Equality: A tool for free-form equation editing Stephen Cummins, Ian Davies, Andrew Rice, and Alastair R. Beresford Computer Laboratory University of Cambridge JJ Thomson Ave, Cambridge, CB3 0FD, United Kingdom [email protected] Abstract—We describe a new tool, Equality, for equation entry will sometimes have access to a keyboard and mouse, but using free-form layout of components drawn from a palette of other times will have only a touchscreen. Similarly, there symbols. Our approach is designed to enable learners to easily are circumstances where use of a stylus (for handwriting manipulate the structure of their equations, to be functional in both desktop and mobile environments, and to minimize the recognition) is infeasible. Equation entry needs to work well in amount of learning required to use the tool. We present the results all of these contexts. The third issue is that of having to learn of a study comparing a prototype of our approach with Microsoft how to use the language or tool prior to being able to use it Equation Editor using a desktop machine. The initial results are properly. Tools which are difficult to learn or frustating to use promising with participants reporting that the mechanism is easy will simply increase the obstacles already faced by students to learn and an easy way to manipulate their equations. We report the results of the study and the views of the participants of STEM subjects. 1 and identify how these will inform the future development of We have attempted to address these issues with Equality , Equality. a new tool which allows users to construct equations through Index Terms—equation entry; equation manipulation, teaching free-form positioning of elements drawn from a palette. mathematics, teaching physics In this paper we describe our prototype implementation of Equality (Section III). We describe a user study comparing I. INTRODUCTION Equality with the Microsoft Equation Editor (MEE, Section The teaching and learning of problem solving skills is IV). The results (Section VI) show that users found it signif- important to many Science, Technology, Engineering and icantly easier to manipulate their equations with Equality but Mathematics (STEM) disciplines. An important skill in these that the lack of keyboard support in our prototype was a large disciplines is the use of mathematical formulæ as a means to impediment to fluency. We explain the threats to validity in describe and reason about physical and theoretical concepts or our study (Section VIII) and highlight our future intentions problems. (Section IX) to meet the needs identified in our study. The increasing use of technology in learning means that it is The contributions in this paper are: becoming more common to ask students to enter and manipu- • a novel equation entry mechanism which we believe is late mathematical equations electronically. There are a plethora well suited to multi-modal use on modern devices; and of tools available to support entry of these symbols including • a demonstration that users find this mechanism easy to A LTEX, Microsoft Equation Editor, and handwriting recognition learn and an easy way to manipulate their equations. tools [1]. Online platforms such as Khan Academy provide specially designed symbolic entry interfaces that require a A. Project background combination of Windows, Icons, Menus, Pointer (WIMP) and Equality has been developed as part of a series of tools to keyboard entry modes. be used in the Isaac Physics project.2 The project is funded However, these existing tools suffer from a variety of by the Department for Education in the United Kingdom and issues. The primary concern is that of the educational process. aims to ‘bridge the gap’ between secondary school physics and Pupils do not (and should not) get everything right first time. Higher Education. In particular, the project hopes to improve Therefore, equation entry tools should make it easy to change problem solving skills by providing technologies and content the structure of an existing answer. As an example, a structural to support physics problem solving. All content produced by change to an equation might involve changing the scope of a the project is available under a Creative Commons license and square root, or replacing the multiplication of two expressions all software developed (including Equality) is open source.3 with an exponentiation. This is a more sophisticated change Equality has been produced to support users of Isaac than simply editing a single element of the formula in- Physics, who may be distance learners, in the entry and place. This is obviously particularly difficult for text-based manipulation of equations to be submitted as part of their tools such as LAT X since they are attempting to serialise E problem solving practice. The target audience is students aged a two-dimensional layout into a one-dimensional character stream. However, any tool which relies on a structured entry 1http://equality.isaacphysics.org approach will also be affected. The second issue is that of 2https://isaacphysics.org input modality. The rise of mobile devices means that pupils 3https://github.com/ucam-cl-dtg/equality between 16 and 18 who are interested in Physics, however the order to focus development, a number of design requirements Equality editor has been developed as a stand-alone module were identified. The final implementation must: and can be used outside of the physics context. • allow unconstrained entry and manipulation of formulæ, II. EXISTING TOOLS • allow students to evolve and change their solutions as Development of equation entry tools typically involves they learn more about the problem, solving a number of problems including: symbol recognition, • be usable without the need for a keyboard, segmentation of the symbols, structural analysis (parsing) and • produce machine-readable representations of the entered ambiguity resolution [2]. formulæ, Use of the keyboard for input makes the stage of symbol • require minimal training on first use, recognition unncessary. Probably the most widely used ex- • be suitable for display on mobile as well as desktop devices, ample of this are the LATEX math commands. The problem with this approach is that it requires learning a defined set • be general purpose and stand alone from the rest of the of keywords which together allow description of both the Isaac Physics project. symbols as well as their layout in 2D space. Taking these requirements into consideration, we elected to Many studies advocate the use of handwriting directly build a user interface based on drag-and-drop of symbols onto onto touch devices as being a natural and efficient mode for a canvas. No restrictions are placed on the user, they are free to entry [3], [4], [5] and manipulation [1], [6] of mathematical select any symbol from a palette and place it anywhere on the equations, whereas our use of a palette instead provides a canvas at any size. In many traditional What You See Is What form of direct symbolic entry. We took this decision since You Get (WYSIWYG) equation editors the user is required to it is uncommon to find a stylus shipped with desktop PCs edit a hidden Abstract Syntax Tree (AST), parts of which are or mobile phones and mobile devices in particular are often sometimes exposed as placeholder boxes. While this can help used in circumstances where accurate writing with a stylus is with accurate entry, it often makes manipulation of formulæ difficult. confusing for users without a mental model of the AST. In Symbolic entry of equations using only the keyboard has Equality, there is no grid system or hidden AST for the user been found to be slower and more error prone when compared to explicitly edit, they need only think about the layout of the to other modes (handwriting, speech, handwriting and speech symbols on the page. The prototype software described here in parallel) [3]. However, it is unclear if the advantage of meets most of the requirements above, but still requires some handwriting based modes remains if one also considers that use of the keyboard for entering numbers. reliable segmentation and parsing into machine readable form One consequence of allowing free-form symbol manipula- is required for a useable system. tion is that an extra parsing step is required. The description Our hypothesis is that the combination of symbolic entry below is split into two sections, the first describing the user with free-form layout will provide a better, more flexible interface and the second describing the parsing engine built solution. Tools that support free-form layout of formulæ and for Equality. the associated parsing of the mathematical expression are User Interface rare but not unheard of [7], [8]. Pollanen et al. introduce the interface for Collaborative Equations (iCE), a tool with In order to achieve maximum cross-platform compatibility, similar intentions as Equality, however it does differ in its the Equality user interface is implemented as a web com- implemention. For a start, the iCE tool introduced requires ponent, written in Javascript and making use of the React server connectivity in order to support parsing of equations framework.4 This user interface, shown in Figure 1, consists of limiting its offline usability somewhat. Equality offers a com- a main canvas surrounded on three sides by symbol palettes, plete client-side tool so after the initial download of the web with an output pane beneath. Symbols can be dragged from page users are able to use it without internet connectivity, the palettes using the mouse or touch and dropped onto the which can be particularly intermittent on mobile devices. canvas. Once placed, clicking or tapping on a symbol selects it From our perspective, we agree with Pollanen et al., that and selected symbols can easily be moved, resized or deleted.