategies above. Do these annotations make the ideas being discussed more clear? Is it more obvious why these strategies work?

Annotation for Strategy 1

Annotation for Strategy 2

Annotation for Strategy 3

Keep a public record

Here is a record of what participants noticed and what their meta-reflections were when I used this task with them.

What participants noticed

What participants reflected on

Having this public record means that if a student’s attention wanders, they can get back into the flow of the class. It also means the information you want students to take away with them remains up for as long as possible. Further, when you move to prompting students to consider why the mathematical strategy works, students’ cognitive load around what the actual strategy being considered is decreased while there is a public record that they can access.

Prompt students to consider each others’ ideas

Unfortunately, usually when people (our students included) listen to each other they listen for what they expect to hear rather than what was actually said.

While it is worth saying that you should actively listen to what students say rather than changing its meaning or filling in the gaps in logic yourself, you will also want to use talk moves (like revoicing, restating, asking a student to restate, asking questions, wait time, etc…) to push your students to actively listen to each other as well.

Use independent think time

When we run professional development sessions, we virtually always incorporate a section on doing the math as this increases the odds that our teachers are able to have meaningful discussions about how to teach the mathematics. If students have already thought about a problem themselves, they will find it easier to understand someone else’s approach more easily.

What are other ways we can make the mathematics in our students’ strategies explicit while simultaneously respecting the thinking that students put into these strategies?

In our project, we organized our work this past year around the use of instructional routines (née instructional activities) with teachers. Our curriculum work has been largely focused on instructional routines, our professional development activities have been focused on instructional routines, our school-based work in some cases has shifted to focus on supporting teams using instructional routines together. Our objective this last year has been to develop teachers in teaching ambitiously through the use of instructional routines that embody this kind of teaching.

Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.

Kazemi, E., Franke, M., & Lampert, M. (2009)

Instructional routines are a little bit like a standard play in sports. If everyone knows the play, then the game runs more smoothly. But just like when in sports plays have to be adjusted due to what’s going on in the game, instructional routines have to be adjusted on the fly due to the ideas that emerge from students in the classroom.

Here is an example of a task intended to be used in an instructional routine called Contemplate then Calculate.

Find the number of grey squares in the next term of this pattern.

The instructional routine Contemplate then Calculate has (roughly) these five steps:

1. Launch: The teacher launches the routine to let students know what, why, and how the class will be proceeding.
2. Noticings: The teacher flashes an images for kids and asks kids to describe what they noticed in the image, share this with a partner, and then records some noticings for the whole room to use.
3. Partner work: The teachers reshares the image with a problem task associated with it, then kids work with a partner to solve the problem given.
4. Share: Selected students share their strategy with the whole class while the teacher annotates the strategy and uses talk moves like restating and probing questions to ensure that everyone understands the ideas being presented.
5. Meta-reflection: Students write reflections based on choosing from prompts given to them by the teacher.

The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.

This year we noticed a number of benefits to using instructional routines that lead us to plan continuing using them next year as well.

1. Instructional routines allow teachers to communicate about classroom practice with each other using a common language and common understanding of what kinds of instructional strategies are being implemented.

Usually conversations about classroom practice are extremely difficult because each teacher’s context is so different and because teachers visit each other so infrequently. My experience suggests that these conversations often devolve from talking about specific decisions that were made and the rationales behind those decisions and into discussions about mathematical topics and what order they should be taught. With a common instructional routine, teachers’ conversations can shift to a more granular level of discussion since so much more of the context can be assumed.

2. Instructional routines can support teachers and students in having access to high cognitive demand tasks by reducing the cognitive demand needed to attend to “what am I doing next”. Since the activity is routine and well-defined, the steps to doing the activity can fade into the background over time for both teachers and students. This allows more of the cognitive load for teachers and students to be potentially focused on making sense of each others’ reasoning and the mathematics of the task at hand.

