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How University Mathematics Instructors Form Groups and How Students Experience Them

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Title: How University Mathematics Instructors Form Groups and How Students Experience Them
Language: English
Authors: Valentin A. B. Küchle (ORCID 0000-0002-2997-7128), John P. Smith III, Jihye Hwang (ORCID 0000-0002-7075-4508), Reshma Menon
Source: International Journal of Mathematical Education in Science and Technology. 2025 56(9):1804-1831.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 28
Publication Date: 2025
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 1835946
Document Type: Journal Articles
Reports - Research
Tests/Questionnaires
Education Level: Higher Education
Postsecondary Education
Descriptors: College Faculty, Mathematics Teachers, Mathematics Instruction, Teaching Methods, Grouping (Instructional Purposes), Ability Grouping, Homogeneous Grouping, Heterogeneous Grouping, Decision Making, Introductory Courses, Mathematical Logic, Active Learning, Classification
DOI: 10.1080/0020739X.2024.2369651
ISSN: 0020-739X
1464-5211
Abstract: 'Group work' is a vague description of an instructional activity, because many factors shape its character and effect on students. One important factor is group formation, that is, how groups are formed by the instructor. In this paper, we sought to better understand the variation of group work with respect to group formation by addressing: How do university mathematics instructors form groups over the course of a semester? To this end, we examined eight instructors' methods of forming groups in one multi-section introduction to proof course. Our findings include a classification of group formation methods and descriptions of how instructors varied their formation methods across the semester. Further, we sought to understand: How do students experience different group formation methods? We analysed interviews with 29 students from the eight instructors' classes and identified central themes among students' experiences of different group formation methods. Finally, we discuss the sometimes conflicting research on (when) which group formation method is most appropriate and offer our thoughts on how the differences between typical undergraduate and K-12 mathematics classrooms may contribute to different recommendations.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1483313
Database: ERIC
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  Value: <anid>AN0187890701;imt01sep.25;2025Sep15.05:03;v2.2.500</anid> <title id="AN0187890701-1">How university mathematics instructors form groups and how students experience them </title> <p>'Group work' is a vague description of an instructional activity, because many factors shape its character and effect on students. One important factor is group formation, that is, how groups are formed by the instructor. In this paper, we sought to better understand the variation of group work with respect to group formation by addressing: How do university mathematics instructors form groups over the course of a semester? To this end, we examined eight instructors' methods of forming groups in one multi-section introduction to proof course. Our findings include a classification of group formation methods and descriptions of how instructors varied their formation methods across the semester. Further, we sought to understand: How do students experience different group formation methods? We analysed interviews with 29 students from the eight instructors' classes and identified central themes among students' experiences of different group formation methods. Finally, we discuss the sometimes conflicting research on (when) which group formation method is most appropriate and offer our thoughts on how the differences between typical undergraduate and K-12 mathematics classrooms may contribute to different recommendations.</p> <p>Keywords: Active learning; group formation; group work; student experience</p> <hd id="AN0187890701-2">1. Introduction</hd> <p>Broad support now exists for using active learning methods in university mathematics and science courses, where lecture has long been the dominant instructional activity (Deslauriers et al., [<reflink idref="bib8" id="ref1">8</reflink>]; Freeman et al., [<reflink idref="bib11" id="ref2">11</reflink>]; Prince, [<reflink idref="bib24" id="ref3">24</reflink>]; Theobald et al., [<reflink idref="bib34" id="ref4">34</reflink>]). One such active learning method is group work (Braun et al., [<reflink idref="bib4" id="ref5">4</reflink>]; Freeman et al., [<reflink idref="bib11" id="ref6">11</reflink>]; Lombardi et al., [<reflink idref="bib22" id="ref7">22</reflink>]; Prince, [<reflink idref="bib24" id="ref8">24</reflink>]), which has been shown to have substantial positive effects (e.g. Armstrong et al., [<reflink idref="bib1" id="ref9">1</reflink>]; Brown, [<reflink idref="bib5" id="ref10">5</reflink>]; Davidson, [<reflink idref="bib7" id="ref11">7</reflink>]; Heller & Hollabaugh, [<reflink idref="bib14" id="ref12">14</reflink>]; Slavin, [<reflink idref="bib29" id="ref13">29</reflink>]; Springer et al., [<reflink idref="bib31" id="ref14">31</reflink>]). Yet, due to many dimensions of variation in group work enactment (e.g. frequency and duration of group work, instructor assistance, feedback and evaluation) (Smith et al., [<reflink idref="bib30" id="ref15">30</reflink>]), 'group work' is a vague description of an instructional activity. The purpose of this article is to examine differences in the enactment of group work with respect to one basic factor of variation, namely group formation. Since researchers have suggested using different group formation methods at different points in a semester (e.g. randomised [Hagelgans et al., [<reflink idref="bib13" id="ref16">13</reflink>]] or randomised and flexible grouping [Reinholz, [<reflink idref="bib26" id="ref17">26</reflink>]] at the start of the semester), we are also interested in studying the variation in group formation methods over time. Thus, we address the question:</p> <p></p> <ulist> <item> How do university mathematics instructors form groups over the course of a semester?</item> </ulist> <p>To this end, we studied the group work enactment of eight instructors, who taught sections of a coordinated undergraduate introduction to proof course. From our semester-long observations, we generated seven different profiles of group formation and examined how instructors changed their group formation methods over the course of the semester. We combined these profiles with Hagelgans et al.'s ([<reflink idref="bib13" id="ref18">13</reflink>]) classification of group formation methods to offer a refinement of that classification that we hope the reader will find more instructionally useful.</p> <p>Further, we were interested in understanding how students experienced these group formation methods, leading us to ask:</p> <p></p> <ulist> <item> How do students experience different group formation methods?