The effectiveness of AR in mathematics education

1.1 Background

1.1.3 The effectiveness of AR in mathematics education

With the advance of emerging technology, representations, which have been widely used in STEM as mentioned above, are more likely to break through the constraints of traditional devices, presenting in novel forms and ways. Different from either concrete or abstract representations that has been used in mathematics for a long time, it is promising to explore the

effects that how mixed reality technology would bring another brand-new type of representations to students’ learning on mathematics concepts. More specifically, augmented reality (AR) could be one of the feasible means to carry out the effectiveness of mixed reality.

Billinghurst (2002) indicated a few benefits that AR brings along, one of which is that AR supports seamless interaction between real and virtual environments. With the aid of AR, students would be able to perceive even more and various kinds of representations which are not possible to realize before (Sarama & Clements, 2009). The effectiveness of AR in developing mathematics abilities was also proved from psychological perspectives, i.e. physical, cognitive, and contextual aspects (Bujak, Radu, Catrambone, Macintyre, Zheng, & Golubski, 2013)

1.2 Research purposes and questions

1.2.1 Assumption of solution to the problems

Lesh (1981) proposed a model that illustrated the translation among various representations, including pictures, manipulative, spoken symbol, written symbol and real world situation (as Figure 1). Due to the structural characteristics of mathematics concepts, learning a concept must involve coordinating a system of relations, operations, or processes.

Lesh also noted that if students were suffered from translating one particular representation to another, then the reverse order instructions could solve the problem. In other words, because mathematics symbols are lack of real life relevance, students often have difficulty in translating from symbol to real world situation. Subsequently, according to Lesh’ s model, it would be helpful for students to learn in the reverse order, i.e. from real world situation onto symbols.

On the other hand, the means of pictures have been used for a long time in mathematics classroom. However, pictures can hardly convey any further information except the static and meaningless appearances. They barely facilitate students to learn in real-life-like situations, which are much more meaningful. Thus, this study hypothesized that it might be beneficial for

students to perceive the concepts in life situation, and then be able to translate from real-life situation to symbol more easily than from pictures, with the aid of technique of Augmented Reality (AR).

Figure 1.The translation between representations

1.2.2 The process of validating the assumption

In order to validate the assumption that whether real-life representations, which facilitated with AR instruments, were more beneficial than pictures, the study designed an instrument called “AR-facilitated Representations tools”. Following are the proposed concepts of AR Representations, shown in Figure 2. First of all, AR Representations shows the virtual boundary line of the targets when students scan objects in reality. Then, dynamic virtual distributing lines appear on the screen to demonstrate the process of sharing equally. Finally, the orange square would eventually combine with blue line, and the assigned amounts are shown in gray-colored little squares.

Figure 2. AR Representations concepts

With AR Representations, the study was conducted as quasi-experiments with 3^rd-grade students. The control group learned with pictures, as the usual instruction commonly seen in standard elementary school classroom, while the experimental group learned with AR Representations. The study was intended to examine research purposes under circumstances of students’ learning fractions, as following:

1. To examine whether representations (picture/ AR) affect students’ understanding 2. To examine difference of representations’ (picture/ AR) effects on students’

understanding.

3. To examine different levels of prior knowledge’s (high/low) effects on students’

understanding

4. To examine whether prior knowledge has an interaction effect with representation on students’ understanding

According to the above, four research questions were further explored:

1. Overall, do instruction treatments (picture/ AR) affect students’ understanding?

2. Is there a significant difference between learning with pictures and AR representations on students’ understanding?

3. Is there a significant difference between high and low prior knowledge on students’

understanding?

4. Whether the level of prior knowledge has an interaction effect with representations on

students’ understanding?

1.3 Limitations

Fraction is one of the most critical concepts closely related to further studies (i.e ratio et al.), and also the competent predicators of the learning performances afterward. Despite the importance of fraction, its’ well-known notorious reputation also leads to students’ struggling in learning. Thus, rather than other mathematics topics, the study aimed at this peculiar topic, in hopes of providing remedy to students’ learning difficulty.

The participants of the study were third graders, aged between 8-9 year-old. They were chosen particularly in accordance with the policy of formal implementation of fraction concept learning starting from the third grade in elementary school. Accordingly, the results might not apply to students of different age groups.

1.4 Definition of terms

1.4.1 AR Representations

The AR Representations is the instrument that designed by the study, in order to assist students to perceive the contexts around the learning concepts in real-life situations with the mechanism of virtual information overlay of concepts.

1.4.2 Prior Knowledge

Two levels of prior knowledge were distinguished with the pretest score. The top 27% of pretest scores were labeled as high prior knowledge, while the lowest 27% as low prior knowledge.