Shifting cognitive demand for teachers and students

Teachers already have many routines they use in their classrooms but those routines may or may not be used by other teachers (see point #1) or they may have too many different routines that they enact for each type of mathematical task they use. In order for the cognitive benefits of an instructional routine to occur, the instructional routines must in fact become routine.

3. Instructional routines have allowed our curriculum team to rapidly develop mathematical tasks to fit into these instructional routines because we don’t need to communicate the routine separately for every task. The routine stays the same (but see point #4) over time while different tasks are enacted within the routine.

4. Since an instructional routine keeps much of the classroom interaction the same, it becomes possible for individual teachers or groups of teachers to iterate on their practice more rapidly. If every day a teacher has to re-invent her practice, then it becomes more difficult to figure out what teaching strategies work in her context, when those teaching strategies work, and why she might choose a different teaching strategy.

I remember my first year teaching. I was unprepared. I didn’t know how to structure lessons. Each day I was floundering. I kept experimenting and trying different activities, different ways of communicating with students, etc… I would have benefited from more support in planning lessons.Note that this benefit supports newer and experienced teachers differently. A new teacher needs a starting place to iterate on their practice from. An experienced teacher who wants to refine her practice needs a tool with which to do so.

5. In professional development settings, we and teachers in our project can model teaching strategies more easily (this is really a combination of point #1 and #4). Since the routine is well-established, when someone does something different within the routine when modeling it with a group of teachers, it becomes easier to focus on the something different.

For example, we used the routine Contemplate then Calculate to model instructional moves intended to facilitate student discussions. We then, as a group, unpacked just that aspect of the routine. This was enabled because instead of everyone participating having to keep all of the teaching occurring in one’s head at one time, the routine aspects of the teaching could be ignored to focus on the non-routine aspects for that day.

6. The instructional routines all include built in opportunities for formative assessment and responsive teaching. We struggled to find ways to staple formative assessment practices on top of existing teaching and mostly failed. Instead the different aspects of formative assessment (as described by Dylan Wiliam) are embedded within an instructional routines, which as it turns out, makes them easier to learn how to use.

There are other more mathematical benefits of these instructional routines, but those depend largely on the specific routine being used.

We intend to continue supporting the two instructional routines we used this year (Contemplate then Calculate and Connecting Representations) and to add one or two more routines to support different mathematical goals teachers may have, because we have seen each of the benefits of these routines listed above play out in various ways across our project.

Is this curriculum?

On the day before I first started teaching, the district coordinator came to me and handed me a piece of paper with twenty questions on it. “Here’s what you have to teach, David. If your students can answer all twenty of these questions by the end of the year, you will be fine.”

Needless to say that this was insufficient curriculum. But what kind of curriculum would have been useful for me at that stage in my career?

A place many math educators, especially in our accountability-driven system, seem to start when teaching mathematics is teaching children how to solve questions. “Here’s how you solve this kind of question.” The connections between different problems are left for students to discover on their own.

Strategy: If the triangles are a different size, the transformation is a dilation.

The advantage to this system is that you can look at the state assessment and check off of all of the question types and feel like you have done your job. For the old NY state exam, this approach works in the sense that students were able to sit down at the exam and feel like they had been prepared for every question since the assessment was so predictable, even if they didn’t always know how to solve the problems .

The problem with this approach is that students now have to remember each question archetype and each solution to each type of problem separately, leading to a relatively unorganized and overwhelming set of problem-solving schema for students. This leads to students forgetting how to solve individual problems, forgetting which solution strategy they should use when, or misapplying strategies to solve problems for which the strategy is not appropriate. Even when students do master all of the problem types, knowing how to solve problem x, y, or z doesn’t help students make connections when they start studying further areas of mathematics.

