Mathematics Apps and Mobile Learning
Beth Bos
Texas State University-San Marcos
USA
BB33@txstate.edu
Kathryn Lee
Texas State University-San Marcos
USA
KL10@txstate.edu
Abstract: The following paper is a literature review of research on mobile learning applications
and their use in teaching mathematics. Mobile learning is described in terms of its evolution to
mobile seamless learning (MSL). Design principles are suggested as we attempt to determine what
is the basis for MSL for mathematics in both primary and secondary. Benefits are detailed and
suggestions are made. The ultimate aim of MSL is to make mathematics more accessible to more
students.
Introduction
The following is a literature review on mobile learning (m-learning) in mathematics and the use of mobile
applications (Apps). Project Tomorrow (2012) statistics show how well situated we are for mobile learning in the
classroom. Project Tomorrow claims that 21% of upper elementary school students have a personal smartphone; a
quarter of middle school students have a personal tablet device; and more than half of high school students access
the Internet outside of school via 3G/4G mobile devices. Traxler (2005) offered an early definition for m-learning
specifically linking educational necessity to a handheld device, and Peng, Su, Chou, and Tsai (2009) positioned the
student within ubiquitous and mobile computing technologies that are convenient and expedient and make
immediate learning possible. Park (2011) defines mobile learning as “the use of mobile or wireless devices for the
purposes of learning while on the move” (p.79). Small, portable computing devices are changing the way we access
information and the nature of the way we use technology to learn. Such iOS devices (e.g., iPod Touch, iPhone, iPad)
are consumer-friendly in ways computers are not. Their ease of use makes these devices efficient with direct
interaction that is naturally engaging and efficient for children as well as adults (Murata & Iwase, 2005; Pelton &
Francis Pelton, 2011).
Wong (2012) offers a learner-centric conceptualization of mobile learning where continual knowledge
construction occurs seamlessly by several dimensions. Wong and Looi (2011) conducted a literature review and
distilled 10 features/dimensions that characterize “seamlessness” in the MSL (mobile seamless learning)
environment. This entails (a) encompassing formal and informal learning, (b) encompassing personalized and social
learning, (c) learning across time, (d) learning across locations, (e) ubiquitous knowledge access (a combination of
context-aware learning, augmented reality learning and ubiquitous access to online learning resources), (f)
encompassing physical and digit worlds, (g) combined usage of multiple device types (including stable technologies
such as desktop computers, interactive whiteboard), (h) seamless and rapid switching between multiple learning
tasks (such as data collection + analysis + communication), (i) knowledge synthesis (prior and new knowledge as
well as multiple levels of thinking kills, and/or multidisciplinary learning), (j) encompassing multiple pedagogical or
learning activity models (facilitated by the teachers). With the proliferation of cloud computing technology a
person’s “learning hub” needs not be associated with a hardware device; indeed it is the device-independent
seamless learning platform that provides a suite of learning affordances (Wong, Chai, Chin, Hsieh, & Liu, 2011). In
this regard Wong (2012) “advocates the combination of a cloud-based ‘learning’ account, a smartphone (24/7
access), and additional notebook/desktop computers as an ideal technical setting for a personalized seamless
learning environment” (p. E21).
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Mathematical Apps
It is a common belief that the incorporation of computer technology into mathematics learning motivates
and engages student (Pierce & Ball, 2009). With mobile devices proliferating the educational environment, a close
look at 4000 mathematical related Apps revealed that most all are simple flashcard, numeric procedures of mobile
textbooks (Pelton & Francis Pelton, 2012). Most do not support sensemaking and fail to support active learning and
integrated visual models (Pelton & Francis Pelton, 2010; Yuan, Chae, Nantwi, & Garg, 2010). There is a danger of
technology driving pedagogy, rather than pedagogy driving technology. As Attard and Northcote (2011) have said,
technology sometimes becomes the focus of the mathematics lessons instead of the mathematics itself. Research by
Samuelson (2007) revealed that teachers who regularly incorporate computers into their lessons tend to use them in
ways that resonate with a didactical, teacher-centered approach. A study by Barry Kissane (2011) pointed to the use
of mobile devices in a mathematics class as a (a) graphing tool, (b) calculator, (c) reference source, (d) measuring
tool, (e) drill and practice device, and (f) access to the Internet. But keep in mind, a typical math class requires
critical thinking skills, problem classification effort, along with recollections of physical facts and relations as well
as experimentation, data collection, and analytical writings (Subramanya & Farahnai, 2012). Often students manage
to complete these tasks with the aid of a scientific or graphing calculator. However, calculators are not employed as
tools to teach mathematical concepts; they are often used as an efficient computational tool. Of those students who
employ technology tools in mathematics, most use a computer with specialized software packages to perform
problem analysis, formulation, and synthesis— to final report writing. The computing power, ease of use, and
portability of mobile devices are clearly more appealing than carrying a laptop and are made possible by the use of
cloud technology (Subramanya & Farahnai, 2012).