</item> </ulist> <p>We addressed this question by analysing interviews with 29 students from the eight instructors' sections. The results of this analysis are themes of student experience of different group formation methods. Finally, we reflect on the implications of these different group formation methods and changing them over the course of a semester.</p> <hd id="AN0187890701-3">2. Background literature</hd> <p>Since the 1970s, research on effects of group work on students' learning has been extensive, in and outside of mathematics. But where group work in mathematics has been a focus in K–12 studies, research in university mathematics classrooms has been more limited. Below, we first focus on group work's impact on students' learning experiences before discussing research on group formation. Finally, we address the use of group formation methods over time (e.g. a semester).</p> <hd id="AN0187890701-4">2.1. Impact of group work</hd> <p>The central result at both the K–12 and university level has been that small-group problem-solving has had productive effects on learning. For example, Springer et al.'s ([<reflink idref="bib31" id="ref19">31</reflink>]) meta-analysis showed substantial positive effects of group work on university students' achievement, attitudes, and persistence in mathematics and other STEM disciplines. These effects held across students' gender, major, and race. Broad positive effects have also been reported in studies in other university disciplines (e.g. Armstrong et al., [<reflink idref="bib1" id="ref20">1</reflink>] [biology]; Brown, [<reflink idref="bib5" id="ref21">5</reflink>] [anatomy and physiology]; Heller & Hollabaugh, [<reflink idref="bib14" id="ref22">14</reflink>] [physics]), as well as in pre-university mathematics (Davidson, [<reflink idref="bib7" id="ref23">7</reflink>]; Slavin, [<reflink idref="bib29" id="ref24">29</reflink>]).</p> <p>A growing body of research, however, qualifies these findings by demonstrating the role identity and power play in group work. For instance, students of colour and White girls may find themselves marginalised during group work in a classroom (Esmonde et al., [<reflink idref="bib9" id="ref25">9</reflink>]). In addition to race and gender, language proficiency (in a group's chosen language of communication) has also been shown to shape group work experiences (Hwang et al., [<reflink idref="bib16" id="ref26">16</reflink>]; Takeuchi, [<reflink idref="bib32" id="ref27">32</reflink>]). We see this research not as a rebuttal of group work, but rather a call for teachers to be purposeful when forming groups and to consider the influence of identity and power on group interactions.</p> <hd id="AN0187890701-5">2.2. Group formation methods</hd> <p>Many factors have been shown to affect group work enactment, such as group goals (Slavin, [<reflink idref="bib29" id="ref28">29</reflink>]), individual accountability (Slavin, [<reflink idref="bib29" id="ref29">29</reflink>]), the character of tasks (Heller & Hollabaugh, [<reflink idref="bib14" id="ref30">14</reflink>]; Liljedahl, [<reflink idref="bib20" id="ref31">20</reflink>], [<reflink idref="bib21" id="ref32">21</reflink>]), group spaces (Cohen & Lotan, [<reflink idref="bib6" id="ref33">6</reflink>]; Liljedahl, [<reflink idref="bib20" id="ref34">20</reflink>], [<reflink idref="bib21" id="ref35">21</reflink>]), and group wrap-up (Cohen & Lotan, [<reflink idref="bib6" id="ref36">6</reflink>]). The factor we focus on, however, is <emph>group formation</emph> – how groups are formed in a class (e.g. randomly, by student choice), in part because this issue has not been widely studied at the university level. Group formation is crucial because it creates a frame within which interactions and social dynamics play out, considerably shaping students' experiences and learning. Of central importance to our article is Hagelgans et al.'s (2001) classification of group formation methods: (a) random selection, (b) pseudo-random selection, (c) instructor selection, and (d) student selection, each of which presents advantages and disadvantages.</p> <hd id="AN0187890701-6">2.2.1. Random selection</hd> <p>As Hagelgans et al. ([<reflink idref="bib13" id="ref37">13</reflink>]) noted, random selection (e.g. by giving students cards from a shuffled deck and asking them to group up by face value) can be a fast way to form heterogeneous groups – in general. Heterogeneous groups have often been considered desirable (Cohen & Lotan, [<reflink idref="bib6" id="ref38">6</reflink>]) because they, for example, generate diverse ideas and approaches to a problem (Hagelgans et al., [<reflink idref="bib13" id="ref39">13</reflink>]), avoid problematic ability grouping (Reinholz, [<reflink idref="bib26" id="ref40">26</reflink>]), and 'teach students tolerance, acceptance, and strategies for working successfully with a diversity of partners' (Lou et al., [<reflink idref="bib23" id="ref41">23</reflink>], p. 111). But Esmonde et al. ([<reflink idref="bib9" id="ref42">9</reflink>]) have raised concerns about whether '"heterogeneous groups" might actually be supporting privileged groups – especially White boys – rather than creating spaces with rich opportunities to learn for students of color and for girls in general' (p. 41). Teachers may address this issue by being more purposeful in their group work implementation (e.g. by using complex instruction) when forming groups heterogeneously (Boaler, [<reflink idref="bib3" id="ref43">3</reflink>]; Esmonde et al., [<reflink idref="bib9" id="ref44">9</reflink>]).</p> <p>Instead of creating heterogeneous groups, random selection may also lead to 'groups skewed in ability, groups with personality conflicts, or groups with incompatible schedules' (Hagelgans et al., [<reflink idref="bib13" id="ref45">13</reflink>], p. 19). Further, in classrooms with a small number of minoritized students, minoritized students will frequently find themselves grouped only with majoritized peers, which may lead to them being silenced or ignored (e.g. Heller & Hollabaugh, [<reflink idref="bib14" id="ref46">14</reflink>]; Hwang et al., [<reflink idref="bib16" id="ref47">16</reflink>]).</p> <p>Liljedahl ([<reflink idref="bib21" id="ref48">21</reflink>]) recommended a particular type of random selection to form groups, namely <emph>visibly random grouping</emph> – forming random groups in class in a way that students can 'see' the randomness. As Liljedahl ([<reflink idref="bib19" id="ref49">19</reflink>]) reported, visibly random grouping led to: (a) students willing to work in whichever group they are placed, (b) social barriers being eliminated, (c) increased collaboration within and between groups, and (d) increased student enthusiasm. As opposed to teachers randomising groups in private (before or during class), 'making it <emph>visibly random</emph> was necessary for the students to both perceive and believe the randomness' (Liljedahl, [<reflink idref="bib21" id="ref50">21</reflink>], p. 44).