以擴增實境表徵學習數學分數概念對學生學習理解之影響

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(1)國立臺灣師範大學圖文傳播學系碩士論文 Graduate Institute of Graphic Communications and Arts National Taiwan Normal University Master’s Thesis. 以擴增實境表徵學習數學分數概念對學生學習理解之影響 The Effects of learning through Augmented Reality Representations on Students’ Understanding of Fraction. 研究生：鐘敏綺指導教授：王健華教授. 中華民國 108 年 2 月.

(2) Chinese Abstract 表徵（Representations）常見於 STEM 教學當中，教師在教學中常以圖示、操作物等表徵，將學生難以理解的抽象概念予以具體化。然而，應用表徵進行學習，學生並非能自然而然習得其對應的抽象概念，甚至應用至真實情境當中。此外，大多數低先備知識學生，因無法掌握所學數學概念與生活間之關聯，導致缺乏學習數學的動機，甚至也傾向只注意表徵表層資訊，而無法更深入注意到關鍵資訊。本研究提出以擴增實境技術所創造的表徵 (AR Representations)，將數學概念以不同於常見具體或抽象的方式呈現，導引學生從真實情境中出發注意關鍵表徵資訊，來解決以上之問題。為檢測此教具之教學效果，本研究採用準實驗法，並有來自北台灣共 101 位國小三年級學生參與本次實驗。學生的前後測試卷分數為學生理解之量化數據，高低先備知識則以前測分數進行區別，同時亦採集質性的訪談資料。研究結果顯示，實驗組學生在使用擴增實境表徵學習後，高低先備知識學生之後測成績並不顯著。總結而言，藉由擴增實境表徵學習，低先備知識學生之理解與高先備知識學生之間的差異呈現彌平的趨勢。此外，透過質化訪談討論，更發現這一群學生比起其他組別能將所學應用至更多元的例子中。本研究亦根據結論，提出教育意涵及未來研究建議。關鍵詞：數位學習、表徵、擴增實境、分數學習. 1 .

(3) Abstract Representations are widely used in science, technology, engineering, mathematics (STEM) fields. Teachers often make use of representations, such as pictures, manipulatives, to demonstrate abstract concepts, which students find hard to understand, in more concrete and tangible ways. However, it is not automatically that students can pick up the target abstract concepts with those representations. Moreover, learning with pictures in textbooks, students have difficulty in translating from pictures to real-life situations. Nevertheless, most low prior knowledge students are lack of learning motivation in face with the mathematics concepts because they are not able to relate what they learned with their own real-life situations, and vice versa; also tend to pay attention to the task-irrelevant information. With the technique of augmented reality, the present study proposed “AR Representations” instructions, which are different from the well-used concrete or abstract concepts. By redirecting students’ attention to the most relevant information, the instructions aimed to solve the above problems. Thus, quasiexperiments were performed to verify the effectiveness of AR Representations instructions. 101 3rd-grade elementary students in northern Taiwan have participated. Pretest and posttest scores were collected as quantitative data of understanding. Levels of prior knowledge were also derived from pretest scores. Results of this study showed there was no significant difference on students’ understanding between high and low level of prior knowledge after learning with AR Representations. In conclusion, With the aid of AR representations, (1) low prior knowledge students’ understanding would equate with the higher ones’, and (2) low prior knowledge students could identify more novel examples other than textbooks provide. Pedagogical implications and future research are also discussed.. Keywords: e-learning, representations, augmented reality, fraction learning 2 .