1.4.3 Understanding

The understanding tests were constructed with the concepts that delivered in the lesson.

Students’ posttest scores, with pretest score as covariance, were defined as understanding in the study.

CHAPTER TWO

LITERATURE REVIEW

The study attempted to explore the effectiveness of the real-world-situation representations facilitated by AR in fraction education at elementary level. The chapter is comprised of six sections. In section 2.1, learning mediated with representations is described.

The effectiveness of representations interrelated with students’ prior knowledge, followed by exploration of understanding in math, are investigated in section 2.2 and 2.3, respectively. AR instruments and applications are examined in section 2.4. Finally, students’ learning fraction circumstances are inspected in section 2.5. The summary of the present study rationale is shown in section 2.6.

2.1 Learning mediated with representations

2.1.1 The effectiveness of representations

Representations play important roles in primary mathematics education. More specifically, learning with pictures, students are able to see the unseen, and understand mathematics concept not only procedurally but also conceptually (Arcavi, 2003). Besides, representations in mathematics education are not limited to mere pictures. Either concrete or abstract models make mathematics ideas more meaningful to students and help them transfer the knowledge to novel situations, through the utilization of each model varies from one another (Fennema, 1972). Among concrete, abstract and pictorial manipulative instructions, long-term uses of concrete manipulatives were proved to be most effective, while abstract and pictorial instructions did not show differences in effectiveness (Sowell, 1989). Nonetheless, the same representations of pictures showed different effectiveness on students’ problem solving abilities, depending on how the pictures depict the spatial characteristics (Hegarty & Kozhevnikov, 1999).

The number of representations also takes different effects on learning. Single representation, as discussed above, is emphasized on how each of them influence students’

understanding in mathematics concepts, individually. Compared with single representations, multiple representations were proved to foster students’ deeper understanding more efficiently (Moreno & Mayer, 1999). A comparison of representation-based and traditional-based instruction on fraction knowledge were drawn by Chahine (2011). The results indicated that students who were taught with representation-based instructions outperformed the counterpart in school assessment tests.

2.1.2 The mechanism of linking representations

Lesh (1981) proposed a model that illustrated how average and learning-difficult students translate among various representations, including pictures, manipulative, spoken symbol, written symbol and real world situation (Figure 1). He also noted that if students were suffered from translating one particular representation to another, then the reverse order instructions could solve the problem. The significant role of manipulatives was also emphasized due to its function of “bridge” the symbols and real world. On the other hand, Lesh addressed the importance of fostering students’ interaction with concrete materials and peers, in order to connect those individual mathematical concepts in particular situations to take on meaningfulness in the process of solving realistic problems.

Despite the evidences that promoting translation between modes of representations are likely to foster students’ conceptual understanding, most students are not able to translate spontaneously. Students were found difficult in translating between mixed pictures and mathematics instructions (Ainsworth, Bibby, & Wood, 2002). Ainsworth (1999) introduced a taxonomy which identified different functions of multiple representations with corresponding multimedia supporting mechanism to maximize learning outcomes. Among three functions which multiple representations can offer, in order to construct deeper understanding, scaffolding of translation is the critical mechanism to be applied.

According to dual-coding theory, students who learned with both pictures and words outperforms those who did not (Mayer & Anderson, 1991). That is, these two kinds of representations can support each other to foster students’ learning. The research also provided evidence of two kinds of connections: representational connections and referential connections.

The former is the connection between same type of stimuli and representations, while the latter is the connection between different type of stimuli and representations, i.e. pictures and words.

In the follow-up research, Mayer and Anderson (1992) indicated that referential connection between different types of stimuli and representations can somehow cultivate students’ problem solving abilities. Furthermore, in addition to pictures and words, some other various representations connection instructions were also proved to be necessary in students’ learning process.

In order to make the abstract concept being meaningful to students, it is necessary to construct the connections between symbols and referents with sequential process: (1) connecting individual symbols with referents; (2) developing symbol-manipulation procedures;

(3) elaborating and routinizing the rules for symbols; (4) using the symbols and rules as referents for a more abstract symbol system. First two steps are to build the individual representations semantic meaning with real-life referents, then the last two are to elaborate and routinize the manipulation procedure with each representation syntax (Wearne & Hiebert, 1988).

Moreover, different degree of concreteness also has effect on students’ learning.

Compared with sole generic or concrete instruction, the fading mechanism from concrete to generic was proved to be effective in promoting students’ transfer ability (McNeil & Fyfe, 2012).

More specifically, based on Bruner’s theory of the enactive, iconic and symbolic modes of representations, the sequence of Concrete-Pictorial-Abstract (CPA) could help students transit different representational seamlessly (Leong, Ho, & Cheng, 2015). Also, CPA were proved to provide remedy to mathematics disabilities effectively (Agrawal & Morin, 2016).

