Types of Technology in Mathematics Education
KeywordsComputers Computer software Communication technology Handheld Mobile E-learning
Terms and Definitions
Many of today’s mathematics classrooms around the world are nowadays equipped with a variety of technologies. By using the term “technology,” we mainly mean “new technology,” as we refer to the “most prominent,” recent, and “modern tool” in the teaching of mathematics that is labeled with terms “computers,” “computer software,” and “communication technology,” according to Laborde and Sträßer (2010), p. 122. Another term “digital technology” which denotes a wide range of devices including a hardware (such as processor, memory, input–output, and peripheral devices) and software (applications of all kinds: technical, communicational, consuming, and educational) is used by Clark-Wilson et al. (2011). This is contrasted with yet another term information and communications technology (ICT) widely used in a variety of educational contexts and describes the use of so-called generic software which means word processing and spreadsheets, along with presentational and communicational tools (such as e-mail and the Internet) (2011).
First Computers and Their Use in Education
The time period after 1950 and till the early 1980s was marked by as rather slow but sure penetration of mainframe and minicomputers in education, including mathematics education. With the main focus on accessibility of such devices for schools (question of costs and space), other questions arose by mathematics educators at that time regarding the purposes of its use and impact on learning. Zoet (1969) pointed at several dilemmas, namely, (1) about the capacity of computers to process data, like in business management to produce bills for millions of customers, on the one side, and to compute data, like in mathematical modelling where scientists need to do large amount of calculations in a short period of time; (2) about the time needed to master a particular part of technology (to solve mathematical problems), which will soon be replaced with a new one; and (3) about the possibility of computer to assist a greater number of students to grasp principles of mathematics, as well as strengthen and broaden students’ understanding, about whether mathematics learned by the students will be more functional, once they see how it is used in computers, or if small computers can be integrated into mathematics programs as the slide rule in the training of engineering students.
In 1984, the NCTM (National Council of Teachers of Mathematics) produced a yearbook entirely devoted to the topic on computers in mathematics education (Hansen and Zweng 1984) portraying newest types of technologies called microcomputers as having endless list of applications available for mathematics teachers and learners which are becoming widely accessible for schools at low cost; it also adds graphics capabilities to support mostly two-dimensional representations (Fey and Heid 1984). Again this technology development interacts with pedagogical use as tutor, tool, and tutee (Fey and Heid 1984, referred to Taylor 1980) with questioning whether “traditional collection of mathematical skills and ideas needs” to be acquired by students to enable them “to operate intelligently in the computer-enhanced environment for scientific work” or one must have “new skills or understandings” to get prepared “for mathematical demands that lie in the twenty-first century” (Taylor 1980, p. 21).
Regarding the format of integration of such technology in the process of teaching, educational institutions usually put computers in one classroom (computer lab) shared by several groups of students, or they can put a number of desktop computers (1–4) in a regular classroom, so teachers and students can work with them individually or in small groups. On those computers, teachers could find general software, including spreadsheets (like SuperCalc, Lotus, or Excel) that could be used in multiple teaching and learning purposes, for example, to conduct probabilistic experiments and simulations (Anand et al. 2012).
The 1980s and 1990s were also marked by widely spread use of educational games, on small floppy disks, and later multimedia on CD-ROMs and DVDs, helping even the very young students to learn basics about numbers and shapes and develop mathematical thinking while playing with patterns. Another kind of the software specifically designed for mathematics classroom based on a constructionist’s ideas leads to the development of dynamic and interactive computer environments in geometry (dynamic geometry systems) and algebra (computer algebra systems). Different types of virtual manipulatives thus become available to teachers to make learning more visual, dynamic, and interactive (Moyer et al. 2002).