(4) Table of Content Chinese Abstract ...................................................................................................................... 1 Abstract ..................................................................................................................................... 2 Table of Content ....................................................................................................................... 3 List of Tables............................................................................................................................. 6 List of Figures ........................................................................................................................... 7 CHAPTER ONE INTRODUCTION ...................................................................................... 8 1.1 Background ................................................................................................................ 8 1.1.1 Representations in STEM education ................................................................. 8 1.1.2 Students’ learning difficulty in mathematics ..................................................... 9 1.1.3 The effectiveness of AR in mathematics education .......................................... 9 1.2 Research purposes and questions ............................................................................... 10 1.2.1 Assumption of solution to the problems .......................................................... 10 1.2.2 The process of validating the assumption ....................................................... 11 1.3 Limitations ................................................................................................................. 13 1.4 Definition of terms ..................................................................................................... 13 1.4.1 AR Representations ......................................................................................... 13 1.4.2 Prior Knowledge ............................................................................................. 13 1.4.3 Understanding ................................................................................................. 13 CHAPTER TWO LITERATURE REVIEW ...................................................................... 14 2.1 Learning mediated with representations .................................................................... 14 2.1.1 The effectiveness of representations ............................................................... 14 2.1.2 The mechanism of linking representations...................................................... 15 2.1.3 External and internal representations .............................................................. 17 2.2 Prior Knowledge ........................................................................................................ 18 2.2.1 The impact of prior knowledge on students’ learning mediated with representations .......................................................................................................... 18 2.2.2 The role of prior knowledge in multimedia intervention ................................ 19 2.3 Understanding ............................................................................................................ 19 2.3.1 Procedural and conceptual understanding ....................................................... 19 2.3.2 The impacts of attention on understanding in learning ................................... 20 2.4 Learning with AR instruments ................................................................................... 20 2.4.1 The benefits of mixed reality........................................................................... 20 2.4.2 AR applications in math .................................................................................. 22 3 .

(5) 2.5 Learning Fractions...................................................................................................... 24 2.5.1 Common difficulties in learning fraction ........................................................ 24 2.5.2 Fraction teaching applications ......................................................................... 24 2.5.3 Learning fractions mediated with representations .......................................... 25 2.6 Summary .................................................................................................................... 25 CHAPTER THREE RESEARCH METHOD ..................................................................... 27 3.1 Research Framework .................................................................................................. 27 3.2 Participants ................................................................................................................. 28 3.3 Instruments ................................................................................................................. 29 3.3.1 Fraction Understanding Test ........................................................................... 29 3.3.2 Interview Questions......................................................................................... 31 3.4 Instructional Design for AR Representations ............................................................. 31 3.4.1 Design principles ............................................................................................. 31 3.4.2 Development of AR Representations .............................................................. 33 3.5 Procedure .................................................................................................................... 37 3.6 Data analysis .............................................................................................................. 39 CHAPTER FOUR RESULTS AND DISCUSSION ............................................................ 40 4.1 Results ........................................................................................................................ 40 4.1.1 RQ1: Overall, do instruction treatments (picture/ AR) affect students’ understanding? ......................................................................................................... 40 4.1.2 RQ2: Is there a significant difference between learning with pictures and AR representations on students’ understanding? ............................................................ 41 4.1.3 RQ3: Is there a significant difference between high and low prior knowledge on students’ understanding? ..................................................................................... 43 4.1.4 RQ4: Whether the level of prior knowledge has an interaction effect with representations on students’ understanding? ............................................................ 43 4.1.5 RQ A1: Is there a significant difference between high and low prior knowledge on understanding in picture representation group? ................................ 43 4.1.6 RQ A2: Is there a significant difference between high and low prior knowledge on understanding in AR representation group? ..................................... 45 4.1.7 Interview data .................................................................................................. 46 4.2 Discussion .................................................................................................................. 47 4.2.1 Multiple representations facilitated by AR Representations ........................... 48 4.2.2 “Concrete-Pictorial-Abstract” sequences of AR Representations ................... 48 4.2.3 High and low level of prior knowledge students’ perception to AR Representations ........................................................................................................ 49 4.2.4 Enhancement of analogical thinking with personal experiences .................... 51 4 .

(6) CHAPTER FIVE CONCLUSION AND SUGGESTIONS ................................................ 52 5.1 Conclusion ................................................................................................................ 52 5.1.1 With the aid of AR Representations, low prior knowledge students’ understanding would equate with the higher ones’ .................................................. 52 5.1.2 With the aid of AR representations, low prior knowledge students could identify more novel examples other than textbooks provide ................................... 53 5.2 Contributions of the study .......................................................................................... 53 5.2.1 Creation of real-life representations by integrating CPA transition and AR technique .................................................................................................................. 53 5.2.2 Non-3D mathematics AR overlay ................................................................... 54 5.3 Limitations of the study.............................................................................................. 54 5.4 Suggestions for Future Research ................................................................................ 55 5.4.1 The applicable scope of non-3D AR Representations ..................................... 55 5.4.2 Discovery learning with AR Representations ................................................. 55 5.4.3 Graphical analogical thinking ......................................................................... 56 REFERENCES ....................................................................................................................... 57 APPENDICES ........................................................................................................................ 64 Appendix A: Understanding pretest ................................................................................. 64 Appendix B: Understanding posttest................................................................................ 67 Appendix C: Pictures treatment instructions .................................................................... 70. 5 .

