Card Sorting : A new pedagogy for understanding challenges in Mathematics during Emergencies and Crises


 Interest in challenges faced by university students during COVID-19 has led to research and development initiatives that include educational, technological, economical and socio-cultural provisions. Despite these initiatives, little is known about the usage of open card sorting, similarity matrix, and Hierarchical Clustering Method - 3D Cluster View algorithm in understanding and analysing mathematics challenges in a regional university during emergencies and crisis. This paper presents findings from a study that explored the challenges encountered by first-year mathematics students in a South Pacific institution. The findings reveal seven challenges: i) financial hardship; ii) motivational challenge; iii) moodle issues; iv) lack of face to face interactions; v) problem with course delivery; vi) internet challenge; and vi) home disturbances. A heptagon model is presented with possible solutions for the challenges identified by participants. The findings point to the complex inter-relationship between the institution’s emergency remote teaching, students’ learning needs, and students’ dynamic socio-cultural environments as important factors for delivering quality mathematics learning during a pandemic. This paper highlights the contribution of card sorting, as a new pedagogy, to the field of educational research as a provider of new learning analytics for desirable learning outcomes in a given pandemic. Decisionmakers and Policymakers of Higher Education Institutions around the world may benefit from these findings while formulating strategies to support first-year mathematics students during the current and future pandemics.


Introduction
The world is currently experiencing and gradually responding to a widespread of transmissible respiratory disease caused by an original coronavirus named COVID-19. The disease was declared a global pandemic by the World Health Organisation (WHO) on March 11, 2020. Over 1.5 billion students in 165 countries are affected by COVID-19 related school closures, equivalent to 87% of the world's student population. To protect this population from the spread of COVID-19 as a contagious and deadly virus, the emergency remote teaching (ERT) has been an overwhelming response of many Higher Education Institutions (HEI) around the world (Mulenga and Marbán 2020).
The global shift to ERT (Bozkurt and Sharma 2020;Hodges et al. 2020;Karakaya 2021;Trust and Whalen 2020;West et al. 2020) and its actual application using online platforms has posed many unexpected emergencies and crises; 5. use of new heptagon model for better comprehending of the complex nature of mathematics students' learning experiences in a South Paci c regional university, calling for a more inclusive, affordable and sustainable ERT. The ndings can be applied in any learning environment with special attention given to learners' dynamic socio-cultural backgrounds that should include contextualisation.
The paper consists of 8 sections, including this introductory section. The next two sections provide a brief description of the research context and the literature relevant to the research topic. This is followed by research strategies in section four then the questionnaire in section ve. Later in section six is the presentation and discussion of ndings followed by demonstrating potential solutions for participants' challenges in a heptagon model in section seven. The nal section concludes the paper.

Research Context
The USP is one of only two regional universities in the world. It is jointly owned by the governments of 12 member countries across the South Paci c region: Cook Islands, Fiji, Kiribati, Marshall Islands, Nauru, Niue, Samoa, Solomon Islands, Tokelau, Tonga, Tuvalu, and Vanuatu. The role of the USP in developing the knowledge economy of the region places the university in a unique position in relation to education and research. The university has campuses in all member countries, and the current study was fully implemented at the main campus, Laucala, in Fiji. After the announcement of the rst case of COVID-19 in Fiji on March 19, the USP was totally closed down for 2 weeks, March 26 -April 9, before the University began its emergency remote classes in the second half of the rst semester for 7 weeks from April to June 2020. The courses returned to normal blended mode (f2f and online) at the beginning of the second semester of 2020.
During the 7 weeks of ERT at the USP, the authors found student's self-discipline at the core of their own success. Mathematics students with high self-discipline could follow instructions and complete work on time, whereas those with low self-discipline would be impossible. When students' learning behaviours cannot be monitored remotely, their lack of self-discipline would greatly impact their learning outcomes. The unexpected shift to ERT also required teaching staff to use new software and unfamiliar online tools without having proper technical support. A signi cant challenge for HEI teachers has been their lack of the pedagogical content knowledge (PCK) (Tondeur et al. 2020;Shulman 1987) needed for teaching online (Kali, Goodyear and Markauskaite 2011;Ching et al. 2018). Accordingly, the use of digital and technological resources for ERT of mathematics has increased sharply. Still, its success can only be guaranteed when the teachers are competent and ready to develop and utilise these resources. This reinforces the need for HEI to be more prepared for the future, emphasising the important contribution the current study can offer to achieve that from a Paci c learning perspective during an emergency.