Computer networks – systems of interconnected computers and systems of their support called intranet and the Internet – emerge and spread out in the 1990s and 2000s. The first (intranet) allows to connect computers with a restraint number of people having access to it; often it is used within an organization, like school or school board or university. The second (the Internet) is open to a much wider audience, in many cases worldwide, although it can serve closed groups/communities built with different purposes. This technology, with the time becoming more rapid (high speed), wireless, and handheld, enhances communication of people or machines with other people or machines to share information and resources in all areas including mathematics. As example of such kind of technology, we will analyze Web 2.0 tools.
E-learning: Web 2.0 Tools and Their Use in Mathematics
Solomon and Schrum (2007) use the year 2000 as a turning point in the development of a new Internet-based technology called Web 2.0. They begin their timeline with year 2000 when the number of web sites reached 20,000,000. The year 2001 was marked by the creation of Wikipedia, the first online encyclopedia written by everyone who wanted to contribute to the creation of the shared knowledge. In 2003, the site iTunes allowed creating and sharing musical fragments. In 2004, the Internet bookstore Amazon.com allowed buying books entirely online. In 2005, the video-sharing site Youtube.com appeared, allowing producing and sharing short video sequences. The authors state that by the year 2005, the Internet had grown more in 1 year than in all the years before 2000, reaching 1,000,000,000 sites by 2006.
The result of this tremendous growth of Internet-based environments and the educational resources generated by them is a transformation of e-learning itself. According to O’Hear (2006), the traditional approach to e-learning was based on the use of a virtual learning environment (VLE) which tended to be structured around courses, timetables, and testing. That is an approach that is too often driven by the needs of the institution rather than the individual learner. In contrast, the approach used by e-learning 2.0 (a term introduced by Stephen Downes) is “small pieces, loosely joined,” as it combines the use of discrete but complementary tools and web services – such as blogs, wikis, and other social software – to support the creation of ad hoc learning communities. Let us look at several features of these tools as we analyze a few examples of mathematical opportunities they create (adapted from Freiman 2008).
Wiki is an Internet tool allowing a collective writing of different texts as well as sharing a variety of information. Everybody can eventually be a contributor to the creation of a web site on a certain topic (or several topics, as it is in the case of the Wikipedia, www.wikipedia.org/).
Podcasts can be used to audio-share mathematical knowledge among a larger auditorium than one with people sitting in a traditional classroom. It can be used as a method of delivering mathematical lectures online as well as for the promotion of mathematics.
Video-casting opportunities are provided by multiple Internet sites, allowing the creation and sharing of video sequences produced by the users. For example, an article published in one local newspaper informs the readers about one university professor who put a 2-min video about a Mobius strip on the Youtube.com site. The sequence was viewed by more than 1 million users within 2 weeks. The environment offers not only an opportunity to view the video but also to assess it (using a five-star system) and to share it with others, as well as publish a comment.
Photo sharing is yet another form of creating and sharing knowledge, available on several dynamic sites with photo galleries like Flickr. Regrouped by categories that can be found by an easy-to-use search engine, the photos can be published and discussed by the members of a community, for example, the community that discusses geometric beauty which numbers almost 5,000 members. Each photo is provided with a kind of ID card that documents useful information such as the date of its publication, the author’s (or publisher’s) username, as well as the list of all other categories to which the photo belongs, the date when the photo was taken, and how many other users added it to their albums.
Discussion forums allow building online communities that talk to each other by posting questions and giving answers. This collective work may enable a student who is struggling with mathematical homework to address other people and ask for help, as illustrated by the following example from the math forum site (mathforum.org). The message posted by one user says that “after having asked a teacher and having read a book,” she “still had a feeling” that she needed more explanation, so she appealed to the whole virtual community asking for help. The discussion on some questions can take the form of multiple exchanges between members.
Blogs may provide multiple educational opportunities as they are built by means of easy-to-use software that removes the technical barriers to writing and publishing online. The “journal” format encourages students to keep a record of their thinking over time facilitating critical feedback by letting readers add comments – which could be from teachers, peers, or a wider audience. Students may use blogs for different purposes: to provide a personal space online, pose questions, publish work in progress, and link to and comment on other web sources.
