Formula Collection Mobile Apps
Realized by Teachers
Paul Libbrecht, Kerstin Schneider
Informatics, Weingarten University of Education
Abstract
Formula collections are books used in several European countries tosupport the practice of mathematics of by its learners and later by its
practitioners. They are compendia of mathematical formulæ that students
employ during their studies, and which may be of use in later professional
life.
This research explores the opportunity of bringing the formula collection to the mobile phones of the school students. It is motivated by thefact that the formula collection may be one of the rare objects that pupils
may refer to later in their lives, when mathematical knowledge needs tobe recalled. Making it a mobile phone app makes it a zero weight object
which is in easy reach. Technically, this app is realized by a set of HTMLfiles compiled into a mobile application. This makes the formula collection
a first class citizen that students can easily access.
More importantly, this makes it an artifact that teachers can create
themselves instead of relying on a publisher: using their notations andtheir selected content. The centerpoint of this research is to enable teachers in creating and delivering such an app to the mobile devices of theirstudents.

1

Motivation: What can a Formula Collection do
and why could it be mobile?

Formula collections are books used by mathematics practicioners, be they pupils,
students, or workers, in order for them to be reminded of the essential formulæ
they can apply. These are generally small books, or sometimes photocopies,
which teachers choose as representing the essential knowledge. Formula collections are generally short and thin, so that they are always at hand anytime
they practice mathematics, in particular, in all exercises and exams. The formula collections follow the spirit that nothing in math needs to be learned byheart and that such collection constitutes the complete set of knowledge that
one would need to refer back to, when trying to remember the mathematics one
has already practiced.

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Figure 1: An example page of a formula collection book

Their use is common in France (where one is called formulaire) and Germany (where one is called Formelsammlung) as can be seen, for example in thenational standard of Germany [Kul04] where it is cited among the tools such as
calculators.
However, the only equivalent books we have found in English speaking countries would be the CRC Handbook of Physics and Chemistry [Lid07] which contains a part dedicated to mathematics formulæ. We have not found dedicatedequivalent works in English, except the concepts of “legal cheat sheets”. The
latter are sheets summarizing all learned formulæ which students’ prepare themselves. While we acknowledge that such cheat sheets are pedagogically effective
for responsible students, the usage of a formula collection is likely to be be safer
and thus more effective for weak scoring students.
An example page of a formula collection is in Figure 1. The photographgives a hint of the many of formulæ and the intensive work that has been donewith this booklet on the desk, but the complete lack of handwritten notes, as it
is allowed in an examination.
The motivation of this research stems from the strong potential of applicationof a ubiquitous formula collection beyond the mathematics learning phase: asformula collections allow a person to be reminded of the facts that he or she
studied in a very efficient way, it may support the person who encounters amathematical task. This person would be reminded of the formula to apply ina way that he or she remembers and is able to apply.
As a simple example, a person needing to measure an amount of paint necessary to cover a given shape, needs to compute the surface of the shape and, for
this, may need to be reminded how he or she did it at school. In such situations,a formula collection is among the easiest references to pull from.
We contend that making such a formula collection an application of a mobile
phone, in short, a mobile app, raises its applicability because it is available on
the personal device with almost no weight and can thus long with the person.
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1.1

Support to Remember

Cognitive psychology has studied the forms of knowledge presentation that permeate most effectively to memory. Among other, the well known multimedia
principles, Meyer and Sweller describe broad principles of the nature of thelearning material, for example that learning is more effective if knowledge isprovided by multiple simultaneous channels (the multimedia principle) or if aconsistent vocabulary is used (the consistency principle). The book of ColvinClark and Meyer [CM02] provides an accessible presentation of these research
results.
From these considerations, one can deduce principles about the design of aformula collection, could it be done afresh for each student:• It should contain the formulæ that the students have learned and canapply, no more and no less.
• It should employ the same vocabulary (words, mathematical notations informulæ and in geometric figures, typographical convention) as the student
has learned.
• It could be able to remind the students how a particular formula was
obtained by a short summary such as a typical figure.
While such principles are very similar to those of a textbook or an exercise
statement, their implication on the formula collections is larger, because of
the potentially large time lapse between the learning time and the lookup (aslong as several years). The abilities for such formula collection to contain justthe information that was presented using the same word leverage better the
spreading activation effect of long term memory retrieval.1