A typical list of units in a math curriculum

To their credit, every textbook author I’ve ever read takes a different approach (to some degree). Textbooks start by dividing the year’s worth of mathematics to be learned into units of study and apportioning mathematical principles into those units. Within each unit of study, specific problems are used to illuminate mathematical ideas and ideally students at the end of a unit can articulate the mathematical ideas they have learned, rather than just the problems to which they apply. Where textbooks often fall down is in making the connections between units and ideas explicit.

When well done, problems become vehicles for teaching mathematical principles. Mathematical representations (like graphs, tables, etc…) are embedded across the units so that students get multiple exposures to these critical representations and can use them to make sense of similarities and differences between different mathematical ideas. Ideally students explicitly learn connections between different mathematical ideas so that they see, for example, how solving a linear equation is related to solving a quadratic equation and how the graph of an absolute function is related to the graph of a linear function.

One way to support students in making these connections is to ask them to answer reflection questions like: How is what we learned today related to what we learned yesterday? How is this problem we solved today like the problems we solved last week? What did we learn today that we can probably generalize and use to solve other problems? At the very least while planning, make sure you can answer these questions yourself.

A major disadvantage of the second approach to teaching mathematics is that it takes much more work to organize curricular resources into the general themes and to make the connections between ideas explicit. Rare is the classroom teacher who has time to do this all herself. This is one of the reasons why I think well-organized curriculum of some kind is always going to be a helpful resource.

There is a gradient between the two approaches to teaching mathematics.

For further reading on a related topic, check out this post on instrumental versus relational knowledge.

When I first started planning lessons, each lesson took ages to plan. I don’t really remember exactly what I wrote except that usually the lessons were based on choosing example problems to go over, producing a worksheet for students to work on, and assigning homework questions. Eventually I finally had some textbooks for students and so these lessons ended up being reduced to references to page numbers and questions in those textbooks.

A sample lesson plan

At some point I decided that being very economical with my planning was the mark of good teaching and so my lesson “plans” ended up being really short. “Teach Pythagoras” was an actual lesson plan I wrote. Of course, “Teach Pythagoras” is not a lesson plan. It’s not even really a topic. It’s a short-hand for pick some examples, tell kids how to solve those examples in an ad hoc fashion, followed by make up some examples for kids to try and solve themselves.

A sample project

When I started using longer-term projects, that meant planning lessons got even easier as I could assign a time-line to the completion of the project and in each day I would support students either with some examples for the whole class to move their work in the project forward or by circulating through the class to help students out.

Up until this stage, any student mathematical discussions that occurred were ad hoc and almost always initiated by students. Not once in my classroom teaching experiences did I plan for student discussion.

About three years ago, I started a new job as a formative assessment specialist. It was then that I first read Peg Smith’s Thinking Through a Lesson protocol. As I read the article, I realized right then that I had spent most of my career planning poorly.

Next I read Dan Willingham’s book on Why Don’t Students Like School? and realized that I had spent most of my career planning tasks for students to do and not planning the thinking I wanted students to do. Tasks prompt thinking but what thinking? Here’s an example of a lesson that could have been one my early lessons. What are children being asked to think about?

A sample anticipation of student ideas

Fortunately, I had an opportunity to test out these new ideas around planning for myself. I started teaching my son and between 6 and 8 other kids close to his age in a Saturday class. I decided to plan the student thinking, to try out the Thinking Through a Lesson Protocol, and most importantly, to ask other people to comment on my plans. Here’s my first lesson plan for this class. It’s by no means perfect but it has far more detail on what I will do in response to what I expect students to do, and the first example I can recall of a lesson where I explicitly planned a whole group discussion.

But who has time to plan lessons at this level of detail five days a week, potentially 4 or 5 times each day? The level of planning linked above is unsustainable for classroom teachers.

A sample task for Contemplate then Calculate

Last summer we started introducing instructional activities to teachers across our project. Instructional activities, as defined by Magdalene Lampert and Filippo Graziani, are “designs for interaction that organize classroom instruction”. Essentially they define a set of moves a teacher makes to position students to talk to each other about mathematical ideas, surface student thinking about those mathematical ideas, and then orchestrate a classroom discussion around the ideas in order to focus students on a mathematical goal.