</p> <hd id="AN0187890701-7">2.2.2. Pseudo-random selection</hd> <p>By <emph>pseudo-random selection</emph>, Hagelgans et al. ([<reflink idref="bib13" id="ref51">13</reflink>]) referred to randomising groups and then making modifications. They recommended making any modifications before class to keep them private.[<reflink idref="bib1" id="ref52">1</reflink>] Cohen and Lotan ([<reflink idref="bib6" id="ref53">6</reflink>]) suggested the opposite, recommending <emph>controlled randomness</emph> or <emph>no hidden agendas</emph> – visibly random grouping followed by public modifications.</p> <p>Like random selection, pseudo-random selection generally increases group heterogeneity. Unlike random selection, it enables teachers to, for example, separate close friends known to socialise instead of work, separate quarrelling students, and – where relevant – to group emergent multilingual students with translators (Cohen & Lotan, [<reflink idref="bib6" id="ref54">6</reflink>]). Further, groups can also be modified to ensure a minimum number of minoritized students in a group (e.g. the teacher could impose the constraint that in groups in which women are present, there should be at least two).</p> <hd id="AN0187890701-8">2.2.3. Instructor selection</hd> <p>As Liljedahl ([<reflink idref="bib19" id="ref55">19</reflink>]) noted, when instructors select groups, they typically do so for both educational (i.e. pedagogy, productivity, peacefulness) and social goals (i.e. diversity, integration, socialisation). One such combination of goals is to ensure heterogeneity, which, as aforementioned, is generally considered desirable (Cohen & Lotan, [<reflink idref="bib6" id="ref56">6</reflink>]; Hagelgans et al., [<reflink idref="bib13" id="ref57">13</reflink>]; Reinholz, [<reflink idref="bib26" id="ref58">26</reflink>]). Rather than leaving the heterogeneity of groups up to chance, by selecting groups, instructors can ensure that groups are gender-balanced or that emergent multilingual students are grouped with a friend (Reinholz, [<reflink idref="bib26" id="ref59">26</reflink>]). Thus, Reinholz ([<reflink idref="bib26" id="ref60">26</reflink>]) advocated for '[i]nstructor assigned heterogeneous groups, with attention to race, gender, and individual needs' (p. 915) – possibly created after soliciting students' input on whom they work with well.</p> <p>That said, several concerns have been raised about instructor selection. First, 'a group with extreme differences of talent or academic background may also have problems' (Hagelgans et al., [<reflink idref="bib13" id="ref61">13</reflink>], p. 19). Second, as Cohen and Lotan ([<reflink idref="bib6" id="ref62">6</reflink>]) warned, students are likely to catch on to any purposeful heterogenizing and may begin to view their group members as stereotypical representatives rather than as individual persons. Third, irrespective of the instructor's goals, students' group goals may not align with the instructor's and possibly cause friction[<reflink idref="bib2" id="ref63">2</reflink>] – thus, an instructor might as well not be purposeful in their group formation (Liljedahl, [<reflink idref="bib19" id="ref64">19</reflink>]). Finally, instructors' implicit or explicit biases (e.g. about who is more or less mathematically competent) may affect their grouping.</p> <hd id="AN0187890701-9">2.2.4. Student selection</hd> <p>As noted by Reinholz ([<reflink idref="bib26" id="ref65">26</reflink>]), '[s]elf-selection can lead to a lack of diversity, groupthink, and excessive homogeneity, which may further marginalize certain students' (p. 907). Furthermore, Cohen and Lotan ([<reflink idref="bib6" id="ref66">6</reflink>]) argued that student selection is problematic because: (a) struggling students may lack the resources for productive group work, (b) groups of friends are likely to be distracted and become unproductive, and (c) some students may become socially isolated.</p> <p>If an instructor opts for student selection, Hagelgans et al. ([<reflink idref="bib13" id="ref67">13</reflink>]) suggested that instructors should provide the students with some constraints (e.g. on group size). An advantage to students self-selecting may be that students are likely to choose group members they already know, allowing them to skip the get-to-know-each-other phase and become productive more quickly (Hagelgans et al., [<reflink idref="bib13" id="ref68">13</reflink>]).</p> <hd id="AN0187890701-10">2.2.5. Other</hd> <p>Some group formation methods may defy Hagelgans et al.'s ([<reflink idref="bib13" id="ref69">13</reflink>]) classification, such as 'sociometric grouping', where groups are determined by the results of some sociometric measure (e.g. a survey students filled out). Some research suggests its benefits over other forms of grouping (e.g. randomised grouping, student selection) (Davidson, [<reflink idref="bib7" id="ref70">7</reflink>]), but the term appears to have gone out of favour. Since sociometric measures could be as simple as asking students whom they want to work with and avoid (Shaw & Shaw, [<reflink idref="bib28" id="ref71">28</reflink>]), we see overlap between sociometric grouping and instructor selection with student input.</p> <hd id="AN0187890701-11">2.3. Group formation over time</hd> <p>Research has rarely considered how instructors change their group formation methods across a semester. That said, the results of the Cooperative Learning in Undergraduate Mathematics Education (CLUME) survey (Baker & Hagelgans, [<reflink idref="bib2" id="ref72">2</reflink>]) showed that of the 101 mathematics instructors who responded to the question 'How were groups formed?', 74 (i.e. 73.3%) used just one group formation method (i.e. either student selection or instructor selection). The remaining 27 responses split into: (a) a combination of student and instructor selection (<reflink idref="bib22" id="ref73">22</reflink>), and (b) other group formation methods (<reflink idref="bib5" id="ref74">5</reflink>). In short, the CLUME survey responses suggest that mathematics instructors may generally be using one method of group formation across a semester.</p> <p>A proponent of sticking to one (particular) group formation method is Liljedahl ([<reflink idref="bib21" id="ref75">21</reflink>]), who insisted on visibly randomising groups every hour or so. He observed that when groups were kept together for longer than an hour, some students would assume passive roles in their groups. Liljedahl's ([<reflink idref="bib21" id="ref76">21</reflink>]) recommendation, however, contrasts with Reinholz's ([<reflink idref="bib26" id="ref77">26</reflink>]) suggestion to move from initial random grouping to instructor selection with student input. As Reinholz ([<reflink idref="bib26" id="ref78">26</reflink>]) and Hagelgans et al. ([<reflink idref="bib13" id="ref79">13</reflink>]) noted, more permanent groups can lead to group members building stronger, more trusting relationships. Thus, after students have benefitted from getting to know their peers in the first few weeks of term (e.g. through random grouping), they could be placed in more permanent and productive groups. For students with disabilities, the Council for Exceptional Children argues for flexible grouping: Depending on lesson goals and objectives, sometimes groups should be homogeneous and at other times heterogeneous (Kennedy et al., [<reflink idref="bib17" id="ref80">17</reflink>]).