A tool called Graspable Math found an audience on Twitter late last week, and a couple of people asked me for my opinion. I’ll share what I think about Graspable Math, but I’ll find it more helpful to write down *how* I think about Graspable Math, the four questions I ask about all new technology in education. [Full disclosure: I work in this field.]

**1. What does it do?**

That question is easier for me to answer with basic calculators and graphing calculators than with Graspable Math. Basic calculators make it easy to compute the value of numerical expressions. Graphing calculators make it easy to see the graphical representation of algebraic functions.

Graspable Math’s closest cousins are probably the Dragonbox and Algebra Touch apps. All of these apps offer students a novel way of interacting with algebraic expressions.

Drag a term to the opposite side of an equality and its sign will change.

Double click an operation like addition and it will execute that operation, if it’s legal.

Drag a coefficient beneath the equality and it will divide the entire equation by that number.

Change any number in that sequence of steps and it will show you how that change affects all the other steps.

You can also link equations to a graph.

**2. Is that a good thing to do?**

No tool is good. We can only hope to figure out *when* a tool is good and *for whom* and *for what set of values*.

For example, if you value safety, an arc torch is a terrible tool for a toddler but an amazing tool for a welder.

I value a student’s conviction that “Mathematics makes sense” and “I am somebody who can make sense of it.”

So I think a basic calculator is a great tool for students who have a rough sense of the answer before they enter it. (ie. I know that 125 goes into 850 six-ish times. A basic calculator is perfect for me here.)

A graphing calculator is a great tool for a student who understands that a graph is a picture of all the x- and y-values that make an algebraic statement true, a student who has graphed lots of those statements by hand already.

A basic and graphing calculator can both contribute to a student’s idea that “Mathematics doesn’t make a dang bit of sense” and “I cannot make sense of it without this tool to help me” if they’re used at the wrong time in a student’s development.

The Graspable Math creators designed their tool for novice students early in their algebraic development. Is it a good tool for those students at that time? I’m skeptical for a few reasons.

First, I suspect Graspable Math is *too helpful*. It won’t let novice students make computational errors, for example. Every statement you see in Graspable Math is *mathematically true*. It performs every operation correctly. But it’s enormously helpful for teachers to see a student’s *incorrect* operations and mathematically *false* statements. Both reveal the student’s early understanding of *really big ideas* about equivalence.

In one of their research papers, the Graspable Math team quotes a student as saying, “[Graspable Math] does the math for you – you don’t have to think at all!” which is a red alert that the tool is too helpful, or at least helpful in the wrong way.

Second, Graspable Math’s technological metaphors may conceal important truths about mathematics. “Drag a term to the opposite side of an equality and its sign will change” isn’t a mathematical truth, for example.

It’s *a technological metaphor* for the mathematical truth that you can add the same number (3 in this case) to both sides of an equal sign and the new equation will have all the same solutions as the first one. That point may seem technical but it underpins *all of Algebra* and it isn’t clear to me how Graspable Math supports its development.

From an educational viewpoint, I would get rid of dragging across the equal sign. I've been teaching Algebra for 38 years and I spend most of my 'solving equation' time correcting incorrect math procedures because students don't know 'why it works', they just do it.

— Tony Riehl (@riehlt) June 11, 2018

Third, Graspable Math may persuade students that Algebra as a discipline is very concerned with moving symbols around based on a set of rules, rather than with understanding the world around them, developing the capacity for conjecturing, or some other concern. I’m speaking about personal values here, but I’m much more interested in helping students *turn a question into an equation* and *interpret the solutions of that equation* than I am in helping them *solve the equation*, which is Graspable Math’s territory.

These are all tentative questions, skepticisms, and hypotheses. I’m not certain about any of them, and I’m glad Graspable Math recently received an IES grant to study their tool in more depth.

**3. What does it cost?**

While Graspable Math is free for teachers and students, money isn’t the only way to measure cost. Free tools can cost teachers and students in other ways.

For instance, Graspable Math, like all new technology, will cost teachers and students time as they try to *understand how it works*.

