The Reflective Educator

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Year: 2018

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.




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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.

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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

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