What Algebraic Symbols Have Been and Might Become
Formal notations present a basic conundrum for mathematics teaching and learning: We all know that algebraic notation is tremendously powerful in the hands of experts. On the other hand, the sad reality is that most students don’t get much out of using them except frustration.
In some sense, it’s not surprising that formal notations are hard to use. Most of our reasoning tools have changed dramatically since the Dark Ages, but the technology of formal notation hasn’t really changed in 400 years. The easiest way to write an equation is still by hand, on paper. While text is now typeset, word-processed, and hyperlinked, the equation remains, more or less, a creature of ink and parchment.
For a few years, a group of us—mathematicians, math educators, psychologists, and computer scientists—have been imagining ways to reconstruct the idea of a formal notation by using digital technology. We see the problem of learning formal algebra as, partially, one of interface design. Once you think of the notation as an interface into deep mathematical structure, you quickly see that the design is horrible—it’s like trying to use MS-DOS to run your brand new PC.
We think it’s time to apply modern design approaches to notations themselves, to build more intuitive, more fluid interfaces to these deep structures. We envision systems that enable early learners to experience some of the power of formal notation, without having to first go through the laborious process of mastering algebra’s extremely clunky interface.
We call this the better paper project.
The Better Paper Project
Our guiding vision is to create systems that serve both the expert and the novice. For the expert, we want tools that unlock the creativity, control, and clarity of formal notations on paper. For the new learners, we want tools that scaffold experiences of algebraic structure, supporting genuine inquiry about how the structure of algebra works and what it means.
The most recent incarnation of this vision is called Graspable Math. This digital system is designed around the idea that the strength of algebraic notation lies not (just) in the internalization of a set of rules, but in the structure of its dynamics: objects like ‘x’, ‘5’, and ‘8’ move from one place to another, split apart, join together. Derivations aren’t abstract sequences of rules—they’re stories that happen to objects.
Here are some of the aspects of algebraic notation we’ve focused on so far:
Symbols are Physical Objects
Wittgenstein famously said that the essential nature of the formal expression becomes clear when we imagine it made up of everyday object—books, tables, and chairs, rather than written symbols. We think that’s about right. In Graspable Math equations and expressions are, well, Graspable—they are objects you can see and touch and move, but in ways that are learnable and relatable to our everyday interactions with objects. In this way, our approach resembles modern operating systems, with their “folders” that you can drag from one place to another, and concrete “icons” that can be clicked to launch a new application.
What does this look like? Graspable Math uses fluid animations to transition from one state to another. Expressions are transformed by simple gestures that resemble actually physical interactions: since you usually pick up an object in one place, and set it down where it goes, this is how you move a term from one place to another. To distribute, you drag objects into parentheses; to factor, you drag them on top of each other and out.
You can interact with a sequence of derivations as if it’s a bunch of objects. You can rearrange the expressions, copy and paste single lines, and delete things you’re done with. You can easily make your own expressions, and equations can interact with each other through substitution, letting a learner solve multiple equations. You can also just draw.
Chess and the Body
But math is supposed to be abstract. Why do we think it’s worthwhile to make algebra more concrete? Well, let’s think about the game of chess. Chess is a lot like mathematical notation: it has rules, and you have to follow them to play chess. And those rules interact to create a really, really complex game. That’s the beauty of chess.
But imagine if you tried to learn chess the way we expect students to learn algebra; instead of a chessboard, with only pen and paper. You would copy out the starting position by hand, and then each time you wanted to make a move, you’d copy a new board, starting over, with just one thing changed. Like this:
This might be a great thing to do once or twice—you might learn to make shortcuts, or learn to imagine things in your head. But we suspect it would be a pretty miserable way to learn if it was all you had. There are just too many opportunities to make simple copying mistakes. You’d waste so much time hunting down the time you forgot to draw that knight, or put that pawn that moved in both its old and its new spaces. These copying errors have little to do with chess—its rules, its structure—and lots to do with the obsolete technology of rote transcription. Learning chess on paper would make stupid mistakes easy and deep learning hard.