Literature Review
Some recent studies on ERT highlight different types of challenges experienced by students during the unexpected educational shift (e.g., (Aguliera and Nightingale-Lee 2020; Bozkurt and Sharma 2020; Hodges et al. 2020;Mohmmed et al. 2020;Mulenga & Marbán 2020). A study that explored students' lived experiences as impacted by emergency shift to remote teaching in the United States of America con rmed that students with existing educational inequality had been exposed to more learning inequalities during the abrupt shift to ERT (Aguliera and Nightingale-Lee 2020;Trust and Whalen 2020). Students had to provide their own learning resources that put them at a great disadvantage in relation to the unsafe learning environment, poor access to the internet and inability to possess electronic devices (Aguliera and Nightingale-Lee 2020;Zilka et al. 2018). Those students who do not have access to laptops or high-speed internet at home would experience more severe learning challenges, which may delay the acceptance of technology-enabled education (Zilka et al. 2018). These resources (non-nancial such as computer, internet access) are crucial to obtain HEI goals and become essential home possessions during emergencies and crises The ERT also comprises of issues including time management, technology illiteracy, students' assessment, communication, and the lack of in-person interaction (e.g., (Karakaya 2021;Steele 2019). Interacting online requires educators to rethink online pedagogy so as to support meaningful (higher-order) learning and its assessment (Gikandi et al. 2011;Ćukušić et al. 2014). Aside from online infrastructure challenges, Bozkurt and Sharma (2020) argue for more attention to the lack of empathetic support for students during the crisis of COVID-19 because "students will remember not the educational content delivered, but how they felt during these hard times" (Bozkurt and Sharma 2020). This presents the importance of motivation and self-esteem protection, during an ERT, for resilience and quality learning. Mulenga and Marbán (2020) explore the perspectives of teachers who were engaged in teaching mathematics online during COVID-19 and found that educators, teachers, and ERT staff need better training and support for using online tools (Trust and Whalen 2020). These challenges give direction to the future, calling for the institution to work in collaboration with stakeholders to offer better solutions in preparation for future interruptions (Bozkurt and Sharma 2020) and to formulate more forward-looking strategies towards improving teaching-learning activities during COVID-19 ERT (Mohmmed et al. 2020). With the rise in the use of online modalities during COVID-19, it is necessary to assess their effectiveness regarding teaching and learning from different stakeholders (Schwartz et al. 2020). The nature of online learning means that working in partnership with numerous digital innovators and instructors who see technology as a method of solving problems and reaching new learners is needed.
The use of card sorting to explore mathematics students' challenges during a pandemic in the South Paci c has been absent. However, the practical guidance on card sorting in a Paci c HEI will help to understand how it can be implemented appropriately in the Paci c context (Paea et al. 2020). Card Sorting is utilised to evaluate the information architecture (IA) of a nding and to design a navigation structure that can offer an exciting variety of content and functionality (Righi et al. 2013). The IA of the current research refers to the challenges that emerged from this study in relation to rst-year mathematics students' learning experiences during the emergency and crisis of COVID-19. It provides insight into users' mental models (Katsanos et al., 2019), revealing how they often implicitly group, sort, and label tasks and content within their own heads according to their understanding. The term card sort applies to a wide range of activities, including ordering, grouping, and/or naming objects or concepts (Paea and Baird 2018). With card sorting, researchers learn how users categorise and label their thoughts to express their feeling and experience in a challenging situation. Card sorting provides a map of users' knowledge of the content and where they might look for the content within the design of an actual interface display.
Card sorting also has applications in visualisation research. For example, it has been used to explore people's mental models of classifying visualisation methods. It has also been used as a way to conduct a task analysis of geovisualisation tools and interactions (Lloyd et al. 2007). Using the card sorting method in this study aims to target the users' understanding of the challenges based on their experiences during the shift from f2f to ERT. This is also the rst time to utilise the card sorting method to generate useful insights about the research question. Efforts to develop, understand, and productively use card sorting data have organised into a eld of educational research.
The need for a better understanding of rst-year mathematics students' challenges during COVID-19 using card sorting approach in a Paci c learning context is highlighted by its absence in the literature. It appears from the ERT literature that students' challenges during the emergency and crisis of a pandemic is complex in a way it is based on the dynamic inter-relationships between students' home learning environments, students-teachers digital literacy, course delivery, internet network access, students' motivation, and the sociocultural contexts in which students live and operate. These challenges set the foundation for exploring the research topic, which offers timely insights to the gap in the literature of interconnectivity between card sorting, quality learning in mathematics, ERT, and Paci c regional HEI.