I encourage you to try to solve a basic linear equation with Graspable Math, something like 2x – 3 = 4x + 7. Your experience may be different from mine, but I felt pretty silly at several points trying to convince the interface to do for me what I knew I could do for myself on paper. (Here’s a tweet that made me feel less alone in the world.)

Graspable Math performs algebraic operations correctly and quickly but at the cost of having to learn a library of gestures first, effectively trading a set of mathematical rules for a set of technological rules. (There is a cheat sheet.) That kind of cost is at least as important as money.

**4. What do other people think about this?**

I spent nearly as much time searching Twitter for mentions of Graspable Math as I did playing with the tool itself. Lots of people I know and respect are very excited about it, which gives me lots of reasons to reconsider my initial assessment.

While I find teachers on Twitter are *very easily excited* about new technology, I don’t know a single one who is any less than *completely protective* of their investments of time and energy on behalf of their students. Graspable Math may have value I’m missing and I’m looking forward to hearing about it from you folks here and on Twitter.

**BTW. Come work with me at Desmos!**

If you find these questions interesting and you’d like to chase down their answers with me and my amazing colleagues at Desmos, please consider applying for our teaching faculty, software engineering, or business development jobs.

**2018 Jun 11**. Cathy Yenca pulls out this helpful citation from Nix the Tricks (p. 54).

**2018 Jun 11**. The Graspable Math co-founders have responded to some of the questions I and other educators have raised here. Useful discussion!

## 30 Comments

## Patty

June 11, 2018 - 10:24 am -It feels like Graspable may be more valuable if it connected the algebraic with a visual model of equation solving. But I’m not convinced that would be better than going low tech and just pulling out the Dixie cups and counters.

## Aaron Bergmann

June 11, 2018 - 11:41 am -Dan,

Thanks for the blog post.

Did you feel the same way about AlgebraTouch’s approach to solving equations? I really liked AlgebraTouch because I felt that it could teach better than any other tool the skill of sequencing algebraic steps.

I’m not yet sold on graspable because it seems to have a much higher learning curve than other tools.

## Dan Meyer

June 11, 2018 - 7:42 pm -Hi Aaron, I’m afraid I don’t have a lot of experience with AlgebraTouch. I remember feeling conflicted about how easily I could make things happen without thinking about them. Tap or drag anywhere, even thoughtlessly, and something would happen. Perhaps that was by design, though.

## Aaron Bergmann

June 12, 2018 - 4:51 am -I can appreciate yours and Tony Riehl’s sentiment that programs like Graspable Math and AlgebraTouch de-emphasize a most of the fundamental properties of Algebra. But I think what is gained is incredible: for the first time in mathematics education history we gain a way to teach the laws of Algebra in a truly exploratory way, the way 3 year-olds learn how to use an iPhone. We can give kids a tool and ask them “what patterns do you notice?” and then generalize student results, refining them into the properties of Algebra. We can start with a puzzle and discover the laws and properties. No where else can you do that.

AlgebraTouch was phenomenal for doing these kinds of discovery activities. I’m not sure that Graspable Math is user friendly enough to not get bogged down in user mechanics, but still trying it out.

## David H Landy

June 14, 2018 - 7:50 am -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)

## Melvin Peralta

June 11, 2018 - 11:43 am -Thanks for this. Based on this post and the comments surrounding how the app treats solving equations, I think I’m going to go so far as to stop using any kind of metaphor related to movement or even isolation while talking about solving equations. I used to teach how adding or subtracting a number from both sides of an equal sign keeps the solution the same, and that we do so *in service* of moving numbers and variables to ultimately isolate the variable.

Lovely low tech solution.## David Petro

June 11, 2018 - 11:57 am -I agree with much of what you have said but there are some features I do like here. For example, I like the Scrub feature to show how changing one value affects the chain of manipulations in real time.

But beyond that, whenever I see tech like this emerge (another example is Photomath), I ask people: “Given that this now exists, how does this change how you value things like solving equations in your class?” That is, If a student can just drag stuff around (with Graspable) or take a picture (with Photomath) does that diminish the importance of teaching methods for solving equations?

