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.

**References**

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. https://doi.org/10.1016/j.ridd.2017.09.002

Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. *Child Development Perspectives*, *3*(3), 145–150. https://doi.org/10.1111/j.1750-8606.2009.00095.x

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. https://doi.org/10.1007/BF00302715