There are two common activities teachers do that have either little to no impact on student learning but which do take teachers a tremendous amount of time, time that could be better spent on other activities.

Grading Student Work

There’s limited evidence that putting marks on students’ papers leads to students either being more motivated to work harder or that these marks leads to increased student learning. In fact, some of the literature on feedback suggests exactly the opposite is true:

When teachers pair grades with comments, common sense would tell us that this is a richer form of feedback. But our work in schools has shown us that most students focus entirely on the grade and fail to read or process teacher comments. Anyone who has been a teacher knows how many hours of work it takes to provide meaningful comments. That most students virtually ignore that painstaking correction, advice, and praise is one of public education’s best-kept secrets.

Source: Dylan Wiliam

But grading student work is extremely time-consuming and so if this effort doesn’t lead to student learning, why do we do it?

• To communicate progress to children and their parents,
• To evaluate students,
• We are expected to grade student work.

In a recent parent-student-teacher interview, the teacher had samples of my son’s work in front of him. He shared directly what he liked about his work and where he thought my son could improve and we never talked about the numbers at the top of the paper at all. It is easier to communicate progress using artifacts of student learning.

Schools, in the interest of focusing on activities which have clear connections to student learning, should stop grading students and focus on time-efficient ways to solve the problems grading is intended to solve. Should teachers still look at student work? Definitely, but this should be as part of their process of planning future lessons and thinking about opportunities for feedback for students.

"When you grade, you help one child at a time. When you plan, you help all kids. Spend your time accordingly." ~ @hpicciotto #NCTMregionals

— David Wees (@davidwees) October 28, 2016

Creating Our Own Curriculum Resources

I spent years writing tasks for each of my classes, borrowing from other teachers when I could, but mostly making my own resources from scratch. These resources were not of much higher quality than what I could get from a textbook but I understood how they were designed and how I intended to use the resources. Writing curriculum took me SO much time.

Now I write curriculum about 50% of the time for my day-job and can see that the curriculum I currently create is far superior to the curriculum I used to create and far more complete and coherent. But it probably suffers to some degree from the same problem that led me to create my own curriculum as a teacher — it is extremely difficult to make sense of someone else’s lesson or task.

One feature of the curriculum I write that I think helps mitigate this problem of understanding curriculum is that it contains instructional routines. Once one knows a particular instructional routine, the task of understanding tasks to go along with that routine is far easier. Teachers who know the Connecting Representations instructional routine can look at tasks and are better able to make decisions about which tasks to use and why.

Evidence on use of curriculum suggests that all teachers benefit from access to high quality curriculum resources. In this experimental study some teachers were given access to Mathalicious  and others were not, and the teachers who had access saw better performance from their students.

What I think teachers should have more time to do is modifying and adapting curriculum for their particular context and their particular students. This is why our curriculum is licensed with an open license and why we share the curriculum in Google Document format — it makes it much easier for teachers to adapt and modify the curriculum rather than having to recreate a document from scratch just because the original is locked in PDF format.

We also wrote our curriculum to be largely sequenced but with lots of opportunities for teachers to make choices within the curriculum or to design their own tasks. Our tasks are aligned to the evidence of understanding we expect to see in students, which means the blueprints for constructing their own tasks are available to teachers as a support – leading to the best of both worlds for teachers: the autonomy to construct their own curriculum while not needing to reinvent the wheel each day.

Teaching Better

If teachers stopped grading all student work and writing all their own curriculum from scratch, then they would have more time for other tasks that contribute more to student learning such as:

• Designing responses to evidence of student achievement (eg. formative assessment),
• Collaborating with colleagues to investigate instructional strategies (eg. micro-teaching)
• Work with individual or groups of students to support their learning.

 
What else could teachers do with their time to support student learning if all of a sudden they had more time available?
 

A number of times over the past few years, I have been confronted by a phenomena I do not understand: children being handed workbooks and, with very little to no instruction, expected to complete the workbooks and develop an understanding of the material contained within.

For some children this works out fine. The workbooks, mostly being designed for independence anyway, are sufficient for children who have some background in the material and see the patterns developed within the workbooks. For many children however, the experience is potentially one of frustration and humiliation: they have been given a task that their teacher clearly expects them to be able to do, and they cannot.