But when it comes to algebra, this is exactly what we make kids do, and what we do ourselves! And yet we all know that, when doing algebra, we spend a huge proportion of our time tracking down little mistakes, errors with little relationship to our true mathematical understanding.
Graspable Math is a chessboard for notations—something that frees learners from the constraints of rote transcription, allowing them access to the power and beauty of the structure.
Also, like any good chessboard, we try to make interactions that are beautiful, elegant, and fun. We want to build a comfortable space in which the user can ask questions about formal inscriptions, explore how they change. And we want something that doesn’t require a lot of learning to get started—at least, it should require less startup cost than the old-fashioned paper and pencil way!
Supporting User Inquiry
Better notations should be better supports for inquiry, both by the expert and the novice. To start, dynamic notations make work easy to share with other students, with teachers, with classes, and with computer-based assessments; in fact, Graspable Math works in any internet browser. But how, exactly, will interactive notations help algebra learners ask useful questions and more fruitfully interrogate their own and others work? We’re still finding out, but here are a few ideas.
I use Graspable Math in a ‘networked’ mode to help my daughter with her homework when I’m traveling. Together, we can manipulate the same equation, at the same time, in the same digital workspace, sharing our understandings and intuitions fluidly. By turning equations into objects, these dyadic interactions become more spontaneous, shared, and collaborative. From what we’ve seen in classrooms, this can lead to fruitful group discussions about properties of equations.
Another tool that we’ve started exploring is ‘derivation mapping’, in which a learner or teacher can illustrate how a particular element of an equation changes and transforms over time. Clicking on different elements while in a special “scrub” mode allows each to be inspected.
Better inquiry can also be served simply by better availability. Many dynamic algebras, including our own previous tools such as Algebra Touch, are ‘siloed’ in their own systems, and are hard to adapt to classrooms. Graspable Math is available in multiple ways: Right now, learners can enter their own expressions in the ‘canvas’ on our website. We’ve also made a plugin for Chrome, which allows a user to drag expressions from Wikipedia or other popular online math repositories into a flexible canvas that sits alongside an open webpage. And we’re working to increase the opportunities: A partnership with Worcester Polytechnic Institute has led to a tool that allows teachers to make problems expressed using Graspable Math for students to do in class or at home, using WPI’s free ASSISTments tool.
The Better Paper Project has focused on the notation of formal algebra, and as a result it may seem to be about arbitrary rules without context. But of course you can’t just think of algebra as a collection of arbitrary rules and expect to understand it. So, we are beginning to ground our dynamic algebra tool in the same sorts of meaningful scenarios that are typically used to ground static, pen-and-paper algebra. Except with Graspable Math, we’re using a more interactive, less laborious and less error-prone interface with the notational world. In most connected curricula, the formal inscriptions sit statically next to a meaningful picture or animation. The connection between drawing and the inscription is often obscure and provides little to aid a learners’ mathematical understanding of the situation. We’ve imagined some ways to allow people to make that connection more physical and more direct, and to unleash the power of the notation.
This interaction can happen in lots of ways: one is pictured below. In this visualization, an equation is dynamically bound to the picture, so that a learner can see through visual inspection how lines of the derivation relate to each other, and how both relate to the visual elements in the diagram. Both the diagram and the derivation are dynamic objects, which can be updated in light of changes to the other. We have implemented a preliminary version of this functionality using graphs.
Algebraic notation is long overdue for updating to the digital age. There are so many possibilities for algebraic notation to be less onerous and less burdensome for learners, or even for it to be replaced by some new system, even more radical than mere better paper. We hope that projects like our Better Paper Project can spur conversations among all stakeholders from elementary school students to experts about what properties we want and should demand in our mathematical tools for thinking.
This post was written by David Landy for LearnTeachLead.
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