(7) List of Tables Table 1 The number of students in each group and answered-all-corrected students ............. 29 Table 2 Descriptive data of two group participants’ pretest score ........................................... 29 Table 3 Independent t-test on pretest score from different group ............................................. 29 Table 4 The construct of understanding tests ........................................................................... 30 Table 5 Reliability of Pretest .................................................................................................... 30 Table 6 Reliability of Posttest................................................................................................... 30 Table 7 Interface designs of AR Representations with function explanation ........................... 34 Table 8 The types of fraction and corresponding objects......................................................... 36 Table 9 The experiment process of control group and experimental group ............................. 38 Table 10 Comparison of single task in control group and experimental group ....................... 39 Table 11 Descriptive Statistics of Picture Group ..................................................................... 40 Table 12 Paired t-test on Pretest and Posttest Scores of Picture Group .................................. 41 Table 13 Descriptive Statistics of AR Representations Group ................................................. 41 Table 14 Paired t-test on Pretest and Posttest Scores of AR Representations Group .............. 41 Table 15 Homogeneity Test of Regression Coefficients on prior knowledge and representations Students’ pretest scores ................................................................................... 42 Table 16 Levene' s Test of Equality of Error Variances ........................................................... 42 Table 17 Descriptive statistics of Representations and Prior knowledge ................................ 42 Table 18 ANCOVA test on Representations and Prior knowledge Students’ understanding ... 43 Table 19 Independent t-test on High and Low prior knowledge students’ pretest scores in picture group ............................................................................................................................ 44 Table 20 Levene's Test of Equality of Error Variances ............................................................ 44 Table 21 Descriptive Statistics of Achievements on High/Low Prior Knowledge in Picture Group........................................................................................................................................ 44 Table 22 ANCOVA test on Prior knowledge Students’ understanding in picture group .......... 45 Table 23 Independent Samples t-test on High and Low prior knowledge students’ pretest scores in AR Representations group ......................................................................................... 45 Table 24 Levene's Test of Equality of Error Variances ............................................................ 46 Table 25 Descriptive Statistics of Achievement on High / Low prior knowledge in AR Representations Group ............................................................................................................. 46 Table 26 ANCOVA test on Prior knowledge Students’ understanding in AR Representations group ........................................................................................................................................ 46 Table 27 Interview transcripts.................................................................................................. 47. 6 .

(8) List of Figures Figure 1.The translation between representations ................................................................... 11 Figure 2. AR Representations concepts ................................................................................... 12 Figure 3. Virtuality Continuum (VC) ....................................................................................... 21 Figure 4. The rationale of the present study............................................................................. 26 Figure 5. Research Framework ................................................................................................ 27 Figure 6. The mechanism of linking representations ............................................................... 32 Figure 7. Stepwise manipulation of AR Representations ........................................................ 33 Figure 8. Three steps of AR Representations ........................................................................... 33. 7 .

(9) CHAPTER ONE INTRODUCTION In chapter one, the introduction is divided into four main sections. First of all, section1.1 briefly outlines background and motivations of the study, including the current trend of using representations in STEM education, and students’ learning difficulties with representations in math and fraction. The assumption of solution with the technique of mixed reality to the problems are also delivered. Furthermore, in section1.2, in order to test the effectiveness of proposed remedy, four research purposes and four research questions were identified, with the process of validating the assumption alongside. In section 1.3 and 1.4, the limitations of the study and definition of terms are included.. 1.1 Background 1.1.1 Representations in STEM education In recent decades, STEM (science, technology, engineering, mathematics) have become core subjects in education domain. Researchers have made lots of efforts on these subjects by investigating how to deliver these subjects more efficiently; also, how to cultivate students with better understanding on them. Among various teaching methods, there has been an enormous wave of interest on the relationship between STEM and representations. Representations have been widely used in STEM due to their effectiveness to transform those abstract concepts into more concrete ideas (Arcavi, 2003). Moreover, researchers have given constant attention to the effects of fostering students to translate among representations, which have been proved to be beneficial to learning and understanding (Fennema, 1972).. 8 .