2.1.3 External and internal representations

In addition to various representations discussed above, representations also can be considered as external and internal within one system. Uttal, Scudder and DeLoache (1997) reconciled the conflict between concrete manipulatives and abstract instructions with the notion of dual representation hypothesis, encouraging teachers to use manipulatives as symbols, but not substitutes. Clements (1999) distinguished two types of concreteness, i.e. sensory-concrete and integrated-concrete. The differences between these two are how meaningful it is to connect ideas and the situations. In essence, external representations stand for what can be seen or manipulated in instructions, whereas the internal representations reflect external ones with students’ personal experiences in their minds. The intertwine of external and internal representations plays a critical role in learning (Goldin & Shteingold, 2001). In practice, it is necessary to decrease students’ attention to the external representation (i.e. object property), in order to facilitate the linking between the concrete and the abstract (Uttal, O’Doherty, Newland, Hand, & DeLoache, 2009).

In addition to the factors of learners themselves, the environment also contributes to learners’ learning effectiveness under the circumstances of learning with representations.

Martin and Schwartz (2005) highlighted the benefits of adaptive environment, which brought about advanced learning. As Martin (2009) noted, the external environment and students’

internal states “coevolve” with each other. The embodied-interaction design framework, which promoted the alignment of spatial-temporal simulated action, was proposed to enhance mathematics learning efficiently (Abrahamson & Trninic, 2011).

The key to bonding external and internal representations to foster mathematics learning lies in meaningfulness. A number of studies have embedded real-life experiences in instructions, in hopes to stimulating students’ real and tangible experiences. Davis (2007) encouraged students to learn with self-invented terminology; however, the research indicated that using students’ own representations led to fragile mathematics understanding. Moreover, MacDonald

(2013) transformed what students knew into modes of representations by asking them to create their own representations. Though these self-draw representations showed students’ meaning-making processes, these were also quite selective in details due to students’ unique personal experiences; thus, intensive encouragement for students to depict more completely about the target concepts were always required. On the other hands, the variety of representations could be confined, for some widely-held prototype graphical representations were commonly shared by the majority of students (Jones, 2018).

2.2 Prior Knowledge

2.2.1 The impact of prior knowledge on students’ learning mediated with representations Prior knowledge is one of most dominant factors that have impacts on students’ learning, especially, when students’ learning mediated with representations. As mentioned earlier, representations aim at assisting students to develop deeper understanding of abstract concepts with more tangible ones. However, this kind of interventions were not always effective (Treagust, Chittleborough, & Mamiala, 2003).

DeLoache (2000) formulated “Dual Representation Hypothesis” by investigating how toddlers perceived pictures and objects. In his studies, he found out that it was difficult for toddlers to tell symbol from its referent. From the perspective of novices, who possess lower level of prior knowledge, just the same as toddlers, it is challenging for them to learn with object to comprehend the meaning beyond. Put simply, students’ domain prior knowledge guides their own attention to what they see; that is, students with different levels of prior knowledge learn in different ways (Uttal & O’Doherty, 2008).

Take two levels of prior knowledge into account, students with higher one are more capable of seeing meaning beyond, whereas the counterparts struggle to comprehend only surface features. For example, expert chess players encoded the entire chess arrangement, instead of just single one (Charness, Reingold, Pomplun, & Stampe, 2001). In terms of school

education, experts detected the conceptually relevant information, while low level ones concentrated on superficial features (Cook, Wiebe, & Carter, 2008). Moreover, experts were able to recognize pattern fast, while novices heavily relied on pictorial representations and stepwise instructions (Moyer-Packenham, & Suh, 2012).

Rau (2018) analyzed two different representation-connection interventions on different level of prior knowledge students. The results showed that low prior knowledge students were only benefited from sense-making one, rather than perceptual-fluency one. It indicated that it was necessary for low level of prior knowledge students comprehend each representation before draw connection between different ones. What matters is to help them draw much attention to task-relevant features.

2.2.2 The role of prior knowledge in multimedia intervention

With the advance of emerging technology, representations could be demonstrated in various forms embedded in multimedia, as well as assisting strategies to help low prior knowledge students deal with their learning difficulties. Signaling principles, which highlight the relevant information in multimedia, was proved to be support low prior knowledge students to learn more effectively (Richter, Scheiter, & Eitel, 2016). Additionally, signaling combined with animated pedagogical agents also helped low prior knowledge students to achieve equal scores with high prior knowledge students in posttests (Johnson, Ozogul, & Reisslein, 2014).

In animations, low prior knowledge students were benefited more with visual cues, while the higher ones were not (Arsla-Ari, 2018).

