The Reflective Educator

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Open Source Curriculum

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.


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.


The Great American Teach-Off

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!


Quiz Banker

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?


Approximating Teaching Practice

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.




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


How to use technology with only one computer

A very common situation in many classrooms is that there is only one computer and it is usually attached to a projector. How can one meaningfully use technology under these circumstances?


An example of an interactive tool


Here is a strategy that may help when you want students to use an interactive tool but either have limited access to devices or do not want to waste a bunch of classroom time handing out devices.


1. Introduce the goal of using the technology to students.

We introduce the goal first so that students have some sense of what they are trying to accomplish. The goal can be somewhat vague so that it doesn’t take any of the magic out of the lesson, but ideally upon reflection students should be able to see either how they reached or did not reach the goal.


2. Have a pair of students come up to use the tool and demonstrate in front of the class.

The pair of students will problem solve with direct access to the computer. They can manipulate sliders, drag things around, etc… and use the interactive tool as designed. Everyone else works with a partner to do the same thinking and discussion about the interactive tool but without the direct access that the one pair has. The rest of the class is relying on the pair at the front of the room to manipulate the interactive tool in ways that are useful to their own learning.

This also frees the teacher up to circulate around the room and listen in on the conversations students have. This will give you some formative data on what students are thinking about during their discussions.

Note that it is best if the pair at the front of the room takes enough time to finish using the tool so that everyone in the room has sufficient time to notice relationships (if that is the goal). Therefore the pair that comes to the front of the room should be a pair of students that you can rely on to move deliberately enough that everyone has access to the range of possible things noticeable via the tool.


3. Have someone else describe their thinking.

Have someone, other than the pair of students already at the front of the room, describe relationships they noticed while the pair already at the computer manipulates the interactive tool under their direction in order to demonstrate the thinking being described. At this stage, it may be helpful to annotate or otherwise record a representation of what is being discussed so that it is clear for all students. You may decide to ask a few different students to present and you may want to select which students to present based on circulating around the room earlier.

During these presentations, you may want to use a variety of strategies to make the thinking of the pair of students clear for everyone.


4. Have students reflect on or apply what they learned.

Students should have opportunities to reflect on experiences that they have, either by trying to apply the ideas to other problems or by writing about those experiences.


Here is a good point from David – if you have a document camera you can use this same principle with anything that you have.


Take a look at this video of an interactive tool being manipulated so that you can experience what it is like to watch something being changed without being able to change it yourself.

(For those who are interested, the tool in the video is available here).


Responding to Student Mistakes

A while ago, I had something very similar to the following shared with me. The student was given the diagram and asked to find the measure of the angle marked with the question mark.

Example of student work

The student has clearly made a mistake. Why did they do it? I asked on Twitter and here are some theories:


I think all of these are possible answers. Only one of them is possibly correct for this particular student but any of them could potentially be reasons a student might do this calculation incorrectly.

But do each of them have the same instructional response? I’m not clear on that. I think we need to know more than what students did right or wrong, I think we need to know what thinking students were doing. And I think we need to know what students are thinking whether a student has done a problem correctly or incorrectly. I also think we should focus on learning, not just individual performances.

If all we know is if students go an answer right or wrong, the best instructional approach we can imagine is to essentially repeat our prior instruction, except maybe slower and louder than before. If we know more about student thinking, then we can focus on experiences that will change the information students are using to make decisions, which I think is far superior feedback to students than x’s or checks on a piece of paper or the digital equivalent via a computer.

If my theory that our responses could vary depending on different student thinking, then adaptive computerized systems have a long way to go before they are really going to meet student need. None of them currently has any hope in gaining better insight into student thinking than a teacher asking the simple question, “Can you explain to me what you did here?”



Online Practice is Terrible Practice

One of the ways computers are being used in math education is to provide students with online practice. There are a bunch of serious problems with most of these programs.


Here is one example from the Khan Academy (apparently at least one of the flaws outlined below no longer applies to the Khan Academy. But that same flaw still applies to IXL. And Prodigy Math. And a thousand other practice apps out there.)


Feedback is terrible or nonexistent

Many programs will, as David points out above, only allow a student to progress after they have gotten a certain number in a row correct. But if a student is struggling to complete an activity and the feedback to the student is terrible, how exactly are students meant to achieve the streaks necessary in order to advance?

Note: Watching a video of a concept isn’t feedback if the learner has already watched that video before. That’s information the learner already has.


Impossible to see patterns

One way that people learn math is by observing patterns in their work or solution strategies as they work on a set of problems in a row.For example, what pattern do you see if you try the following exercise:

5 x 3 = ?
4 x 3 = ?
3 x 3 = ?
2 x 3 = ?
1 x 3 = ?
0 x 3 = ?
-1 x 3 = ?
-2 x 3 = ?
-3 x 3 = ?