I have seen workbooks handed out with minimal to no instruction in two very different kinds of circumstances. In one case, the teacher sees a class with a diverse needs and although all of the students had the same overall goals for the class, students clearly have different levels of preparation and interest in the course. The other case is when teachers have a multi-age classroom (which also contains students with different levels of preparation and interest) and no time to actually teach anything at all. Both of these are compounded by the preparation of the teacher leading the classroom to deal with varied needs.

I have been in the first case, leading a group of students with varied needs but with the same goal, many times during my career. The best solution I have learned is the use of instructional routines with designs for interaction to support a varied set of needs, while working together towards the same mathematical goal.

As for the second case, when teachers have a multi-age classroom, I don’t know for sure what I would do. If I found myself teaching in such a classroom,  I think I would do two things; teach mini-lessons (using a variety of formats depending on the mathematical goals, but again instructional routines seem like a good fit) and assign problems from a somewhat independent workbook, and then while some students are working in the workbook, engage the other students in a mini-lesson on a different topic. I would have to organize the space so that students who were working on similar material sat together and I would need to develop classroom culture so that students knew to talk to each other after first trying the math themselves and to wait until I was available to assist. I would also look for places in the mathematical goals for the year where potential alignment between different grade levels occurs and use these opportunities to engage the entire classroom. For some students the activity might be review and for other students the activity might be somewhat new material, but both of these seem like helpful ways to use classroom time.

Imagine you are teaching a classroom with 20 students roughly evenly distributed from grade 2 through 7 (age 7 through 13). What would you do to meet the needs of these learners?

According to Adding It Up  (H/T Dan Meyer):

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized.

This image represents someone who understands some isolated facts.

And this image represents what conceptual understanding looks like:

I propose that those line segments between the points, representing the connections between isolated knowledge, are themselves a type of knowledge, consequently our second image is really more like the first image.

What is clearly missing in this analogy is that conceptual understanding is knowledge that is well-organized, but I claim that organization of knowledge is itself another type of knowledge. Also missing is knowing why the knowledge is important and the contexts in which an idea are useful, but again these also seem just like other types of knowledge.

Let’s take a look at a problem I had to solve recently and see if we can use it to unpack what conceptual understanding is.

According to the solution key, the writers of this question are assessing students’ ability to solve quadratic equations.  However, that likely requires applying the quadratic equation to all four equations given, which would be time-consuming and potentially error-prone.

When I solved this problem, I saw four quadratic functions, each transformed from y = x², and based on those transformations, I was able to eliminate (1) and (3) as options since they have real solutions. I also know that if 1 – i is a solution, then so is 1 + i, and that the sum of the roots of a quadratic function are related to the coefficient of x in the equation. I could not remember if the sum of the roots is equal to the coefficient of x or to -1 times the coefficient of x, so I multiplied out (x – a)(x – b) to double-check and determined that if a and b are the roots of a quadratic equation, then -(a + b) is the coefficient of x in that equation (assuming a lead coefficient in the equation of 1). 1 – i  + 1 + i = 2 so (4) must be the answer.

My actual solution strategy doesn’t matter all that much, what matters is that in order to produce it, I needed to know some smaller things and that those smaller things could be seen as connected together to create a larger idea. Instead of seeing this as the difference between an expert and a novice, I prefer to think of knowing things as being a gradient between knowing a little bit and knowing more than a little bit.

Conceptual understanding is both knowing ideas in an interconnected and organized way and knowing more ideas along the novice-expert spectrum.

The reason why conceptual understanding is an important goal is because otherwise we might be tempted to rely on teaching kids tricks instead of mathematics.

What mathematics does a child learn from learning this procedure for adding fractions?

Source: Pinterest

Very little! It is extremely difficult for a child to connect this procedure to other mathematics that they know, consequently using a trick like the butterfly method is likely to lead to students knowing an isolated idea and not much else. Note that I am not opposed to memory aids, I just think they should be based on solid mathematical reasoning and they definitely should not be taught instead of the mathematics.

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.

***

There are many cultures of mathematics, a question is, which one dominates k-12 education and why?

The majority of those two cultures are likely aligned to one of these two possibly overlapping goals of teaching mathematics.

1. To enable students to be able to solve a specific set of mathematical problems.
2. To teach students a body of mathematics.

If you look at what is assessed in mathematics, the majority of assessment focuses on students being able to solve specific mathematical problems. Here are some examples of problems that New York State feels are so important for students to be able to solve that they gave these questions to the tens of thousands of student who sat for the June 2018 Algebra I Regents examination.