1.2

Leveraging the Freedom of Teachers

While teachers often employ publishers’ formula collections, they have to remainclose to the vocabulary used there, explain the use of the book, and ensure it isbrought in action regularly.
The digital world allows much more individualization since publishing andcopying operations are easily done at all possible scales (from individual throughclassrooms to entire countries). The opportunity to craft an individualized
formula collection can be used by teachers who often publish simple documentson the web such as PDF-encoded exercise pages.
This opportunity is well exercised if users re-use others’ source documents sothat only a small set of adaptations is necessary to obtain a formula collectionthat can be made accessible to the students. The world of open educational
1 Spreading activation is a model for the retrieval of small chunks of knowledge from the human memory; a short description is provided in http://en.wikipedia.org/wiki/Spreading_activation which illustrates well the utility of the exact retrieval in order to activate easier
related knowledge items (e.g. the ability to apply a formula having found it it back).

3

resources whose common licenses stimulate the re-use and remix of resources
showcases this approach well.2
The goal of this research is to empower teachers, using technical advices andusing prepared sources, so that they can create and publish a formula collectionthat works on mobile devices of the students.

1.3

Use of Mobile Phones at School

Mobile phones tend to permeate all layers of the society of industrialized countries. And indeed, the recent JIM report [FKR13] showed that in the south ofGermany in 2013, almost the complete population between 12 and 19 years had
mobile phones and 88% of them have an internet capable mobile phone. However, schools have, thus far, offered strong resistance against the use of suchdevices in their courses because of the inherent distraction, the strong danger ofabuse they expose and the management challenges they may create for teachers(of technical, social, and legal nature). This has lead to the general prohibitionof switched-on mobile phones in the regular school times.
In this research, as reported by many school principals, we consider that thisprohibition is a short-term measure. We expect that conditions can be devisedso that a trusted productive application of personal mobile phones can be negotiated and applied in schools. This follows the mobile learning trend, scheduled
to happen in the next year by the the Horizon Report 2013 K12 [JAC+ 13]. It
also follows the bring-your-own-device trend (BYOD) which is found there andin multiple other places as an important challenges that infrastructure teams
need to face currently.
We contend that pedagogically, the applicability of personnal mobile phones
in school settings is an effective way to bring the school business closer to theteenager’s population instead of leaving the mobile phones to gaming and fun
communication alone.

1.4

Works related to the Formula Collection Mobile App

Many research works can be mentioned that approach the objective of a formula
collection mobile phone application, but each in a very different setting.
Paper-based formula collections remain the major source of inspiration
for their content coverage and their use by mathematics teachers. Example
include [DFH+ 07]. The ubiquity of these books, and their anchoring in thepractices is what motivated our research.
Clearly, online references are most commonly used to be reminded of amathematical formulæ. The first and foremost example is the Wikipedia encyclopedia (http://wikipedia.org) which is widely used on mobile phones,
for online and offline use. Similarly, the Digital Library of Mathematical Functions (DLMF, http://dlmf.nist.gov) is an important online reference for (ad2 The Open Educational Resources form the body of resources relevant to knowlege who
can be freely shared empowered by a license such as Creative Commons. More information
at http://creativecommons.org/education.

4

vanced) mathematical formulæ which includes a formula search. Two aspects
differentiate our objective from online encylopedias:• Their universality makes each page of them not completely relevant to thelearners: the pages present a lot of knowledge that quickly overwhelms
a casual appliers of mathematics and include knowledge manipulations
that do not apply the way such a person has learned. As result, while itis possible for many persons to understand and apply the knowledge toa problem, such a process is time consuming and is thus often rejected.
Moreover, it only leverages the person’s memory in a shallow fashion.
• Their wealth make them difficult to access. To the authors’ experience,these resources are generally accessed by search which is an error prone
process and may have near misses where two articles with similar titles
provide slightly different information sets between which a casual user isunable to differentiate.
From these considerations, we contend that a smaller reference that is close to
the person’s knowledge has more chances to support the memory activation thatallows a fast application in the current problem solving tasks.
Formula collections mobile apps are available in a large variety, somefree some not, some very complete, some lightweight... The diversity of the offerthat one can find on such application delivery channels as the Android Play Store(http://play.google.com/) or the Apple App Store (http://itunesstore.
apple.com/) is a good sign, we claim, that such formula collections are desirable
in many different flavours so as to support best the learners. While observing
the available apps, we were unable to notice one whose content was similar, forexample, to the book [DFH+ 07], neither bigger nor smaller.