The learning model that can be extracted from our examples features three major educational trends related to the Web 2.0 technology: knowledge building/co-constructing, knowledge sharing, and socialization by interacting with other people. Moreover, further development toward semantic web (Web 3.0) technology has a potential to enhance self-learning, critical thinking, and collaborative and exploratory learning.
There is a clear need of in-depth study on social media in mathematics education. As latest developments, it is worth to mention a working group at the Canadian Mathematics Education Study Group 2017 conference (see for details at https://judylarsen.ca/2017/06/03/cmesg-2017-wgc-social-media-and-mathematics-education/ and in the WG report, Larsen et al. 2017).
M-Learning: Anytime, Anywhere with Laptops and Other Handheld Devices
According to Burrill et al. (2002), the first type of handheld technology mentioned as a part of the secondary school curriculum in 1986 was a Casio fx-7000G model. Even if the appropriate role of it in mathematics classroom was at that time (and still remains) debatable, it supported the creation of new visions for mathematics education while calling for broader access to deeper mathematics for all students (Burrill et al. 2002). Regarding the newest development of this type of technology, Burrill (2008) sees its potential to combine various learning environments like computer algebra systems (CAS) and dynamic geometry computer software, such as dynamic geometry Sketchpad or Cabri: “new technologies such as TI-Nspire bring together both of these environments in one handheld, providing the opportunity to create an even wider variety of dynamic linked representations, where a change in one representation is immediately and visibly reflected in another” (http://tsg.icme11.org/document/get/218).
Several laptop studies report about a variety of teaching and learning opportunities to use 1:1 portable technology for several subjects including mathematics. Freiman et al. (2011) developed and implemented problem-based learning (PBL) interdisciplinary scenarios (math, science, language arts) to measure and document students’ actual learning process, particularly in terms of their ability to scientifically investigate authentic problems, to reason mathematically, and to communicate. In a rapidly changing world of technology and infinity of educational applications, mathematics teachers can now try to integrate newest technology, like iPads, in mathematics lessons. While only few research are available, first pilot studies, like one reported by HMH (2010–2011, http://www.hmheducation.com/fuse/pdf/hmh-fuse-riverside-whitepaper.pdf), seem to have a positive impacts on students’ performance. In this study, individual iPads were used along with the HMH Fuse: Algebra 1 programs. The application helped students use its multimedia components whenever and wherever they saw fit, regardless of Internet availability. In addition, students could take the device home and “customize them,” adding their own music, videos, and additional applications (Freiman et al. 2011).
Among other types of technologies to be mentioned are interactive whiteboards which, according to Jones (2004), might encourage more varied, creative, and seamless use of teaching materials, increase student’s enjoyment and motivation, and facilitate their participation through the ability to interact with materials. While the whiteboards support and extend whole-class teaching in a more interactive way, haptic (in-touch) devices have a potential to enhance multimodal learning in 3D spaces, on the individual base, or working in small groups, as the technology becomes less costly and more flexible in terms of usability, with better feedback options, allowing for better merging with other mathematical learning environments, such as dynamic geometry (Güçler et al. 2013).
One of the latest volumes of the book series Mathematics Education in the Digital Era co-edited by Calder et al. (2018) does provide an in-depth investigation at the last development in research on mobile technology and its use for mathematics teaching and learning which reveals, among others, its potential for app developers, mathematics educators, and researchers of “identifying and enacting opportunities for enhancing mathematical thinking” but also eventually changing the nature of mathematical activity by evoking a “variety of understanding and ways of thinking mathematically” (Calder et al. 2018, p. 1). For instance, one of the contributing authors to this book, Attard (2018), analyzes potential of a specific kind of programs that educational institutions trying to implement the so-called BYOD (bring your own device), thus targeting improvement of students’ engagement and eventually learning outcomes.