</p> <p>In conclusion, university mathematics instructors may be leaning toward using one method of group formation across a semester. Research appears at odds over whether to use one or more group formation methods across a semester.</p> <hd id="AN0187890701-12">3. Method</hd> <p>To learn more about different group formation methods, we observed eight instructors – each teaching different sections of the same coordinated introduction to proof (ITP) course – as they implemented group work. We first describe this ITP course to provide context for our results before outlining our data collection and data analysis.</p> <hd id="AN0187890701-13">3.1. Context: the introduction to proof course</hd> <p>The ITP course was offered by a research-focused U.S. mathematics department. It introduced basic proof methods (direct, cases, induction, contradiction, and contrapositive) as applied to elementary statements in set theory, logic, real analysis, and number theory. Each semester, five small sections (≤ 25 students) were typically offered, each taught by an instructor and a teaching assistant, most of whom were mathematics graduate students. Typically, ITP teaching assistants would become ITP instructors in the subsequent semester. Teaching assistants were assigned to the ITP course if they met at least one of the following requirements: (a) they had previous teaching experience, (b) they were recommended by a faculty member, or (c) they had specifically included the ITP course as a teaching preference on their yearly teaching preference submission alongside a rationale for this preference.</p> <p>Sections of the course generally met three times a week for 80 min. Each week, instructors presented new content on one day (or the equivalent on multiple days), and students worked in groups for most of the remaining time (the equivalent of two days). Group work began on the first day of the course.</p> <p>A mathematics faculty member coordinated the course and led weekly instructor meetings where issues of content, pacing, and assessment were discussed. The coordinator required extensive group work in all sections, suggested groups of three or four students, but did not specify how groups should be formed. Students' work in groups was not assessed; the course grade was determined by attendance, homework, and exams.</p> <p>Most students in the course were intending mathematics majors or minors who had completed two or three semesters of calculus and would subsequently take two or more proof-based courses. More than one-third of regularly attending students in all sections were women. Almost half of regularly attending students in all sections were international students from East Asian countries, predominantly from China.</p> <hd id="AN0187890701-14">3.2. Data collection</hd> <p>In two consecutive semesters, eight (of eleven) ITP sections were observed.[<reflink idref="bib3" id="ref81">3</reflink>] Of the approximately 45 class meetings each semester – of which the equivalent of 30 class periods were typically dedicated to group work – between 12 and 20 were observed. In both semesters, more class sessions were observed in the first two weeks of the semester to understand how instructors introduced group work and how group work norms were presented and established.</p> <p>Mathematics and mathematics education graduate students and one mathematics education faculty member completed the observations. The authors carried out most of these observations. Observers sat in the back of the classrooms for the duration of each class session and paid attention to how the instructional team (i.e. the instructor and their [graduate] teaching assistant) presented content and formed groups. During segments of group work, observers focused on how students interacted with group members and how instructors interacted with groups. Observers also used diagrams to document where and with whom students sat during group work. All observers used the same observation template developed by the team (see Appendix A). One challenge observers faced while seeking to follow group work interactions was hearing the dialogue in distant groups.</p> <p>Several instructors were also willing to talk to us and clarify whether an observed method was random (private), pseudo-random (private), or instructor selection (without student input). As these three methods typically involved the instructor writing down groups on the board, these discussions were helpful in distinguishing among them. But overall, we refrained from engaging instructors about their instructional practices because our focus was on student experience and access to instructors' classrooms was possible in part because we were not interested in assessing instruction.</p> <p>The student data were gathered as part of a larger longitudinal project that sought to understand students' experiences and their development of agency and autonomy over the course of their degree. Thus, in each of the eight observed ITP sections, there were two to five participants for a total of 29 participants (see Table 1). All but one participant participated in two semi-structured interviews during their ITP experience: one a month into the course, and one toward the end of the course.</p> <p>Table 1. Participant overview.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>Instructor</td><td>Participants</td></tr></thead><tbody><tr><td>1st Semester</td><td>1</td><td>Kaleb (K), Koffi (O [<italic>sic</italic>]), Melanie (M), Gamon (G)</td></tr><tr><td>2</td><td>Jun (J), Leo (L), Eva (E), Benjamin (B), Iris (I)</td></tr><tr><td>3</td><td>Christian (C), An-Shi (A)</td></tr><tr><td>4</td><td>Henry (H), Noah (N), Falon (F), Dani (D)</td></tr><tr><td>2nd Semester</td><td>5</td><td>Lei (L2), Amelia (A2), Dorothy (D2)</td></tr><tr><td>6</td><td>Jia (J2), Broxton (B2), Hao (H2), Casey (C2)*</td></tr><tr><td>7</td><td>Flora (F2), Nick (N2), Emily (E2)</td></tr><tr><td>8</td><td>Galen (G2), Kellen (K2), Ingrid (I2), Paul (P2)</td></tr></tbody></table> </ephtml> </p> <p>1 Note<emph>.</emph> The asterisk denotes that Casey was the only of the 29 participants who did not complete both interviews. Although we typically refer to our participants by their pseudonyms, we believe that in this article, the use of letters will greatly enhance the reader's experience. There were no participants M2 or O2.