These instructional activities have the advantage of bounding the scope of decisions a teacher can potentially make when planning a lesson while focusing the decisions that are made ahead of the lesson on planning for the thinking students will do, and then in the lesson enactment, allowing teachers and students the space to think about and respond to each others’ thinking. The routineness of an instructional activity, if the same structure is used many times, allows thinking about roles, what’s coming up next, to fade into the background so that more thinking can be focused on the mathematical ideas.

When planning one of these instructional activities, I find myself choosing an appropriate task based on some understanding of anticipated student thinking, then imagining how students might approach the task and what they will think about, then considering how to sequence the different strategies student might use toward a big mathematical idea, and then creating the resources to enable me to use the instructional activity in the classroom. This level of planning is sustainable.

As I reflect on my own development of planning over the course of my career, it seems to me that I would have benefited from knowing about planning routines that other people use. I would have benefited from learning at least a few instructional activities so that I didn’t need to plan every aspect of my teaching. I would have benefited from access to tasks where student thinking was anticipated for me.  I don’t think a highly scripted curriculum would have developed me as a professional (but maybe my students would have benefited). I would have benefited from seeing how other people sequenced mathematical ideas.

In short, I would have benefited from more explicit teaching of how to plan lessons.

A recent analysis by David Blazar has stirred up some interest about Ambitious Teaching. But what exactly is Ambitious Teaching?

According to Elham Kazemi, Megan Franke, and Magdalene Lampert:

“Ambitious teaching requires that teachers teach in response to what students do as they engage in problem solving performances, all while holding students accountable to learning goals that include procedural fluency, strategic competence, adaptive reasoning, and productive dispositions.”

While this defines some aspects of Ambitious Teaching, this is not a sufficient definition such that any two observers would agree that an episode teaching observed was Ambitious.

In this same paper, E. Kazemi et al. go on to talk about instructional activities they are teaching to pre-service teachers which:

In his recent analysis, D. Blazar uses a research instrument which he separates into two aspects, one of which called “Ambitious Mathematics Instruction” positively correlates the following teaching practices with student performance on an assessment given to those students.

From this we can infer that these labels describe teaching practices which are part of Ambitious Teaching. We can generalize these practices to other content areas, although the practices will look different in those different content areas.

Dylan Kane unpacks one aspect of Ambitious Teaching, teaching to big ideas, which is about making connections between different ideas rather than treating each lesson as being in isolation from each other lesson.

Photo of me leading a whole class discussion while annotating student strategies.

One key element that distinguishes Ambitious Teaching from other teaching is that ideas that emerge in class are built up and extended directly from student thinking rather than the converse. This one element leads to added layer of complexity to teaching as suddenly all of the ideas in the classroom have to be surfaced and some subset of those ideas needs to be selected to talk about in public. Authentically listening to children’s ideas while leading a class with a specific objective and trying to move all of the students within that class toward that objective leads to complexity that most teaching does not contain — and so we call the teaching that requires this at its core to be Ambitious. The name of this kind of teaching is not a rhetorical device, it is a reminder that learning to teach is challenging.

Another added layer of complexity is that when ideas are shared or presented to the whole group, not only does a teacher need to be conscious of how their decisions in this theater impact every child, they need to make sure that whatever idea that is being discussed is clear for every child. In a mathematics classroom this can be done through use of thoughtful representation, asking clarifying questions, asking students to listen to and restate each others’ ideas, adding on explanations to the student explanations, etc… but all of these individual practices have the same goal — ensure that everyone understands what is being discussed while treating students as sense-makers.

Based on this photo (which has a portion of the strategy cut-off on the right hand side), what mathematical idea is being discussed here? How do you know?