I’ve tried this with students and most of them notice that they are subtracting each time to find the next product, and so then they make a leap and decide that -1 x 3 must be -3.

But if an online practice program only ever shows one question at a time and the numbers for these questions are selected randomly, there will be very limited opportunity for students to notice and subsequently use any patterns that emerge.


Blocked practice

Except for a handful of studies, there is a lot of research that suggests that for most people, if the goal is to remember some mathematical idea, practicing topics in blocks will take longer than if different topics are interleaved together. Almost all of the programs out there focus on students practicing discrete topics. Caveat: I did read a study recently that suggested that for students entering a course with weak prior performance, while interleaved instruction was beneficial, interleaved practice was less effective for these students than blocked practice. Further caveat: I cannot find the link to this study.


Too easy or too hard

For some students the exercises are too easy. Sometimes this is because kids select easier problems for themselves, sometimes this is because students already know a bunch of mathematics and do not need this particular practice activity. Either way, needing to work through a streak of 5 or 10 problems just to be able to move on is ineffective for these students.

For other students the exercises are too hard. A student who really doesn’t know a particular area of mathematics doesn’t benefit from practice in that area – they need teaching or access to information.


Inappropriate medium

For many, many math problems, the best choice of a medium to work on the problem is a piece of paper. Or maybe the best choice for working on a particular problem is a programming language.

These online systems offer neither. This means students are often working in a possibly unfamiliar medium without the most useful tools available for them to work.

This also restricts the people who design questions for the system as they end up likely severely restricted as to what kinds of questions they can ask if they need the answer to the problem assessed by a computer.


It obscures information from teachers

If you are a teacher and you are using one of these online systems for your students to practice, there is usually a dashboard you can look at to see how well your students are doing with a particular exercise. But these dashboards truncate an enormous amount of information about the progress of learning and actually make it harder for you as a teacher to gather the information you need to be able to act to improve your students’ learning.

They also are likely to lead to teachers looking for students making mistakes instead of looking for student conceptions, which promotes a deficit view of students instead of treating students as sense-makers.


It can lead to bad practice

Virginia Tech has an online remedial math program where students go to sit at a terminal and watch videos on math and then take quizzes on what they learned, over and over again. There is a Facebook post where almost all of the students complain about how much they hate this mathematics class. If the online practice programs did not exist, neither would this course.

Teach to One uses a computer online practice program to inform teachers when small group instruction should occur. But in this middle to upper class neighbourhood, parents revolted and the program was scrapped. But what about districts where parents have less power?

Dan Meyer outlines the many problems with Rocketship Learning Labs, another personalized learning model, in this post.


It isn’t really mathematics

If you ask a mathematician or anyone who uses mathematics regularly what mathematics is, literally none of them will answer “it is a series of multiple choice questions or short response questions asked and answered on a computer screen.”  While practicing mathematics is a decent way to get better at what you know how to do, it isn’t really the goal of teaching students mathematics.

If answering a series of problems is the only experience of math that students have, they are likely to end up with a very limited definition of what mathematics actually is.

Note, I’m not opposed to students practicing math at all. This is obviously an example of good practice and there is plenty of research to support this claim. I’m opposed to this being the primary experience of math that students have.



If you can possibly avoid it, don’t use these programs. Or at least try the program yourself for a couple of hours to see what the experience is really like for students. And if you are a designer of one of these online computer practice systems, for the love of God please do a better job than the industry currently is.



10 things that might actually disrupt US education

There’s a list being shared of ten things that will disrupt US education and I agree with Dan Willingham.


In no particular order, here are ten things that might actually disrupt US education.


Teachers being afforded respect as a profession by policymakers and others

You do not enact law like No Child Left Behind if you fundamentally believe that teaching is a profession. You know who primarily regulates lawyers, engineers, and doctors – that’s right, in many countries they do that themselves.


Teachers, especially elementary school teachers, having adequate time to plan

In some states elementary school teachers teach all subject areas and have a total of 45 minutes to plan AND are paid so little that many of them need second jobs.


Provide curated resources to teachers

Although increased planning time may reduce this tendency, designing ambitious curriculum is difficult and extremely time-consuming, so most teachers would benefit from curated resources that they can modify and adapt using their professional judgement. Surprisingly, many teachers have to use Pinterest and/or Google to find resources for their classroom because of a lack of curriculum resources aligned to their new state standards.


Paying teachers enough that they do not need second jobs and can afford to live in the communities that they work in

One way to make getting into teaching competitive would be to pay people enough that it makes teaching an attractive choice. It would also mean fewer people leaving the profession to find more lucrative careers and leaving vacancies, especially in harder to fill content areas.


