Un-structuring mathematics

In two of my courses this semester, we’ve spent some time talking about well-structured domains (WSDs) and ill-structured domains (ISDs) and the ways in which beneficial instruction might look different for each. A well-structured domain is one in which all concepts and procedures can be readily delineated and described. By contrast, “[i]ll-structured domains are characterized by being indeterminate, inexact, noncodifiable, nonalgorithmic, nonroutinizable, imperfectly predictable, nondecomposable into additive elements, and, in various ways, disorderly” (Spiro & DeSchryver, 2009, p. 107). As examples, Spiro and DeSchryver (2009) describe the idea of muscles bending a joint as a complex, but ultimately well-structured domain. There are many facets to this process, but in the end all those facets can be well described and the way in which muscles work is consistent across human beings. By contrast, the concept of justice is an example of an ISD because its application and meaning across instances will vary considerably.

Spiro and DeSchryver’s (2009) principle argument is that highly guided and direct instruction may well be the most effective approach for learning material in WSDs, but such approaches are ineffective in ISDs. Providing definitions and a discrete set of examples of justice, for example, serves to provides students with a narrow understanding of the term and can promote significant misconceptions. Spiro and DeSchryver argue that instruction in ISDs should therefore facilitate “a nonlinear criss-crossing of knowledge terrains to resist the development of oversimplified understandings” (p. 115). Students should be given experiences that help them apply existing pieces of knowledge about justice, for example, in new ways, so that they can think flexibly about the concept.

For the most part, I buy into this idea. I agree that we tend to teach kids to think too rigidly about concepts. Here’s my problem with the discussions surrounding this, though: While history and philosophy are the typical examples given for ISDs, mathematics is almost always used as a go-to example of a WSD. For mathematics, the narrative says, direct instruction is just fine!

This deeply bothers me.

I’m not going to try to make the claim that structure doesn’t play a huge role in mathematics. I completely understand why people think of mathematics as well-structured. Being a mathematician is, in large part, about seeing structure in an ill-structured world. My worry is that references to mathematics as a WSD justify the perpetuation of mathematics instructional practices that are both problematic and deeply entrenched in our instructional system.

Take for example, the standard algorithms for the four basic operations. Claims that direct, highly guided instruction is optimal in mathematics would suggest that explicit teaching of the standard algorithms is fine. Yet we’ve known for years that rote teaching of algorithms, without accompanying opportunities to invent algorithms, is harmful to kids’ number sense and understanding of place value (Kamii & Dominick, 1997). And what about word problems? If direct instruction is suitable for mathematics, it would seem that all those superficial word problems we place at the end of lesson problem sets are just fine. Yet we’ve know for years that students’ school experiences with word problems leads them to dissociate school mathematics with sensemaking — they don’t take context into account when solving word problems (Silver, Shapiro, & Deutsch, 1993).

This has gotten me thinking harder about whether or not I believe mathematics is really well-structured in the way that Spiro and DeSchryver (2009) describe. Does addition, for example, have a straightforward definition? If we’re thinking about it as an operation on abstract numbers, then maybe it is. But if we’re thinking about how it applies to contexts, then I do think it can have many means. Addition, after all, has multiple use cases. It’s useful when you want to find a total (4 blue fish and 3 red fish, how many all together?), make a change that results in more (4 blue fish and 3 more blue fish come, how many blue fish now?), understand a comparison (I have 4 blue fish and 3 more red fish than blue fish — how many red fish?), and so on (Usiskin & Bell, 1983). Given this, is addition any more easily described to students than the ill-structured concept of justice? I am not so sure.

I like Spiro and DeSchryver’s (2009) call to help students “criss-cross” ill-structured domains to avoid oversimplified understanding. I just wish that mathematics were not so often casually discussed as the domain where it does not apply.


Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16(1), 51-61.

Silver, E. A., Shapiro, L. J., & Deutsch, A. (1993). Sense making and the solution of division problems Involving remainders: An examination of middle school students ’ solution processes and their interpretations of solutions. Journal for Research in Mathematics Education, 24(2), 117–135.