Design Principles
Various experts in the field oppose using Apps in mathematics only for drill and practice or as a calculator,
or as a repository for stored facts and believe instead that they need to support sensemaking and active learning
(Pelton & Francis Pelton, 2011; Subramanya & Farahnai, 2012; Yuan, Chae, Nantwi & Garg, 2010). Design
principles have been suggested for mathematics apps that include both an elementary and secondary education
focus. Keep in mind that “best practices” in mathematics encourages teachers to help children understand
mathematical concepts concretely with language first, then pictorially, and finally symbolically. Pelton & Francis
Pelton (2011) offer four design principles. First, keep it simple. A consistent and intuitive interface helps to
minimize the learning curve (threshold) and the effort required to decode and attend to the important information as
it is presented (friction). Distractions such as such as flashing elements, timers, background noise, etc., should be
minimized, while useful affordances and resources such as navigation elements, progress indicators, and instructions
should be simple and intuitively placed. The focus on a single topic/process reduces the amount of time required to
understand the purpose and potential of the resources and the amount of support/scaffolding required to help
learners make sense of the challenge presented. The second design principle is to provide meaningful models and/or
manipulatives, and provide scaffolding support where needed. Models and manipulatives are intended to allow the
students to explore and master the concepts. The third principle is to support individual needs. Apps need to be selfpaced, multilevel, success-oriented, user controllable, and low stress. Fourth, provide support for parents, teachers,
tutors, and researchers (Pelton & Francis Pelton, 2011). Quesenbery (2002) believes that the “Five Es” of usability
of product apply to Apps as well. A product is useable when it is effective, efficient, engaging, easy to learn, and
error tolerant. It is effective when it affords goal accomplishment with minimum effort; efficient when it allows rapid
completion with few errors; engaging when it offers enjoyable day-to-day operation; easy to learn when it supports
rapid initial skill acquisition and expanded skill development with experience; and error tolerant when it prevents
errors and supports error recovery.
After examining secondary mathematics/engineering Apps in terms of (a) content and simplicity, (b)
special features, and (c) usefulness, Subramanya and Farahani (2012) identified “elements that are generally needed
in an App for learning Math/Engineering concepts” (p. 179). The elements are as follows:
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Text lecture – This component contains parts of relevant screen shots of formulas and equations,
figures, graphs, definitions, etc. This is predominantly alphanumeric information.
Video lecture – This consists of video clips of the actual lecture, explaining concepts. This also
consists of derivations of complex equations or concepts. A text layout of derivations is static and
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usually will be hard to follow in most cases. On the other hand, a dynamic demonstration of the stepby-step derivation along with the explanation of the actual equation or concept will be much easier to
follow and can be played over and over again.
Simulation – In many mathematics or engineering courses, laboratories are expensive and time
consuming to maintain. With the advances in computer hardware and software, simulations have
become a cost-effective alternative. The simulation element in the mobile App is expected to provide
the same learning experience of concepts and working systems as a lab.
Calculator/computational – Computations are the core of the working of mathematical and engineering
systems. The computational element facilitates the easy computation of various formulas and solving
of equations.
Animation - Animation and graphics facilitate the ease of understanding complex concepts,
computations, and data interactions. The availability of powerful processors in a smartphone would
enable the implementation of this component.
Game – Although optional, a well-designed game element would provide the learner with an
entertaining and engaging experience, while at the same time focusing on the relevant learning of the
concepts. The availability of powerful graphic processors, adequate memory, multi-touch screens, and
sensors (proximity sensor, accelerometer, etc.) enable this element to be effective.
