All the world may never know

One of my course projects this semester focuses on reviewing and synthesizing literature on the use of digital manipulatives in elementary mathematics. To keep the number of articles I’m reading to a manageable number, I have tried to focus on articles that go beyond asking the question of whether or not digital manipulatives are beneficial for learning. I’m interested in learning about ways that researchers have explored the more nuanced questions of when they might be beneficial and for whom. One paper caught my particular attention because it has really pushed my thinking about the role of tools in learning.

Moyer-Packenham and Suh (2012) examined the different ways that students of varying achievement levels interacted with the same digital fractions manipulatives. In general, the students classified as high-achievers used the manipulatives to quickly recognize patterns. They used the manipulatives early on, but then tended to rely on them less as they applied the patterns they had noticed to later problems. By contrast, the students classified as low- or average-achieving tended to use the manipulatives to consistently and methodically work through the problems. The authors mapped these differences onto particular affordances of the manipulatives. The high-achieving students seemed to be benefiting from the efficient precision of the manipulatives, or their ability to generate precise representations quickly. The precision made it easier for the high-achieving students to recognize patterns. The average- and low-achieving students seemed to benefit from the focused constraint of the manipulatives, or the way the manipulatives focused attention on particular aspects of the task. This set of manipulatives focused attention on the idea that fractions can only be added if they have a common denominator.

What struck me about this finding was the idea that the exact same manipulative, used for the exact same task, could be uniformly beneficial while still being used in different ways. When I first started thinking about the question “for whom?”, I somewhat naively expected the answer to indicate that a particular digital manipulatives are good choices for some students, but not others. This finding is much more nuanced and interesting. The same manipulative was good for all students for different reasons.

Another article really helped to clarify this point for me. Tucker, Moyer-Packenham, Westenskow, and Jordan (2016) examined how second-graders interacted with a set of digital manipulatives targeting skip-counting and place-value concepts. They found that the ways in which students interacted with the manipulatives, and the outcomes of their interactions, varied  with students’ approaches to the tasks and their mathematical abilities. In short, these researchers pointed out that the existence of an affordance of a tool does not mean it will be taken up by students, and that even when the affordances are taken up, the outcomes of their use will not be the same across students.

Writing them out now, these results seems somewhat obvious, or at least intuitive. Of course different kids will use different aspects of the tools for different purposes. Of course the outcome of using the tools will be different for different kids. And yet it wasn’t until I read these results in articles that I really started to think about them. I started the semester by thinking that the question, “Are digital manipulatives good for learning?” was too simplistic a question, and set out to learn what we know about what I thought were better questions: “When are digital manipulatives good for learning, and for whom?” I’m realizing now that those questions only take microsteps toward the kind of nuance that characterizes real learning in real classrooms.

This realization was discouraging at first, as I felt like it led me to this conclusion: “There is a whole bunch of stuff going on and we’ll never really figure it out entirely.” But after a while I chose to take a more optimistic stance. Every classroom is full of kids who differ in their backgrounds, learning styles, and needs, and one of the hardest challenges of teaching, in my opinion, is to find a way to help them all learn. In that sense, it is actually encouraging to think about the fact that the same tools can be used in many ways for many purposes, even when students are tackling the same overall task.


Moyer-Packenham, P. S., & Suh, J. M. (2012). Learning mathematics with technology: The influence of virtual manipulatives on different achievement groups. Journal of Computers in Mathematics and Science Teaching, 31(1), 39–59.

Tucker, S. I., Moyer-Packenham, P. S., Westenskow, A., & Jordan, K. E. (2016). The complexity of the affordance–ability relationship when second-grade children interact with mathematics virtual manipulative apps. Technology, Knowledge and Learning, 21, 341–360.


Papert v. Shulman

As I mentioned in last week’s post, I recently read a seminal work I’ve been meaning to read for years: Mindstorms by Seymour Papert. I have read bits and pieces of it in the past, but reading it start-to-finish was absolutely worth the time it took. Papert (1980) describes his vision for learning and for school eloquently and using many viewpoints and examples. I really got a feel for Papert’s philosophy by spending a couple days reading this book.

One of the things that struck me the most was Papert’s characterization of typical school mathematics and physics as perpetual learning of prerequisites:

Most physics curricula are similar to the math curriculum in that they force the learner into dissociated learning patterns and defer the “interesting” material past the point where most students can remain motivated enough to learn it. The powerful ideas and intellectual aesthetic of physics is lost in the perpetual learning of “prerequisites.” (Papert, 1980, pp. 122-123)

He goes on to argue, in some detail, that the material we teach in schools, largely laden propositional knowledge and equations, is not really prerequisite to understanding the powerful ideas of the disciplines. Indeed, the more qualitative, less precisely described ideas were developed first historically. Presenting a hypothetical dialogue between Aristotle and Galileo, Papert (1980) argues that these great thinkers worked by manipulating intuitions, not equations. Why shouldn’t students be asked to do the same? Papert does not believe there is any reason why children should not be asked to think like physicists and mathematicians.

