## Response from the Graspable Math Co-Founders

Thanks for your interest in Graspable Math! We agree that no tool is inherently good or bad—tools have a role to play. No tool can do it all. We believe that Graspable Math is a powerful tool for student learning, when applied in the right contexts. We are a little disappointed that you seem to have misunderstood the role it can play.

The biggest mistake you make in the post is in characterizing who we designed our tool for. You suggest that it’s for “novice students early in their algebraic development”. We built GM as a tool that can support thinking at many levels*, but which is especially aimed toward situations where a teacher and student(s) have moved past basic operations and are putting those pieces – with understanding – together into larger situations that involve proofs or real-world contexts. We don’t intend Graspable Math as a tool that replaces understanding at any point. Of course, a tool as powerful as this can be used without understanding – as can paper and pencil algebra. We believe that both can also support understanding.

Sometimes a teacher wants to focus on the basic building blocks of algebra; sometimes they want to focus on the strategy, or use algebra to understand a situation. We think that at times like these, it’s often best to get the struggle with notation out of the way. That lets the student reveal higher level confusions and understandings, and lets the student and teacher focus on common misconceptions. One might be able to correctly report and describe distributivity in an area model, for instance, but struggle to apply it when trying to understand and connect different representations of a linear relationship. Similarly, trying to work with the law of gravity in an introductory physics context often devolves into chasing down a -1, or struggling to get the right steps executing correctly.

At times like these, the interesting errors are not the typos and failures of retrieval–they are the conceptual understanding of the system of relations that comprises so much of algebraic thinking. Our system is not a solver: students make many errors with it –but most of those errors are errors of strategic understanding that reveal deeper misconceptions. If a student knows with confidence and ease that they’ve done the steps without small calculation errors, they can focus on the exciting challenge that connects the algebraic work to a real situated context. (Of course, sometimes you’re trying to focus on algebraic building blocks and related errors! At those times, don’t use GM).

You also worry that GM will take too long to learn and use, and students will pay a cost in time. The concern is important, and it’s one of our biggest areas of improvement. Our ambition is to make an algebra that takes less time than paper and pencil to learn, and is faster once it is mastered. It’s part of what sets us apart from older tools like Dragon Box and Algebra Touch (disclosure: our David L. was part of that project, too); those tools really take longer, for experts, than paper & pencil algebra; their aim was different. One of our focuses in making GM has been to make a user interface that, for experts, is faster than algebra*. We’ll let you and your readers decide if we succeeded yet – but we’re constantly improving, and that is certainly our goal**. We’re making a world where algebra is fluid and easy, once you’ve learned to use it.

You also seem to imagine a situation in which students work on their own, with no outside structure. We understand; similar tools like Algebra Touch and DragonBox were designed with those environments in mind. More powerful tools–like Geogebra, Desmos, and Graspable Math–work best when combined with appropriate situations and examples–and great teachers). Even when we think GM is useful for inspiring guesses and hypotheses before they’ve been taught, we don’t imagine the main value as being in a teacher-free zone. We are excited to see creative lessons that use what we’re providing in contexts–these contexts can make use of or illicit different optional features, like dragging across the equals sign. Sometimes it’s not right, and you should instruct your students not to do it (yet!). In our experience, sometimes student discoveries of shortcuts–of many kinds–provide opportunities for conversations about deep principles: conversations that can be enabled by algebraic discoveries. Our features like term tracing are designed to facilitate that sort of ‘discovery then principle’ approach. That said, we are testing that allow teachers to set their own ‘allowed operations’, so that teachers can be empowered to set what’s right in their class at that moment. It’s not ready yet–we’re a small company, and not every feature is ready yet. That’s why we’re looking for constructive conversations with teachers and users, while we’re just starting out.

You call this ‘four questions I always ask’. We think it would have been great if you had asked us those questions earlier – for example in March, when we reached out to you and you declined. Of course we don’t have all the answers and are not the authority on whether our tool is ‘good’ or ‘bad’ in any particular context, but a conversation could have been quite enlightening for both sides. We appreciate that you are taking the time now and look forward to the thoughts of your readers as we continue to grow and improve.

You said at the start of your post that it was hard for you to see what GM was useful for, and to characterize what it does. We have clarified some places we think GM can be helpful. We don’t mean to say that GM is only useful in these contexts. We’ve been astounded by the creative things we never thought of that are appearing on twitter and other places. We are excited for a future where the uses expand to include many more of the places that a dynamic algebraic notation is used and useful. Thanks again for your time and interest, and a constructive conversation!

*Here’s a link to a paper that shows some of our progress on this front.

**Indeed, one of our founders David Landy has used it personally to derive published models in mathematical psychology.