Spiro, R., & DeSchryver, M. (2009). Constructivism: When it’s the wrong idea and when it’s the only idea. In S. Tobias & T. Duffy (Eds.), Constructivist Theory Applied to Instruction: Success or Failure (pp. 106–123). Mahwah, NJ: Lawrence Erlbaum.

Usiskin, Z., & Bell, M. (1983). Applying arithmetic: A handbook of applications of arithmetic. University of Chicago.

Making the Leap

I just returned from my first time at the annual SITE (Society of Information Technology in Teacher Education) conference. I had a wonderful time learning about all sorts of ed tech research and connecting with new colleagues.

My contribution to the conference was a brief presentation about the ways in which we (meaning the research team I’m a member of here at Michigan State) have been thinking about bringing computational thinking (CT) into teacher education (Rich & Yadav, 2018). Hopefully, I’ll be able to point to a few publications about these ideas soon, but in the meantime, the crux of our argument is as follows:

  • CT ideas are present in many subjects — not just computer science (e.g., Barr & Stephenson, 2011).
  • Teachers need to understand CT in the context of what they teach (Yadav, Stephenson, & Hong, 2017).
  • Thus, we need to bring CT into methods courses in preservice teacher preparation programs.
  • Starting out with unplugged CT activities (that is, activities that do not involve any tech), and building first to low-tech and then high-tech activities, will build on what teachers know and are comfortable doing (National Research Council [NRC], 2011).

I talked about a series of fractions activities to illustrate our proposed no tech to low tech to high tech continuum:

  • No tech. As an entry point, preservice teachers (PSTs) could think of a fraction, such as ⅔, as an abstraction — a symbol that represents a lot of coordinated ideas, including two different numbers of parts and the ideas of equal-sized pieces (Confrey & Maloney, 2015). They could use decomposition to break down the task of identifying examples of a particular fraction into a series of progressive sorts of fraction cards. This would be an entirely unplugged activity that highlights key CT ideas.
  • Low tech. A next step might be using virtual manipulatives to help students explore connections between different representations of fractions. Many available virtual manipulatives use simultaneously linking of representations (Moyer-Packenham & Westenskow, 2013), so that one representation automatically updates in response to changes to another. Using such linked representations can support students in making mathematical generalizations (Anderson-Pence & Moyer-Packenham, 2016). Discussing with PSTs (and helping them think through ways to discuss with their students) how the affordances of technology were helpful can start to connect CT to the power of technology. We think of these kinds of uses of precreated technology as low-tech activities.


  • High tech. After thinking through the connections between CT practices and technology, PSTs and their students could move on to creation of technology, via programming in any of the available student-friendly programming languages. For example, they could create an algorithm that compares a fraction to 1 based on the value of its numerator and denominator.


My presentation of our proposed no tech → low tech → high tech progression was the last presentation in a symposium on CT in teacher education. The conversation with the attendees, after the talks were completed, focused in on a key difficulty in bringing CT to teacher education: making the leap from unplugged to plugged contexts for applying CT is very challenging for teachers. Several attendees shared experiences that made this challenge apparent.

I really appreciated this discussion because it made me realize something that has been incomplete, or even backward, in my thinking about integrating CT in other disciplines. As I wrote about a few weeks ago, I think a key challenge in bringing CT to kids via integration with mathematics is figuring out ways to help student apply the CT ideas already embedded in mathematics to computer science. That was the leap that concerned me. When thinking about teachers, on the other hand, my concern was reversed. Perhaps because I first learned about CT in the context of computer-science-specific initiatives, I pictured teachers learning about CS/CT in isolation, and then needing help seeing the CT in the context of other disciplines.

In short, for kids, I thought about the leap from CT in math to CT in computer science. For teachers, I thought about the leap from CT in computer science to CT in math. I realize now that this separation is artificial. Either way, the leap is going to be difficult. What we’re talking about is transfer — something that is notoriously difficult in education no matter what the context.

