The Reflective Educator

Education ∪ Math ∪ Technology

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Handing out workbooks is not teaching

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.

A variety of math workbooks


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?


What is Conceptual Understanding?

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.

Isolated facts as separated points

And this image represents what conceptual understanding looks like:

Points connected with line segments

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.

Connected dots transforming into disconnected dots


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.

Algebra II August 2016 Regents Exam, Question #1

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 (xa)(xb) 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 + = 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?

Butterfly Method for Adding/Subtracting 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.




Two Cultures of Mathematics Education

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.

The first 4 problems from the June 2018 Algebra I Regents Exam

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:

The goals of a Tim Gowers Math Course

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.



  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.



Magical Hopes: Technology and the Reform of Mathematics Education

In 1992, Deborah Loewenberg Ball wrote an article called Magical Hopes: Manipulatives and the Reform of Mathematics Education. This article is intended to draw some connections between our use of manipulatives and our use of technology in math education, and hopefully offer some suggestions for improving the use of technology.

There is a similar magical hope that technology can be used to reform mathematics education and I think that some of this reform is misguided, and this is in fact why I no longer work as an educational technologist. Without reforming work we as educators do to make links between the resources we use and the learning outcomes we hope to see borne out by children, there will be little effective change in the overall learning experiences for children.


Here is a paragraph from Dr. Ball’s article that stood out to me. For some context, Dr. Ball starts her article describing a student explaining how they understand odd versus even numbers. For further detail, I recommend reading her article.

Some teachers are convinced that manipulatives would have been the way to prevent the students’ “confusion” about odd and even numbers. This reaction makes sense in the current context of educational reform. In much of the talk about improving mathematics education, manipulatives have occupied a central place. Mathematics curricula are assessed by the extent to which manipulatives are used and how many “things” are provided to teachers who purchase the curriculum. Inservice workshops on manipulatives are offered, are usually popular, and well attended. Parents and teachers alike laud classrooms in which children use manipulatives, and Piaget is widely cited as having “shown” that young children need concrete experiences in order to learn…


Here is an updated paragraph with the current reform efforts focused on utilizing technology in mathematics education.

Some teachers are convinced that [technology] would have been the way to prevent the students’ “confusion” about odd and even numbers. This reaction makes sense in the current context of educational reform. In much of the talk about improving mathematics education, [the use of technology has] occupied a central place. Mathematics curricula are assessed by the extent to which [technology is] used and how many “things” are provided to teachers who purchase the curriculum. Inservice workshops on [technology] are offered, are usually popular, and well attended. Parents and teachers alike laud classrooms in which children use [technology], and [Papert] is widely cited as having “shown” that young children need [access to technology] in order to learn…


Here is an example of technology intended to be used in mathematics education. This is a virtual manipulative I created which is intended to draw students attention to how function rules can be represented as visual sequences. Try changing the sliders and keeping track of what you notice and what you wonder.

Now consider, what do you think children might notice when looking at this manipulative? What will they wonder? And most importantly, what will they learn as a result of using it?

If you have time, I recommend actually trying this activity out with a variety of children at different age levels. What I have learned is that children rarely see things the same things as adults (especially if those adults are mathematicians and/or mathematics teachers). In particular most children with whom I have used this particular manipulative do not attend to the differences between terms in the same way that adults do, and very few children even notice that the manipulative above includes an equation. They tend to focus almost entirely on the elements of the virtual manipulative over which they have control (the sliders) and the elements that change the most (the visual pattern) and in particular generally ignore features not immediately visible (the difference between terms) or recognizable (the equations).

Technology that relies on children independently making mathematical discoveries is likely to fail for some children. Given that one’s current mathematical knowledge is a large factor in how easily one makes mathematical discoveries, this use of technology may increase educational inequity rather than decrease it.


Recommendation: Utilize instructional strategies intended to make the sharing of mathematical ideas explicit for all students.


Here is another paragraph from Dr. Ball’s article:

Manipulatives –and the underlying notion that understanding comes through the fingertips– have become part of educational dogma: Using them helps students; not using them hinders students. There is little open, principled debate about the purposes of using manipulatives and their appropriate role in helping students learn…


Fortunately, there is tremendous debate about the use of technology in mathematics education, but most of this debate has centered on grain sizes of discussion that are far too broad to be helpful. Instead of questions like, “Is technology helpful in mathematics education?” we should be asking ourselves much more specific questions that have the same form as “is this use of this technology helpful for these students in this situation?” And then when we come up with a first draft answer for that question, we should make sure that our second draft includes some evidence from our students that the learning experience in question was actually improved by the use of the technology.