New Technologies and Literacies for the Twenty-First-Century Classroom: Robotics, 3D Printing, Virtual Reality, Minecraft, Scratch, and More
Today’s mathematics classrooms are increasingly becoming equipped with novel technologies which are closely connected with two new trends in education: (1) STEM (science, technology, engineering, mathematics) or STEAM (science, technology, engineering, arts, mathematics) focus stressing interdisciplinary connections and integrative teaching approaches and (2) twenty-first-century soft-skills. At the end of the day, the both trends appear to be closely connected (Freiman et al. 2017). For example, Ardito et al. (2014) have found that robotics-based challenges using LEGO Mindstorms, Robotics, Programming kits “can be used to reshape the classroom environment in terms of student collaborative work and problem-solving skills” (p. 85). While looking into the assessment issue within this novel type of tasks, Savard and Freiman (2016), while pointing at difficulties to assess students’ cognitive development, raise “questions about developing critical thinking and meta-cognitive skills when implementing complex STEM tasks” (p. 110). LeBlanc et al. (2017) studied several school makerspaces where new types of technologies might afford the development of higher-order mathematical processes (such as mathematical reasoning, communication, and problem-solving).
Higher levels of thinking, innovation, and creativity were also mentioned by Huleihil (2017) referring to the context of the use of 3D printing technology which is becoming increasingly popular in STEM education while providing important connections to teaching geometry applications. By applying design thinking, learners use computer-aided design software to create their own objects and watch the action on the screen, thus giving the students a sense of ownership of the work and increasing their understanding.
Another novel technology, still underused in our schools and under-researched, called geographic information systems (GIS), according to Edelson (2014), can, along with the development of fundamental spatial reasoning skills, also contribute to deepening of such very important and challenging concepts in mathematics as scale and density which, according to the author, are essential to the natural and social sciences.
Minecraft game-like web-based environment that can be installed on different types of mobile devices can stimulate students’ interest, curiosity, and creativity (Dawley and Dede 2013, cited by Bos et al. 2014). One of the mathematical connections analyzed by Bos et al. (2014) is exploring area and perimeter by primary students from Grade 3.
Within the increasing attention to the augmented reality (AR) technology, such as tabletop system, researchers and practitioners target students’ motivation, along with opportunity to differentiate mathematics instruction according to students’ needs and capacities, as is recently shown by Cascales-Martínez et al. (2017). The researchers used a set of interactive 3D educational materials related to the European monetary system developed specifically for a tabletop system. In a context of learning about the European Monetary System, the application affords visualization and manipulation of all coins, association of coins, and notes with the corresponding amount, all within a scenario where students can solve mathematical problems in a virtual shopping simulation game (Cascales-Martínez et al. 2017, p. 359).
Interaction between STEM-related types of technologies and different sets of soft-skills draws attention of mathematics educators to new types of literacies, namely, computational thinking, financial literacy, and data literacy while introducing new learning spaces (Freiman and Chiasson 2017). A recent boom in introducing computer programming and coding to the school curricula (including mathematics curriculum) returns to the debates on the 1980s where Russian computer scientist Ershov (1981) called for considering computer programming as “second literacy” which, being combined with the traditional (or first) literacy, contributes to forming a “new harmony of human mind.” Later on, in 1996, Papert used again the term “computational thinking” when analyzing an approach used by Wilensky and Resnick in building a geometric model of a Rugby game to investigate the question: Where should the kick be taken from to maximize the chance of a score? At the time, Wilensky and Resnick were using the StarLogo environment, which was an extension of LOGO. In Papert’s interpretation, this was an illustration of how geometric thinking can use computational thinking “to forge ideas that are at least as “explicative “as the Euclid-like constructions (and hopefully more so) but more accessible and more powerful” (Papert 1996).
Recently debated at the 2017 Symposium on Computational Thinking in Mathematics Education (http://ctmath.ca/computational-thinking-in-mathematics-education-symposium/), researchers looked in how computer languages are used to connect the ideas of computational thinking and learning and teaching mathematics. Based on their work with Scratch programming blocks, Brennan and Resnick (2012) define three dimensions of computational thinking (CT): computational concepts, computational practices, and computational perspectives.