His ideas are compelling, and as per usual, this fact has  made them controversial. I have read a number of arguments against Papert’s model of learning. Most were not worth reporting, but one got me started on an interesting thought experiment. In a lengthy argument against all forms of “minimal guidance” learning (Papert’s constructionism included), Kirschner, Sweller, and Clark (2006) argued,

Another consequence of attempts to implement constructivist theory is a shift of emphasis away from teaching a discipline as a body of knowledge toward an exclusive emphasis on learning a discipline by experiencing the processes and procedures of the discipline …  Yet it maybe a fundamental error to assume that the pedagogic content of the learning experience is identical to the methods and processes (i.e., the epistemology) of the discipline being studied and a mistake to assume that instruction should exclusively focus on methods and processes. (p. 78)

In support of this argument, Kirschner, Sweller, and Clark (2006) point to Shulman’s (1986) oft researched, oft cited construct of pedagogical content knowledge (PCK). Shulman separated content knowledge, or knowledge of a discipline, from pedagogical content knowledge, or knowledge of how to teach a discipline. Kirsch, Sweller, and Clark argued that to treat learning of a discipline as the same as practicing a discipline was to ignore the existence and importance of pedagogical content knowledge.

Overall, the argument did not hold a lot of water for me. Now that I have read Mindstorms in its entirety, it’s clear to me that Papert’s ideas have been wildly oversimplified and mischaracterized as they were discussed and debated. Papert does not argue that children should learn physics by attempting to engage in problems exactly as a physicist does. Rather, he advocates that under the right conditions, children naturally think the way physicists think, and shouldn’t be precluded from doing so.

So, the argument is moot. But still, I found it very interesting to ponder the following question: Would use of a “pure discovery” or fully “unguided learning” approach in a classroom necessarily mean there is no place for pedagogical content knowledge? By presenting students with a task and leaving them be, are educators who use such approaches failing to apply any PCK?

After a bit of pondering, I decided the answer was no for this reason: Educators who use such approaches still have to choose a task. Task development and selection is a huge part of pedagogy. Is asking students to solve a decontextualized fraction problem the same as asking them to reason about how to share a snack fairly among their friends? Is asking them to summarize a novel the same as asking them to write a letter from the perspective of one of the characters? Definitely not.

Papert’s constructionism is more than just setting a task. But even so, it’s also true that pedagogical content knowledge is more than guidance in completing a task. It’s also about choosing a task to be completed.

I guess the cage match between Papert and Shulman is draw.


Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work. Educational Psychologist, 41(2), 75–86.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc..

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14.


Getting to Know a Triangle

One of the first experiences that convinced me the use of technology in mathematics education had enormous potential was interacting with dynamic geometry software. With programs like Geometer’s Sketchpad and Cabri, a user can create a 2- or 3-dimensional shape and then manipulate it. The very idea of having some control over a shape, rather than being restricted to looking at a static representation on a printed page, was intriguing to me. But what really blew my mind was not the freedom, but the restrictions on the manipulations. I could drag the corners and the sides around, but the way the shape responded would always maintain the defining properties of the figure.

Take a square, for example. If I used the software’s square tool to draw a shape, I would get a square, for sure. And then if I made a side longer, all the sides would get longer too. If I dragged a vertex, the sides might change in length and the orientation of the shape might change, but all four angles stayed right angles. It was amazing. Even though I was a 20-something holder of a B.A. in mathematics who felt she knew what there was to know about squares, the experience pushed my understanding of a square.

The whole idea became even more interesting when I learned about research showing that young children often struggle to apply the names of shapes beyond prototypical examples (e.g., Verdine, Lucca, Golinkoff, Hirsh-Pasek, & Newcombe, 2016). Young kids tend to think the triangle on the left — the one they see most often in the world — is indeed a triangle, but the shapes on the right are not.

When it came to 2-dimensional shapes, exposing kids to more and different examples of non-prototypical shapes seemed feasible to me without the use of dynamic geometry software. All sorts of triangles and rectangles, for examples, are embedded in everyday objects. Still, there seemed to be something special about manipulating a single example as opposed to seeing multiple examples.

Two pieces I’ve read in the last few weeks have helped me to reflect on why the manipulation of a dynamic triangle might be different and more powerful than examination of many different triangles. The first is a book that I’ve been meaning to read for some time: Seymour Papert’s Mindstorms. In it, Papert describes the LOGO programming environment and the ways in which it could be used to support the development of new ways of thinking and learning. This particular passage really struck me:

Working in Turtle microworlds is a model for what it is to get to know an idea the way you get to know a person. Students who work in these environments certainly do discover facts, make propositional generalizations, and learn skills. But the primary learning experience is not one of memorizing facts or of practicing skills. Rather, it is getting to know the Turtle, exploring what a Turtle can and cannot do. (Papert, 1980, p. 136)

The notion of getting to know an idea the way one gets to know a person brought me back to the dynamic triangle. The reason that manipulating a single triangle, rather than creating or examining multiple examples, feels like a more powerful experience is because the manipulation allows for a student to get to know what it means to be a triangle. They get a feel for what the entirety of the triangle category looks like, rather than deciding whether particular examples fall into it. I really love Papert’s emphasis on qualitative thinking, and thinking about it in relation to dynamic shapes was powerful for me.