I do not know if it is possible to assess students on their understanding of mathematical principles without giving them some type of problem to solve, but I have never seen an assessment that did not involve solving mathematical problems, so it is either systemic in the way mathematics is assessed or it is impossible to assess mathematical principles apart from problems they can applied to solve.

I have occasionally seen teaching that focused on students building understanding of mathematical principles rather than attempting to solve specific problems, but even in this case the basis of the activity could be considered a mathematical puzzle or problem. However, almost every time I have observed a class the focus is on students being able to solve a fairly specific set of problems with little to no opportunity to generalize from the experience.

According to Tim Gowers, in his essay Two Cultures of Mathematics, these two same cultures apply in the world of mathematicians as well, except the culture that dominates is reversed; most mathematicians focus on learning more mathematical principles and eschew pure problem solving as a goal.

As Sir Michael Atiyah (Atiyah, 1984) puts it:

MINIO: How do you select a problem to study?

ATIYAH: I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.

So if mathematicians value mathematics as a set of organizing ideas and mostly prefer to study the organizing ideas rather than the problems they can be used to solve, then why does almost all of k-12 mathematics focus primarily on problems to be solved and much less so on general mathematical principles?

Maybe the problem lies with the standards that kindergarten to grade 12 teachers are expected to focus on? Let’s take a look at the Common Core clusters each of the questions listed above is supposed to correspond, which are A-REI.B, F-IF.A, A-APR.A, and A-SSE.B respectively.

The text of those clusters is as follows:

• A-REI.B: Solve equations and inequalities in one variable.
• F-IF.A: Understand the concept of a function and use function notation.
• A-APR.A: Perform arithmetic operations on polynomials.
• A-SSE.B: Write expressions in equivalent forms to solve problems.

It certainly seems like 3 out of 4 of those are focused on students being able to use what they know to solve specific kinds of problems with verbs like solve, perform, and write. So the fault could lie with how standards are written for courses.

Let’s contrast these standards with the objectives from one of the courses Tim Gowers teaches to see how they compare:

One obvious difference between the standards and Gowers’ syllabus is the inclusion of verbs in the standards, which besides being the intended foci of a high school algebra course are also intended to be a map for people writing assessments, and you cannot write an assessment without specifying what students will do in order to demonstrate understanding. Gowers’ syllabus is just a list of topics, but I note that I have no idea based on his syllabus what kinds of problems these topics would be useful for and maybe the absence of verbs is why.

There is a way of teaching that breaks this dichotomy between solving problems and learning mathematical principles and uses problems to teach mathematical principles but this style of teaching seems incredibly rare and is somewhat difficult to learn and somewhat difficult to do. The best tool I have found for learning this kind of teaching is through the use of instructional routines, particularly an instructional routine like Connecting Representations that focuses on students making connections between two different representations of the same mathematica principle.

I’m not really sure why this difference between the work of mathematicians and the goals of k-12 mathematics teachers exists. One possibility is that it’s a lot easier to see how mathematics can be applied to solving problems than it is to see the mathematical principles being used and very few k-12 mathematics teachers are as mathematicians.

References:

1. T. Gowers, The Two Cultures of Mathematics, Mathematics: Frontiers and Perspectives (2000)
2. F. Atiyah, An interview with Michael Atiyah, Math. Intelligencer 6 (1984), 9-19.

I received this email from a teacher I know and with their permission, I am posting it and my response here. Identifying information has been redacted from these emails.

So I tutor this Junior from Stuyvesant in Algebra 2. But her text book is College Algebra & Trigonometry.

Her parents feel the materials I bring are entry level or at level at best, but fall short of the Sty expectation. I mentioned to you before she’s doing limits & pre-cal/cal topics as an 11th grader & not honors. She got a 60 on the last exam because she miscalculated a few limits & recurring series.

When I asked the parents what can I do if they feel my supplements are minimal skills at best, they told me the teacher said they are getting their juniors college ready, not only in practices but content. By the end of the Junior year, they should be proficient in material covered in the SATs, which includes some Cal & Discrete Math.

When I said but she’s rocking all the Regents material, her father replied, that’s one test, what about what comes after?

So are we short changing our kids at [school redacted]? Have we lowered the bar/expectation from being jaded by the system & have given up on the challenge of keep them learning? In my own practice, I have strong students & I felt like this semester I may have served them a disservice by not teaching basic limits & next level mathematics as an extension.