As key computational concepts, the authors give the following list: sequences, loops, parallelism, events, conditionals, operators, and data. When defining components of computational practices, Brennan and Resnick (2012) used data from interviews with children about the strategies they adopted while developing interactive media. From the variety of such strategies, four main categories were identified: being incremental and iterative, testing and debugging, reusing and remixing, and abstracting and modularizing. The third dimension of CT, one of gaining computational perspectives, describes the different roles people can play when working with interactive media, roles that go beyond “pointing, clicking, browsing, and chatting.” Although this role of being consumers is important for learners, it is not sufficient in terms of the development of CT. According to Brennan and Resnick’s (2012) framework, working with design tasks encourages more active roles, which emphasize expression, connection, and questioning.
While arguing for a more equitable focus on each of the STEM disciplines, English (2017) sees particular value in the approach to CT adopted by Gadanidis et al. (2016). This approach focuses on investigating, depicting, and learning “from cases of ‘what might be’ (or ‘what ought to be’), to disrupt common conceptions of what CT and mathematics are accessible to young children, how they might engage with it, and how CT affordances may affect mathematics teaching and learning.” In the case of mathematics, this approach can open the door to higher-level mathematics even to a very young learner. Extending these sentiments to the broader scope of STEM education, English (2017) suggests that equal access to a high-quality STEM education that integrates CT is a “key issue for future research, not only with respect to socioeconomic, gender, and ethnicity factors, but also in terms of capitalizing on and extending the capabilities of all learners.” As concrete example of how such ideas can work within a mathematics modelling course for teachers, we can cite the study of Broley et al. (2017) who asked their students to select an open mathematics conjecture or a question of interest (to themselves) and then use programming to investigate whether it may hold true or not.
Another literacy which can also be incorporated into mathematics curricula (also connected to mathematical modelling) is so-called data literacy, which relies, among others, on a relatively new concept of big data technology. According to Gil and Gibbs (2017, p 168), the use of Gapminder and iNZight technologies within an interdisciplinary collaborative activity can contribute to the development of secondary school students’ understanding of big data and scaffold the development of skills needed to create meaning from complex data. According to the authors, this type of learning provides a promising response to numerous challenges in extracting useful information from big data, related to volume, velocity, variety, and veracity, thus contributing to the development of modelling capacity and covariational reasoning, which are important for educating statistically literate twenty-first century citizens.
Along with computational and data literacies, digital financial literacy makes its way into the twenty-first-century classrooms as challenges of “managing in a cashless (or near cashless) world, maintaining online security, navigating digital currencies such as bitcoin, utilizing alternative online banking/lending platforms, or understanding digital wallets, asset and securities tokenization, and a multitude of other new digital technologies” need to be addressed by educators while preparing students to deal not only with financial industry but in many others, rapidly adopting automation and blockchain technology (https://dnotesedu.com/2018/04/financial-literacy-needs-to-include-knowledge-on-cryptocurrency/). While quite an advanced mathematical cryptography is involved in conceptualizing digital cryptocurrency (Dlahaye 2017), some ideas o blockchain technology can be (and should be) integrated already at the primary school level (Hunter and Pillai 2018).
One of the latest developments of educational innovations that appeal to three literacies mentioned above is related to the Internet of Things (IoT), new types of technology consisting of smart connected devices which could use data to transform the way we live (McKinsey Global Institute, 2015, cited by Davis, 2017). As an example of how this technology can be used in a primary STEM education, Davis (2017) conducted a project engaging Grade 4 students in inquiry and creation of a plant pot, a traditional engineering design challenge “that is grounded in collecting, analyzing, and communicating data.” According to the author IoT took the challenge to the next level by “adding data to the mix--create a smart plant pot” (Davis 2017).
It is clear that technologies and skills discussed in this section need to be investigated in more depth, both as research topics and innovative practice.
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