Population, setting, and sample
The target population is the rst-year or 100 level students who studied mathematics at the USP in semester one (February -June) of 2020. Targeting rst year f2f mathematics students is essential for understanding their learning needs as new entrants to the university during a pandemic and how best to support them towards persistence or successful completion of mathematics courses. This also provides a more realistic insight into the challenges that mathematics students faced while going through unprecedented change to teaching and learning during the university's COVID-19 lockdown.
Since it was impractical to explore the topic with the complete USP population across the university's regional campuses, the research team chose the Laucala campus in Suva, Fiji, because this setting recruits the highest proportion of rst-year face-to-face mathematics students. It is also the most central setting considering COVID-19 restrictions on regional travels in the South Paci c. The respondents in the target populations were recruited by convenience sampling facilitated by advertising on MA111 moodle page. The study recruited a total of 32 (16 men and 16 women) rst-year mathematics students who are currently studying at the USP and citizens of the university's country members. Such a recruitment re ects the diverse realities shaping the formation of learning at the USP, and the complexities involved with direct and indirect impacts of pandemic on students' learning experiences. The participants' ages ranged from 18 to 28 (M = 20 and SD = 2.9).
The authors' decision to recruit 32 participants is based on their experience as card sorting researchers and from reviewing of previous card sorting studies that have examined the adequate sample size needed to produce high-quality representation (Optimal Workshop; Tullis and Wood 2004;Wood and Wood 2008). Tullis and Wood (2004) found that a sample size of 20-30 participants explains 90-95% of the true information space structure, with diminishing returns in explanatory power as the sample size increases beyond 30. Wood and Wood (2008) con rmed that recruiting as few as 25-30 participants would likely yield results similar to those of several hundred provided these participants are representative of actual users and are familiar with the domain being considered. Rosas and Kane (2012) indicated the variability in each study's stress values was dramatic when about 15 or fewer participants were included. As the number of participants for each of their ve studies reached about 35 participants, substantial improvements in stress (i.e. lower stress values) were observed. However, beyond 40 participants, only marginal improvements in stress were detected. These ndings suggest between 20 and 30 participants is warranted to maximise the consistency of t in the concept mapping representation by minimising the variability in the stress value found with smaller groups of participants. For open card sorting, the ideal sample size is between 30 and 50 participants as supported by Card sorting 101 (Optimal Workishop). Completed card sorts within this sample interval would be easy to identify ideas and consensus (Optimal Workshop). Also, keep in mind that the more participants you have completed your card sort, the potential for more complexity in your analysis increases as well. This is simply because narrowing down the most effective structure from 40 different suggested categorisations will probably be easier than 200 various suggestions. Also, Research suggests a minimum of 15 users to obtain robust data from open card sorts (Katsanos 2018;Nielson 2004;Tullis and Wood 2005), thus our studies had adequate sample size. Hence the authors decided to recruit 32 participants as the sample size.
Participants were recruited through a variety of means including personal contacts, referrals and voluntary.
The authors then administered the research announcement to the students who were enrolled in MA111. The Moodle message, course announcement via Moodle and email distribution were used to inform the students about the research and encourage them to participate voluntarily. Participants were also recruited using an informal snowball process that was based on researchers' cultural knowledge and skills of recruiting Paci c participants through networking and relationship building (Paea et al. 2020). This type of recruitment is important for building trust and respect amongst participants and the researcher because Paci c people can willingly partake when they trust the researcher; and know their contribution is recognised and valued (Paea et al. 2020).
Prior to the day of the actual card sorting, a card sorting demonstration video and an information sheet were sent to participants beforehand. This is to provide participants with relevant information about the research objectives, how to do card sorting, and how it would affect them during and after the eldwork.

Card size and names
A total of 44 physical card names were drawn from the university's in-house report on the challenges facing rst-year mathematics students in the process of a sudden shift to emergency remote learning. The 44 cards are related to each other in the sense of challenges, but they clearly comprehend the group's own cluster by the participants. This was supported by two mathematics lecturers' experience in the current research team to generate a Paci c solution-based framework to strengthen students' resilience and retention during a pandemic. Card names represent the challenges, problems, or issues that hinder students' ability to achieve during the unforeseen shift. The names were made in a lower-level meaning for participants to understand.
For instance, the card name 'internet data is expensive' means that students cannot access to online learning because of nancial hardship and the card names are related to each other. The physical cards are chosen to align with what most studied recommended (Optimal Workshop; Tullis and Wood 2004).