Don’t get me wrong here, I think that algebraic thinking is hugely important for students but solving equations may not be. I’ll take that to it’s logical (or illogical) conclusion: does it matter at all what we teach students? Like is it the end of the world if someone doesn’t ever learn the quadratic equation or how to calculate slope? Most of the content that we teach (not just in math) is just stuff and is only useful (the individual tools) to a very small handful.

I think more than anything we need students to become good problem solvers. Math offers a good vehicle for that but so can other subjects. So, yup, maybe Graspable isn’t really useful for teaching students how to solve equations (and may even reinforce poor rules) but maybe this now allows us to focus on some other aspect of math that might be better for students to learn.

## STEVE PHELPS

June 11, 2018 - 12:54 pm -As I have wrote on twitter, I am thinking of how this can be used to support, for example, my seniors next year who are behind their peers a year (or more) in math, who should be taking algebra 2, but would get CRUSHED in that class (after having “learned” all the solving equations tricks and they still don’t get it).

I am also thinking of this as a way to support those students who may be working under a math-specific IEP.

I see this tool as a supporting tool. I am not convinced on how it can be used a teaching tool. I do think it could help students with bad algebra experiences make sense of things that have not made sense for them.

## Melvin Peralta

June 11, 2018 - 1:15 pm -Check out Melvin!## STEVE PHELPS

June 14, 2018 - 8:01 am -@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

## Rachel

June 11, 2018 - 1:12 pm -I am still feeling out how this tool can be used. I was immediately kind of amazed by the slickness, I admit.

I thought of one student who has pretty serious reading and writing disabilities. Writing, even steps in math, for them is extremely laborious. The cognitive load takes over all of the thinking around what they are doing. I can see this being a cool tool for them to be able to do some of the math while short circuiting some of the labor of writing. But it certainly short circuits some of the thinking, too. I will have to be cautious about when to introduce it if we choose to use it.

I can also see it being useful in supporting things like deriving the quadratic formula, which is very symbolically difficult for some students, even if the concept makes sense.

This is “just” pushing symbols around, which is definitely not what I want my algebra class to be about. But it is something that shows up in my state standards and exams, so I can’t just ignore it, either.

There’s another newish tech tool, casios class pad. I’m not sure where that fits in my current ti/desmos/other tech mosh pit.

## Jen Silverman

June 11, 2018 - 4:08 pm -I see value in the symbolic/graphical connections you can make. It has a seamless interface with a GeoGebra graphing window. Check it out at https://youtu.be/DfC9K_jKP0E.

Fwiw, I had trouble getting it to do what I wanted it to do, too. (I wanted to “move” 2x.)

## Dan Meyer

June 11, 2018 - 7:45 pm -Yeah, the GeoGebra integration is really nice. I’m used to dragging sliders to see a function change and I liked dragging the numbers themselves. It removes a layer of abstraction.

## Jen Silverman

June 12, 2018 - 2:24 pm -After just a few minutes with the tutorials linked in the Graspable Math “help” tab, I was able to grab that 2x, no problem. I regret being curmudgeonly before spending more time exploring. Sorry, GM. https://graspablemath.com/learn/gesture-library, https://graspablemath.com/learn

## Tony Riehl

June 11, 2018 - 5:34 pm -If you are using this to solve equations, what makes this better than paper and pencil? Knowing how to do the process is a necessary skill and understanding the ‘properties of equality’ is important at the Algebra 1 level.

## Joanne Robert

June 11, 2018 - 5:34 pm -Maybe I’m nuts but when teaching 7th graders algebra, students seem to understand the use of the additive inverse to create zero pairs and the need for balance which requires “BS”- both sides.

Featured CommentThe research says kids retain more with paper pencil anyway. I’d rather see it make mistakes and see if students can spot them and make sense of them.

## Rachel

June 11, 2018 - 6:13 pm -I think that is true for many if not most students. But there are some students for whom the writing is a major impediment. And making an error can become an almost insurmountable obstacle to fix, especially early in a skill. I can see a tool like this being helpful, there.