In order to make better choices about what problems and what representations of those problems to use, teachers need to develop specialized expertise about how students understand, and often misunderstand, their subject area in order to surface those important understandings and support students in transitioning to other understandings.

What mathematical ideas are being connected here? Why might that be helpful? What’s missing in these representations?

Now to be clear, I cannot easily describe Ambitious Teaching in a single blog post. Magdalene Lampert wrote an entire book about it and she only described this kind of teaching in the context of a single 5th grade mathematics classroom. All I can hope to do is to generate questions about teaching and to hopefully suggest that there is greater complexity to even typical teaching than the selection of the three worked examples for the day.

This entire short talk by Richard Feynman is worth watching, but if you look at his point at about 5 minutes into the video, you’ll hopefully understand my next point better.

[youtube id=”EYPapE-3FRw” parameters=”rel=0 start=300″]

Most of the time when we talk about teaching, we do in such vague terms that our conversations slide past each other. This includes educational research which claim to show that this method of teaching is better than this other method of teaching without carefully defining what the methods being compared actually are and what the goals of either particular style of teaching might be. One goal of my work this year is to make certain teaching practices explicit for teachers in our project so that when we talk about teaching we can be reasonably sure that everyone understands the decisions being made and why someone might make one decision or another based on the objectives of the teaching for the day.

Nothing you can do can guarantee that every child actively participates in your math class but there are some things you can do to increase the odds.

Source: Wikipedia

In a typical classroom a teacher asks a question, a student responds, the teacher indicates whether the response is incorrect or correct, and this is repeated until the class discussion is over. If this is the only type of interaction a teacher has with their students, this can be problematic.

1. It’s very difficult to keep track of who has participated and who has not which often leads to skewed participation in the conversation.
2. It’s impossible to know how each child understands the mathematical ideas being discussed.Here’s an example Dylan Wiliam uses to illustrate this point: If a teacher asks four questions in a row of four different children and the responses to the first three questions from children are incorrect but the fourth is correct, the only thing that can be concluded is that the fourth child was able to answer that question but what is typically concluded is that the class understands and so the teacher moves on. Unfortunately I myself am guilty of this many, many times. All too often we stop questioning when we get a correct response.

Fisher and Frey (2007) say the following on this matter:

“I’ll ask the questions,
a few of you will answer
for the entire class,
and we’ll all pretend
this is the same thing as learning.”

But what can we do differently? How can we minimize this kind of interaction? Note that it is probably not possible to completely eliminate these types of interactions, nor would it necessarily be advisable to do so, but there are some ways to reduce their impact on a classroom.

Turn and talk:

Prompt students to talk to each other about a question while you circulate and listen for responses (and hold students accountable for talking about what you want them to talk about). Come back to the whole class and ask a few people to share responses. This has the benefit of increasing the chance every student shares an idea with at least one person and increases the amount of time students have to process and think about the question.

Wait time:

The average amount of time teachers wait for a response after asking a question is less than one second. Just delaying this to five seconds (or even more sometimes) increases the amount of students who have time to process the question, think about it, and then formulate a response.

Whiteboards:

Instead of posing a question and waiting for one response, pose a question and ask students to, either individually or with a small group, construct a response that they can share on their whiteboards. This way you can walk around and see what ideas students have and every student has an opportunity to take their time to think about the question.

Use student generated questions:

There is some evidence that students do better when they own their learning. Having students generate questions they have and allowing students to answer each other’s questions (perhaps initially mediated by you) or you to answer questions they have, gives students more control over their own learning. Of course, students are often not used to this kind of interaction and will likely need support of some kind in generating questions.

Start with the problem:

It is extremely common to start class with a somewhat interactive description of how to solve a problem, give students opportunity for guided practice with lots of feedback, followed by independent practice, followed by a summary by the teacher or a student of what was learned that day. In this case, any questions that are asked within the context of the entire class can only really happen either during the initial describing of how to solve the problem at the beginning of the class or during the summary at the end.