2.3 Understanding

2.3.1 Procedural and conceptual understanding

As for the understanding, the emphasis has been placed on both procedural and conceptual understanding. Byrnes (1992) defined conceptual knowledge as “knowing that,”

essentially, the relationally linked knowledge. In contrast, procedural knowledge has been characterized as the “knowing how” to do something, as “goal directed action-sequences.”

2.3.2 The impacts of attention on understanding in learning

As for the educational interventions, it seemed visual cueing can guide attention, but other factors were also important in determining the effectiveness of visual cues on learning (de Koning, Tabbers, Rikers, & Paas, 2010) Among a number of factors that direct attention in learning, prior knowledge and the type of representations could have great impact on attention.

Students’ prior knowledge guided their own attention to what they see (Uttal & O’Doherty, 2008). Furthermore, in regards to representations, different representations had different influences to learners’ attention. (Uttal, et al., 2009).

2.4 Learning with AR instruments

2.4.1 The benefits of mixed reality

Milgram & Kishino (1994) defined Mixed Reality (MR) between real and virtual environments on Reality-Virtuality (RV) Continuum, as Figure 3. One of the major effectiveness of mixed reality is that it provides users with pervasive experiences, which blurs the boundary between real and virtual environment (Montola, 2011). As one of the means to carry out the effects of mixed reality, the technique of “Augmented Reality (AR)” has been researched for decades. Billinghurst (2002) indicated three benefits that AR brings along: (1) Support of seamless interaction between real and virtual environments; (2) The use of a tangible interface metaphor for object manipulation; (3) The ability to transition smoothly between reality and virtuality.

Figure 3. Virtuality Continuum (VC)

Note. From ”Augmented Reality: A class of displays on the reality-Virtuality Continuum.” by P. Milgram, H. Takemura, A. Utsurmi, F. Kishino, 1994, Telemanipulator and Telepresence Technologies, SPIE. 2351, p.283.

Moreover, the technique of AR enables conventional teaching strategies even more effective in teaching and learning. For example, the research suggested that utilizing scaffolding teaching strategies with the media of AR, MR’s affordance of merging the real and the virtual could offer (1) a unique level of educational scaffolding, and (2) an improved learning-transfer from abstract to concrete domains (Quarles, Lampotang, Fischler, Fishwick, & Lok, 2009).

Radu (2014) also reviewed benefits and drawbacks of applying AR in education, including (1) Increased content understanding; (2) Long-term memory retention; (3) Improved physical task performance; (4) Improved collaboration, and (5) Increased student motivation, while the drawbacks are (1) Attention tunneling; (2) Usability difficulties; (3) Ineffective classroom integration, and (4) Learner differences.

Among various applications of AR in education, a classification was created to distinguish different affordance of AR in diverse subjects, three categories were identified: (1) roles, (2) locations, and (3) tasks (Wu, Lee, Chang, & Liang, 2013). Different categories were emphasized on particular instructional approaches. Distinguished from the former two categories, which focus on participatory simulations and physical environment, respectively, the “tasks” AR applications usually implemented with problem-based approaches, and

emphasize the authenticity. In practice, it is noted that when implemented in classroom setting, several design principles are mandatory: (1) Integration, (2) Empowerment, (3) Awareness, (4) Flexibility, and (5) Minimalism (Cuendet, Bonnard, Do-Lenh, & Dillenbourg, 2013).

With the development of mobile devices, AR techniques has become more competent in fostering students’ learning; especially, its abilities to contextualize students in corresponding learning environments (Chang, Wu, & Hsu, 2013). The handheld AR devices were also directed to focus on how embodied cognitive elicited by physical body movement with five methods:

(1) Perspective Change Through Movement, (2) Exploration Through Physical Action, (3) Reenactment Through Physical Action, (4) Interaction with Abstract Concepts, and (5) Embodying New Entities (Radu & Antle, 2017).

In practice, as Mayer & Moreno (2003) suggestions, nine multimedia design principles are required in order to reduce students’ cognitive load, (1) Modality effect, (2) Segmentation effect, (3) Pretraining effect, (4) Coherence effect, (5) Signaling effect, (6) Spatial contiguity effect, (7) Redundancy effect, (8) Temporal contiguity effect, (9) Spatial ability effect.

2.4.2 AR applications in math

With the advance of emerging technology, Kaput (1986) noted that it could transform mathematical representations into new forms, and even provide automatic linking or mechanism to connect between, the functions that traditional paper-and-pencil instructions can hardly achieve. Though teachers often use concrete manipulatives to transform abstract concepts to be more concrete, it is not necessary that concrete experiences only stem from collaborating with concrete manipulatives. Even computers or any other virtual manipulatives can offer the concrete learning experiences with several mechanisms to achieve what the

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