## Learning Tools

In 1981, after ten years in K-5, I switched to teaching high school math. In some ways, this felt like starting a whole new career: the math was more involved, the relationship with students less like parenting, and tradition weighed a lot more heavily on the profession. Still, in other ways, teaching is teaching, and much of what I learned in K-5 still applied. For example, “Michelle’s table is ready!” works a lot better than “Stop talking and listen to me!” Also: puzzles and learning tools, which had dramatically enhanced my math lessons in elementary school, could also be used to improve high school classes.

Yet high school teachers didn’t use manipulatives much, if at all. The dominant idea seemed to be that clear explanations followed by silent practice on paper was the gold standard. I knew from experience that this was not a particularly effective approach on its own, and started to complement it with traditional elementary school materials such as pattern blocks, geoboards, tangrams, and pentominoes. I was thrilled at how well that worked. Encouraged by this early success, I designed new manipulatives for use in math classes: the Lab Gear (for algebra), a well-thought-out geometry drawing template, the CircleTrig Geoboard, (both for *Geometry Labs,) *and supertangrams. I also had the amazing privilege to work with George Hart on the *Zome Geometry* book.

(Find links about manipulatives in middle and high school here.)

This interest in manipulatives does not stem from a naive belief that if students can touch it, they will understand it. (“Magical hopes”, as Deborah Ball put it back in the day.) The usefulness of manipulatives depends on how they are used: as with any approach, reflection, discussion, connections, and generalization are crucial. But that’s the point: with proper teacher leadership, manipulatives can offer a concrete environment that increases student interest, levels the status playing field, facilitates communication, and stimulates mathematical thinking.

Likewise, rich computer learning environments provide opportunities to raise the level of intellectual engagement — as long as they are used well. (Alas, it is easy to misuse electronic tools… See my article on this topic.) By “rich environments” I mean electronic platforms that support teacher and student creativity: Logo and its descendants such as Scratch and Snap, electronic graphing, interactive geometry, and the like. True, screens can be hypnotizing and isolating, but the Desmos team has found ways to make their graphing software part of lively classroom discussion, and it seems like GeoGebra is trying to catch up on that front.

(Over the years, I’ve written a lot about learning tools. See especially For a Tool-Rich Pedagogy which features both a philosophical argument and dozens of relevant links.)

## Virtual Manipulatives

Virtual manipulatives attempt to bridge the gap between high-tech and low-tech learning tools. That attempt has not always been successful. In some cases, the tools were just poorly designed, perhaps because the designer didn’t understand how the manipulatives are intended to be used. (Two recent examples: tangrams whose parallelogram cannot be flipped; pattern blocks that rotate in 5° increments, so that it takes six clicks for the smallest useful rotation.) In other cases, the design was decent, but did not use the power of the computer, so that switching to an electronic version of the manipulatives led to a loss in classroom interaction, with nothing gained.

However, in a time of remote instruction, virtual manipulatives cannot be compared with physical manipulatives, since those are typically not available in students’ homes. The comparison is with not using manipulatives at all. I don’t have a lot of experience teaching remotely, but I’m pretty sure that virtual manipulatives are better than no manipulatives. As is my wont, after a little bit of research on what’s “out there”, I started making my own. This is now a lot easier than it would have been even a few years ago, as there are graphical and computational environments that put this within the reach of an amateur like myself. I am thinking of Google online apps, and GeoGebra, but I’d love to hear about more (preferably free) options.

I will use this post and the next to think about virtual manipulatives, and to let you know about my attempts along these lines.

## Lab Gear

Some years ago, I used GeoGebra to create applets to explore and discuss multiplying binomials and squaring a binomial, using the Lab Gear model. What I liked about the applets is that if a teacher could project their laptop screen, they had a way to discuss these ideas with a whole class. The applets could also be used by students, for example to create figures and illustrate a report. Another applet, completing the square included specific questions that students can answer to show and consolidate their understanding of an algorithm that pre-manipulatives was exceedingly difficult to learn.

That’s all well and good, but these uses only make sense after the students have had plenty of experience with the physical blocks. And moreover, these applets each had one single purpose. The physical manipulatives can be used in a huge range of algebra topics: signed number arithmetic, polynomial factoring, equation solving, and so on (as you can see in the Lab Gear books). Clearly, useful as they are, these applets do not qualify as virtual manipulatives.

To address that, and to respond to a teacher’s request, I used Google Drawings to make Virtual Lab Gear. What’s great about those is that several students can work together in one drawing, and one person (teacher or student) can demonstrate something for all to see. In addition to live manipulation, the drawings allow for the creation of Lab Gear illustrations by anyone — a long-standing request of Lab Gear users. In turn, those illustrations can be copy-pasted into other applications.

I was pretty sure that this Virtual Lab Gear would make it possible to carry out all the activities in the Lab Gear books. However, no sooner had I posted the drawings that a correspondent pointed out that it was not possible to use them for “Face to Face” (Activity 1-1D in the *Algebra Lab Gear: Algebra 1* book). This is an activity I like, because it previews a key idea: common dimensions correspond to common factors. It gets at that by asking students to place the blocks on rectangles that represent the various possible faces.

Here is an alternative which can be done using Virtual Lab Gear. The idea would be to make “trains” or “towers” of blocks. Each would consist of all different blocks. (No repeats within a train or tower.) The blocks would connect face-to-face, only on congruent faces. The assignment would be to do this for 1 by 1 faces, 1 by x, x by x, and so on.

Here is a 1 by 1 train I created using Virtual Lab Gear:

Here is an x by x tower:

and so on. I suspect that this is in fact an improvement over the original activity, whose instructions for some reason were hard to communicate to students. (I really did not expect that the virtual manipulatives would suggest an enhancement over the physical version!)

If I create such a worksheet to substitute for 1-1D, I’ll post it on the site. And if you do, send it to me! In fact, if you use the Virtual Lab Gear in creating PDFs, or slides, or whatever, I’d love to see what you did, and I might share it on my website. (Crediting you, of course.) Also, as always, I would appreciate any feedback on this.

Well, that’s all for now. To be continued! I’ll share some more thoughts and some more virtual manipulatives in my next post.

— Henri