Face-to-face open card sorting
The study was implemented through open card sorting (OCS) in f2f mode because the nature of sharing responsibilities and co-constructing meaning between participants and the researchers have strengthened the quality of their relationship and the ndings (Paea et al. 2020).
On the day of f2f card sorting, one of the researchers welcomed participants by acknowledging their presence, time, and contribution. Participants were allowed to introduce themselves including their Paci c originalities because it gave them a sense of belonging to the card sorting context. The researcher went on to brief participants about the research objectives, informed them that their personal details would be kept con dential, and allowed them to ask questions about the research before they signed the consent form.
A pile of 44 physical cards was placed on the table. Participants were asked to sort the cards into groups of similarities and labelled groups according to the challenges they experienced during the unexpected change from f2f to ERL. The participants performed the card sort individually to assure independence of grouping strategies. For each participant, the authors took a photo of the nal card sorting and audio-recorded their verbalised thoughts. Additionally, audio recordings were transcribed and analysed to provide valuable insight, detailed data pre-processing and elaborate them to rich visualisation. The participants' details were kept anonymous and the responses were only used for analysis purpose.
The actual time of card sorting varied from 30-70 minutes to complete. Some participants created just four categories, while others created more complex classi cations involving up to 10 categories (M = 7, SD = 1.5). There were no signi cant differences between the number of categories formed by males (M = 7) and females (M= 7), t(7) = 0.88, ns, and the number of categories formed was unrelated to age (r = -0.23, ns). Once participants had grouped the challenges, they named each grouping they had formed to help explain commonalities between the challenges contained within the grouping. Figure 1 illustrates participants' pathway through the f2f OCS during an active card sorting performed by one of the participants. It shows how f2f card sorting is conducted using physical cards in an OCS. During the actual performance, participants were allowed to move cards to ensure their experiences were consolidated.
The blue sticky papers on top of each column represent the group numbers with unknown category names, and sorted cards are presented under each blue coloured paper. The category names were numbered for ease of reference.

Card sorting analysis
There were two main phases to the analysis. The rst phase was the construction of similarity matrices to test how strongly the group elements of challenges are related to each other. The second phase was a 3DCV analysis of the card-sort data (Optimal Workshop ; Paea & Baird 2018;Paea et al. 2020). To ensure usability and simplicity for better comprehension of participants' challenges, the category labels were revised. For example, in Group 3 of Table 1 (p 14), there are 3 commonly shared categories based on participants' data: nancial issues (82%), challenges with rolling expenses (80%), and family nancial background (78%). Since all these categories represent participants' nancial di culties, the category name ' nancial hardship' was used. While these steps can be seen as modifying the original data, it enhances data consistency and clarity without changing the meaning of participants' original data. This process of co-constructing meaning between participants and the researcher(s) is acceptable in the Paci c Way of carrying out card sorting research (Paea et al. 2020).

Phase 1: Similarity Matrix
A a similarity matrix was constructed to represent the raw card sort data. A matrix was created for every participant, indicating whether each pair of strategies was placed in the same grouping ("1") or in a different grouping ("0") in the participant's card-sort solution. These matrices were then aggregated cell-by-cell to create a matrix with cell values ranging between 0 (if no participants had placed a particular pair of challenges in the same group) and 32 (100%) (if all participants had placed the pair in the same group).
The similarity matrix is used to interpret how strongly the group elements of challenges are related to each other. Since the research has considered card names as representation of participants' learning challenges during an emergency, the similarity matrix in Figure 2 is a straightforward representation of cards combinations. It intends to give insights into the challenges that participants pair together in clusters, which also identify pairs of closely related challenges by assigning them higher similarity than those that are distantly related. The similarity is measured between two individuals in the cards of challenges, with the similarity matrix being formed by combining this information for all pairs of challenges. For instance, the rst column of the matrix shows that 87% of participants put the challenges 'expensive to buy relevant software' and 'cannot afford to buy the textbooks' in the same group; meaning that both cards are interconnected highlighting participants' nancial di culties as the lead cause of their learning challenges during the pandemic. Looking further down the same column, 'expensive to buy relevant software' and 'not working hard enough on assessment' were never placed together. This means that participants' nancial and motivational challenges are not related.
Accordingly, the strongest pair is positioned at the top left corner, grouping them with the next associated strongest pair that either of those challenges have, and then the process is repeated for that new pair. This way, groups of challenges that are strongly related to each other appear together in the same shade of blue on the similarity matrix. The darker the blue shaded areas where two challenges intersect, the more often they were paired together by the participants (Optimal Workshop; Paea & Baird, 2018; Paea et al., 2020).
It can be seen from Figure 2 that the blue shaded areas have an inconsistent pattern. The authors subsequently display the data that are positioned along the right edge in Figure 2 as a line graph (Figure 3) to identify possible major challenges of clusters. The technique can assist in nding a suitable analytical method to analyse the nding. For instance, in Figure 3, the red colour can be potential clusters. The black colour is threatening to decide if these challenges belong to a particular cluster due to low participant agreement. With the inconsistent patterns and low participant agreement, the 3D Cluster View (3DCV) is utilised to visualise the data clearly in Figure 2 by grouping the challenges in 7 clusters, as demonstrated in Figure 5. The next section describes how the authors' nalised the total number of categories in 7 clusters.