The second reading that pushed my thinking on this issue was Stephen Kosslyn’s response to The Edge’s Annual Question in 2010: How is the Internet changing the way you think?” In his response, Kosslyn (2010) says, “The Internet has made me smarter, in matters small and large. For example, when writing a textbook it’s become second nature to check a dozen definitions of a key term, which helps me to distill the essence of its meaning” (5th para.). As we discussed this passage in one of my courses, we came to the idea that the ability to quickly and easily see many instantiations of something leads to a softening of your definition of that thing. Children appear to have a rigid definition of triangle when they only consider horizontal-based, equilateral examples as triangles. Manipulating a dynamic triangle, however, allows them to explore all the ways in which they must soften this definition to come to what defines a triangle mathematically. Examining multiple individual examples can work toward this, of course, but it is the all-ness that seems most important. With dynamic triangles, children can directly confront the properties that they think define a triangle, whatever and however many of those things there may be.

That’s what makes dynamic manipulation so powerful. Cool.

As with most connections I made on this blog, I suspect that in both cases, I am not the first to realize this. If you know of pertinent papers to read on these ideas, please let me know in the comments!


Kosslyn, S. M. (2010). A small price to pay. Retrieved from

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc..

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2016). The shape of things: The origin of young children’s knowledge of the names and properties of geometric forms. Journal of Cognition and Development, 17(1), 142-161.

Interest update

Hello, everyone. Happy Friday.

It happens to be my birthday today, so I hope you’ll forgive this short and rather self-indulgent post. I always consider my birthday to be an opportune time for self-reflection. I’ve taken a little bit of time today to think about how my research interests are evolving.

Throughout last semester, I thought about my interests in terms of three overlapping circles: kids’ use of digital tools to learn math in K-5, teachers’ use of digital curriculum materials, and the intersection of computational thinking and mathematics. I was able to pursue two of those in some depth last semester.

When it comes to CT, through my assistantship, spent a good chunk of time last semester exploring how to help G3-5 teachers integrate computational thinking into their mathematics and science instruction. This work spurred a particular interest in research on abstraction, which in turn led to a manuscript about how my advisor and I think a framework for teaching abstraction developed by some CS educators could be applied to problem solving in K-5 math. We’re continuing to work on that manuscript this semester and hope to submit it within a couple of months! Also, in last semester’s Educational Inquiry class, I developed a research proposal to interview some elementary school teachers about the prospect of teaching mathematics through computational thinking ideas. Happily, that study is also becoming a reality this semester. So, the CT thread remains connected from the last semester to this one, and should do so into the future as well. The main issue I’m negotiating in this work is trying to make sure I keep the mathematics alive as much as I can. I consider myself a mathematics education researcher with an interest in CT rather than a CS education researcher, per se.

I was able to explore teachers’ use of digital curriculum materials last semester mostly through my Proseminar in Mathematics Education course. My culminating project in that class was a paper synthesizing 12(ish) scholarly studies of teachers using various digital curriculum materials ranging from single manipulatives to digital textbooks. The main finding of that paper was that technology is more likely to lead to new kinds teaching in classrooms when teachers are involved in the design of the curriculum resources, or at least invited to play a role. This was unsurprising to me, though admittedly not something I had thought about much before writing the paper. I’ve carried this interest over into this semester through my current Qualitative Methods course. I’m working on designing a study that would explore how teachers’ sense of agency interacts with use of curriculum materials. If this becomes something I pursue in greater depth, I’m hoping such a study and its follow ups could lead to some design principles for the creation of digital curriculum materials.

The third area, kids’ use of digital tools, got the least attention last semester. I find this a little odd, upon reflection, considering that this area is the crux of what brought me to graduate school in the first place. To remedy this, my Ed Psych and Ed Tech Proseminar course project is going to be a critical integration paper based on the last 10 years of research on digital manipulatives. Often, this paper becomes a component of students’ practicum projects (the small-ish research projects that we do before embarking on comprehensive exams and a dissertation proposal). I don’t know for sure where this would go, of course, but dynamic representations (often via digital manipulatives) and how they might influence kids’ thinking is of definite interest to me.

Interestingly, as mentioned above, some of the studies I read about teachers’ use of digital curriculum materials involved digital manipulatives. The flexibility of a single manipulative, as opposed to a comprehensive curriculum program, made them an interesting context for involving teachers in the design of the materials — hence, there’s a chance that a practicum could involve studying both teacher and student use of a manipulative, combining two of my areas of interest. Similarly, there is some research showing an association between use of digital manipulatives and abstraction abilities, so there could be a CT overlap, too. The trick of it will be keeping the scope manageable.

In sum, I’d say my overall areas of interest have not changed much, but some of them are becoming more focused. I am still rather resistant to purposefully letting go of any of these lines of inquiry (I hate giving stuff up!), but I’m more hopeful now that a productive path will become clear to me as I progress.

Thanks again for indulging me in this post and for your continued readership. Hopefully this next year will be full of interesting ideas and insights to share with you.

