Hello, again. Welcome to the third installment of my musings on virtual manipulatives.

To briefly recap:

Two weeks ago, I shared a piece of a course paper that used ideas from Seymour Papert and James Gee to explain why virtual manipulatives seem to have an edge over their physical counterparts for promoting mathematical learning.

Last week, I discussed research (DeLoache, 2000) showing that when a representation is perceptually rich, young children have a harder time thinking of it as a representation of something else, rather than just an interesting object in its own right. I discussed the implications of this for use of manipulatives to model word problems, and put an idea on the table about how virtual manipulatives might be designed to overcome some of the issues associated with physical manipulatives. In short, perceptually rich manipulatives seem to help kids better makes sense of problems, but bland manipulatives seem to help kids focus on the mathematics (McNeil, Uttal, Jarvin, & Sternberg, 2009). I suggested the development of a virtual manipulative that fades and restores its perceptual richness during the appropriate stages of the mathematical modeling process.

But what about when manipulatives are used outside of contextualized problem solving? What about the cases where manipulatives are used as a model of mathematical concepts but not of real-world contexts? Consider, for example, when we ask students to use base-10 blocks to add whole numbers, or to use fraction circle pieces to add fractions. We are, in essence, hoping that the blocks and pieces function as representations of mathematical ideas, and nothing else. Today, I’m going to discuss the implications of the research on perceptual richness for that.

In my admittedly brief literature search on this topic, I did not find a study that removed context entirely, but I did find one with a problem context that I don’t think rises to the use of mathematical modeling. Peterson and McNeil (2013) asked preschoolers to complete a counting task using manipulatives with varying perceptual richness. Children were told they needed to count the objects to help out a puppet character—to either check whether the puppet’s counting was correct, or count out the number of objects the puppet asked for. So, there was an element of context, but only to set up the mathematical task.

In addition to varying the perceptual richness of the manipulatives, Peterson and McNeil (2013) also varied the familiarity that the children had with the objects. So, some of the perceptually rich manipulatives were objects found in their classroom, like miniature animals. Others were objects that they had not seen before, like glitter pom-poms from a craft store. Similarly, the bland manipulatives varied according to student familiarity; craft sticks were considered bland but highly familiar, and wooden pegs were considered bland but unfamiliar.

The results of the study were, in short, that perceptual richness depressed students’ performance on the counting task when the manipulatives were familiar, but enhanced their performance with the manipulatives were new (Peterson & McNeil, 2013). I don’t think I would have predicted this, but the results do make intuitive sense to me. If children have used perceptually rich manipulative for a different purpose—for example, if they have used toy animals in imaginative play—they have a hard time seeing those manipulatives as representations of the mathematics. This jives with the results of the DeLoache (2000) study, particularly the experiment where DeLoache asked students to play with the 3D model of the room before asking them to use the model as a representation of the real room. The play led students to think of the model as a toy in its own right, and then they had a harder time using it as a representation (DeLoache, 2000).

The more interesting result, though, is that when students’ first use of a perceptually rich manipulative is to use it as a mathematics manipulative, the perceptual richness might actually be beneficial (McNeil & Peterson, 2013). This at first seems to counter the DeLoache (2000) study, as the 3D model was new to the children in that study and they still had trouble using it as a representation. However, even though the specific 3D model was new to children, it’s quite possible (maybe even likely) that the children had seen a dollhouse or other miniature scene like it before. In that case, the children might have been “familiar” with the model in a way that led to that familiarity depressing their ability to use the model as a representation.

So, returning to the original question about students using base-10 blocks and fraction circle pieces to add and subtract, what do the results of the McNeil and Peterson (2013) study suggest? Overall, the results made me feel a little better about the brightly colored fraction circle pieces I see used in many classrooms. After first reading the DeLoache (2000) piece, I wondered if we were doing kids a disservice by making them so colorful (i.e., perceptually rich). Because kids are not likely to have seen them before using them to think about fractions, maybe the color actually helps. I do wonder, though, whether giving students a period of open exploration of materials before using them to work with fractions—a common practice among teachers, in my experience—might be doing some harm to the ability of kids to use them as a fraction representation. It’s an interesting empirical question.

When it comes to virtual manipulatives, I wonder what the implications for this line of research are for manipulatives that are embedded within apps. Watts et al. (2016) studied students’ use of a variety of virtual manipulative apps, some of which had little in the way of context or familiar elements, and others that involved a game-like setting with lots of familiar elements, like fish, fruit, and dogs. Watts et al. took into account many aspects of the apps, such as whether the tasks were open or closed ended. However, the issue of perceptual richness wasn’t treated as a variable, and I wonder what kind of influence it had on students’ use of the apps. That’s another interesting empirical question.

Putting together the thoughts from last week’s post and this one, then, here is where I end up:

- The potential of virtual manipulatives to vary their perceptual richness based on the point in the modeling process is exciting, but…
- That also means we might need to be more careful, both when designing and studying virtual manipulatives, to think about the effects that perceptual richness
*and*students’ familiarity with the elements of richness might have on the ability of the manipulative to serve as a mathematical representation for kids.

In short, perceptual richness of manipulatives is more easily controlled in a virtual environment than a physical one, and we should use that control to our advantage to support students’ use of manipulatives as mathematical representations.

Thus ends our series on virtual manipulatives. Hope you found it interesting! Next week will start a new 3-part series. Stay tuned for the topic. 🙂

**References**

Deloache, J. S. (2000). Dual representation and young children’s use of scale models. *Child Development*, *71*(2), 329–338.

McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. *Learning and Instruction*, *19*(2), 171–184.

Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers’ counting performance: Established knowledge counts. *Child Development*, *84*(3), 1020–1033.

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. https://doi.org/10.1016/j.chb.2016.07.029