1.5

Applicable Technologies

In order to make the realization of formula collection
applications an easy task, we have attempted to ellicit
technologies that can easily be leveraged.
Simple ePaper formats, such as PDF or pictures,
as well as e-book formats such as epub can carry thecontent of a formula collection. However, it was decided
to avoid this avenue because there is no way to create an
icon as a startable application for such. Moreover, PDFs
and pictures would not present any form of adadptivity
to the display size other than scaling, while a responsive
approach (which would avoid columns in particularly
narrow displays, for example) is desired. Thus we set tosearch for a solution that could generate full-fledged
mobile apps, hopefully cross-platform.
From the market observation, it was clear that theHTML language was the primary carrier of content for
5

Figure 2: ...readily
accessible from theuser’s start screen,
here in Firefox OS.

mobile phone apps. Multiple authoring techniques were applicable to our
project with a favour towards mathematics-oriented authoring tools and thus towards LaTeX-like formula input and simple graphics inclusion for HTML. While
MathML was a clear contender for formula encoding, its need for a dedicatedinput editor made it less desirable as an input format.
Generation of HTML pages from other formats has appeared as possible but
not very necessary since a fine control of the HTML layout was desirable andsince HTML-editing was deemed a very accessible task for teachers.
For these two reasons, and for the lack of availability of a MathML-fromLaTeX generator that was easily installable on multiple platforms including
Windows, MacOSX, and Linux, it was decided that editing HTML web-pages
with LaTeX inside was a comfortable approach. This approach is well supportedby the MathJax library, which also allows edited HTML files to be tested in thedesktop while being developed.
This choice makes the input of mathematical formulæ rather comfortable
for a teacher that knows LaTeX. Moreover, the text-like nature of the source
files, as opposed, for example, to the Oasis Open Document format or theMicroSoft Word format, make them good candidates to participate in a textbased versioning system (such as GitHub.com currently used for the project’s
code hosting or any other versioning server).
Another facet that is important when aiming at the mobile phone is the great
variety of display configurations that the mobile phone market offers: from 20”
tablets to 2” mobiles, from double resolutions, to the simplest 320 × 240 display.
To support a diversity of appropriate display, the HTML design approach called
responsive design has been coined which consists of a strategy of HTML andstylesheet markups that allow, for example, two cells aside of each other, tobe stacked below each other on narrower screens. A few simple steps of the
responsive design have been applied in the FormelSammlung app.
Packaging techniques from HTML web-pages to a mobile app are multiple
and of multiple feature sets and license. Our choice has been made on a foundation which hosts code projects for the very long term development under avery open license. The Apache Foundation’s offer of the Cordova library3 has
thus been our choice.
The resulting app is relatively easy to install on one’s own mobile phone anddisplays the web-pages using the traditional MathJax effect: first the LaTeX
formulæ then a rendering of them. The display takes an amount of memory andtime to load (as bad as 3-4 seconds on relatively old mobiles).

2

Realization of the Formula Collection App

The Formelsammlung mobile phone application is realized following a technicaltutorial which teachers are invited to follow. The tutorial, in German, can befound at
3 Apache Cordova is the successor of Adobe Phonegap, it converts a set of HTML and
JavaScript files into mobile phone applications. See http://cordova.apache.org/.

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Figure 3: A few sample views of the current FormelSammlung sample app:the intro screen (Android), the parallelogram formulæ (Android), the main listof shapes (iOS), and the explanation animation for the parallelogram formulæ
(Android).