Wayfinding through Hypermedia

This week, I’m still thinking about curriculum materials and how they could be made more flexible and useful. I had two disparate experiences this week that ended up tying together to advance my thinking a bit from last week.

First, I read a study that traced the effects of a highly scripted curriculum resource on a preschool teacher’s instructional practices (Parks & Bridges-Rhoads, 2012). Despite the resource in question being a literacy curriculum, the highly scripted nature of the teacher’s interactions with students trickled into her math instruction. The article details the ways in which the scripting led to some missed opportunities to explore students’ mathematical thinking.

None of this was surprising to me, really. I had become convinced before that scripted curriculum can reduce a teacher’s opportunities for responsive teaching. (I want to emphasize the opportunities part here — neither the authors of the article nor I saw this research result as reflecting poorly on the teacher. If anything, it reflected poorly on the curriculum materials.) The problems with scripted curriculum don’t just live in early childhood, either. Developers of curriculum materials for students in middle school have recommended a shift from scripting to steering (Hoyles, Noss, Vahey, & Roschelle, 2013), especially in materials that are digitally based.

Dr. Amy Parks, the lead author of the preschool article, is now a faculty member at MSU. I had a chance to talk to her about this article, and we brainstormed some metaphors for navigating through a lesson that might be more conducive to responsive teaching than reading a script. One metaphor she had heard recently from a colleague was wayfinding, a kind of navigation that doesn’t involve use of a map, but rather reference to visible markers as you go. (It’s a cultural practice of Polynesians that recently got some attention via Disney’s Moana.) I really latched onto this. It made sense to me to think of the process of teaching as knowing where you want to go, and trusting that there will be markers along the way that give clues about how to get there.

This brought me back to the importance, and difficulty, of the number of in-the-moment decisions required in teaching. But it didn’t really get me any farther in thinking about how curriculum materials could be re-envisioned to better support teachers in their wayfinding. I understand that teachers don’t have the luxury of stopping to consult materials each time the classroom conversation takes an unexpected turn. Still, if we know that scripted materials are associated with less flexible teaching (Parks & Bridges-Rhoads, 2012), doesn’t it seem that there must be a different format for curriculum materials that could associate with wayfinding in teaching?

Enter my second relevant experience this week: delving into the history of research on educational technology. In his defense of research studying the effects of media on learning, Kozma (1991) highlighted research on nonlinear, hyperlinked text as an example of a then-emerging field of research of multimedia. He points to other research suggesting that use of hypertext supports cognitive flexibility, as navigation through the text supports making connections among concepts and themes: “[H]ypertext facilitates this cognitive flexibility because it allows a topic to be explored in multiple ways using a number of different concepts or themes” (Kozma, 1991, p. 202).

Exploring a topic in multiple ways feels similar to navigating through a lesson in multiple ways, don’t you think? Perhaps creating hyperlinked versions of curriculum materials, wherein teachers make decisions about what to read and in what order, is a way to move away from scripting and toward wayfinding.

There is a lot of exploration to be done on what links would be productive and useful for teachers, as well as how to deconstruct a curriculum into parts that can effectively be linked. Still, I think the idea could be a powerful one for giving teachers agency over curriculum decisions while still providing useful information via curriculum materials. I’m looking forward to delving into more hypermedia research to discover if anyone’s applied it to curriculum.


Hoyles, C., Noss, R., Vahey, P., & Roschelle, J. (2013). Cornerstone Mathematics: Designing digital technology for teacher adaptation and scaling. ZDM – International Journal on Mathematics Education, 45(7), 1057–1070.

Kozma, R. B. (1991). Learning with Media. Review of Educational Research, 61(2), 179–211.

Parks, A. N., & Bridges-Rhoads, S. (2012). Overly scripted: Exploring the impact of a scripted literacy curriculum on a preschool teacher’s instructional practices in mathematics. Journal of Research in Childhood Education, 26(3), 308–324.

Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform‐based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467-500.


Thinking Flexibly about Flexibility

Hello, everyone. Happy new year! I’m back at the blog after a much-needed winter break.

This week I’ve been thinking more about digital curriculum materials and adaptations (ideas I talked about in this post and this post). When I talk about this area of my research interests to new people, I usually frame it as a question of how a digital medium might be used to create curriculum materials that are more flexible, thereby helping teachers to make adaptations according to their needs and contexts.

A natural follow up to such a description is, what does it mean for curriculum materials to be flexible? Oddly enough, I don’t think I’ve ever been asked that, despite giving that description a lot. It is a question I asked myself a few days ago, though. What are the kinds of flexibility that would be useful to teachers?

To this point, I’ve always been pretty focused on the idea of teachers changing the order in which the content is presented. Maybe kids need to be able to measure to the nearest half-inch to do an art project next week, for example, and so a teacher wants to teach a lesson on that. If she moves the lesson a unit earlier in the sequence, what are the repercussions of that? In what other ways would the sequence need to be adjusted to make sure it still follows a coherent progression? I’ve always thought — and still think — that is must be possible to design a system that gives intelligent feedback to teachers about potential impacts of sequence changes. It’d be a way to reveal the curriculum designer’s intentions, as described by Hoyles and Noss (2003) (and much later and less thoroughly by me, in this post).

