In a Math 2.0 email group conversation about screencasting and tools like vZome, Brad Hansen-Smith of WholeMovement posed this question:

Can you explain how using this virtual zome tool will give students a better understanding of polyhedra than actually building it from scratch for themselves? I have the same question about any virtual experience when compared to actual experience of doing something. I assume you have done a lot of model construction and it is easy for you to understand having the experience, but what understanding do students get with only virtual experience?

Here was my reply:

It is better to have both experiences. The reason is that they are different. In particular, and to answer your question, there are three major features of virtual tools that physical tools don’t have.

1. EASY SHARING
Virtual constructions can be uploaded to the web and emailed around. I can’t directly email you the construction of the lopsided origami dragon I made yesterday, though I am attaching a photo of the end product (and I could take a video, for sure). But it’s not as easy as with virtual objects, and you don’t get the perfect copy of the real thing, but a representation of it. I remember our exchange of many emails about me trying to replicate one of your constructions. It took quite a lot of work to share.

2. EASY STEP REVIEW & UNDO
Speaking of the dragon, I would love to rewind the construction step-by-step and find where I made the extra fold: the wings look different. It’s somewhere around step 9 of 21. I don’t feel like finding the mistake in my paper version: it will ruin the dragon completely, and I am not sure I will trace the mistake anyway. Repeatable step-by-step review, analysis and changes are hard to do by hand, especially for young students whose memory works differently and has fewer registers than adults have.

Step review works wonders with sharing. A student can send the whole construction (often animated, or a screencast – easily made!) and ask peers or mentors to analyze steps, or post questions like, “What would you do differently in Step 5?” With some environments, they can then all share their fully interactive constructions that are answers to that question.

3. EASY DYNAMIC LINKS AMONG REPRESENTATIONS
You can dynamically link formulas, graphs and constructions, which support depth of mathematics. It provides a certain holographic view on the essence of math, metaphorically speaking. GeoGebra is probably a better-known example of this, with algebraic representations linked with geometric constructions. Check out DGS (dynamic geometry software) systems in Paul Libbrecht’s i2geo series (more coming up, stay tuned) at Math Future for beautiful examples:
http://mathfuture.wikispaces.com/JSXGraph_DGS
http://mathfuture.wikispaces.com/CaRMetal_DGS
http://mathfuture.wikispaces.com/i2geo

The word “easy” here is the difference between thousands and millions doing the three activities I described above.

—

I posed the same question to Katherine, my daughter, who added two items to the list:

In virtual constructions, you can see infinity. (In particular, I am thinking of fractals – MD).

It really helps to change a variable and see what happens to the construction as a result. It is very hard to do in physical space.

I am adding another key item that came to mind: modularity. Once you build a module in a virtual space, you can copy and paste it whole. In physical space, you have to repeat all constructions step-by-step at all times.