</p> <p>In the first semester's early-semester interviews (ESIs), we wanted to learn about students' previous experiences with mathematics and their initial impressions of the ITP course. As part of this, we asked students to compare the ITP course to other mathematics courses they had taken, and we followed up with a question about whether the teaching methods were similar or different to their past mathematics courses. In every interview, students discussed group work. In the second semester's ESIs, we asked explicitly about group work: 'What do you think about the group work in ((ITP course number))?' In both semesters' late-semester interviews (LSIs), which were used to learn about students' experiences across the semester and also involved tasks, we asked 'How did working in groups during class work for you?' (alongside several follow-up questions). Per the nature of semi-structured interviews, we followed up with our participants in participant-specific ways.</p> <hd id="AN0187890701-15">3.3. Data analysis</hd> <p>To address our first research question, we completed a 'bottom-up' analysis of 116 observations – 83 of which included segments of group work. For each of these 83, we carefully attended to how instructors formed groups. Upon finding multiple instances of an approach to group formation, we created a profile for that approach – a profile consisted of a name and a detailed description.[<reflink idref="bib4" id="ref82">4</reflink>] After several rounds of revising our profiles and recoding, each of the 83 observations had been coded by at least two of the authors, any conflicting codes were resolved, and we had generated seven profiles of group formation methods. Upon comparing our methods to those identified by others, we noted that our seven methods complemented Hagelgans et al.'s ([<reflink idref="bib13" id="ref83">13</reflink>]) four methods. Thus, we relabelled our profiles and adopted their terminology, thereby offering a refinement of Hagelgans et al.'s ([<reflink idref="bib13" id="ref84">13</reflink>]) classification of group formation methods.</p> <p>To answer our second research question, the first author used structural coding – coding used to identify large segments of data similar in content (Saldaña, [<reflink idref="bib27" id="ref85">27</reflink>]) – to identify all interview segments in which group work was discussed. He then used structural coding on these group-work related interview segments to identify all (sub)segments about students' experiences related to the group formation method. During this second coding pass, he included all references to: (a) the group formation method (e.g. 'I wish that the groups were like maybe buddy system every once in a while [instead of randomized groups each week]'), (b) the group composition (e.g. '[...] I'm with my friend ((friend's name)) and then someone else.'), (c) the group duration (e.g. '[...] now we have a fixed group [...]'), and (d) how groups worked together (e.g. 'I'm usually the one who volunteers first [...] Usually goes me, I think ((Name 1)), then ((Name 2)), then ((Name 3))'). While the first three types of references are clearly concerned with the group formation method, we included 'how groups worked together' to attend to Liljedahl's ([<reflink idref="bib21" id="ref86">21</reflink>]) observation that groups kept together for longer than an hour led to 'roles within the group calcify[ing] into their active and passive states' (p. 44). In line with this observation, when structural coding for 'how groups worked together', special attention was paid to discussions of 'positions' – positions for subjects to take up in relation to other people (Hollway, [<reflink idref="bib15" id="ref87">15</reflink>]; van Langenhove & Harré, [<reflink idref="bib35" id="ref88">35</reflink>]) – such as students positioning themselves as smart (or less so), dominant (or less so), extroverted (or less so), the group's writer (or not), a teacher (of their peers), or a student (who asks questions of their peers). The first author then themed the data (Saldaña, [<reflink idref="bib27" id="ref89">27</reflink>]), whereby an extended thematic statement (i.e. a several sentence-long summary) was added to each previously identified segment about students' experiences related to the group formation method. Finally, the first and third authors used pattern coding – coding that consolidates extensive material into a smaller number of themes (Saldaña, [<reflink idref="bib27" id="ref90">27</reflink>]) – to identify larger themes among our thematic statements.</p> <hd id="AN0187890701-16">4. Results</hd> <p>We first offer a refinement of Hagelgans et al.'s ([<reflink idref="bib13" id="ref91">13</reflink>]) classification of group formation methods and then describe each of the group formation subtypes before characterising instructors' use of them through the semester. Subsequently, we focus on students' experiences of these group formation methods via our identified themes.</p> <hd id="AN0187890701-17">4.1. Group formation type taxonomy</hd> <p>Our analysis generated seven profiles of group formation methods, which we found to complement Hagelgans et al.'s ([<reflink idref="bib13" id="ref92">13</reflink>]) four group formation types. Thus, we offer a refinement of Hagelgans et al.'s ([<reflink idref="bib13" id="ref93">13</reflink>]) classification in Table 2, adding one hypothetical group formation method that we did not encounter in the data (i.e. <emph>pseudo-random (private) selection</emph>). We include this hypothetical method to acknowledge that whether an instructor uses <emph>pseudo-random selection</emph> in a public or private fashion has been identified as relevant; although using different terminology, Cohen and Lotan ([<reflink idref="bib6" id="ref94">6</reflink>]) argued for <emph>pseudo-random (public) selection</emph>, whereas Hagelgans et al. ([<reflink idref="bib13" id="ref95">13</reflink>]) argued for <emph>pseudo-random (private) selection</emph>.</p> <p>Table 2. Hagelgans et al.'s ([<reflink idref="bib13" id="ref96">13</reflink>]) group formation types alongside our identified subtypes.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Group formation type</td><td>Group formation subtype</td><td>Group formation subtype description</td></tr></thead><tbody><tr><td>Random selection</td><td>Random (public) selection</td><td>The instructor publicly uses a random process to form groups (e.g. the instructor distributes cards from a shuffled deck of cards and groups students according to matching face value).</td></tr><tr><td>Random (private) selection</td><td>The instructor privately uses a random process to form groups and shares the outcome with the students (e.g. the instructor generates random groups with Excel at home and then writes these groups on the board in class).</td></tr><tr><td>Pseudo-random selection</td><td>Pseudo-random (public) selection</td><td>The instructor either publicly uses a pseudo-random process to form groups (e.g. lining up students by height and grouping them accordingly) or the instructor uses a random selection method and then adjusts the groups in public.