Phase 2: Number of Clusters
One important challenge that arises in quantitative analysis of card sort data is deciding the optimal number of clusters. In the initial solution, the number of clusters is equal to the number of cards included in the study, that is 44 physical cards (see Table 1). This paper uses the approached by Katsanoe et al. 2008 based on the widely used eigenvalue-one criterion to identify the optimal number of clusters. Every cluster has an eigenvalue representing the amount of variance accounted for by a given cluster. Usually, the rst variables have the greatest eigenvalues. The method identi es the optimal number of clusters in terms of variance explained by implementing an eigenvalue analysis of the challenges' similarity matrix ( Figure 2) and keeping only the eigenvalues greater than 1 (see Table 1). Table 1 shows that only the rst seven components have the eigenvalue greater than one.
Another method used for factor extraction is the analysis of the scree plot (Cattell 1966) or elbow criterion.
According to this criterion, the signi cant factors are disposed like a cliff, having a big slope while the trivial factors are disposed at the base of the cliff. This is achieved by plotting the eigenvalue against the number of clusters (see Figure 4a). Also, plotting the percentage of variance explained against the number of clusters (see Figure 4b). In Figure 4 a and b we can appreciate that starting with the seven-factor the slope of the curve is relatively small and these factors could be excluded from the model. Nevertheless, Figure 4 is very subjective because the curve's cut-off point is sometimes not very clear. Then we compare the result from Table 1 and gure 4 with the Optimal Workshop 3D Cluster View algorithm. This is calculated simply by taking the average (mean) of the number of categories created by participants in the survey. 3D Cluster View algorithm also provided seven clusters and the result agrees to the nding in Table 1 and Figure 4. Then the author's concluded the optimal number of clusters is seven.

Phase 3: Stress and Goodness of Fit
The goodness of t of the multidimensional scaling (MDS) results was shown by stress values and squared correlation (r-squared), as displayed in Table 2 and Figure 5. In order to select the best tting model data, the t values of stress and R-squared were examined. The stress in three dimensions for the output shown in Figure 5a) is 0.288. MDS literature suggests lower stress values are preferred and re ect better congruence between the raw data and the processed data (Davison 1983;Kruskal 1964). The stress values found in the dataset are typically higher than those recommended in the literature on MDS. Several reasons for the discrepancy have been presented by Trochim (1993) and Kane and Trochim (2007). The r-squared ( ) against dimension is plotted to assist us in choosing the best dimensions. Figure 5b shows that as the number of dimensions increases from three to four, r-square values observed converge and begin to level off. The three dimensions squared correlation (r-squared) value (0.928) approaching 1 (100%) indicates that the MDS model can be said to be good (Redell 2019;Seok 2009). This study found that the three-dimensional solution was the appropriate model for the card sorting datasets. To establish whether the participants' underlying structure supported the theoretical grouping, we subjected the similarity matrix in Figure 2 to hierarchical cluster analysis. Hierarchical cluster analysis arranges objects (in this case, challenges) into relatively homogeneous groups (Aldenderfer and Blash eld 1984;Antonenko et al. 2012), thus allowing researchers to recognise the commonalities and distinctions relevant to the participant group as a whole. An agglomerative clustering method was selected, whereby the most similar challenges (in this study) were successively merged to produce non-overlapping hierarchically clusters of increasing inclusiveness (Davidson and Ravi 2009;Zhao and Karypis 2005). The average linkage rule (Sokal and Michener 1958), also referred to as the within-group linkage method, was used. This rule joins the two most similar cards together in a cluster and then calculates the average similarity of a card with all other cards within and outside the cluster. A card only joins a cluster if a given level of overall similarity is achieved.
The 3DCV is utilised to visualise the data in Figure 2 and Figure 3 more clearly. The main output is a 3Dplotted, which graphically represented how the 44 challenges group into hierarchical clusters. This method reduces the di culty in interpreting a plot that contains too much data, long labels, and inconsistent patterns. Figure 6 shows seven group of challenges as proposed by 3DCV method from the dataset given in Figure 2, with each cluster shown in different colours. Each point in the visualisation represents a distinct challenge. Challenges that are closer together were more frequently sorted into the same category. Polygons show the group of challenges that are clustered together. Each of these groups can be interpreted as a potential category within an IA. Placing the cursor over any colour ballpoint will highlight the card name that the ballpoint represents. As shown at the bottom right-hand side of Figure 6, the category labels and the challenge names will be highlighted when the cursor is placed over any polygon. To reduce this complex representation to a more simpli ed and meaningful solution, we also visually inspected it, to determine the number of clusters at each polygon shown in Figure 6. We then interpreted the clusters produced at each polygon shown in Table 3. Figure 7 presents a bar graph of Figure 6 to show the number of cards in each cluster for clear visualisation.
The bar graph reveals the hidden meaning of the challenges in each category. The number of cards indicates how serious the challenges are, meaning that most students encountered a learning challenge during COVID-19. The higher the number of cards in a cluster, the more frequently the challenge is being faced by students during the COVID-19 crisis. The two category names that contain the highest number of cards are nancial issues and students' motivational challenges. This indicates that the majority of students have considered their ' nancial hardship' and motivation as major causes of their challenges during the emergency shift. The seven category names are related to each other. In the next section a second data collection method of online questionnaire has been applied to rate the category names.

