Hello again, all.

I’m trying something a bit different with my blog this year. As I noted last week, I’m hoping to do a better job this year of staying with ideas a bit longer instead of dabbling in lots of different things all the time. I think a good way to facilitate that might be to have sets of themed blog posts. Sets of three related posts seems a reasonable place to start.

So, welcome to virtual vs. physical manipulatives, part deux.

Last week I shared a piece of one of my course papers that discussed a few reasons why virtual manipulatives might have an edge over their physical counterparts in terms of instructional effectiveness (at least in certain contexts). I read an article for a course this week that led me to another possible reason. It’s going to take me a little while to get to the physical versus digital comparison, but stay with me — I promise there are some really interesting tidbits along the way.

As an assignment for my cognitive development class, I read a piece by Judy DeLoache (2000) about young children’s ability to use scale models to reason about real objects. Researchers showed 2.5- to 3-year-old children where a stuffed toy was hidden in some sort of model of a real room (more on the models in a minute). After kids were shown where the toy was hidden via the model, they were let into the real room and asked to find the stuffed toy. Researchers recorded whether the kids found the stuffed toy in the first place they looked (in which case it was likely they understood how the model of the room mapped onto the real thing) or searched more randomly.

The main independent variable of interest was the physical or perceptual salience of the model. Sometimes, the model kids saw was a 3D physical model similar to a dollhouse. Sometimes, it was a photo of the room. Still other times, it was a 3D model, but a piece of glass was in front of it so children could not interact with the model. Consistently, in this study and several related studies, children were more likely to find the stuffed toy immediately when they saw a less perceptually salient model. That is, kids were more successful when they saw the flat photo of the room or the 3D model behind glass than when they saw the 3D model with no glass.

This result was very counterintuitive to me. Why would it be easier for kids to map a photo of a room onto the real room than to map a 3D model? The 3D model is a closer match! DeLoache (2000) theorized that the 3D model was more interesting to children as an object in itself, and so they had a harder time using it as a representation of something else. To further support this theory, the researchers ran another experiment, asking some children to play with the 3D model before showing them where the stuffed toy was hidden. Children who physically manipulated the model performed worse on the task than those who did not. Assuming that playing with the model made children more interested in the model in its own right — a reasonable assumption — this result is supports the theory that increased perceptual salience of an object makes it more difficult for children to use the object as a representation for something else. Once children think of the model as a toy, for example, they have a harder time thinking of it as a representation of the real room.

This theory got me thinking more about manipulatives. We give young kids manipulatives all the time in early mathematics classes, and sometimes those manipulatives are quite interesting in and of themselves. Counters are often shaped like animals, and fraction pieces come in bright colors. By making the manipulatives visually appealing, are we actually making it harder for kids to think of them as representations of mathematical objects?

I couldn’t help but recall an episode in a first-grade classroom several years ago, when I was working with a child to create combinations of 10 using red and green counters. The goal of the task was for children to generate as many different combinations as they could — 1 and 9, 4 and 6, and so on. This particular child spent most of the time I was with him (maybe 10 minutes?) making the same combination of 5 and 5, but arranging the colors differently. First he make a row of 5 red and a row of 5 green. Then he alternated them in a single line – red, green, red, green, and so on to 10. I tried a couple of tactics to get him to see that these were all the same combination, but I didn’t get there. He insisted they were different. After reading the DeLoache (2000) piece, I realized that he couldn’t see the 5 and 5 combinations as the same because he wasn’t seeing them as representing 10. He was seeing red and green counters — just that! — and those counters were in arrangements that were meaningfully different to him.

What might this mean instructionally? Should we make manipulatives as bland as possible? Would that help kids see them as representations of mathematical concepts? Unsurprisingly, I am not the first person to think about this, and also unsurprisingly, the answer is not simple. In one study, researchers compared fourth-grade students’ performance on money-based word problems according to whether they used coins and bills that closely resembled real money or paper rectangles and circles simply labeled as dollars and coins (McNeil, Uttal, Jarvin, & Sternberg, 2009). Students who used the simpler manipulatives answered more problems correctly, suggesting (at first glance) that less perceptually rich manipulatives are better. However, the authors also analyzed the errors made by students in each group and found that students using the realistic bills and coins made fewer conceptual errors. That is, students who used the realistic bills and coins were less likely to completely misinterpret the problems — their errors tended to be arithmetic mistakes. Students using the bland manipulatives were more likely to misinterpret the problem conceptually (e.g., use the wrong arithmetic operation), but overall had better performance.

As a potential explanation for this finding, McNeil et al. (2009) theorized that the two kinds of manipulatives are good for different purposes. The realistic bills and coins helped kids make sense of the word problems, perhaps because the students more readily see them as representations of real money. The bland versions helped students carry out the mathematics once the problems had been correctly interpreted, perhaps because students more easily see them as representations of decimal numbers. So, each manipulative might be better, depending on what you’re hoping that they’ll help kids model — the connection to the real world, or the mathematics itself.

The trouble is, when we ask kids to solve word problems, we’re asking them to connect all the way from the real-world context to the mathematics. And neither version of the manipulative is getting them all the way there.

Oy. Is anyone else feeling like education is impossible? We can’t ask kids to use two different manipulatives in the course of solving one word problem. I can feel all the elementary school teachers in the world laughing derisively at that idea right now.

But… What if one manipulative could start out perceptually rich and then have its perceptual details temporarily fade away?

We could do that with a digital medium. And now I’m super curious whether it would help kids better understand mathematical modeling.

In one of the books I used to work on as an editor and curriculum developer, there is a diagram of mathematical modeling that looks something like this*:

What if there were a virtual manipulative where kids could choose when to toggle between the real world and the world of mathematics? They could mess around with realistic coins and bills until they decide how to proceed mathematically. They could click an arrow or button to enter the world of mathematics, and the details of the manipulatives could fade away, making it easier for kids to focus on the mathematics. Then they could come back to the real world, and the perceptual details would come back, helping them recall the context and interpret the answer.

I have no idea if this would work. Part of me thinks it might be just as complicated as asking students to use two different manipulatives. But for the common contexts for word problems — money for decimals, sharing pizzas for fractions, counting toys or animals for whole numbers — I don’t think the manipulative itself would be hard to make.

Anyone want to make it for me so I can study it? Think about it and get back to me.

Believe it or not, I have even more to say about this. Kids use manipulatives for more than just contextualized problems, after all — they also use them strictly for modeling mathematical concepts without the context issue. What do these issues of perceptual salience suggest about physical and digital manipulatives used for that purpose?

I’ll be writing about that next week. Come back for manipulatives, part 3, and let me know what you think.

**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.

Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual representation and the linking of concrete and symbolic representations. *Child Development Perspectives*, *3*(3), 156–159.

*Figure adapted from:

University of Chicago School Mathematics Project. (2007). *Everyday Mathematics 3: Teacher’s Reference Manual.* New York: McGraw-Hill Education.

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