I’m not the only one interested in the idea of building and maintaining coherent sequences of content — other researchers have talked about ways to do that in both mathematics (Confrey, Gianopulos, McGowan, Shah, & Belcher, 2017) and science (Shwartz, Weizman, Fortus, Krajcik, & Reiser, 2008). I haven’t lost interest in that idea. In recent days, however, I’ve been thinking about other kinds of changes and adaptations teachers might want to make to curriculum materials, and realizing that my thinking about flexibility, to this point, has been pretty inflexible.

In a recent conversation with some curriculum development colleagues, we were discussing one of the most pervasive and ongoing tensions in curriculum development work. On one hand, we aim to always provide adequate features and information to support all teachers in implementing lessons. On the other hand, we want to communicate that the suggested discussion questions, samples of student work, and other information are intended to support teachers in thinking about the lesson — not to script it. More information is — in some ways — more support, and yet it can also seem like an attempt to take the agency away from teachers, which is something we don’t want.

This led me to wonder whether there were certain types of information that are potentially useful to some teachers, but perceived as scripts or mandates to others. Feiman-Nemser (2001) points out that teachers have different professional development needs across the course of their careers. Newer teachers, for example, may benefit from more information about how a lesson might play out than more experienced teachers. Could the number of sample discussion questions shown be something teachers would adapt to their needs? What about whether or not information about how to support to English-language learners is shown? This could be a whole different class of adaptations that has nothing to do with how content is sequenced.

I was also reminded a few weeks ago that there is research showing that teachers interact with curriculum materials in different ways at different points of the planning and implementation process (e.g., Sherin & Drake, 2009). I have been pretty focused to this point on adaptations that teachers might make while planning, but limiting my thinking to planning activity disregards the importance of in-the-moment decisions that make up so much of a teacher’s work. Listening to students’ thinking can lead a teacher to change a task or learning goal in the moment. Such a change is an adaptation to curriculum, but could it also necessitate an adaptation to the curriculum materials? If so, how could curriculum developers support such an adaptation?

All in all, I find myself thinking in a whole new way about the idea of how curriculum might be made more flexible. What other kinds of flexibility have I not thought about yet?


Confrey, J., Gianopulos, G., McGowan, W., Shah, M., & Belcher, M. (2017). Scaffolding learner-centered curricular coherence using learning maps and diagnostic assessments designed around mathematics learning trajectories. ZDM – Mathematics Education, 49(5), 717–734.

Feiman-Nemser, S. (2001). From preparation to practice: Designing a continuum to strengthen and sustain teaching. Teachers College Record, 103(6), 1013–1055.

Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education. In Second International Handbook of Mathematics Education (pp. 323–349).

Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform‐based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467-500.

Shwartz, Y., Weizman, A., Fortus, D., Krajcik, J., & Reiser, B. (2008). The IQWST experience: Using coherence as a design principle for a middle school science curriculum. The Elementary School Journal, 109(2), 199–219.


Five Things I Learned in My First Semester as a PhD Student

Hello, everybody.

Tomorrow is the last day of my first semester as a PhD student. I survived! And I’m really happy that I’ve managed to post a blog entry every week. I hope to keep doing so next semester and throughout my graduate school career.

To close out the semester, before taking a couple of weeks off for the holidays, I thought I’d share a list of some of the things I’ve learned this semester. Not the ed psych or the perspectives on math ed or the research methods, but things I’ve learned about myself and about being a thoughtful and productive scholar.

Here goes.

1. Reading academic articles is not only about making sure you understand what they say, but also understanding what you think about them. I’ve explored the difference between content and practices in relation to mathematics education for a long while, but the distinction has become clearer to me in graduate school. Graduate study not really about content at all. The content is just a useful mechanism for learning how to think. Learning to clearly articulate what you think about the work of others is a major component to becoming a better academic writer.

2. Sometimes the key to making connections among articles is reading them with a lens the author did not necessarily intend. Academic work does not build on itself in a linear way, but rather in a widely sprawling way. Synthesizing studies that were all approached in the same way often does not lead to anything interesting. Making a unique contribution to the field can happen through empirical work, but also through making connections between seemingly different pieces by viewing them each through the same lens.

3. Citation management software is an academic’s best friend. If you’re in grad school and not using it, start. It will change your life. I spent 12 years reading academic articles that I promptly forgot about because I didn’t catalog them. In just one semester, I’ve built up a library that I regularly draw upon rather than starting from scratch in each new pursuit. That’s all thanks to my citation software.

4. One of the best pieces of advice I got before beginning graduate school was that I needed to treat it like a job. That means getting up at a regular time and getting myself to campus during working hours. This habit has not stopped the work from bleeding into my evenings and weekends, of course. But overall it has made the lifestyle manageable.