Questionnaire
The second part of the research methodology is an online questionnaire provided to the 32 participants. Since, this is the rst study done on the use of card sorting and the students' challenges during the covid-19, the authors decided to use an online questionnaire to collect the data to identify how the 7 challenges ( Figure   5 and 6 and Table 2) were rated by participants. Using online questionnaire methodology enabled the authors to collect information regarding the students' attitude and satisfaction in naming the seven challenges for their learning. The students were given an online questionnaire which was designed using a 7-points in descending order. The question is "Rate the 7 challenges from the highest to least challenges, that is, top challenge = 1 and least challenge = 7. The authors then administered the questionnaire to the participanrs who were participated in the study.
The email distribution was used to inform the 32 participants about the questionnaire. The questionnaire was open to students for one week. A student took a maximum time of seven minutes to ll in the questionnaire. The students' responses to the online questionnaire were automatically saved in author's email. The participant's responses were con dential. All responses were compiled and analysed as a group.
The participants' details were kept anonymous and the responses were only used for analysis purpose. The simpli ed solution to the participants' classi cation is displayed in Table 3. The table also lists the most prototypical challenges from highest level category.

Findings And Discussion
The key ndings from card sorting analysis are presented in Table 3 with the primary level group number in the rst column, group labels in the second column, proposed group label in the third column and the list of card challenges in the nal column. For instance, row 1 of Table 3 shows that 82% of participants label primary level Group 1 ' nancial issues', 80% label it 'challenges with rolling expenses', and 78% label it 'family nancial background'. This result suggests that ' nancial hardship' can be the proposed category label for primary level Group 1 as determined by the list of similar related category labels and similar related card challenges displayed in the third column. A similar application can be repeated for the rest of the proposed group labels in Table 2.  Since the research sets out to understand rst-year mathematics students' learning challenges during the unexpected shift from f2f to ERT, this section discusses participants' ndings against the literature in order of the seven categories of challenges presented in Tables 3 and 4. The primary purpose of Table 3 is to show how the participants rated the category names that they found affecting their studies the most during the shift. As seen in Table 4, nancial hardship is the biggest challenge and the home disturbance is the least challenge during the unexpected shift from f2f to ERT. Table 4 is also re ected the hidden meaning showing in Figure 7. The number of cards indicates how serious the challenges are, meaning that most students encountered a learning challenge during COVID-19. The higher the number of cards in a cluster, the more frequently the challenge is being faced by students during the COVID-19 crisis.

Financial hardship
Financial hardship is one of the major challenges faced by the majority of participants during the COVID-19 lockdown. The ndings indicate the effects of job displacement on students' nancial situation during an emergency. As reported, domestic workers have suffered from job loss and/or a drop in working hours as one of the negative impacts of COVID-19 (International Labour Organisation, 2020). The issues of parental unemployment and job displacement during COVID-19 have put many families around the world in nancial crisis, making it very di cult for them to take care of everyday needs including education. It is evident in this study that participants have identi ed nancial di culties as the leading cause of their learning challenges which inter alia lead to human stress during COVID-19.

Motivational challenge
Studying from home commonly requires greater self-discipline and motivation to follow through online lessons, particularly in the earlier period when students are getting used to the new system, which might affect the feeling of an increase in study obligations. On the other hand, lecturers' unfamiliarities and incompetencies with the new mode of delivery could overload their students with study materials and assignments adding to the demotivate students feeling toward the course (Aristovnik et al. 2020). An emergency switch from f2f to ERT makes the learning experience entirely different and challenging to maintain intrinsic motivation in students which is even more pronounced in mathematics learning environment The list of challenges that participants did consider under this category has highlighted the importance of putting an effective learning support system in place and the importance of protecting participants' motivation and willingness to learn during a pandemic crisis. Self-motivation is an indispensable requirement for online learning; however, it seems to be absent from many online students which is a commitment to be ful lled by the institution (Bozkurt and Sharma 2020). Also many online students feel disconnected to their studies (Choudhury and Pattnaik 2020). After shifting to ERT, many students fall behind and give up as problems in handling a technological medium also seem di cult due to the lack of relevant ICT competencies (Aguliera and Nightingale-Lee 2020; Reddy et al. 2020). Therefore, the attitude change and technological literacy would help them gain con dence in order to succeed in their ERT courses with a lively atmosphere.