## Christa Amezcua

June 13, 2018 - 2:30 pm -I never teach to “pass actos”, but for some reason, there is always (at least) one student that brings that trick to the classroom. He heard it from a sibling, his parents or another teacher. Maybe this technological tool could help the student see why it is NOT a good idea. It actually shows that dividing by the coefficient affects all the terms, but it is not allowed to subtract a term before the distributive property is used, for example. 2(x-5) should “pass” the -5 as a positive 10.

## Martin Flashman

June 11, 2018 - 6:37 pm -What does the manipulation of the symbols in solving a simple linear equation such as ax+b=c add to understanding the mathematics that underlies the procedures?

How does it help a student make sense of the problem and its solution?

This approach with graspable fails to connect many concepts either through its process or a visualization.

This is precisely what I have been developing with GeoGebra using mapping diagrams to visualize the function concepts used in solving equations: composition and inverses. See my links to a recent presentation at the Sacramento Valley Community College Mathematics (SVCCM) Conference, American River College (Sacramento),

“Solving Polynomials: Mapping Diagram Visualization”, Saturday, March 3, 2018.Links: http://users.humboldt.edu/flashman/Presentations/SVCCM2018.MD.LINKS.html

## Christa Amezcua

June 13, 2018 - 2:48 pm -It could make sense in a problem. Example: I had x dollars and my mom gave me five more. Now I have eight. How much did I have before adding five, well, five less.

I can solve problems undoing the operations, but taking care of the order.

## Tom Harris

June 12, 2018 - 1:41 am -I noticed Graspable Math just the other day, and then today saw your post. Yes, I also wondered where this tool could be useful, and where less so. What I can say is that your post got me thinking about what I’m teaching these days – computer programming — here’s what your post inspired:

https://talkaboutquality.wordpress.com/2018/06/12/computer-programming-the-rules-of-the-game/

## Audrey McLaren

June 12, 2018 - 5:49 am -But there are other options, like if you hold down the equals sign, you can tell GM what operation to do to both sides: https://drive.google.com/file/d/1HIC4YdeE0xYdCnKS2LF6eym1qc8LGnW3/view?usp=sharing

You can see all the identities being used, you can control the order of operations, you can SEE the division as a fraction. When I use it next year, which I can’t wait to do, that’s how I’m going to instruct my students to show their steps.

## Dan Meyer

June 13, 2018 - 6:15 am -I think that’s a really interesting feature, one that has the potential to develop a novice’s understanding of the properties of equality. (Though the “E” that pops up is a bummer IMO.)

## David Landy

June 14, 2018 - 7:47 am -We agree about the “E”–we tried it without, and you run into all kinds of weird situations (how do you raise both sides to a power? How do you distinguish multiplying on the left and right? Many of our younger students really didn’t like the idea of just writing 2* and leaving it blank). We’re not in a perfect place yet–Suggestions very welcome! (Disclosure: I’m a founder of GM, and of Algebra Touch).

## Michael Eiseman

June 12, 2018 - 7:12 am -[

Disclosure: Michael helped build Algebra by Hand.]Tools like GM and Algebra By Hand(TM) (https://www.algebrabyhand.com/teacherFAQ#infoVideo) are useful for teaching ADVANCED PROCEDURAL FLUENCY in algebra by reducing the students’ cognitive load. They do NOT assist with conceptual understanding.

Advanced procedural fluency (APF) is the ability to manipulate algebraic equations by moving terms and factors in your head. Every STEM professional possesses APF. Students need APF before moving on to calculus. In the US, we have always forced our students to learn it on their own. Why not teach it to them AFTER they have acquired conceptual understanding? These tools can be used to do this while reducing teachers’ workload. THIS is what technology SHOULD be used for in the classroom.

## David H Landy

June 14, 2018 - 7:54 am -[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).

## Michael Eiseman

June 14, 2018 - 10:08 am -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 Hand™. 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.

## David Landy

June 14, 2018 - 6:33 pm -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!