Interactivity–The interaction component is crucial since it should facilitate self-learning in the absence
of interactions with the instructor. A well-designed interaction would enable exploration and scenario
evaluations that are essential in Engineering, since most designs are developed iteratively (Subramanya
and Farahani, 2012, p.179).
Benefits
Two meta-analyses discussed by Pelton & Francis Pelton (2011) point to the benefits of computer
technologies in mathematics education and accredit part of the success to pragmatic constructivist teaching
approaches (e.g., problem-based, inquiry-oriented and situated cognition) and suggest that appropriately designed
computer technologies can play a role in supporting and encouraging students as they learn (Hattie, 2009; Li & Ma,
2010). In mathematics, games are constructivist by nature when built on guided discovery, strategy seeking,
problem solving, and classroom discussion. According to a recent Pew study (Brenner, 2013) 97% of teenagers, age
12-17, play computer, web, portable, or console games. Additionally,
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50% of teens played games "yesterday."
86% of teens play on a console like the Xbox, PlayStation, or Wii.
73% play games on a desktop or a laptop computer.
60% use a portable gaming device like a Sony PlayStation Portable, a Nintendo DS, or a Game Boy.
48% use a cell phone or handheld organizer to play games.
Game playing is ubiquitous among Americans teenagers. Fully 99% of boys and 94% of girls report
playing video games. Younger teen boys are the most likely to play games, followed by younger girls and older
boys. Older girls are the least "enthusiastic" players of video games, although more than half of them play. Some
65% of daily gamers are male; 35% are female. However, only relatively recently has there been a significant focus
on the potential of instructional educational video games and their design (Hirumi, Appelman, Rieber, & Van Eck,
2010; Kebritchi & Hirumi, 2008). The danger in preparing mathematics games is using practice problems with no
sensemaking value.
Smartphones can effectively supplement traditional course materials and enhance learning based on the
following assumptions:
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Well-designed smartphone/tablet Apps are quite intuitive and easy to use, and there is really no
learning curve.
Tools are/would be available to create content without too much effort for mobile Apps.
The Apps for learning could be made interactive and engaging using commonly available technologies.
Thoughtfully designed Apps can help consolidate important concepts of a course.
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The Apps could facilitate customization/personalization to cater to different learning styles.
They facilitate ‘any-time, any-where’ learning due to their inherent mobility and Web access.
The Apps cater to short attention spans by presenting a logical unit of content one screen at a time.
The Apps can be designed to make learning fun using judicious use of text, graphics, audio, video, and
animation.
They could be designed to make them exploratory by incorporating numerous related content and
providing support for non-linear navigation (Subramanya and Farahani, 2012).
Major benefits of mobile apps for learning concepts in math and engineering include supplemental learning
aids, anytime/anywhere use, self-paced learning, and reinforcement of abstract concepts. Mobile apps also enhance
retention while being entertaining and engaging. They encourage the use of multiple (rich) media, immersion in
interaction, self-assessment, and provisions for exploration and experimentation. Mobile apps have an element of
fun and exploration leading to extended learning. They have been credited with positive feedback and increased
motivation while being cost-effective, customizable, and time-effective (Subramany & Farahani, 2012, p. 174).
Another potential benefit of mobile Apps for learning is the repetitious use of content that will enable the
reinforcement of abstract concepts. A well-designed App could be entertaining and engaging and facilitate
exploration and experimentation. A student performing below grade level may use an App repeatedly to learn or
reinforce difficult concepts. Learning can be personalized to meet the individual needs of each student. The potential
of well-planned Apps is beginning to be realized. In general, educational Apps provide a new and more efficient,
interesting, and convenient approach to learning by delivering instructional modules directly to students’ mobile
devices.
Content Presentation Factors
As shown in Figure 1, content presentation is a very crucial part of the user interface of the App and
directly addresses learner styles in the following ways:
 The physical level consists of physical effort required for the interactions, namely, the number and
ease of multi-touch gestures and the ease and extent of alphanumeric input.
 The low-sensory level consists of font types, font sizes, colors, textures, contrast, borders, etc.
 The upper-sensory level consists of the screen layout structure – the relative positioning of screen
objects, menus, icons, graphics, navigation, feedback, etc.
 The cognitive level consists of the use of appropriate contexts, intuitiveness of the interaction model,
minimal cognitive load, and use of appropriate real-world metaphors (Subramanya, 2012; Subramanya
& Yi, 2005).