Moodle issues
The effectiveness of online learning depends on the designed and prepared learning material, the lecturer's engagement in the online environment, and lecturer-student and/or student-student interactions (e.g., Bao 2020; Wu and Liu 2013). In consideration of the list of challenges compiled by participants under this category, it suggests that without having a proper design of the moodle page and a proper training of the moodle page with students-lecturers as end users, the full potential and purpose of establishing such a platform cannot be reached, hence having negative impacts on students' success. The majority of participants believe that poor audio and lecture/tutorial videos as well as too much information in the course shell can cause confusion. The challenge of writing mathematics online using moodle features is still huge, and the option of alternatives such as uploading snapshots of write-ups is also not feasible keeping in mind the intermitant internet facilities and costly mobile data during emergencies and crises in the South Paci c.
Arguably, moodle is expected to be a friendly learning environment for students, teachers and course instructors to develop, disseminate learning materials, and share knowledge through multiple online activities such as forums and chats (Kumar et al., 2020;. The authors' experience has con rmed the need for moodle designers to measure usability because it determines the moodle's success based on students' learning needs. Many teachers have no previous experience in online teaching and although teachers received various types of training during the outbreak, the short-term effects of such training remains arguably minimal. Therefore, the instructor and students should be aware of both the advantages and potential pitfalls of using the latest technological advances like moodle during the shift to ERT.

Lack of 2f2 interaction
There is broad agreement that teachers play a key role in providing high-quality learning opportunities to students and fostering students' learning (e.g., König et al. 2021). Most HEI in the Paci c region rely heavily on f2f mode for sharing and distributing knowledge, hence, the capacity of the institution to handle the circumstances of unprecedented change to ERT can be a real challenge. For instance, the majority of participants believe that lack of f2f learning interactions and assessment strategies can make things hard for them to succeed during the ERT. Similarly, educators found that a lack of student f2f engagement is a primary problem that can cause distraction in effective learning. To overcome this inadequacy, a range of f2f learning support systems should put in place to improve students' interest in emergency online learning. This includes visual communication such as video group discussion, live and recorded tutorials; strengthen the insitution's online learning culture and policies; and allowing academics to be heavily involved in facilitating effective online learning activities that have direct positive impacts on students (Beetham and Sharpe 2007). 6.5 Problem with course delivery Participants were concerned with the lack of support from teaching staff during the ERT, which can link to the delivery mode's remote nature. This includes little time for inter-personal, poor communication, and being less active in the online activities. Such challenges imply a speci c pedagogical content knowledge associated with the designing and organising of healthier learning experiences and distinctive learning environments with the help of digital technologies. The pedagogical readiness of university teachers who have little experience in online teaching has become an integral part of any virtual learning. As reported, some of the main di culties facing university teachers concerning web-based courses arise from the complexity of the instructional situation and shortcomings in planning and organisation (Ching et al. 2018;Ocak 2011). 6.6 Internet challenge The sudden closing-off of face-to-face educational work, in response to the COVID-19 pandemic, gave teachers and students a strong sense of the difference between f2f and ERT, arguing that online learning can work more effectively in digitally developed countries (Basilaia & Kvavadze, 2020) such as the United States of America and not in most developing countries. In the Paci c Island countries, online learning (as well as blended learning) is sometimes ineffective due to the lack of access to fast, affordable and reliable internet connections Sharma et al., 2020) or even a lack of electricity. This hinders the process of online learning, especially for those who are living in rural as well as marginalised communities (Aristovnik et al. 2020;Wains and Mahmood 2008). Students who access the internet through smartphones are sometimes powerless to take advantage of online learning because a vital amount of online content is not accessible via smartphones. Low-income families mean that access to the internet is occasional provides poor internet connection and poor-quality internet.

Home disturbances
Due to the COVID-19 lockdown, students were required to study from home at the time of social distancing and lockdown. Students had no choice but to accept this unprecedented change in response to ght against the spread of the virus. The challenges under this category reveal that study from home during an emergency or crisis would be much more di cult and challenging for students who live in villages, extended families and crowded houses without any study-friendly environment. This is a common situation with Paci c students and families from the rural and low socio-economic backgrounds. Therefore the availability of different kinds of home infrastructure is needed to ensure e cient study. The potential solutions for all the seven challenges emerged from this study based on participants' card sorting ndings are demonstrated in Figure 7.