(10) 1.1.2 Students’ learning difficulty in mathematics Among STEM, mathematics is the fundamental subject. However, according to TIMSS 2015, Taiwanese 4th grades students showed low level of motivation in leaning math, for what they have learned in class are not relevant to their daily lives. Nonetheless, within the math fields, the majority of students are suffered from learning fraction due to the difference from whole number concepts and invisible essence in daily lives. Thus, the relationship between mathematical concepts and daily lives is worth more attention. Representations have been particularly influential in mathematics education, especially fraction. In every mathematics textbook, there are pictures of pizza and cakes, to demonstrate the concepts of part-whole and equal sharing. Teachers use these kinds of concrete representations in hopes to represent the intangible abstract concepts with students’ experiencerelated ideas. Also, with the means of these real-life-like representations, teacher look forward to enhance students’ learning motivation and develop their deeper understanding of abstract concepts. However, it’s not spontaneously that students are able to connect various representations. Students’ difficulty in translating representations has also been found (Sowell, 1989). Furthermore, in terms of students’ acquirement of domain knowledge, the level of prior knowledge has impacted student’ understanding when learning with representations. More specifically, novice learners tend to heavily rely on pictures by only perceiving surface features, while experts can extract task-relevant information from learning materials (Chi, Feltovich, & Glaser, 1981). 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 9 .

(11) 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 10 .

(12) students to perceive the concepts in real-life situation, and then be able to translate from reallife situation to symbol more easily than from pictures, with the aid of technique of Augmented Reality (AR).. Figure 1.The translation between representations Note. Adapted from “Applied mathematical problem solving” by R. Lesh, 1981, Educational Studies in Mathematics, 12(2), 246. Copyright 1981 by Springer.. 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.. 11 .

(13) Figure 2. AR Representations concepts. With AR Representations, the study was conducted as quasi-experiments with 3rd-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 12 .

(14) 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. 13 .

(15) 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). 14 .

(16) 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. 15 .

(17) 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). 16 .

(18) 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 17 .

(19) (2013) transformed what students knew into modes of representations by asking them to create their own representations. Though these self-draw representations showed students’ meaningmaking 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 18 .

(20) 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,” 19 .

(21) 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.. 20 .

(22) 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 21 .

(23) 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 22 .

(24) 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.. 23 .

(25) 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. 24 .

(26) 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 25 .

(27) 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. 26 .

(28) 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? 27 .

(29) 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).. 28 .

(30) Table 1 The number of students in each group and answered-all-corrected students No. of students No. of Students answered all correctly. No. of effective Samples. Group. School. Class Code. Pictures. A. C. 21. 2. 19. B. E. 24. 2. 22. A. A. 17. 0. 17. B. B. 25. 0. 25. A. D. 21. 3. 18. 108. 7. 101. AR Representations. Total. Table 2 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 Pretest score of two groups. t. df. p.. .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. 29 .

(31) 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. 30 .

(32) 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 kinds of design principles, including (1) representations, (2) fractions, and (3) multimedia instructional design. First of all, the pedagogical sequences of concrete-picture-abstract is proved to enhance students’ math learning outcomes effectively (Leong, et al., 2015). Among the transition of these three kinds of representations, McNeil & Fyfe (2015) indicated that fading mechanism is beneficial for learning. In general, decreasing students’ attention to the external representation is favorable for students to connect the external representations with the internal ones (Uttal, 2009). The linking mechanism is shown in Figure 6.. 31 .

(33) Figure 6. The mechanism of linking representations As for fraction, three main fraction concepts: part-whole, equalization, and units were highlighted throughout the entire manipulation process. Also, the AR Representations illustrated three kinds of common fraction representations: 1D continuous quantity, 2D continuous quantity, and discreteness. Several instructional principles were considered to avoid cognitive overload while receiving the AR Representations. Segmentation effect, pre-training effect, coherence effect and signaling effect were considered according to Mayer and Moreno’s (2003) claim. Following describes how they worked: (1) Segmentation effect: Display stepwise on students’ tapping; (2) Pretraining effect: Deliver introduction before starting; (3) Coherence effect: No irrelevant information will be included; and (4) Signaling effect: the learning concept in each steps will be colored. Integrated with design principles of representations, fraction, and multimedia, 3 steps manipulation of AR Representations were designed. 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. The processes of manipulation of AR Representations are showed in Figure 7. More specifically, Figure 8 further demonstrates these steps with an object. 32 .