5. There are a variety of dimensions along which researchers arrange themselves. Positivist or interpretivist? Behaviorist, cognitivist, or socio-culturist? Theorist or practitioner? Choosing an area of research focus is much more intimately connected to developing a personal and professional identity than I ever could have imagined. Because of that, I’m having commitment phobia. It is going to take me a while to convince myself I’ve found the area where I want to stake my claim.


Thanks for taking the journey with me this semester. Happy holidays, and see you in January.

In the Moment

I have always been a little bit of an anomaly in my career field. Although I’ve worked on various education projects and materials for over ten years now, but I’ve never been a classroom teacher. I was a math major as an undergraduate student, with little interest in education at the time. It was only through a random search for “mathematics” in a job search engine upon graduation that I ended up with a temporary contract position in educational publishing. After that, I went to graduate school to get a masters in learning sciences, then to the university for ten years of work in curriculum development, and finally, this year, I started a PhD in educational psychology and educational technology. I taught a couple of recitation sections of a math course as an undergraduate senior, and I’ve taught brief lessons in various classrooms on occasion, but other than that, I have no teaching experience.

People ask me sometimes why I didn’t choose to go through a teacher ed program and teach for a few years once my interest in education started to develop. There are a couple of related answers to that question. One is that it felt somehow insincere to enter teaching with the explicit intention of leaving it after a couple of years. Another is that I didn’t (and still don’t) think I’d be particularly good at it. I find it difficult to develop good rapport with kids, and I am not a quick thinker. When I’m in a conversation that doesn’t go the way I expect, I am almost never able to respond intelligently in the moment. It takes me a while to work through things. That’s not possible when 30 students are sitting in front of you, waiting for reasonable answers to their questions.

The constant need for in-the-moment decision-making, in my opinion, is among the most difficult aspects of teaching. The very thought of it is one of the major reasons I never pursued teaching. The willingness and ability to handle the constantly changing course of conversation in classrooms is the thing I admire and respect most about teachers. I have always been too scared of the idea to even try.

As a researcher interested in processes of learning, I’ve always been curious about the ways in which teachers learn to be agile and responsive in their teaching. I know that it’s something that develops over time with practice and experience, but are there ways of helping that development along? How does that play out in teacher education?

This week I read a couple of pieces that answer that question, at least in part. Crespo, Oslund, and Parks (2011) asked pre-service teachers to imagine and write out narrative scripts of how they thought their lesson plans would play out in their classrooms. What would students give as answers to particular problems? What would the teacher say in response? How would the conversation progress toward the mathematical goals of the lesson?

The researchers explored the idea of having preservice teachers write out enactments, rather than focus exclusively on lesson plans, because they felt that the former would allow a different window into teacher thinking. “Consider the difference between writing out a teacher action plan and acting out that plan (even if on paper). Although both of these serve to map the intellectual journey and destination of a math lesson, they do so from different perspectives. The first takes a bird’s eye view perspective, while the second takes a ground level viewpoint” (Crespo, Oslund, & Parks, 2011, p. 121).

I found this contrast fascinating, because it clearly articulated the difference between planning lessons and teaching lessons in a way that starts to get at the challenges of in-the-moment thinking that happens while teaching. Writing out scripts of possible enactments is an interesting way to prompt teachers to think about the possibilities of that in-the-moment work, in a context that doesn’t have the same time pressure for thinking through responses.

The Crespo et al. (2011) article discusses script-writing as a method for gaining a window into teacher thinking. That is, they discuss it as more of a research tool than a teacher education tool. When I brought up the prospect of using scripting as a technique for helping teachers develop skills in responding to students in the moment, my classmates and instructors rightfully brought up a concern that writing out a script might make teachers less apt to deviate from that script if something plays out differently than they expect in the classroom. I understand this, and so don’t really believe that hypothetical scripting of lesson should be part of daily lesson planning. I do, however, still wonder if making the creation of multiple possible scripts for the same discussion, done in the right context at the right time, could be beneficial to developing flexible thinking, that in turn helps teachers to be flexible in moment-to-moment thinking.

The second article I read this week that spoke to this particular issue was Lampert et al.’s (2013) study of rehearsal as a teacher education technique. Lampert et al. explain that rehearsal “can involve novices in publicly and deliberately practicing how to teach rigorous content to particular students using particular instructional activities” (p. 227). During rehearsals, preservice teachers teach a lesson to their peers and teacher-educators, with the teacher-educators calling out feedback and sometimes interrupting and pausing the lesson to discuss what just happened and ways a situation might have been handled differently. Rehearsal, like scripting, was appealing to me because it represents a way to examine and develop in-the-moment thinking with a built-in capacity to give teachers a bit of extra time to think things through and consider possibilities.

What do you think? Is one of the keys to developing skills in responsive, in-the-moment decision-making providing ways to slow down time a bit, as in scripting and rehearsal? What other ways of developing skills have you heard about or experienced?

And if you’re a teacher: Does it get easier over time? What has helped you feel prepared for the unknown trajectories of conversation in a classroom?


Crespo, S., Oslund, J. A., & Parks, A. N. (2011). Imagining mathematics teaching practice: Prospective teachers generate representations of a class discussion. ZDM – International Journal on Mathematics Education, 43(1), 119–131.