I don’t need a response, just venting on a reflection. But to have that face to face with a parent who has specifically asked me to raise my level of academic material sucked some life out of me. Are these conversations lost in our own population with parents who struggle to get their kids to 1) show up then 2) complete the minimum assignments & work.

I’m not sure where to go from here…..

For reference, Stuyvesant is one of the highly selective public schools of New York City which currently selects students based on academic achievement on an entrance exam. The school my friend works at is not one of the selective schools.

Here is my response:

Hi [Redacted],

This is a powerful and tough reflection on your practice.

One of the reasons we started the a2i project was to improve mathematics instruction in NYC and one element of mathematics we hoped to improve was the quality of the mathematics content to which students were exposed. That’s likely part of the reason we used the SVMI materials as a starting place and why we are in the middle of writing our own curriculum; because we aren’t super happy with what’s out there and available for our schools.

On the other hand, it seems clear that if you have a bunch of kids who are already struggling, it is counter-intuitive that what you want to do is raise the bar, so we introduced instructional routines with the goal of supporting classroom instruction, and in particular, teaching teachers instructional components they can include in their teaching which increase access to rigorous and challenging mathematics for all students, consequently allowing teachers to raise the bar.

In your situation, you’ve noticed the contrast between two Algebra II courses from two different schools. This contrast is part of the reason that parents and students work so hard to get into the selective schools. However, this contrast is a result of the system in which you work and not so much because of your own personal fault.

Kids select Sty because they want to be challenged. Kids choose a neighborhood school sometimes because they don’t want school to take up too much of their time. At Sty, kids are engaged in academic discourse and push and support each other to handle content that might otherwise be too challenging for them. At your school, much of this peer support may not be in place, and so as a community, your learners are not as able to handle more challenging materials.

Of course, this inequity of experience isn’t really fair. How can we help all schools develop the kind of academic community that Sty and the other selective schools have that push kids to be better than they are?

I wrote this post a while back and it may be relevant. Basically it can be summarized to, kids are more likely to have higher expectations for themselves and meet those expectations when the adults that work with them have high expectations. But expectations are part of a system-wide bias that exists that is hard to look outside of and even harder to change. Even if you turn around tomorrow and start trying to tackle more challenging material with your students, they’ll still be students who spent the previous 10 years not having to work or think as hard.

David

Here is their response to my email.

I appreciate the response. As I explained the situation & context to my [partner], I’m still torn & feel responsible for “creating” a finite course that terminates with a Regents rather than raise the bar & teach “mathematics” to where the regents is absorbed along the way. With 32 instructional days left, I’m curious to see how I can elevate my current practices to which I can start higher next September.

What other advice or support would you offer this teacher?

I know of people who are proud that they do not use a textbook and that they eschew all formal curriculum resources. I used to be one of those people but no longer.

If we define curriculum broadly as a collection of physical and digital resources that are used to support teachers with students in their classrooms, then every teacher has curriculum. The quality and quantity of that curriculum just vary.

A collection of resources found via Pinterest

The primary problem with a lack of access to curriculum is that every teacher in this situation is then left to invent their own resources to use with students. While I think many teachers are capable of doing this, almost no teachers actually have enough time to create really high quality resources for every lesson. I have been working on a set of interleaved, spaced, retrieval practice assignments aligned to our Algebra I curriculum and after a dozen or so days working on these assignments, I am about half-way done. These resources are for one small part of a collection of resources intended to support students across a year of Algebra I and are by no means perfect. How long do you think it would take a teacher to create all of the resources necessary to teach Algebra I? And why do we expect thousands of people who teach Algebra I to do so much duplicate work?

Further, almost all resources made benefit from additional eyes looking at them. About half the time when I share a resource via Twitter, someone finds some way of improving that resource. Here is an example of me sharing a collection of resources via Twitter and asking for feedback.

Teaching a quadratic functions unit in Algebra I? We have a complete unit available here: https://t.co/tUQgncyWlQ #alg1chat #MTBoS

We'd love feedback if you try out at least some of our resources. And please feel free to ask any questions you have. pic.twitter.com/LscbzlAmHl

— David Wees (@davidwees) February 1, 2018

A few people who have used these resources have offered suggestions or found minor errors and we use that information to iterate on and improve the original collection of resources. If you can imagine this effort scaled up so that thousands of teachers are each iterating on and improving the same original set of curriculum resources, very quickly the diversity and quality of those resources would far outstrip what any individual teacher could create.