https://github.com/Kerstin91/FormelsammlungApp/blob/master/tutorial.md

where it is explained how to create an app for the Android (and soon the FirefoxOS) operating systems.
The steps involve installing the NodeJS runtime, the Apache Cordova mobile
development framework, one or several platform development kits (such as theAndroid or FirefoxOS development kits) and configure them within the path ofaccessible commands for the command-line. The few steps of the compilation,build, possible simulator run, and deployment to an own web-space are sketchedin this tutorial. They allow teachers to create the necessary files to deliver totheir students:• For Android, within their own web-space (as an .apk file, provided it isexplained to the recipients how they temporarily allow alternate sourcesof applications) or, after registration, at the Google Play store.
• For FirefoxOS, within their own web-space (as a package web-page withmanifest) or, after registration, at the FirefoxOS Marketplace.
• For other platforms supported by the Cordova platforms, for example
Apple iOS or BlackberryOS, similar deployments are intended, not all
being free.
The resulting app displays pages of HTML content is a simple full-screen
fashion. The web-pages are depicted in Figure 3.Thus far, the content developed
has been sufficient as a simple proof of concept with just geometry elementary
formulæ:• a list of graphically recognizable pages documenting each of the classicalgeometric figures by their properties,7• a web-page for each figure, each made of a picture and a few properties,• for each of the area computations, users can see from each figure an animated explanation providing a justification (hence a reminder) of the area
formulæ.
While this content is rather modest, it has allowed us to provide a fairly complete
proof of concept.
These two facets support well the potential teachers who wish to modify thepages to create their own formula collection: creating new pages involves only
duplicating and editing example pages and their references in the page list.
The sample Formelsammlung app, packaged as an Android or FirefoxOS
application can be found at:
http://www.hoplahup.net/FormelSammlung-Download
Its source code, under the Apache Public License, can be found at:https://github.com/Kerstin91/FormelsammlungApp.

3

Technical Challenges

This realization appears rather simple, and this is a good quality of the approach, it is nonetheless paved with little traps some of which we describe here
as remaining to solve.
Offline with MathJax MathJax is a JavaScript library made for the web. It
can be linked into HTML pages both as self-hosted or through a content distribution network. However, thus far, our attempts to include MathJax within theapplication package and link in web-pages have failed on both tested platforms
without visible reasons.
The workaround is to let the first-time users launch the application andwatch at least one page with formulæ as soon as hey install it. Subsequent runs
work fine, as long as the caches are not emptied, even after a restart.
Testing for realistic Devices Testing on multiple platforms is, in theory,
possible using simple emulators but we seem to lack good lists of realistic devices
that one should consider as important for particular populations. The few
devices we have tested on are devices close to what students typically have(contemporary smartphones with fairly large screens) which is not very close
to what pupils at schools typically have; more often smaller screens and older
operating systems.
The responsive design approach allows us an amount of adaptations to beplanned in advance but only testing, for example, allows us to observe the lackof display of a particular button because it is towards the bottom of the page.

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4Conclusions

In this paper we have proposed an approach for teachers to package the formula
collection that they expect their pupils to use regularly in and after the learning
life as a mobile phone application so that the mathematical knowledge practiced
in class remains lifelong at hand. The proposed approach has been realized as atutorial which teachers can easily apply so as to deploy a custom mobile phone
app to their students’ mobile phones.

4.1

Open Questions

While the current state of the research provides a concrete approach that canbe directly applied, a few open questions remain:
For exams? We have described approaches to make it normal to use themobile phone in classrooms during exercise and explanation sessions. These
approaches are not yet well established but have a good chance to penetrateschools. An area, however, where personal mobile phones are likely to be prohibited for a longer time is the examinations, where students respond to questions
on paper or in front of PCs in a formally isolated environment. The personal
mobile phones are here inappropriate because it seems to be impossible to apply
such measures as frequency blocking so as to prevent them communicating tothe outside world. However, a formula collection is allowed (and encouraged) inexams. Is the solution to replace this formula collection by devices of the schoolthat the teachers can control? Is the solution to distribute paper versions of theformula collection? While pocket calculators run a similar risk, their market
has been sufficiently small that no uncontrollable device and no non-resettabledevice is allowed in exams, even though all devices are personal. Would there bea possibility to ensure that personal phones become exam-ready by automating
its control and communication features?
The Making of a Formula Collection as Classroom Project? One of
the gained advantages of having a teacher-produced formula collection is, clearly,
the freedom that teachers exercise for their own mathematical definitions, visualizations, and notations. Another may well be the adoption of an inquiry
learning approach where the realization and update of the formula collection isthe business of the complete classroom.
This would reach a learning effect somewhat close to that of writing one’s
own cheat-sheet but would be less subject to the insecurity of personal errors.
Indeed this is the pedagogical practice of inquiry learning whose process often
follow a classroom discussion and whose outcomes are recorded on a blackboard
and are expected to be pencilled in each students’ theory book.
Such an approach would also contribute to the education in the employmentof mobile phones, discussing such issues as deployment techniques, productive
usage in everyday situation, and trust to knowledge providers. Such an education is part of computer-science-education which, in the German state of9