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.


Are we removing a barrier, or just shifting it?

About six months after completing my master’s degree, I went to a holiday party thrown by one of my former classmates. In my particular program, master’s students take many classes with first-year Ph.D. students. The attendees at the party were a mix of the Ph.D. students I knew — now in their second year of study — and new master’s students that I had not met before that night. At some point, one of the Ph.D. students, who was my officemate the year before, made a comment to one of the master’s students that I still remember.

“We have ten weeks to finish the work of a quarter, right? Katie was always done in seven.”

While finishing three weeks ahead was an exaggeration of my general pattern, I couldn’t deny that I was usually ahead of the pack in terms of finishing course requirements. It wasn’t that I was faster than everyone else when it came to reading or writing. I may be a bit above average on writing speed, but I would bet that I’m below average on reading speed, and usually slow to organize my thoughts, as well. Speed wasn’t the issue. And I really struggled with a lot of the coursework in grad school. Ease versus difficulty wasn’t the issue, either.

The reason that I tended to finish early, and still tend to do this, is this: I live in perpetual fear of running out of time.

While I do have isolated bouts of procrastination, it definitely isn’t my biggest vice. I start projects the day they are assigned just in case I end up making a false start and need more time than I anticipate. If I happen to have extra time during one week, I tend to read ahead in my courses just in case something big comes up the next week. I love to be done early. That’s the only time I feel like I can be justified in not working on something.

I have long known this about myself, but I started to think about it in a bit of a different way after revisiting the work of Carol Dweck (1999) and Jo Boaler (2012).

Dweck’s (1999) work focuses on the idea of fixed versus malleable conceptions of intelligence. Student who believe that intelligence is malleable and they can become more intelligent over time are more likely to choose and persist at challenging tasks. Student who believe that intelligence is a fixed tend to avoid challenging tasks. Among Dweck’s more prominent findings is the fact that the kind of praise given to students when they succeed can have a strong effect on their conceptions of their own ability and their reactions to failure. Students who are praised for being smart develop a fixed theory of intelligence, and tend to react to failure by starting to believe they are not smart after all. Students who are praised for their effort develop a malleable theory of intelligence, and tend to persist in the face of challenge and try again upon failure. Building on Dweck’s theory and findings, Boaler (2012) has conducted studies of interventions aimed at helping students develop a malleable view of intelligence, or a growth mindset.

Dweck and Boaler are among my academic heroes. A big part of my reason for starting a career in education research is my desire to help put an end to the prevalent I’m just not a math person narrative, and hopefully help to prevent a similar I’m just not a computer science person narrative from becoming as common. Growth mindset interventions have a lot of promise in this regard.

At the prompting of one of my instructors earlier this year, I took some time to think about whether I have a growth mindset. I was pleased to come to the conclusion that I think I do. There isn’t much that I don’t think I could learn, if I took the initiative to do so. What I found interesting about this reflection, though, was the caveat I felt I had to add:

I could learn just about anything…. as long as I had enough time. Some things might take me a very, very long time.

This was interesting to me from both personal and professional perspectives. On a personal level, it helped me to explain my perpetual anxiety about running out of time and how that coexists with a solid sense of my own academic ability. I think I’m perfectly able. I just think that there’s always a risk of things being slow and difficult. Not impossible, but slow and difficult. I’ve never articulated that belief before. It’s helped me to reconcile my belief in malleable intelligence with the existence of sports stars, musical prodigies, and the like. Everyone can learn anything they like, with sufficient time, resources, and motivation — but this does not mean that each person will have to do the same amount of work to get there. There is such a thing as a natural proclivity. The power of growth mindset is that it shifts this idea of natural proclivity from separating the world into cans and cannots — something that is false — to identifying the those who have a quicker path to being highly skilled among a huge population of capable people.

From a professional perspective, this realization helped me to think more carefully about the problems that growth mindset work does and does not help to solve. I think many students carry around a belief that they are not good — and cannot be good — at mathematics, and that particular issue can be addressed with growth mindset work. Even so, the interventions won’t remove all potential anxieties about mathematics. Even if a student comes to believe he or she can indeed learn mathematics, that won’t necessarily lead to the belief that it will be easy, enjoyable, or worthwhile.

On one hand, perhaps this is ok. The goal has never been for all students to become mathematicians — just to help them attain a certain level of mathematical fluency and efficacy. One can hope that any overpowering anxieties about slowness and difficulty could be kept at bay through K-12 education. At that point, perhaps a greater number of students will pursue mathematics-related careers, and the rest will have reached a point of mathematical literacy that makes them effectively able to interpret and use mathematics to understand day-to-day issues.

On the other hand, there’s something about this line of thinking that bothers me. Is it a slippery slope? Is “Everyone can do math, but some people can do it more easily and faster than others” really a better narrative than “Some people just aren’t good at math”? I’m not so sure. My perpetual assumption that I need to plan for things to be difficult and time consuming isn’t something I want all kids to be carrying around.