(34) Figure 7. Stepwise manipulation of AR Representations. Figure 8. Three steps of AR Representations. 3.4.2 Development of AR Representations The AR Representations app was developed with UnityTM, with VurforiaTM facilating aumented reality function. The AR Representations app was installed in ACER Iconia One 7 pads, samples of interface design are listed below in Table 7.. 33 .

(35) Table 7 Interface designs of AR Representations with function explanation Demonstration. Funtion explanation 1. Welcom page. 2. In the process of scan, the screen would show the scanning outline for students to scan objects more easily. 3. Once the device scan the target object, the first overlay would pop out ont the screen. As the picture shows, it pops out a rectangle for students to aware the whole object.. 34 .

(36) 4. Then tap on the screen, it would show the addtioanl equalization overlay. As the picture shows, it equalizes corn into 4 pieces, as the meaning of denominator. 5. And then tap on the srceen again, it show how many pieces are chosen, indicating the meaning of numerator. During this process, the concept of minimal unit is stressed.. In total, 6 tasks were embedded in the experiment, including two 2D continuous quantities, two 1D continuous quantities, and two discreteness. The types of fraction and corresponding objects are listed below in Table 8. Students scanned 6 simulated object made of plastic to go thorough 3 steps, (1) whole-part, (2) equalization, (3) minimal unit, as mentioned above.. 35 .

(37) Table 8 The types of fraction and corresponding objects Object. Type of fraction 1. Tomato- 2D continuous quantity. 2. Onion- 2D continuous quantity. 3. Cucumber- 1D continuous quantity. 4. Corn- 1D continuous quantity. 36 .

(38) 5. Beans- Discreteness. 6. Cookies in a box - Discreteness. 3.5 Procedure The experiments were conducted within 80 minutes, with additional 20 minutes for individual interviews. In the main experiments, 4 sessions were included, 25-minute warm up, 5-minute pretest, 45-minute activity, and 5-min posttest, as shown in Table 9. In warm up session, because participants were 3rd-grade elementary students, the relationship-building was required, as well as the basic concepts of fraction. In pretest session, understanding pretest was assessed. In posttest session, understanding posttests were carried out. Interviews were conducted with 12 random-chosen students, 4 from pictures groups and 8 from AR Representations groups. Within 45-minute activity, teacher handed out tablets and worksheet to each group of students, and lead students to complete 6 tasks. Students went through 3 steps and got the answer.. 37 .

(39) Table 9 The experiment process of control group and experimental group. The differences of instruments between control group and experimental group were drawn on how they perceive and manipulate fraction concepts (as shown in Table 10). Throughout the same process of part-whole, equalization and units, the control group perceived the targets as pictures. On the other hand, students in experimental group saw the object targets with the aid of dynamic virtual overlays displaying the process. In part-whole stage, the control group got pictures with equalization lines on them, while the experimental group first identified the boundary of target with virtual overlay. In equalization stage, the control group saw divided pictures, whereas students in the experimental group saw the dynamic dividing lines on the screen. Finally, in last stage, the control group had a single unit of pictures, while the experimental group saw the unit embedded in the targets. The instructional materials of picture group are attached in Appendix C.. 38 .

(40) Table 10 Comparison of single task in control group and experimental group. 3.6 Data analysis Both quantitative and qualitative data analyses were performed. For the quantitative data, two kinds of statistical analyses were conducted. First of all, paired t-tests were performed in order to examine picture/AR Representations’ effects on students’ understanding. Secondly, difference on student’s understanding between AR Representations and picture representations, and prior knowledge’s effects were analyzed by using a two-way ANCOVA with the pretest as covariance. As for the qualitative data, all interviews were audio-taped and transcribed by the researcher and analyzed. Interviewees from control group and experiment group were random chosen and coded according to the class they belong; capital letters, A-E, for the class code, and the numbers representing their identity.. . 39 .