Baden-Württemberg, is currently under the responsibility of the mathematics
teachers even though, thus far, very few teachers are trained to be able to deliver
such an education (less than 5%).
Distribution through an Application Store? Our sketch of deployment
suggests that it is natural for teachers distribute the mobile phone formula
collection through their own web-space as it is made only for the members of
the selected classroom(s). However, we have not been able to articulate if it wasgood practice and how it could be published in traditional application market
stores.
On the one hand, publishing to the application market store is likely to be
slower and less frequently updated, and, in the case of iOS at least, to create
additional costs. On the other hand, such distribution methods are the only
ones that do not need to lower the default security levels of the students’ mobile
phones: settings such as Allow applications from untrusted origins can be aback-door to the installation of malware from arbitrary web-pages. Moreover,because the procedure of obtaining a code-signing certificate and perform thesignature is pricey and complex, only debug-certifcates are deployed by thecurrent tutorial. This can be compensated for by an explanation to the users,
so that they revert the checkbox of allowing untrusted sources. But, similarly
to the necessity of starting the application once with network, it is likely to beforgotten by careless students. Such issues of trustworthiness and authority arepart of the business of distributing apps so as to claim a reliable informationdelivery.
It is not clear yet, whether alternate application providers will emerge as
a viable solution in the long run in the mobile world or not. It would not
be surprising that an identity certification service necessary for creating publiccertificates could be the service of the media support teams that are allocatedto each school board such as the Kreismedienzentren or that application storesdedicated to the learning goals are introduced and well supported as is suggestedby the architectural vision of [Wor13].
It could be that making the creation of mobile phone applications morecommon practice, as is the the intent of this project, may put a pressure on theapplication market store makers and operating systems’ makers, because theflood of submitted apps make it harder and harder for end users to differentiate,
say, the following two applications’ titles:
Formelsammlung Klasse 9 A, Herr Schmidt
Formelsammlung Klasse 9a, Hr Schmitt

Acknowledgements
We thank the insightful comments of the MathUI reviewers for their impulse
and reactions.

10References[CM02]

Ruth Clark and Richard. Mayer. e-Learning and the Science of Instruction. Pfeiffer, San Francisco, 2002.

[DFH+ 07] Hans-Jerg Dorn, Hans Freudigmann, Manfred Herbst, Günther
Reinelt, Manfred Schwier, Peter Sibert, and Sven Stötzer. Formelsammlung Mathematik für Realschule / Sek. I. Ernst Klett Verlag
GmbH, 2007.
[FKR13]

Sabine Feierabend, Ulrike Karg, and Thomas Rathgeb. Jugend,
information, (multi-) media. Technical Report JIM 2013, Medienpädagogischer Forschunsverbund Südwest, 2013. Available fromhttp://www.mpfs.de/?id=613.

[JAC+ 13] L. Johnson, S. Adams Becker, M. Cummins, V. Estrada, A. Freeman,and H. Ludgate. The 2013 Horizon Report. K-12 edition. The New
Media Consortium., Austin, Texas, 2013. See http://www.nmc.org/
publications/2013-horizon-report-k12.
[Kul04]

Kultusministerkonferenz. Bildungsstandards im fach mathematik
für den mittleren schulabschluss, 2004. Available from http://www.
kmk.org/fileadmin/veroeffentlichungen_beschluesse/2003/
2003_12_04-Bildungsstandards-Mathe-Mittleren-SA.pdf.

[Lid07]David R. Lide. CRC Handbook of Chemistry and Physics, 88th Edition (CRC Handbook of Chemistry & Physics). CRC Press, 88 edition, June 2007.

[Wor13]

Working group on Education of the Open Source Business Alliance.
Digitale Medien, Bildungsplattformen und IT-infrastruktur an
Schulen auf Basis offener Systeme und Standards. Available at
http://www.osb-alliance.de/fileadmin/Downloads/OSBA_WG_
Education_Whitepaper.pdf., November 2013.

11