</td></tr><tr><td>Pseudo-random (private) selection</td><td>The instructor either privately uses a pseudo-random process to form groups (e.g. grouping students by last name [without making this selection process public]) or the instructor uses a random selection method and then adjusts the groups in private.</td></tr><tr><td>Instructor selection</td><td>Instructor selection (with student input)</td><td>The instructor forms groups by asking for and considering student input (e.g. students submit group member preferences via a survey and the instructor uses the results to create groups).</td></tr><tr><td>Instructor selection (without student input)</td><td>The instructor forms groups – typically in line with some educational and/or social goals – without asking students for their input.</td></tr><tr><td>Student selection</td><td>Student selection (with constraints)</td><td>The instructor asks students to select with whom they will work, subject to some constraint(s) (e.g. constraints on group size, constraints on group membership).</td></tr><tr><td>Student selection (without constraints)</td><td>The instructor asks students to select with whom they will work. The instructor voices no constraint.</td></tr></tbody></table> </ephtml> </p> <p>In addition to Table 2, we offer brief descriptions of each group formation subtype below.</p> <hd id="AN0187890701-18">4.1.1. Random selection</hd> <p>In our observations, we encountered instructors using <emph>random selection</emph> publicly and privately.</p> <hd id="AN0187890701-19">4.1.1.1. Random (public) selection</hd> <p>With <emph>random (public) selection</emph>, instructors use a random process to generate groups in public (i.e. in the classroom).[<reflink idref="bib5" id="ref97">5</reflink>] In our observations, only one instructor used this method. Each week, he distributed cards from a deck of playing cards – with the number of cards matched to the number of students – and asked all students with the same face value to group together. The instructor prepared the deck so that students were in groups of threes and fours.</p> <hd id="AN0187890701-20">4.1.1.2. Random (private) selection</hd> <p>With <emph>random (private) selection</emph>, instructors use a random process to generate groups in private (typically before class) and then share the results with their students in class. At least one instructor's teaching assistant repeatedly used Excel to randomise groups privately that the instructor then displayed on the board.</p> <hd id="AN0187890701-21">4.1.2. Pseudo-random selection</hd> <p>Hagelgans et al. ([<reflink idref="bib13" id="ref98">13</reflink>]) described <emph>pseudo-random selection</emph> as instructors using a random selection method and then making modifications to the resulting groups. We expand the meaning of pseudo-random selection to encompass another way of making not-quite-random groups: using a not-quite-random (i.e. pseudo-random) selection method, such as asking people of similar heights to group together. We encountered one type of pseudo-random selection in our observations – <emph>pseudo-random (public)</emph> – and hypothesise the existence of a second type: <emph>pseudo-random (private)</emph>.</p> <hd id="AN0187890701-22">4.1.2.1. Pseudo-random (public) selection</hd> <p>When using <emph>pseudo-random (public) selection</emph>, instructors compose groups that are not quite random either because they modify the results of a random selection or use a pseudo-random selection method. In our observations, we noted three of the latter: (a) asking students to line up by height and segmenting the ensuing line into groups of four, (b) counting off students modulo <emph>n</emph>, and (c) forming groups by order of birthdays.[<reflink idref="bib6" id="ref99">6</reflink>]</p> <hd id="AN0187890701-23">4.1.2.2. Pseudo-random (private) selection</hd> <p>Although we did not encounter a <emph>pseudo-random (private) selection</emph> method in our observations, it is possible to imagine an instructor using pseudo-random selection but doing so in private. For instance, an instructor may use a random (private) selection method to generate groups and then (slightly) modify the groups before coming to class.</p> <hd id="AN0187890701-24">4.1.3. Instructor selection</hd> <p>We identified two ways in which instructors grouped students together without resorting to a (pseudo-)random method: <emph>instructor selection (with student input)</emph> and <emph>instructor selection (without student input)</emph>.</p> <hd id="AN0187890701-25">4.1.3.1. Instructor selection (with student input)</hd> <p>When using <emph>instructor selection (with student input)</emph>, instructors assign students to groups based on their sense of who will work well together combined with students' indicated preferences. Two instructors used this method after their students worked for some weeks in groups formed by <emph>random (private) selection</emph>. Thus, students had time to gain experience working with numerous peers before sharing their preferences with the instructor. To learn whom students wanted to be grouped with, the instructors asked students to fill out a confidential e-mail survey.</p> <hd id="AN0187890701-26">4.1.3.2. Instructor selection (without student input)</hd> <p>When using <emph>instructor selection (without student input)</emph>, instructors assign students to groups based only on their sense of who will work well together. We judged that one instructor's teaching assistant used this method once. He wrote groups on the board – groups we (and a participant) judged to be novel and non-random – and asked students to get into them.</p> <hd id="AN0187890701-27">4.1.4. Student selection</hd> <p>We found two distinct ways in which instructors asked students to form their own groups: subject to some stated constraint(s) (e.g. group size) and subject to no constraints.</p> <hd id="AN0187890701-28">4.1.4.1. Student selection (with constraints)</hd> <p>When using <emph>student selection (with constraints)</emph>, instructors direct students to work in groups subject to some constraint(s). We observed two types of constraints: (a) constraints on group size (i.e. an upper and/or lower bound), and (b) constraints on group constituency (e.g. 'try getting into new groups', 'work with the people around you'). Only a lower-bound constraint on group size (≥ 2) explicitly precludes working alone.</p> <hd id="AN0187890701-29">4.1.4.2. Student selection (without constraints)</hd> <p>When using <emph>student selection (without constraints)</emph>, instructors direct students to work in groups (e.g. 'Everyone, let's get into groups') but provide no guidance how the groups should be formed. Students are thus free to choose to work with whomever they want. Further, from our observations, it appears that choosing to work by oneself may be an acceptable outcome of this group formation method. Sometimes, instructors even explicitly endorsed working alone with statements like: 'Work in groups or by yourself'.