Will thinking about progressions like the one I outlined above help with transfer? I don’t know, but I don’t think it can hurt. I’m really excited to start exploring these ideas more.


Anderson-Pence, K. L., & Moyer-Packenham, P. S. (2016). The Influence of Different Virtual Manipulative Types on Student-Led Techno-Mathematical Discourse. Journal of Computers in Mathematics and Science Teaching, 35(1), 5–31.

Barr, V., & Stephenson, C. (2011). Bringing computational thinking to K-12. ACM Inroads, 2(1), 48.

Confrey, J., & Maloney, A. (2015). A design research study of a curriculum and diagnostic assessment system for a learning trajectory on equipartitioning. ZDM – Mathematics Education, 47(6), 919–932.

Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35–50.

National Research Council (NRC). (2011). Report of a Workshop of Pedagogical Aspects of Computational Thinking. Washington, DC: The National Academies Press.

Rich, K. M. & Yadav, A. (2018, March). Unplugged approaches to CT: Embedding computational ideas into teacher education. In J. Voogt (Chair), Learning and Teaching Computational Thinking – Challenges for Teacher Education. Symposium conducted at the annual meeting of the Society for Information Technology in Teacher Education (SITE), Washington, DC.

Wing, J. M. (2006). Computational Thinking. Communications of the ACM, 49(3), 33–35.

Yadav, A., Stephenson, C., & Hong, H. (2017). Computational thinking for teacher education. Communications of the ACM, 60(4), 55–62.


Shifting identities

During my middle- and high-school years, I was what I called a “two-school kid”: I attended a large public high school in the mornings, and after lunch I bused or drove across town to a learning center called the Center for the Arts and Sciences (CAS). Attendance at the CAS required choosing a specialization. Each student spent a full morning or a full afternoon in classes in language arts, global studies, 2D or 3D visual arts, voice and keyboard, dance, theater, or, in my case, math and science. For the most part, this was an amazing experience. I took two semesters of college-level calculus in high school and also got to do some amazing hands-on exploration in science that I have since learned are not at all common in U.S. schools. I don’t have many complaints about my experiences at the CAS. Still, there is one aspect of my time there that bothered me then and still, now and again, bothers me when I think about it now.

Although most CAS students chose their own specialization, some of us still felt… sorted. It often felt like our choices served as ways to separate us. Specialization allowed the time and space to explore topics in great depth, but it also led to some “us” versus “them” mentality. This was particularly pronounced in the relationship between the math/science department and the theater department. I had a few friends — great friends! — in the theater program, but this didn’t mean that I was unaware of the general narrative that math/science students were serious, studious, and nerdy, while theater students were creative, fun-loving free spirits. It wasn’t the oversimplification that bothered me. It was the separation. I didn’t see either of the characterizations as inherently better than the other. It wasn’t that I ever wanted to switch to the theater program and felt like I couldn’t. It was more that sometimes I wanted to be seen as someone who loved theater, creativity, and fun. But as long as I was a math/science student, that wasn’t a persona I was able to put on.

Fast-forward (eep!) 16 years to a few days ago, when I was reading James Gee’s (2013) Good Video Games + Good Learning. In the first chapter of the book, when discussing his idea of affinity spaces, Gee says,

We are never, none of us, one thing all the time. Sure, the world continuously tries to impose rigid identities on us all of the time. But it is our moral obligation — and one necessary for a healthy life — to resist this and to try to create spaces where identities based on shared passions or commitments can predominate. (p. 7)

I was really struck by his first sentence: We are never, none of us, one thing all the time. His point isn’t that we can be more than one thing, which is the argument I usually make about identities. My issue in high school wasn’t that I wanted to be a math/science student or a theater student. It was that I wanted to be both. The multiplicity was what I was focused on. But Gee takes this a step further. Not only can you be more than one thing, but you don’t have to be all of those things all at once and all the time.