(41) CHAPTER FOUR RESULTS AND DISCUSSION In chapter four, quantitative data were statistically analyzed. Furthermore, interviewing data were qualitatively sorted into two categories according to student’s interview transcripts. The results of data analyses are as follows.. 4.1 Results 4.1.1 RQ1: Overall, do instruction treatments (picture/ AR) affect students’ understanding? In order to ensure the effectiveness of the instruction, paired-samples t-tests were conducted to examine the difference of the mean scores between pretest and posttest scores in both picture and AR group. In picture group, the paired t-test results showed a significant difference on scores of pretest (M= 4.66, SD= 1.24) and posttest (M= 5.24, SD= 1.51); t (40) = -3.23, p = .002< .005., as shown in Table 11 and Table 12. Table 11 Descriptive Statistics of Picture Group Score. Mean. N. SD. Pretest Posttest. 4.66 5.24. 41 41. 1.237 1.513. 40 .

(42) Table 12 Paired t-test on Pretest and Posttest Scores of Picture Group Pretest and posttest score of picture groups. t. df. p.. -3.227. 40. .002**. **p < .01. Besides, the t-test results of AR group also exhibited a significant difference on the scores of pretest (M= 4.38, SD= 1.67) and posttest (M= 5.20, SD= 1.57); t (59) = -4.21, p= .000< .005., as shown in Table 13 and Table 14. The t-test results indicated that both pictures and AR Representations were effective.. Table 13 Descriptive Statistics of AR Representations Group Score. Mean. N. SD. pretest posttest. 4.38 5.20. 60 60. 1.678 1.571. Table 14 Paired t-test on Pretest and Posttest Scores of AR Representations Group t. df. p.. -4.214. 59. .000***. ***p < .001.. 4.1.2 RQ2: Is there a significant difference between learning with pictures and AR representations on students’ understanding? Homogeneity of regressions were first examined and satisfied (representation, p=.679>.05; prior knowledge, p=.965>.05), following ANCOVA processes were executed. Refer to Table 15 for the details. Afterward, Levene' s Tests of Homogeneity of variance were performed and satisfied (p= .353 > .05), as shown in Table 16.. 41 .

(43) Table 15 Homogeneity Test of Regression Coefficients on prior knowledge and representations Students’ pretest scores Source. SS. df. MS. F. Sig.. Representations * pretest. .243. 1. .243. .173. .679. Prior knowledge * pretest. .003. 1. .003. .002. .965. Table 16 Levene' s Test of Equality of Error Variances F. df1. df2. Sig.. 1.112. 3. 50. .353. The ANCOVA with the pretest score as the covariance was performed to examine statistically significant difference between representations (picture/ AR Representations) and prior knowledge (low/ high). Descriptive statistics of representations and prior knowledge are presented in Table 17. The ANCOVA results showed that no significant effect of representation was found on posttest scores, F(1, 49) = 1.206, p= .277 > .05, as shown in Table 18.. Table 17 Descriptive statistics of Representations and Prior knowledge Prior knowledge Low Adj. Mean. Representations. SD. Adj. Mean. SD. Picture. 5.31. 1.69. 5.22. 1.04. AR Representations. 6.06. 1.37. 5.21. 0.96. 42 . High.

(44) Table 18 ANCOVA test on Representations and Prior knowledge Students’ understanding Source. SS. df. MS. F. Sig.. Representations Prior knowledge Representations* Prior knowledge Error Total. 1.657 .410 1.660 67.295 1762.000. 1 1 1 49 54. 1.657 .410 1.660 1.373. 1.206 .299 1.209. .277 .587 .277. 4.1.3 RQ3: Is there a significant difference between high and low prior knowledge on students’ understanding? Furthermore, Table 18 also depicts prior knowledge’ s effects. The ANCOVA results exhibited no significant effect, F (1, 49) = .299, p= .587 > .05, indicating that no differences on understanding were found between students of high and low prior knowledge 4.1.4 RQ4: Whether the level of prior knowledge has an interaction effect with representations on students’ understanding? Also refer to Table 18 , the ANCOVA results still exhibited no significant interaction effect of prior knowledge and representation on posttest, F(1,49) =1.209, p= .277 > .05, indicating that no inter-effects on understanding were found between levels of prior knowledge and two different representations Results from the above ANCOVA indicated that none of them were significant. Therefore, two additional analyses were made to acquire more detailed information. 4.1.5 RQ A1: Is there a significant difference between high and low prior knowledge on understanding in picture representation group? An independent-samples t-test was first performed to statistically analyze pretest of high and low prior knowledge in picture group. Results indicated a significant difference on the pretest between high and low prior knowledge of picture group, t (17.329) = -12.393, p= .000 < .005 (Table 19). The statistics indicated that students’ pretest scores of high and low prior 43 .