Potential Solutions For Students' Challenges
The novel seven categories of challenges articulated in this research give direction to the types of support that are relevant for addressing issues as a basis for participants' success in the sudden shift to ERT during the emergency and crisis of COVID-19. Figure 8 presents two heptagons with the top outlining the seven challenges and the bottom presenting the potential solutions. The colour is used to identify the patterns of relationship between the two pentagons. For instance, the ' nancial hardship' challenge in the top heptagon and the potential solution 'provide nancial support' in the bottom heptagon are both coloured in light black to show their relationships. While the model governs the challenges and potential solutions for students studying mathematics, it is equally relevant to students from other disciplines.
When we look at these major categories from an a rmative perspective, it means that the institution's level of preparedness during the ERT must be strengthened. For instance, in terms of ' nancial hardship', the ndings indicate that the institution must be ready to compensate students' internet and technological needs in an emergency. As witnessed in this study, participants considered ' nancial hardship' and 'motivational challenge' as the lead causes of their learning challenges during the erupted shift to ERT due to COVID-19. It emphasises the need to improve the affordability and availability of free access to learning support during the nancial and psychological crisis of a pandemic. The ndings also reinforce the signi cant role of moodle literacy, f2f interaction, quality course delivery, quality internet network access, and having a friendly learning environment at home to participants' success. Figure 8 highlights a new learning support model designed from the insights of the ndings of this research.
The model suggests that the appropriate way to support rst-year Paci c mathematics students learning challenges during any crisis or emergency must be understood from the dynamic interplay between their nances, motivation, online learning literacies, f2f interpersonal interaction, course delivery, internet access, and home environment within a given socio-cultural learning context. This leads to the understanding that participants' learning challenges are a complex system, meaning that participants' learning challenges during ERT is made up of different related parts that must be understood within the socio-cultural context in which it is understood and experienced. As demonstrated by bottom heptagon of Figure 8, participants' challenges can be appropriately addressed in a more interconnected and multidimensional system, It means that people within the respective context, whether at home or in HEI, should be the catalyst for change and driver of students' success during the pandemic crisis and emergencies.

Conclusion
Overall, this study shows that the unexpected shift from f2f to ERT due to COVID-19 has affected rst-year mathematics students to experience a range of challenges including nancial hardship, motivational challenge, moodle issue, lack of f2f interaction, course delivery problem, internet challenge, and home disturbances. These ndings emphasise that if these students' challenges are not fully understood in the context of their own socio-cultural contexts, it may erode their motivation and con dence which can impact their overall academic performance. The paper's arterial strength lies in its ability to effectively introduce card sorting in the eld of educational research as a provider of new learning analytics for quality learning in mathematics and desirable learning outcomes from a range of known and unknown inputs.
The use of card sorting, for the rst time, to explore the topic adds new insights into the literature on the important connection between card sorting, HEI, mathematics, quality learning, and student retention during a crisis or emergency such as COVID-19 pandemic. Findings from this study also offer, for the rst time, a contribution to the understanding of the challenges faced by rst-year Paci c mathematics students as a complex system within a given socio-cultural learning context. This provides a fresh perspective into the complex way in which rst-year mathematics students operate in their own Paci c learning contexts in a given pandemic. In particular, the combination of students' social and cultural learning backgrounds together with the institution's learning support system must go hand in hand for best results. The study highlights the signi cance of using card sorting as a new methodology in the eld of educational research to understand the students' challenges from their own perspectives and to design a more student-oriented learning support model for emergencies and crises. From a wider perspective, the unique methodology has a good scope in the eld of educational research and can be utilised in a number of areas such as design of online courses and other contextualised learning resources and support models.
The ndings from this study would bene t our understanding of the challenges faced by rst-year mathematics students at a HEI in the Paci c. It highlights the need for more in-depth future exploration of the seven aspects of the Paci c learning support system proposed in the study. Such a study can explore and evaluate each challenge's main causes and discovery measures from students' own perceptions and experiences. The use of Paci c research approaches, such as 'talanoa' (talking) (Paea et al. 2020) can produce a lot of rich and deep knowledge about the research topic using qualitative data. An extension of the research setting to the wider Paci c region via the USP's regional campuses would enhance understanding the topic from the dynamic nature of the South Paci c cultural diversity.

Declarations
Data availability statement The datasets produced for this study are available on request to the corresponding author.

Author Contributions
The authors of this paper who agreed to be accountable for all aspects of this study in con rming that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved honesty. We provided substantial contributions to the design of this study, clarifying and interpretation of the data, and revised it critically for intellectual content.

Funding
Funding information is not applicable / No funding was received. Participant pathway through f2f OCS with physical cards in real-time.

Figure 2
The similarity matrix displays how many participants agree with each pair combination of cards. The algorithm attempts to cluster similar cards along the right edge of the matrix.

Figure 3
Shows the portion along the right edge of Figure 2 and indicates how many participants agree with each paired combination of cards.

Figure 4
Determining the optimal number of categories. a) The scree plot for the initial variables. B) The scree plot analysis.
Page 29/31 Figure 5 The graph of stress against the number of dimensions (a) and R2 versus the number of dimensions (b) Figure 6 The 3DCV of the clustering results in Table 1. Using 3DCV produced seven-cluster solutions with each cluster shown in different colours. These polygon groups can be interpreted as possible categories for IA.

Figure 7
For clear visualisation, Bar graph of Figure 2 with the number of cards in each polygon.

Figure 8
Potential solutions for participants' challenges