This seems like a subtle and maybe obvious point, but it changed my thinking about the identity I’m most interested in from a research perspective: that of being a “math person.” It bothers me greatly the number of students and adults who will outright tell me that they are “not math people.” I think the pervasiveness of this narrative is a reflection of the inflexible way mathematics has always been taught. I often talk about one of my career goals as working toward a world where every K-12 student will say, “I am a math person!”

Most of the self-proclaimed “non-math people,” upon hearing this from me, remain skeptical, despite my passionate narratives about the work of people like Carol Dweck (1999) and Jo Boaler (1989). You just had a bad experience! I say. You are a math person, I swear! They never believe me.

Gee’s (2013) comments have me thinking that there might be a way to make my message more accurate, and therefore believable to even the skeptics. Perhaps people (adults, and younger students) can’t imagine taking on “math person” as an identity to wear all the time. But maybe they can imagine putting on a “math person” hat on occasion. Maybe the fluid nature of identities can help people realize that just about anything — even math — is within their realm of possibilities.

Being a math person doesn’t have to mean fundamentally and permanently changing who are. That’s my new thought for the day.


Boaler, J. (1989). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.

Dweck, C. S. (1999). Caution—Praise can be dangerous. American Educator, 23(1), 4–9.

Gee, J. P. (2013). Good Video Games + Good Learning. New York: Peter Lang Publishing.


Refusing to Recognize

In one of my courses this semester, we have an ongoing assignment as we work through the reading list. As we read, we are to pull out quotations that can be unpacked multiple ways. Rather than looking for passages that succinctly sum up one point, our goal is to pull out passages that can be connected to other aspects of the piece (and to other readings from the course) in multiple ways. To point out the connections we see, we’re tagging these multi-spoke quotations with themes.

One of our early readings in the course was Seymour Papert’s Mindstorms. There is one passage I picked out that I’ve been mulling over ever since:

Dynaturtles can be put into patterns of motion for aesthetic, fanciful, or playful purposes in addition to simulating real or invented physical laws. The too narrowly focused physics teacher might see all this as a waste of time: The real job is to understand physics. But I wish to argue for a different philosophy of physics education. It is my belief that learning physics consists of bringing physics knowledge into contact with very diverse personal knowledge. And to do this we should allow the learner to construct and work with transitional systems that the physicist may refuse to recognize as physics. (Papert, 1980, p. 122, emphasis added)

I really liked this passage for a lot of reasons. First, I noted the bold phase, because it illustrates the way Papert argues against the separation between play and learning — a separation I believe we as educators need to keep pushing against every single day. I also liked his reference (shown in italics) to “very diverse personal knowledge,” which I interpret as attention to equity. He doesn’t just mention personal knowledge. He mentions very diverse personal knowledge. I think this reflects a theme that runs through his book: he believes every student can and should find a personal connection to content.

Even though I like this quotation for multiple reasons, it’s the last highlighted phrase, in bold italic, that has been niggling in the back of by brain for a while. At the end of the passage, Papert argues that students and educators should take advantage of “transitional systems that the physicist may refuse to recognize as physics.” The word refuse is what first caught my eye, I think. Papert doesn’t say that physicists cannot or do not recognize work with dynaturtles as physics. He says they refuse. There’s an intentionality in that word. An implication of a decision to deny any connection between dynaturtles and physics.

I have not worked with any physicists in any educational pursuits, but I’m willing to believe that refuse was an appropriate word choice. In mathematics education, at least, one only has to look as far as the “Where’s the math?” debates (Heid, 2010; Martin, 2010; Battista, 2010; Confrey, 2010) to realize that there is evidence of refusal to recognize certain experiences as mathematics learning. While I do not go so far as to claim that disciplines have no boundaries, I do worry about refusals to recognize experiences as related to mathematics. As the incomparable Stephen Hawking (may he rest in peace!) once said, “People have the mistaken impression that mathematics is just equations. In fact, equations are just the boring part of mathematics” (Overbye, 2018). I think that such misconceptions stem from the refusals of professional mathematicians to recognize informal pursuits as mathematics — and I further believe that these misconceptions have contributed making the “I’m not a math person narrative” so prevalent and socially acceptable.

