Oh interesting. I don’t spend nearly enough time digging into the TOS but I’m glad you’re watching out for the teachers and students you support. Thanks for the comment. I featured it in the main post.

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2x = 4x + 10

I would like to drag the 2x over to subtract from the 4x (or vice versa), but I can only drag the 2 individually or the x individually. (I don’t think 2=4 + 10/x is the most helpful? Oh, I can move the 4 in front of the x so it says x*4 now instead of 4x. Still not helpful.) How do you keep going?

Hi Michael, Good to hear from you! I actually think it’s a great idea! Erik Weitnauer and I have just been kicking around the idea of a sort of ‘meeting of the minds’ of people who have developed gestural algebras. I think a standard might be premature (and hard to enforce!), but I agree that some some discussions about what we think works well and works poorly might help us all align a little, and that might help teachers with the learning curve. We’ll be in touch!

Thanks, David. I LOVE GM and have been advertising it to math teachers at regional NCTM meetings because it is the BEST smart board tool for algebraic manipulations – and it’s FREE! I admit that I made an over-simplification that tools like GM and ABH do not assist with conceptual understanding. In examples like the demonstration you reference, it does ASSIST with conceptual understanding, however, only in the sense that it removes the bookkeeping and arithmetic chores from algebra so that students can focus on higher concepts. Desmos does this as well. We need to be careful that teachers don’t use tools like GM and ABH to SKIP OVER the concept of applying equivalent arithmetic operations to both sides of an equation.

I am so happy to see these public discussions and get this great feedback. I think Dan has a great comment about the TIME required for teachers to learn how to use these tools. As I expect that these tools will become more commonplace, it makes me think that one way our industry can be of service to teachers would be to adopt some “standards” for gestures across the tools, so teachers would not need to learn different gestures for different tools. Do you think this is too ambitious? It might require third-party interface design experts, and instructional designers to drive to a standard.

And David is mostly correct when he wrote that I helped create Algebra By Handâ„¢. Actually, it’s more accurate to say that I created Algebra By Hand. Every line of code. I did it as a labor of love for the minority students that I was volunteer tutoring. I wanted to give them the gift of advanced procedural fluency to improve their accuracy, increase their speed, and boost their confidence when doing simple algebra. I realized that other teachers might want to do the same, so I’ve published the work to make it available to others. So I think GM and ABH have very compatible objectives.

@Melvin. I never wrote or meant to suggest that GM be reserved for struggling students or students on math IEPs. But the more I play with GM, the more I think it would be an appropriate support for those students…or any student for that matter.

“could we be doing harm by reinforcing the idea (which they may already hold) that math is about arbitrary symbolic manipulations? ”

My seniors who are a year behind have already have this idea that mathematics is just arbitrary symbol manipulation. I think GM could be used in an action/consequence manner to (perhaps) helps these kids make sense of something that has not made sense to them before.

“I’m also concerned that giving this perpetuates a pattern where certain groups of students are regularly exposed to ways of doing math that fail to engage them in deep mathematical exploration. ”

I think GM could be used to engage all students in deep mathematical explorations, not just kids who are behind or kids in IEPs, but especially those kids who are behind or are on IEPs

[Disclosure: David helped build both Algebra Touch and Graspable Math]

Algebra By Hand is a great project, and I agree that it could potentially help build procedural fluency! We hope Graspable Math can do the same. One difference, that is both a strength and a caution, is that GM is a more open space, which means that students can also explore properties of equations and derivations, modify them, combine them with other representations (like graphs or geometric structures), and so on. That means that there are lots of interesting ways to use GM to do things that go beyond procedural fluency. http://www.kurtsoeser.at/2018/06/14/graspable-math/ has a great example of this kind of thinking (in German).

Thanks Aaron! I definitely agree that the ‘learning curve’ in GM is higher, but we think the tradeoffs on the positive side are real too. There’s even more room for pattern-finding, noticing, and creative exploration that students can use to enrich their conceptual understanding. But maybe not 3-years olds ;-) (Disc: I am a founder of GM and AlgebraTouch)