Recommendation: The links between what we hope children will learn from a particular activity and the evidence that they have learned should be made clear and investigated.


Here is another insightful quote from Dr. Ball:

My main concern about the enormous faith in the power of manipulatives, in their almost magical ability to enlighten, is that we will be misled into thinking that mathematical knowledge will automatically arise from their use. Would that it were so! Unfortunately, creating effective vehicles for learning mathematics requires more than just a catalog of promising manipulatives. The context in which any vehicle–concrete or pictorial–is used is as important as the material itself. By context, I mean the ways in which students work with the material, toward what purposes, with what kinds of talk and interaction. The creation of a shared learning context is a joint enterprise between teacher and students and evolves during the course of instruction. Developing this broader context is a crucial part of working with any manipulative. The manipulative itself cannot on its own carry the intended meanings and uses.


I sometimes see the same kind of thinking from mathematics educators (myself included); that just because a particular Desmos activity or Geogebra applet contains some mathematical idea, that children will learn the mathematical principles that are potentially generative in the activity. Sometimes some children see the mathematical principle, we have them present their work, and then we either hope or assume that this sample of children is representative of the whole class. Unfortunately, selection of students to present is a challenging activity that tends towards sharing strategies deemed as correct rather than surfacing a mathematical idea (like Sean’s observation in Deborah Loewenberg Ball’s article) that is fruitful to present and discuss even though an outside observer would consider the idea incorrect.


Recommendation: Before students engage in a technological activity, anticipate how we think they may consider the mathematical features. While students engage in a particular activity, listen to students talk, watch them work, and ask them questions to probe at their actual mathematical thinking. Use this information to potentially revise the activity.


Dr. Ball continues with this argument:

If we pin our hopes for the improvement of mathematics education on manipulatives, I predict that we will be sadly let down. Manipulatives alone cannot and should not–be expected to carry the burden of the many problems we face in improving mathematics education in this country. The vision of reform in mathematics teaching and learning encompasses not just questions of the materials we use but of the very curriculum we choose to teach, in what ways, to whom, and in what kinds of classroom environments and discourse. It centers on new notions about what counts as worthwhile mathematical knowledge. These issues are numerous and complex. For instance, we need to shift from an emphasis on computational proficiency to an emphasis on meaning and estimation, from an emphasis on individual practice to an emphasis on discussion and on ideas, reasoning, and solution strategies. We need to alter the balance of the elementary curriculum from a dominant focus on numbers and operations to a broader range of mathematical topics, such as probability and geometry. We need to shift from a cut-and-dried, right-answer orientation to one that supports and encourages multiple modes of representation, exploration, and expression. We need to increase the participation, enthusiasm, and success of a much wider range of students. Manipulatives undoubtedly have a role to play in these aims, by enhancing the modes of learning and communication available to our students. But simply getting manipulatives into every elementary classroom cannot possibly suffice to fulfill these aims.


This leads to some questions that we educators can ask ourselves before we embark on using some new technology with students (in the same way that at the time of Deborah Loewenberg Ball’s article was published that manipulatives were a new technology for many teachers).

  • Does using this technology help my students learn mathematics that they can use without the use of this technology?
  • How will someone who does not yet know the mathematics embedded within this technological tool see the mathematics?
  • Does this technology focus solely on the acquisition of a limited set of mathematical knowledge or is it possible for students to use deliberate practice to identify patterns across different problems and acquire new mathematical ideas?
  • Does this technology make it harder for my students to interact with each other and with me?
  • How will I learn how my students understand the mathematical ideas that are the focus of this lesson?
  • Who is the audience of this technology?
  • Does this technology exacerbate existing inequities in mathematics education?


Some of these questions were originally in this post.


I don’t have answers to all of these questions unfortunately but I think as a community of mathematics educators, we should be at least trying to answer them. I also have not unpacked some really large scale and potentially damaging initiatives around online learning and personalized learning. Stay tuned for a Part 2 to this post focused more on these uses of technology.


A Teacher Reflects on Their Teaching

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.



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?


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).