When I first read the Papert quotation above, some part of my mind thought it might be connected to computer science education, but I was not able to articulate the thought right away. A few days ago I finally put my finger on it.

The ongoing efforts to bring computational thinking into K-12 education have led to some discussions about whether or not the CT-related practices already happening in mathematics and science courses can rightfully be considered CT. As I wrote on this blog a couple of weeks ago, I do think there is CT all over elementary mathematics. I also think that explicit connections between these math-embedded CT ideas and CT as applied specifically to computing need to be made for kids. But in order to effectively develop instruction that facilitates those connections, I think we need to recognize CT — everywhere that it lives, in all the “diverse personal knowledge” (Papert, 1980) kids bring to and gain in school — as CT.

Why do I worry that we’re not doing that? Consider this passage from a recent study:

“[I]n our study, the result of student classroom observations, interviews with teachers, and participatory engagement rarely led to activities that fit the prototype of CT if CT means using elements easily recognizable to computer scientists as computer science, that is, creating algorithms and meta-level descriptions of code, coding, engaging in explicit acts of structure creation, structured top-down problem-solving, and so forth. Instead, we found important opportunities to address proto-computational thinking (PCT). PCT consists of aspects of thought that may not put all the elements of CT together in a way that clearly distinguishes them from other human intellectual activity.” (Tatar et al., 2017, p. 65)

At first, I was on board with “proto-CT” as a label for the mathematical and scientific ideas that have elements of CT. Now I wonder, though, if calling that knowledge by a different name is a refusal to recognize it as CT (as Papert would say). And if it is a refusal, I think we need to examine the implications of that refusal for bringing computer science to all.


Battista, M. T. (2010). Engaging Students in Meaningful Mathematics Learning: Different Perspectives, Complementary Goals. Journal of Urban Mathematics Education, 3(2), 34–46.

Confrey, J. (2010). “ Both And ”— Equity and Mathematics: A Response to Martin, Gholson, and Leonard. Journal of Urban Mathematics Education, 3(2), 25–33.

Heid, M. K. (2010). Editorial: Where’s the Math (in Mathematics Education Research)? Journal for Research in Mathematics Education, 41(2), 102–103.

Martin, D. B., Gholson, M. L., & Leonard, J. (2010). Privilege in the Production of Knowledge. Journal of Urban Mathematics Education, 3(2), 12–24.

Overbye, D. (2018). “Stephen Hawking Dies at 76: His Mind Roamed the Cosmos.” The New York Times. Retrieved from https://www.nytimes.com/2018/03/14/obituaries/stephen-hawking-dead.html

Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books.

Tatar, D., Harrison, S., Stewart, M., Frisina, C., & Musaeus, P. (2017). Proto-computational Thinking: The Uncomfortable Underpinnings. In P. J. Rich & C. B. Hodges (Eds.), Emerging Research, Practice, and Policy on Computational Thinking (pp. 63–81). Cham, Switzerland: Springer.


The new curriculum material

I’ve been delving into research on virtual manipulatives this semester, and recently I’ve asked myself this question: Are virtual manipulatives curriculum materials?

From what I know of print curriculum-materials research (e.g., Choppin, 2011; Drake & Sherin, 2009; Remillard & Bryans, 2004), physical manipulatives haven’t generally been considered as curriculum materials in their own right. On the other hand, in the last few months I’ve read at least two studies (Hansen, Mavrikis, & Geraniou, 2016; Trgalová, & Rousson, 2017) that treat a digital manipulative as a curriculum material, studying their use quite similarly to the way that use of curriculum guides have been studied.

Why the contrast? I think there are at least two reasons.

