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.

**References**

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.