(45) knowledge were varied. Table 19 Independent t-test on High and Low prior knowledge students’ pretest scores in picture group High and Low prior knowledge students’ pretest scores. t. df. p.. -12.393. 17.329. .000***. ***p < .001. A one-way ANCOVA with pretest scores as covariance was then performed to examine a statistically significant difference between high and low prior knowledge on posttest in picture group. Levene's Tests of Homogeneity of variance were also performed and satisfied (p= .555> .05), as shown in Table 20. Descriptive statistics of representations and prior knowledge are presented in Table 21. Results exhibited no significant effect of prior knowledge on understanding, F(1,19) = 1.608, p=.220 > .05., as shown in Table 22. The statistics indicated that, after the picture treatments, high and low prior knowledge student’s understanding tended to be similar.. Table 20 Levene's Test of Equality of Error Variances F. df1. df2. Sig.. .361. 1. 20. .555. Table 21 Descriptive Statistics of Achievements on High/Low Prior Knowledge in Picture Group Prior Knowledge Adj. Mean. SD. N. low high. 1.69 1.04. 11 11. 6.38 4.44. 44 .

(46) Table 22 ANCOVA test on Prior knowledge Students’ understanding in picture group Source. SS. df. MS. F. Sig.. Prior Knowledge Error Total. 2.352 27.797 707.000. 1 19 22. 2.352 1.463. 1.608. .220. 4.1.6 RQ A2: Is there a significant difference between high and low prior knowledge on understanding in AR representation group? In AR group, an independent-samples t-test was also first performed to analyze pretest of high and low prior knowledge. Results indicated a highly significant difference between low prior knowledge and high prior knowledge, t (28.832) = -13.900, p= .000 < .005., as shown in Table 23. The statistics indicated that students’ pretest scores of high and low prior knowledge were varied.. Table 23 Independent Samples t-test on High and Low prior knowledge students’ pretest scores in AR Representations group t High and Low prior knowledge students’ pretest scores in AR group. df. -13.900 28.832. p. .000***. ***p < .001.. A one-way ANCOVA with pretest scores as covariance was then performed to examine a statistically significant difference between high and low prior knowledge on posttest in picture group. Levene's Tests of Homogeneity of variance were also performed and satisfied (p= .308 > .05), as shown in Table 24. Descriptive statistics of representations and prior knowledge are presented in Table 25. The ANCOVA results showed that the effects of prior knowledge on understanding were nonsignificant, F (1,29) = .003, p= .954 > .05., as shown in Table 26. The 45 .

(47) statistics indicated that, after the AR treatments, high and low prior knowledge student’s understanding tended to be similar even much closer than the ANCOVA results of picture group (Table 22).. Table 24 Levene's Test of Equality of Error Variances F. df1. df2. Sig.. 1.076. 1. 30. .308. Table 25 Descriptive Statistics of Achievement on High / Low prior knowledge in AR Representations Group PriorKnowledge Adj. Mean. SD. N. High Low. .964 1.37. 16 16. 5.50 5.56. Table 26 ANCOVA test on Prior knowledge Students’ understanding in AR Representations group Source. SS. df. PriorKnowledge .004 1 Error 36.324 29 Total 1055.000 32. MS. F. Sig.. .004 1.253. .003. .954. 4.1.7 Interview data In order to further investigate notable phenomena that the quantitative data did not tell, short interviews were carried out on random-chosen students in both picture and AR group. In the interview data, as few interesting findings were spotted, including manipulation and various analogical thinking related to life experiences. Most of the participants in AR group reported that the manipulation process was beneficial for them to understand mathematics concepts more easily, while students in picture group merely copied what they saw in the textbooks. Furthermore, when asked how they would 46 .