2.1 Learning mediated with representations
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
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(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
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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,”
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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.
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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
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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 concrete manipulatives cannot do in math education (Sarama & Clements, 2009), (1) Bringing mathematical ideas and processes to conscious awareness; (2) Encouraging and facilitating complete, precise explanations; (3) Supporting mental ‘‘Actions on objects’’; (4) Changing the very nature of the manipulative; (5) Symbolizing mathematical concepts; (6) Linking the
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concrete and the symbolic with feedback; (7) Recording and replaying students’ actions.
Furthermore, within the virtual manipulatives, AR technique has been widely implemented in diverse subjects, including science, technology, engineer, and mathematics (STEM). AR provides unique affordance that assist students to develop spatial-related abilities, and guides their attention concisely to knowledge that requires contextual awareness. The effectiveness of AR in developing mathematics abilities have been proved from three psychological perspectives, (1) physical, (2) cognitive, and (3) contextual (Bujak, et al., 2013).
Several applications in math are discussed below. For instance, Kaufmann and Schmalstieg (2003) improved students’ spatial abilities in geometry with AR. Conics went well with AR in promoting the opportunity to interact with abstract concepts that was unlikely to take place with traditional paper-and-pencil instructions (Salinas & Pulido, 2017). AR were also proved to compliment learning in quadratic equations (Barraza Castillo, Cruz Sánchez., &
Vergara Villegas, 2015).
As for the elementary education, Radu, McCarthy and Kao (2016) collaborated with elementary school teachers and AR designers to explore the educational potential of AR technology. In their study, the suitability of several math topics with AR was examined. The criteria of matching were (1) visualizing the mathematical content in three dimensions; (2) visualizing the content through multiple representations at the same time; (3) physically interacting with mathematical topics, and (4) having in-context access to additional information.
Results showed that Fractions and Number Lines was only match with AR in medium degree, while the teaching difficulty was high. According to the authors, it seemed the pedagogical design flaws leaded to ineffectiveness. In the current study, the effectiveness of utilizing AR in fraction was investigated, in order to promote one of the most critical mathematical areas which has huge impacts on elementary school students hereinafter.
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2.5 Learning Fractions
2.5.1 Common difficulties in learning fraction
Elementary school students have been reported that most of them share common difficulty in learning fraction. The main cause of this difficulty is their “whole number bias.”
Different from the concept of whole numbers, fraction possess “part-whole” meaning, which is somehow contradict to what students have learned before. When recognizing or doing arithmetic, students tend to consider the numbers appear in fractions as the regular whole number, ignoring these fractional numbers stand for “part-whole”, in essence (Ni & Zhou, 2005). The fraction competences are fundamental for further mathematical learning; thus, it’s extremely urgent to figure out what imbeds students’ fraction-learning.
2.5.2 Fraction teaching applications
With the aid of technology, when learning fraction, students are able to aware their own misunderstanding or correct themselves when making mistake. Reimer and Moyer (2005) used virtual manipulatives to teach 3rd-graders learn about fraction. Results showed that the virtual manipulatives assisted students in (1) providing immediate and specific feedback, (2) much easier usability than paper-and-pencil methods, and (3) enhancing enjoyment. Suh, Moyer and Heo (2005) identified four characteristics of virtual manipulatives in learning fraction: (1) Allow discovery learning through experimentation and hypothesis testing; (2) Encourag students to see mathematical relationships; (3) Connect iconic and symbolic modes of representation explicitly; and (4) Prevent common error patterns in fraction addition.
Nevertheless, aside from the effectiveness such as effective feedback and enjoyment, other mechanism have been arranged in virtual manipulatives as well. For example, assessment was implemented when students learning with virtual manipulatives in game-based virtual instruments (Ninaus, Kiili. McMullen, & Moeller, 2017). Stoyle and Morris (2017) noted that blog may support students in providing opportunity to explain the process of thinking, and to clarify the misunderstanding.
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2.5.3 Learning fractions mediated with representations
Representations have been widely used in teaching fraction, especially pictorial representations. Three common fraction representations are circle, rectangle, and number line.
When designing the instruments, the interleaving arrangement of these three types of fraction representations leaded to better learning results (Rau, Rummel, Aleven, Pacilio, & Tunc-Pekkan, 2012). Different fraction representations have been used to convey different underlying conception of various forms of fractions. However, students are not able to read the underlying meaning when learning from those unfamiliar representations. Rau, Aleven, and Rummel (2017), with intelligent tutoring system, examined two types of representation competences:
sense-making and perceptually fluency. The results showed that sense-making-first teaching strategy were more beneficial in supporting learning, compared to perceptually fluency. In order to enhance students’ abilities in fraction, the pedagogical methods of helping students to see the underlying meaning would be the top priority.
2.6 Summary
Take real-world and abstract symbol as two ends of spectrum, more rightward would be more abstract to students, as Figure 4 . In this spectrum, all the concepts within that represent real world are called representations.
Among these representations, the concept of pictures is much more abstract than the manipulatives. The differences between these two representations are whether it could be manipulated by hand or not. In other words, manipulative would be much more real than pictures to students. However, according to Clement’s definition on concrete teaching material, the present study tended to categorize either picture or concrete and virtual manipulative as sensory-concrete (orange part).