</p> <hd id="AN0187890701-30">4.2. Observed group formation method use across the semester</hd> <p>The complete coding of our 116 observations (with 83 days of group work) is given in Table B1 in Appendix B. Table 3 below provides a summary of the prevalence of different group formation methods. As suggested by the table, we most frequently observed student selection methods: (a) in 31.3% of the 83 observations with group work we observed <emph>student selection (without constraints)</emph>, and (b) in 19.3% we observed <emph>student selection (with constraints)</emph>. In other words, in more than half of the observations with group work, instructors used a student selection method. Following the student selection methods in popularity were the random selection methods: (a) in 18.1% of the 83 observations with group work we observed <emph>random (private) selection</emph>, and (b) in 13.3% we observed <emph>random (public) selection</emph>. Less frequently, we observed the instructors use instructor selection methods: (a) in 9.6% of the 83 observations with group work we observed <emph>instructor selection (without student input)</emph>, and (b) in 4.8% we observed <emph>instruction selection (with student input)</emph>. Least frequently, we observed pseudo-random selection methods. Only in 3.6% of the observations with group work we observed <emph>pseudo-random (public) selection</emph>, and we did not observe <emph>pseudo-random (private) selection</emph> at all.</p> <p>Table 3. Summary of group formation method use across the semester.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Group Formation method (GFM)</td><td>Weeks 1–5: Percentage of instances of given GFM out of 46 group work instances (# of given GFM instances) [# of instr. using given GFM]</td><td>Weeks 6–10: Percentage of instances of given GFM out of 21 group work instances (# of given GFM instances) [# of instr. using given GFM]</td><td>Weeks 11–15: Percentage of instances of given GFM out of 16 group work instances (# of given GFM instances) [# of instr. using given GFM]</td><td>Weeks 1–15: Percentage of instances of given GFM out of 83 group work instances (# of given GFM instances) [# of instr. using given GFM]</td></tr></thead><tbody><tr><td>R (Pb)</td><td>10.9% (5) [1]</td><td>19.0% (4) [1]</td><td>12.5% (2) [1]</td><td>13.3% (11) [1]</td></tr><tr><td>R (Pr)</td><td>23.9% (11) [2]</td><td>19.0% (4) [3]</td><td>0.0% (0) [0]</td><td>18.1% (15) [3]</td></tr><tr><td>PR (Pb)</td><td>6.5% (3) [2]</td><td>0.0% (0) [0]</td><td>0.0% (0) [0]</td><td>3.6% (3) [2]</td></tr><tr><td>PR (Pr)</td><td>0.0% (0) [0]</td><td>0.0% (0) [0]</td><td>0.0% (0) [0]</td><td>0.0% (0) [0]</td></tr><tr><td>I (I)</td><td>0.0% (0) [0]</td><td>9.5% (2) [2]</td><td>12.5% (2) [2]</td><td>4.8% (4) [2]</td></tr><tr><td>I (woI)</td><td>10.9% (5) [2]</td><td>14.3% (3) [2]</td><td>0.0% (0) [0]</td><td>9.6% (8) [3]</td></tr><tr><td>S (C)</td><td>30.4% (14) [5]</td><td>4.8% (1) [1]</td><td>6.3% (1) [1]</td><td>19.3% (16) [6]</td></tr><tr><td>S (woC)</td><td>17.4% (8) [5]</td><td>33.3% (7) [4]</td><td>68.8% (11) [5]</td><td>31.3% (26) [6]</td></tr></tbody></table> </ephtml> </p> <p>2 Note<emph>.</emph> The group formation methods are abbreviated as: R (Pb) = Random (PuBlic); R (Pr) = Random (PRivate); PR (Pb) = Pseudo-Random (PuBlic); PR (Pr) = Pseudo-Random (PRivate); I (I) = Instructor selection (with student Input); I (woI) = Instructor selection (WithOut student Input); S (C) = Student selection (with Constraints); and S (woC) = Student selection (WithOut Constraints).</p> <p>If we look at how many of the eight instructors used each method, we can see that student selection methods remain most popular – both <emph>student selection (without constraints)</emph> and <emph>student selection (with constraints)</emph> were used by six instructors. The student selection methods were followed by <emph>instructor selection (without student input)</emph> and <emph>random (private) selection</emph>, each used by three instructors. <emph>Instructor selection (with student input)</emph> and <emph>pseudo-random (public)</emph> selection were each used by two instructors, and <emph>random (public) selection</emph> by only one. <emph>Pseudo-random (private) selection</emph> was not observed.</p> <p>Finally, looking across the changes in how instructors formed groups across the semester, we make several observations. First, student selection methods were very popular throughout the semester (47.8% in weeks 1–5, 38.1% in weeks 6–10, and 75.0% in weeks 11–15). That said, whereas <emph>student selection (with constraints)</emph> had been the main student selection method at the start of the semester, by the final five weeks of term, <emph>student selection (without constraints)</emph> had become the standard method for five instructors. In other words, they increasingly conferred the choice of group members to their students as the semester progressed. Second, seven of the eight instructors used at least three different group formation methods during the semester. The eighth continuously used <emph>random (public) selection</emph>. Third, two instructors appeared to want students to get to know each other through <emph>random (private) selection</emph> in the first half of the semester, before then soliciting students' input on whom they wanted to work with and using <emph>instructor selection (with student input)</emph> to form groups in the second half of the semester. In summary, one instructor consistently used <emph>random (public) selection</emph>, five tried out different grouping methods before landing on <emph>student selection (without constraints)</emph> by the end of the semester, and two used <emph>random (private) selection</emph> before then switching to <emph>instructor selection (with student input)</emph>.</p> <hd id="AN0187890701-31">4.3. Student experience of the group formation methods</hd> <p>In this section, we offer several student experience themes for random selection, student selection, and instructor selection. We have no findings regarding the student experience of pseudo-random selection as students did not discuss this (rarely observed) group formation method. An overview of the themes can be found in Table 4.</p> <p>Table 4. Student experience themes of group formation methods.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Group formation method</td><td>Student experience themes</td></tr></thead><tbody><tr><td>Random selection</td><td><list list-type="Bullet"><list-item><p>Participants' overall experience in randomised groups was positive or neutral.</p></list-item><list-item><p>By working in multiple randomly formed groups, participants met many of their peers, which had several positive consequences.</p></list-item><list-item><p>Participants also identified downsides of random selection.</p></list-item><list-item><p>Participants identified advantages and disadvantages of randomising groups <italic>each week</italic>.</p></list-item><list-item><p>New interactions and positions in new randomised groups.