First, virtual manipulatives have a lot more elements that can be intentionally designed. Actions on virtual manipulatives can be constrained (Manches, O’Malley, & Benford, 2010; Moyer-Packenham & Westenskow, 2013), sound and visual effects can be chosen (Moyer-Packenham et al., 2016; Paek, Hoffman, & Black, 2016), representations can be dynamically linked to each other (Moyer-Packenham & Westenskow, 2013; Sarama & Clements, 2009), and so on. Any sort of deliberate choice — of which there are many — by a designer is likely influenced by a particular imagined use.

Second, it’s common for virtual manipulatives to be embedded in apps or websites that either directly provide or are highly suggestive of specific tasks (e.g., Watts et al., 2016). This is in contrast to most physical manipulatives, which are generally sold and stored independently of any particular scheme of use. Activities (within curriculum materials or elsewhere) often assume their use, but the manipulatives themselves don’t assume a task the way that tasks sometimes assume use of physical manipulatives.

In all, virtual manipulatives seem much harder than physical manipulatives to separate from the tasks they’re used to accomplish. And tasks are at the heart of curriculum. While physical manipulatives have not generally been considered as curriculum materials, I think virtual manipulatives should be.

Why does it matter?

One of my biggest worries related to technology trends in classrooms is the overuse of adaptive-tutor-like programs that (claim to) use sophisticated algorithms to diagnose students’ exact position along learning paths and walk them forward. I think the desires for automated assessment and personalized learning are, to a point, understandable, but I worry that overuse of such programs removes the teacher from having a role in making instructional decisions. It speaks to a trend of deprofessionalization of teaching, which I think is a mistake. My hope would be that if we treat virtual manipulatives as curriculum materials — if we take seriously the teacher’s role in their use (Remillard, 2005) — it might prevent the potential of virtual manipulatives as teaching tools from morphing into attempts to turn virtual manipulatives into replacements for teachers.


Choppin, J. (2011). Learned adaptations: Teachers’ understanding and use of curriculum resources. Journal of Mathematics Teacher Education, 14(5), 331–353.

Drake, C., & Sherin, M. G. (2009). Developing curriculum vision and trust: Changes in teachers’ curriculum strategies. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics Teachers at Work: Connecting Curriculum Materials and Classroom Instruction (pp. 322–337). New York: Routledge.

Hansen, A., Mavrikis, M., & Geraniou, E. (2016). Supporting teachers’ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative. Journal of Mathematics Teacher Education, 19(2–3), 205–226.

Manches, A., O’Malley, C., & Benford, S. (2010). The role of physical representations in solving number problems: A comparison of young children’s use of physical and virtual materials. Computers and Education, 54(3), 622–640.

Moyer-Packenham, P. S., Bullock, E. K., Shumway, J. F., Tucker, S. I., Watts, C. M., Westenskow, A., … Jordan, K. (2016). The role of affordances in children’s learning performance and efficiency when using virtual manipulative mathematics touch-screen apps. Mathematics Education Research Journal, 28(1), 79–105.

Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35–50.

Paek, S., Hoffman, D. L., & Black, J. B. (2016). Perceptual factors and learning in digital environments. Educational Technology Research and Development, 64(3), 435–457.

Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.

Remillard, J. T., & Bryans, B. (2004). Teachers ’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352–388.

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

Trgalová, J., & Rousson, L. (2017). Model of appropriation of a curricular resource: a case of a digital game for the teaching of enumeration skills in kindergarten. ZDM – Mathematics Education, 49(5), 769–784.

Watts, C. M., Moyer-Packenham, P. S., Tucker, S. I., Bullock, E. P., Shumway, J. F., Westenskow, A., … Jordan, K. (2016). An examination of children’s learning progression shifts while using touch screen virtual manipulative mathematics apps. Computers in Human Behavior, 64, 814–828.


Computationally conscious

Through my work on the Learning Trajectories for Everyday Computing (LTEC) project and the Computational Thinking for Education (CT4EDU) project(1), I spend significant amounts of time these days thinking about this question:

How can I bring computational thinking (CT) into teaching and learning of elementary school mathematics?

The natural starting point for this exploration is to think about existing connections — or whether there are any. So, first I asked:

Are kids thinking computationally while they do elementary mathematics? If so, when?