Here is an open-source content management system that has 23362 modules and 1642 themes each one representing many dozens of hours of work from individuals. As a collection, this project represents millions of hours of effort devoted to one project with the fruit of that labour available for free anyone who wants it. Where is the similar effort for curriculum?

Illustrative Mathematics and New Visions for Public Schools are creating curriculum licensed under a Creative Commons license but neither yet has a good mechanism that allows sharing of modifications of curriculum back to the greater community. I’m not even sure exactly what they would look like.

If you were designing a system to allow users to build curriculum collaboratively in the same way the open source software movement allows for thousands of people to collaborate on software, what would it look like? What would you want it to be able to do?

Here are some thoughts I have so far:

1. It would be nice if formatting of the resources was a consideration of the technology. We have our resources created in Google Docs, which allows for easy formatting and sharing but Google Docs is proprietary and given Google’s tendency to turn off services, even popular services, this could be problematic.
2. People need to be able to easily create their own copies of resources (or even branches of curriculum) and share them back to the community and these revisions should be easily visible for people looking at a particular resource. Benjamin suggests some additional detail around this idea here.
3. People should be able to comment on resources, either to share their experiences using a particular resource or to suggest modifications.
4. It would be nice if resources could have additional or supplemental resources added to them, like videos of a resource being used in a classroom or pictures of student work. Obviously this raises issues around student privacy which suggests that this community would need some agreed on rules of how student work is anonymized or scrubbed of identifying student information.

I’m part of the design team for Chalkbeat’s Great American Teach-Off and I’ll be coaching one of the pairs of math teachers.

From Chalkbeat:

The event, to be held in March at the SXSW EDU conference in Austin, Texas, will build on live-format shows that celebrate the hidden craftsmanship in other professions — think Top Chef, Project Runway, and The Voice — minus the competition. You can read more about the Teach-Off here.

The goal of this event is to highlight teaching as an intellectual activity and to make visible the invisible decisions that teachers constantly make when they teach.

If you wanted insight into teaching decision-making, who would choose? Which of these pairs of teachers would you like to learn more about their teaching?

Check out these really reflective teachers and help decide who will get their decision-making made visible for the world!

Last year, I created a prototype of a tool that takes Google Documents linked from a spreadsheet and merges them together. During the summer, Frandy and Erik from our Data and Systems team along with some other members of the Cloudlab team at New Visions for Public Schools upgraded the tool into a Google Sheets Add-on. We gave it the name Quiz Banker.

The goal of this work was to take a repetitive task that almost all NY State public school math teachers do, which is to merge and typeset items from Regents exams, and greatly reduce how long this task takes, thus saving teachers time to do other more important tasks. We can easily typeset Regents questions centrally at a fairly low cost, and then a tool like Quiz Banker makes it easier for teachers to work with those typeset questions.

During the summer we asked teachers how long it would normally take them to take all of the Regents questions associated with a particular domain of mathematics and typeset them into the same document. Answers from teachers ranged from 5 minutes to 8 hours with most teachers estimating about an hour. When we demoed Quiz Banker, it took 2 minutes to accomplish this task, including the time spent installing the add-on.

During the summer suggestion I told teachers, “If it used to take you 40 minutes to create a quiz and now it takes 2 minutes, use those 38 minutes you saved to make sure that quiz assesses what you want it to assess.” As Patrick Honner notes frequently on his blog, not every Regents question is of equal quality.

Having a question typeset also means you can easily modify a multiple choice question into a more open-ended question, modify the language of a question to get a slightly different mathematical idea, or just increase the font-size so that students with differences in visual processing are able to read the question.

QuizBanker also includes meta-data like what Common Core Domain, Cluster, and Standard to which each question is aligned as well as alignment to the Units and Big Ideas in the New Visions’ Math Curriculum. This further reduces how long it takes teachers to aggregate those questions usefully.

Quickly filtering for question type

More broadly I believe that if teachers are going to work on changing their teaching, this takes time, but time is a fixed quantity. The cheapest way to give teachers more time to work on improving their teaching is to take repetitive and time-consuming tasks they do and change the amount of time spent on these tasks either by eliminating those tasks altogether or by reducing how long the task takes to do.

What other tasks do you see math teachers doing frequently that could save time if there was a tool that made that task easier and faster to do?