The simplistic solution is to just change the narrative to “Everyone can do math,” and leave it at that. The decision of whether or not to pursue mathematics is then up to the student, and hopefully any decisions made won’t be based on worries about fixed intelligence.  I just worry that the second part, “but some are better than others,” will be carried around after it, implicitly, if we don’t address it head on. Then we won’t really have fixed the problem — just shifted the barrier a bit.

I am sure you can tell by my rough writing here that I have not worked out all of my thoughts on this. But it’s something I’ve been thinking about, particularly when considering how I can use lessons learned from mathematics education to help to prevent a not a CS person narrative from taking hold as CS is introduced into K-12 schools.


Boaler, J. (2013). Ability and mathematics: the mindset revolution that is reshaping education. FORUM, 55(3), 143-152.

Dweck, C. S. (1999). Caution—Praise Can Be Dangerous. American Educator, 23(1), 4–9.

Digital, Dimensional, or Dynamic?

Yesterday I had a conversation with some fellow curriculum developers about the potential impact of translating hands-on activities into a digital environment. It was quite interesting and engaging — and like most interesting conversations I have these days, it left me much more confused than I had been when the conversation started.

We started by talking about 3-dimensional geometry, and the potential perils of studying it 2-dimensional representations on paper or on a screen. Though none of us had research at our fingertips that spoke to this (although I’m reasonably certain such research exists), at first we generally agreed that there are likely important benefits related to manipulating physical, 3-dimensional objects when studying 3D geometry and volume.

Then someone mentioned the idea of a digital environment with a 3D rendering of a physical object. What about a cube, for example, that can be rotated and viewed from any angle? Does this have the same benefits as handling a physical cube?

In other words, is 3D versus 2D the difference that matters? Or is it dynamic (and thus manipulable) versus static? Or is the really important difference the one that started this conversation in the first place: physical versus digital?

If the critical factor is 3D versus 2D, and true 3D representations are what work best for learning, then certainly the prospect of designing a curriculum that is delivered entirely digitally is a problem. This sort of argument is often used against the trend toward digital curriculum products in classrooms, especially for young kids. But this conversation with my colleagues helped me articulate something that has always bothered me about that argument. It’s this: if 3D versus 2D is the critical difference, than digital curricula aren’t the only curricula with problems. Traditional print curricula so often rely on 2D representations of 3D objects — static ones, on paper. So if there are going to be 2D representations anyway, aren’t the digital, dynamic, 2D representations just as good as the paper, static, 2D representations — and perhaps even better?

This chain of reasoning led me into thinking about digital alternatives to other physical objects used in classrooms. If the object of study isn’t something 3-dimensional, do the same kinds of concerns apply? Take base-10 blocks, for example. The physical blocks are 3-dimensional and the digital blocks are 2-dimensional, but that difference doesn’t seem to matter in this case, since neither version is a representation of a 3D object, but rather of an abstract number. Both versions are manipulable, but the particular manipulations are different. With the physical blocks, if children want to think of 1 ten as 10 ones or vice-versa, they must exchange a long (or ten block) for a cube (or one block). In many of digital versions of the blocks, if children want to think of 1 ten as 10 ones or vice-versa, they are able to break apart 1 long or put together 10 cubes. In that sense, the digital blocks are more dynamic than the physical blocks.

Sarama and Clements (2009) further point out that the action of breaking a long is closer to the mental actions children must make on numbers to add and subtract than the action of exchanging a long is. Since the first step toward abstract understanding of a concept is typically interiorization, or the ability to carry out a process through manipulations of mental representations (Sfard, 1991), this “closeness” to the mental manipulations of numbers could be important.

A study by Bouck et al. (2017) gives another example of how a digital tool accelerated the process of interiorization, in this case for students with mild intellectual disabilities. Previous research had shown that leading these students through a concrete → representational → abstract sequence was effective in supporting learning of addition of fractions with unlike denominators. The concrete phase involved physical manipulatives, the representational phase involved creating drawings, and the abstract phase involved symbol manipulation. This study found, however, that when digital manipulatives were used, the representational phase could be skipped entirely. For three out of four students in the study, a virtual → abstract sequence was effective. This suggests that the digital manipulative helped students to interiorize the process of fraction addition.

My point in giving these examples, I suppose, is to illustrate why I’ve come to think that the debates about digital versus physical mathematics manipulatives are placing too much emphasis on physical versus digital as the difference that matters. In some cases, digital versions have affordances that physical versions do not. I’m sure it’s also that case in reverse.

I’ve been struck, in the past, as I’ve noted that there really aren’t many studies that pit physical and digital versions of manipulatives against each each other in terms of effectiveness. It seemed like such an easy and straightforward area of inquiry. Now I think I’m coming to understand why so little research is formulated that way. Studies that show a difference between physical and digital tools for learning would only open more questions about which affordances of each version led to the differences. Digital versus physical isn’t really what matters.


Bouck, E. C., Park, J., Sprick, J., Shurr, J., Bassette, L., & Whorley, A. (2017). Using the virtual-abstract instructional sequence to teach addition of fractions. Research in Developmental Disabilities, 70(June), 163–174.

Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150.

Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22, 1–36.