The answer to this question depends a lot on how you define computational thinking, and also on how kids are engaged in mathematics. But based on the definitions of CT I’ve seen emerging in the CS education field and my understanding of reform mathematics teaching, I’ve come to be pretty solid in my belief that the answer to the above question is – or at least can be – yes.

Consider, for example, these two excerpts from relevant documents:

Generalization is associated with identifying patterns, similarities and connections, and exploiting those features. It is a way of quickly solving new problems based on previous solutions to problems, and building on prior experience. (Bocconi, Chioccariello, Dettori, Ferrari, & Englehardt, 2016, p. 18)

Mathematically proficient students look closely to discern a pattern or structure… Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. (Common Core State Standards Initiative [CCSSI], 2010, p. 8)

The first excerpt is from a recent European Commission report on the state of CT in compulsory education. The report defines generalization as a component of CT. The second excerpt is from the Common Core State Standards for Mathematics (CCSS-M), specifically, the Standards for Mathematical Practice. It’s hard not to notice the parallels. In both CT and math practices, kids identify patterns and exploit them in solving later problems.

Because of similarities like this, I’ve been pretty consistent in my belief that kids do CT when they do math. They develop their own algorithms for counting, adding, subtracting, multiplying, and dividing. They decompose problems into easier parts — for example, adding the 10s and the 1s separately. They abstract important features of problems when they write number models for word problems. They debug their own thinking and problem solving processes when they use estimation to check whether their answers make sense. They think carefully about the impact of sequencing when they use parentheses. They apply conditional reasoning when they sort shapes.

I could continue, but I think I’ve made my point. Once I started looking for it, I saw CT everywhere in elementary school mathematics. And I know there are lots of amazing teachers and wonderful resources out there that help to make this kind of thinking happen every day in elementary school classrooms.

At first, this made it seem like my job — namely, to think about how to bring CT to elementary mathematics — was basically done for me. I don’t have to do anything! It’s already there! I can just kick back and watch it happen!

But then when I stepped back and thought about why CT has become an area of focus in education, I quickly recalled that a major part of the argument for bringing CT into compulsory education is to get kids ready to be literate citizens in an increasingly computer-influenced world. And I had to stop and ask myself:

Is the kind of mathematics teaching and learning that includes CT ideas — discovery-oriented, constructivist, reform-based, practice-focused, whatever label you like — getting kids ready for the increasingly computational world?

And relatedly:

If it is, why are we talking about bringing CT to education if it is already there?

I’m skeptical that the answer to the first question is yes. My admittedly anecdotal argument for that skepticism is that I had the benefit of a mathematics education experience that taught me a lot of about abstraction, pattern generalization, and decomposition, and also the benefit of a decade of thinking about how to make those things a bigger focus in classrooms in my professional life. But it wasn’t until a few years ago that I made any connection between those experiences and computers. So why would I expect kids and teachers to do that spontaneously?

So now we come full circle to how I started this post. For several years, I’ve thought of my work as focused on figuring out how to bring CT to elementary mathematics. Recently, though, I’ve been considering whether I’ve been asking the wrong question. CT is already there. I think perhaps the more helpful questions include:

At what point does conscious awareness of the CT embedded within mathematics become important to supporting students’ transfer of these skills to computing?

What are the key differences between the CT embedded in math and CT applied directly to computing?

How do we give kids experiences that facilitate use of their mathematics practices to computing?

I’m not sure these are exactly right. But I do think these are more difficult, and more important, than the questions that have guiding my thinking to this point.


Bocconi, S., Chioccariello, A., Dettori, G., Ferrari, A., Engelhardt, K. (2016). Developing computational thinking in compulsory education – Implications for policy and practice; EUR 28295 EN.

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/Math/

  1. This is my disclaimer that any views expressed in this blog post don’t necessarily reflect the views of these project teams!

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. https://doi.org/10.1207/s15326985ep4102

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 https://www.edge.org/responses/how-is-the-internet-changing-the-way-you-think

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.