Based on Clement’ s definition once again, in addition to sensory-concrete, the notion of integrated-concrete was identified (blue part). The distinction between these two definition lies
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in meaningfulness. Lots of researcher have been doing research on this topic. But it seems that in order to make the learning more real, the advantages of students’ widely-held representations could be a feasible approach.
As a result, composing shared representations with meaningfulness which derived from learner’s life experiences, the study intended to fill the gap between pictures, which are most common teaching material in classroom, and real-world. With the technique of augmented reality, the study expected to create brand-new types representations that are different from any other concrete and abstract representations. Subsequently, the effects of this new kinds of representations were further explored to ensure the effectiveness.
Figure 4. The rationale of the present study
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CHAPTER THREE
RESEARCH METHOD
Chapter 3 is composed of six main sections. The research framework of the study is provided in section 3.1, with rationale that how variables relate to each other and the specific research questions deduced from research objectives. In section 3.2 and 3.3, participants and instruments are described. The instructional design of AR Representations and implementation are elaborated in section 3.4 and 3.5, respectively. The data analyses are outlined in section 3.6.
3.1 Research Framework
The study aimed to examine AR Representations’ impacts on students’ understanding, with students’ prior knowledge, as another independent variable. Thus, this study hypothesized that prior knowledge level might have an intervening effect with independent variable (Representations) and dependent variables (understanding). Research framework, depicting the relationship between variables, is shown Figure 5.
Figure 5. Research Framework
The research structure leaded to 4 research questions listed below:
1. Overall, do instruction treatments (picture/ AR) affect students’ understanding?
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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?
3.2 Participants
In order to align the AR Representations system with the existing curriculum guidelines, the participants in this study were third-grade students from five different classes in two schools in northern Taiwan. The study carried out in quasi-experiment. Both school were assigned both picture and AR Representation treatment classes. These students, aged between 8-9 year-old, are new to fraction concept, as elementary educational guidelines proposed by Ministry of Educations in Taiwan.
Control group learned with pictures, while experimental group with AR Representations.
To noted, because the present study aimed at examining how different representations affect students learning fraction, students who got full score in pretest were eliminated from the analysis due to the lack of power of discrimination. The corresponding data were presented in Table 1 . After eliminating full-score students, the total number of participant were 101, 41 in pictures and 60 in AR Representations group, respectively, as Table 2 shows. In order to examine whether two groups are similar, independent t-test of pretest scores were also performed. As Table 3 indicated, the two group were equal (p = .345 >.05).
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Table 1
The number of students in each group and answered-all-corrected students
Group School Class
Descriptive data of two group participants’ pretest score
Representation N Mean SD
Pictures 41 4.66 1.237
AR representations 60 4.38 1.678
Table 3
Independent t-test on pretest score from different group
t df p.
Pretest score of
two groups .948 98.381 .345
3.3 Instruments
3.3.1 Fraction Understanding Test
Fraction understanding tests were composed based on the fundamental concepts of fraction, which were delivered in the experiments. The test was conducted in paper-and-pencil format. It includes eight items that assessed students’ principled understanding of fractions.
Each task and its’ construct were presented in Table 4 below.
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Table 4
The construct of understanding tests
No. Fraction concepts Types of fraction Representation translation 1 Part-whole 2D continuous quantity Situation -> Symbol 2 Part-whole 1D continuous quantity Symbol -> Picture
3 Part-whole Discreteness Picture - > Symbol
4 Equalization 2D continuous quantity Picture - > Symbol 5 Equalization 1D continuous quantity Symbol -> Picture
6 Equalization Discreteness Situation -> Symbol
7 Minimal Unit 1D continuous quantity Picture - > Symbol
8 Minimal Unit Discreteness Symbol -> Picture
The pretest had a fair reliability with Cronbach’s α of .611, as Table 5. The posttest was a parallel form of the pretest with different quantity arrangements. It also received a similar reliability with Cronbach’s α of .637, as shown in Table 6. Both pretest and posttest test sheets are placed in the Appendix A and Appendix B.
Table 5
Reliability of Pretest
Cronbach's Alpha N of Items
.611 8
Table 6
Reliability of Posttest
Cronbach's Alpha N of Items
.637 8
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3.3.2 Interview Questions
At the end of the experiment, semi-structured interviews with the students were conducted in order to gain in-depth understanding of the participants’ experiences. The interviews contained a pre-determined set of questions as followed:
1. Overall, how did you feel about the course that I just delivered? Did you find it interesting? Did you feel concentrated in the course?
2. What did you think about using pictures/ AR representations to learn? What are pros and cons of pictures/ AR representations?
3. What’s the difference between the concepts of fraction and number?
4. Take 2/5 as example, could you give me any circumstances in daily lives that you would use this fraction?
5. After using pictures/ AR representations to learn fraction, what kinds of objects on which you will apply fractional concept?
3.4 Instructional Design for AR Representations
3.4.1 Design principles
The AR Representations was developed with UnityTM, and designed according to three
The AR Representations was developed with UnityTM, and designed according to three