Policies intended to improve teaching not teachers

As Jim Stigler and notes in the Teaching Gap, much of US policy is engineered at supporting individual teachers at getting better and that as soon as these teachers retire or quit, their professional knowledge leaves with them and the profession of teaching in the US remains relatively unchanged. It’s a good thing for individual teachers to get better at their practice, it is better that the professional benefits from what they learn.


Equitable funding across US schools

In some school districts, schools spend $9000 per student while a few miles away in a suburban district, schools spend $26,000 a student. While this inequity exists, resources are unevenly distributed across US education and in most cases the students who need the most support to be successful receive the least amount of funding.


Equitable access to teachers across US schools

In almost all large urban areas, it pays better to work in the suburbs than it does to work in the city. This results in teachers leaving the cities for high paying, lower stress jobs outside of the cities and in uneven amounts of teacher experience across the schools in the city.


Design school structures which are coherent and communicate across all levels of education

Imagine a system where the person who teaches teachers never sets foot in a school, the person who runs a school has no time to read research or even see their teachers teach, the person who runs research has never taught, the instructional coach who supports a teacher has their own idiosyncratic teaching style, and a teacher who has to listen to all of these people give them different advice on teaching. This is considered normal for teachers across the United States. But it does not have to be that way! It is possible to design systems where all of these people work collectively rather than individually.


End economic inequality in the United States

Income inequality in the United States is increasing and given that we know already that there exists a relationship between income and educational achievement, any shift toward more economically equitable society is likely to result in improvements in education for most students.


The end of systematic oppression of people of colour

The United States has a long dark history of oppressing people of color in various ways. One way this occurs in the US school system is that the schools attended by children of color are much more likely to be closed and/or labelled as failing than other schools. Ending this systematic oppression would transform the United States educational landscape.


A Conference Experiment

My colleagues have long been frustrated sharing our work at conferences primarily because the work we do is complex and hard for people to understand thoroughly within the constraints of a conference session where we only have at most 75 minutes to work on an idea.

So we contacted the organizers of the two NCTM regional conferences and proposed a possible solution. Instead of running one session, we will run 4. Instead of 4 separate sessions, we will plan those 4 sessions to connect together. Given how closely my colleagues and I work, we were each able to be the lead speaker on a different proposal. Both the NCTM Orlando and NCTM Chicago conference organizing teams agreed to this proposal and scheduled our sessions both so they do not overlap and also sequentially as requested.

So although we have 4 separate workshops listed in the program guide, these sessions are actually one of our day-long workshops divided into four sessions. Our hope is that some participants will experience one workshop and be no worse off than before – they will still learn something even if it is not the complete picture – but that participants who attend multiple sessions will have more insight and ability to use our work.

Here is a video of Contemplate then Calculate in action, with Kaitlin Ruggiero as the teacher and some teachers from one of our courses playing the role of students.

If this teaser intrigues you, our four sessions are:

  1. Experiencing Instructional Routines: 

    In this session participants will experience the same instructional routine three times with three different tasks to consider what elements of the teaching that occurs are part of the routine and what elements probably depend on the task and the students.

  2. Unpacking Instructional Routines: 

    Next, participants will experience the routine again (this will give access to people for whom this is their first session) and name the parts of the routine, why those parts are helpful, and what questions they have about the routine.

  3. Planning and Preparing Instructional Routines: 

    There is good evidence that a new teaching idea sticks better for participants if they have an opportunity to incorporate it into their existing teaching by planning and preparing to use the idea, so that will be the primary focus of this session. This will also connect the planning process for the instructional routines to the 5 practices for orchestrating productive mathematical discussions.

  4. Rehearsing Instructional Routines: 

    There are two goals of this session. First that some participants will have an opportunity to apply what they have learned and actually practice using the instructional routine before trying it out with students. Second, rehearsal of teaching is a useful way to norm around teaching practice and to try things out in teaching in a lower pressure situation than with a group of students.


Here are the times and locations for these sessions in Orlando:


And here are the times and locations for our sessions in Chicago:


Further reading about instructional routines:


Questions about Curriculum

Here are some questions that I ask myself whenever I read through a mathematics curriculum:


• Does this curriculum assume that children will forget ideas over time?
• Does this curriculum provide instructional supports that increase the odds that all children have access to it?
• Does the curriculum assume all students are capable of learning and doing interesting mathematics?
• Are the connections between different mathematical ideas made explicit, both for me as a teacher, and for students who will experience the curriculum?
• Is it possible, based on the license and format of the materials, for me to extend / adapt / modify the curriculum based on student need?
• Does the material make it easier for me to use formative assessment practices each day?


If the answer to all of these questions is not yes, I don’t want to use that curriculum. A curriculum which is no more than a collection of tasks is no more useful to me than my ability to search for resources in Google.


What other questions do you ask yourself when reviewing curriculum?