When someone is learning a new practice, it is common to isolate that practice from other elements of the greater body of work they are also learning. For some areas of learning, this is easier to do than others. For example when learning how to play the piano, one can reasonably easily practice scales and parts of songs and then integrate those parts into the whole.

In learning teaching however, since every practice is connected to other teaching practices, it can be extremely challenging and potentially unhelpful to isolate individual teaching practices. For example, you cannot really get better at the 5 Practices (summary, book) without considering how those 5 Practices interact with each other. If you anticipate student ideas for an upcoming lesson, you will only get feedback on that anticipation if you also monitor what students do.

One strategy to reduce the complexity of learning to teach is to approximate teaching practice in various ways. Instead of teaching a whole class of students, one can teach at one table. Instead of teaching five classes of students a day, one can teach one class. Instead of teaching on one’s own, one can co-teach with a mentor teacher.

Another approximation of teaching we have found helpful in our work is the use of an instructional rehearsal, which is where one teacher (or perhaps a pair of co-teachers) leads the group in a teaching experience with everyone else playing the role of students. At either strategic instances or on request of the teacher(s) leading the experience, the action stops and everyone considers teaching practice either together or in small groups.

Rehearsals: A common practice in many disciplines

It is helpful for one person to act as the facilitator or coach for this experience, and for the rest of the participants to switch between playing the role of students while the teaching experience is in action, and to discuss the teaching as teachers when it is not. If each rehearsal has a different focus, then one can learn different elements of teaching over time, while still maintaining the complexity of teaching. The goal is for the core practices of teaching to become integrated rather than overly isolated. The Teacher Education by Design website has more details and resources for instructional rehearsals here.

A further design element of instructional rehearsals is that the activity to be rehearsed should be fairly clear for participants. We use instructional routines as the frame for our rehearsals because they constraint the scope of potential decisions to be made and subsequently discussed but are still complex enough examples of teaching to allow for different foci or teaching practices to be discussed in different rehearsals. We typically model an instructional routine a few times for teachers, unpack it collaboratively, then teachers plan around a task for the instructional routine, and then rehearse the instructional routine one or two times as a whole group.

Rehearsals can be places to discuss planning processes and protocols that might be necessary pre-steps to improve the enactment of a performative teaching practice. For example, while considering how to annotate a student strategy during a rehearsal, participants will likely realize that practicing different annotation strategies in advance of a lesson would be helpful and that in order to do this, one should first anticipate the student strategies that are likely to emerge for a particular.

We have found rehearsals to be helpful for teachers at all stages of their career, since all teachers have room to grow and to learn. The foci of the rehearsals for pre-service, early career, mid-career, and late career teachers may be different but the overall process is the same.

One other key idea of rehearsals: the goal is rarely to give the teacher leading the rehearsal feedback although that often happens but to collaborate together to consider teaching. The goal is to collectively improve teaching practice not individual teachers.

Rehearsal is not a replacement for working with a mentor teacher over time to learn ways to communicate with parents and other critical aspects of the role of a teacher. Some elements of teaching practice are hard or potentially impossible to rehearse. However the performance aspect of teaching is where most teachers will spend at least half of their time, and rehearsals are a good strategy for developing performative teaching practice.

References:

Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 12-30).

Knowing Teaching from the Inside Out: Implications of Inquiry in Practice for Teacher Education. (1999). In G. A. Griffin (Ed.), The education of teachers (pp. 167-184). Chicago, IL: NSSE.

Lampert, M. (1990). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29-63. doi:10.3102/00028312027001029

Lampert, M. (2009). Learning Teaching in, from, and for Practice: What Do We Mean? Journal of Teacher Education, 61(1-2), 21-34. doi:10.1177/0022487109347321

Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., . . . Crowe, K. (2013). Keeping It Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching. Journal of Teacher Education, 64(3), 226-243. doi:10.1177/0022487112473837

Lampert, M., & Graziani, F. (2009). Instructional Activities as a Tool for Teachers’ and Teacher Educators’ Learning. The Elementary School Journal, 109(5), 491-509. doi:10.1086/596998

Mcdonald, M., Kazemi, E., & Kavanagh, S. S. (2013). Core Practices and Pedagogies of Teacher Education: A Call for a Common Language and Collective Activity. Journal of Teacher Education, 64(5), 378-386. doi:10.1177/0022487113493807