An alternative to this approach is to start with a problem where you can be reasonably sure all students understand what the problem is and what they are trying to find out. Now when you initiate any full group discussion, students are much more likely to have questions and the answers to these questions are much more likely to be understood by everyone.

Note that I’m not a fan of cold calling students. With some students, having a teacher who cold calls raises their anxiety, making it more difficult for them to think about the mathematics. It might be a useful technique once you know every student feels comfortable and has something to say.

What other strategies can you use to encourage participation (even non-verbal participation) in your classroom?

Why should students talk to each other in math class anyway? I was asked this question recently and I’m trying to avoid a tautological answer (eg. it’s important because it’s important).

In a classroom where students speak to each other about mathematics, the ideas of those students are valued instead of ignored or potentially marginalized. This gives students agency in their learning. It also allows new ideas the students learn to extend from the existing ways they understand the world.

Supporting students speaking to each other means that mathematics is much more likely to become a way of knowing and being rather than just a body of existing knowledge (although the value of mathematics as a set of tools that have been developed over time should not be marginalized). As students develop their understanding of what mathematics is and what it is useful for, they are more likely to insert themselves in the role of the mathematician rather than imagining this to be someone else, potentially from another culture. They can see themselves being part of a mathematical community.

In order to completely understand the language we know, we have to use it, either in writing or ideally in conversation, and hear other people using the same language. So from a practical perspective, students need to talk in order to develop their use of language (mathematical or otherwise), and rather than students talking in serial, one at a time mediated through a teacher, it is far more efficient for them to talk in parallel, to each other.

We remember what we think about. When students construct ideas and communicate them to each other, they necessarily have to think about these ideas, which means that they are building memories. While this occurs no matter what students do, the focus is more likely to be on the thinking with student discourse rather than the activity (eg. completing a task).

Finally, students talking and writing to each other also provides their teachers with more information about the ways they are thinking which makes it easier for the teacher to orchestrate productive whole group discussions and to plan activities that respond to the ways students are actually thinking. It is difficult to plan lessons that build off of student knowing if you don’t know how and why students think the way that they do. When students talk to each other, their teacher can gather formative assessment information about not only what they understand but ideally how they understand it.

This should not diminish the importance of students having independent time to work quietly on mathematical problems by themselves. Students are better positioned to work together when they have had time to think about ideas themselves first. Also, some students find working with other students really difficult for a variety of reasons, so in some cases the benefits of students working together may be outweighed by the challenges some students face with this activity.

What would you add as reasons students should talk to each other in a mathematics classroom?

Imagine four teachers each of whom teaches in different schools in a different context. Even if they all teach the same course, their individual teaching looks different.

If these four people come to talk together, they will find it challenging to have a conversation since the way they are teaching is so different from each other. Each person potentially has a valuable perspective but they may have so little in common that it is difficult for them to talk to each other.

Now imagine instead these four teachers are all focused on working on the same instructional activity, perhaps even with the same mathematical tasks within the activity. Their teaching is still different as they still teach in different schools in different contexts but their conversation about teaching becomes much more coherent as they have far more in common to talk about. Instead of talking about their individual teaching they can talk about teaching practice.

Over the last two weeks we launched the instructional activity, Contemplate then Calculate with 100 or so teachers for this exact purpose; to make our shared conversations about teaching focused and coherent. After two days of professional development, which included rehearsing the instructional activity together, virtually all of the teachers indicated that they were excited to try out this activity and then come back together in October to talk about its impact on students.

At one point during the two weeks, some participants and I ended up in a whole-group discussion about when exactly we should annotate our students’ strategy sharing. It was the most specific conversation I’ve ever had with a group of teachers and I feel fairly certain everyone understood the point that was being debated and why it might matter one way or another.

All of these teachers are going to go back to their individual schools and teach according to what they know with their individual contexts and their individual students but now, hopefully, at least one aspect of their teaching will be similar enough that they can come back and talk about the differences.