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Figure 1. The different layers and factors to be considered in mobile content presentation.
Pedagogical and Technological Considerations
Table 1 provides an example of the pedagogical and technological requirements involved in the design of
an m-learning application based on teaching concepts of spatial geometry related to pyramids (de Lima et al., 2011).
Table 1.
Pedagogical and technological requirements
Requirements
Learning Theory
Implemented Solution: Pedagogical proposal employed in a collaborative learning approach and based on
constructivist elements. Such reasons is based on the possibility of cooperative interaction between students and
the machine, which must always encourage discussions that foster teamwork and the building of new knowledge.
Requirements
Content Presentation and Quality
Implemented Solutions: The content provided in the applications makes use of text, questioning, different images,
animations, and video with good picture quality and sound. Moreover, there is a concern about minimizing the use
of the scroll bar and making the most use of the screen space. The animations are presented in a loop for students
to have continuous access to the construction of geometric shapes.
Requirements
Content organization
Implemented Solution: for the application on pyramid shapes, the content is organized from the most general to
the most specific, following the assumptions of Ausubel et al. (1980). First a video with narration about the
pyramids of Egypt is presented. Followed by an animation that conceptualizes a pyramid from a mathematical
point of view. Later basic elements of a pyramid are presented.
Requirements
Support and feedback to the Students
Implemented Solutions: In all applications, messages tell students of success and error situations. In case of error,
the application returns to the theoretical content providing access to the theory of the studied geometric figures.
Thus, students can reflect with colleagues about the problems faced during the tasks.
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Conclusion
Due to the tremendous growth of wireless mobile device ownership by American teens and the multiple
affordances these dynamic devices allow, we purport that the integration of these devices can accelerate
mathematics learning and increase science, technology, engineering, and mathematics (STEM) achievement.
Engaging students in learning by using the devices they own and are invested in using may hook students into
authentic mathematics learning. Of course, technology alone cannot transform secondary mathematics learning;
teachers must be supported in the implementation of mobile devices within a rich inquiry-based learning
environment. Even though m-learning is a relatively new topic, many groups are researching it in a variety of ways.
However, as educators, we acknowledge that effective technology integration is reliant upon discovering specific
pedagogical strategies that amplify specific context (e.g., content area, grade level appropriateness).
Searching for mathematics related Apps that attend to sensemaking and intuitive design, among the other
factors discussed, has posed a challenge. Finding research to support the many claims made in this paper was also
difficult based in part to the difficulty of implementing a “bring your own device to school” philosophy.
Implementation challenges not addressed in this review include the type of school network configuration and
acceptable use policy that best serve mobile learning. School administrators and teachers are often afraid of what
might happen within the classroom that may be recorded and used for other purposes than the intended task.
Future work would address usability constraints within the classroom, as well as the various pedagogical
and different learning styles in mathematics and their incorporation in the App development. The study of the
effectiveness of learning and of assessment, including the development of more elaborate sets of rubrics for the App
is essential. Developing a multidimensional (feature set, cost, topics level) ‘guideline’ matrix to “pick and choose”
various features would be beneficial to the implementation of mobile learning in the mathematics classroom.
Determining the methodological bases of the specific needs of an App for mathematics learning would also benefit
App development.
Furthermore, we need to pursue the seamless integration of technologies into existing courses (Wong, et al.,
2011) and develop a feedback system. Developers, classroom teachers, administrators, and we researchers need to
gather statistics for evaluating the effectiveness of teaching/learning with Apps as tools for learning. We need to
think about how best to manage the implementation of mobile devices as learning tools in classrooms and address
the current barriers of learning with mobile devices in schools. Chris Dede, seeing the challenges of m-learning that
lay ahead, placed a priority on states and districts reinventing their curriculum and the way it’s taught. “For a district
to do this well, they have to look at how the device is going to help change the pedagogy,” he said. “It’s important to
understand that the device itself isn’t the innovation. People can’t just digest worksheets and stick them on a cell
phone and think that’s going to be some kind of breakthrough. Good curriculum has to be rethought—not so much
transferred, but more creatively redesigned. It’s a question of our being willing to be creative and recognize the
opportunities available, to look at this small computer as something that has new potential to redesign education”
(Puente, 2012, p. 62).
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