</p></list-item></list></td></tr><tr><td>Student selection</td><td><list list-type="Bullet"><list-item><p>Participants offered a mixed picture of group work experiences in student-selected groups.</p></list-item><list-item><p>Student selection typically led to participants working 'with' their neighbours for the semester (subject to minor adjustments).</p></list-item><list-item><p>When given the option, some participants chose to work alone.</p></list-item><list-item><p>Fixed interactions and positions in student-selected groups.</p></list-item></list></td></tr><tr><td>Instructor selection (with student input)</td><td><list list-type="Bullet"><list-item><p>Not all participants submitted group preferences, but those who did asked to work with peers with whom they were comfortable.</p></list-item><list-item><p>Being in a permanent group created by the instructor with student input made many, but not all, participants happy.</p></list-item></list></td></tr></tbody></table> </ephtml> </p> <p>3 <emph>Note.</emph> The random selection themes are heavily shaped by the students' experience of <emph>frequent</emph> randomised grouping. We posit that the random selection themes would have drastically differed if the instructors who used random selection had only used it once at the beginning of the semester and kept the groups fixed for the rest of the semester.</p> <hd id="AN0187890701-32">4.3.1. Random selection</hd> <p>11 participants from three classes (i.e. instructor 1, 6, 7's) spoke to us about random selection. That instructor 1 used <emph>random (public) selection</emph> whereas instructors 6 and 7 used <emph>random (private) selection</emph> was not raised by the participants (e.g. they did not voice concerns about whether instructors 6 and 7's methods were truly random). We identified five themes for random selection more broadly.</p> <hd id="AN0187890701-33">4.3.1.1. Random selection theme #1: participants' overall experience in randomized groups was...</hd> <p>Of the 11 participants who spoke to us about random selection, eight told us that they liked group work or found it helpful (i.e. O, M, J2, B2, H2, F2, N2, E2), two told us that group work was 'okay' or they did not mind it (i.e. K, C2), and one (G) found it initially helpful but less so by the semester's end. Generally, participants felt that randomised groups functioned well, even when several reported that they had worked at least once with a group that did not. Consider the following quote by F2, which captures a generally positive view of group work:</p> <p>I haven't really had any group that stands out as really bad or any group that stands out as really good. [...] It's been a constant with the group work. It's always a positive factor for me. I like doing group work. (LSI)</p> <p>Several participants noted that often one or two group members did not contribute, but this did not ruin their group work experience. When confronted with non-contributors, some participants tried to improve the situation by 'put[ting] everyone together' (O, ESI) and getting non-contributors involved by asking questions, like 'What do you think?' (N2, ESI). As N2 (ESI) noted about his question-posing approach: 'I haven't ever been completely shut down before'.</p> <hd id="AN0187890701-34">4.3.1.2. Random selection theme #2: by working in multiple randomly formed groups, participan...</hd> <p>Students in randomly formed groups generally stayed in these groups for only a week. Thus, given that instructors 1, 6, and 7 randomised across many (instructors 6 and 7) or all weeks (instructor 1) of the semester, students worked at least once with many of their peers. The participants indicated multiple positive outcomes. First, students were able to get to know many of their peers. Although it may appear that knowing one's peers is a natural consequence of being in a class (not just one with randomly formed groups), the participants expressed that they had not gotten to know their peers in past classes (O, M, G, F2, E2), even when they were in small classes and there was group work. M (ESI) shared, '[in] high school I felt like I definitely had a bond with my other math kids that I feel a similar bond to the math kids in ((ITP course number))' and C2 (ESI) noted, 'it's not really like a class, it's like a group of friends hanging out and talking and cool math stuff. That's what it feels like'. Getting to know their peers allowed them to find friends (G, K, C2, F2, N2, E2). As G summarised in the ESI: G:[...] I work with my friend. [...]Interviewer:Did you meet when you came to class, or did you know each other before?G:Not known [<emph>sic</emph>] very well, but I was seeing him in my Calculus 2. First semester I remember him, and Calculus 3 I think I've seen him in the class, but I haven't talked to him until I came to this class. And I saw him in class, and I remember him, I remember his face, but I never know [<emph>sic</emph>] him before. Apart from these social benefits, three participants (M, H2, N2) also highlighted the benefit they received from hearing different perspectives. Interviewer:Do you feel like you've learned directly from anybody in class?M:Yes [...] I know that ((Name 1)) usually uses contradiction work and I use direct proofs. His way of doing it is always different. Then ((Name 2)), like I said, just has a completely different way where he brings in other variables if he needs them [...] (LSI) The final two points raised by participants were that changing groups allows students to: (a) 'know how fast, or how slow I am compared to a part of the class' (H2, ESI), and (b) determine 'who I work with better or who I found to be more productive' (N2, LSI).</p> <hd id="AN0187890701-35">4.3.1.3. Random selection theme #3: participants also identified downsides of random selectio...</hd> <p>Despite the participants' generally positive or neutral experience with random selection, several participants also identified downsides. J2, N2, and F2 all noted how in randomly formed groups, language could be a barrier and communication a struggle. (In some classes around half of the students were international students whose first language was not English.) For an in-depth look at two students' experiences with this challenge (i.e. J2, F), see Hwang et al. ([<reflink idref="bib16" id="ref100">16</reflink>]). That said, F2 reported a way she and her group tried to mitigate language issues: '[...] you kind of work through it together on the board so it's more visual, because numbers are universal' (LSI). The following four downsides to random selection were each raised once:</p> <p></p> <ulist> <item> In randomly formed groups, women who are alone in a group with men may feel uncomfortable (M, LSI).</item> <p></p> <item> When (randomly) grouped with friends, students may be less productive due to banter (K, LSI).</item> <p></p> <item> Randomly formed groups do not allow for separation of people who do not work well together (M, LSI).</item> <p></p> <item> Randomly formed groups might be too homogeneous in ability, that is, everyone is stuck at the same time and there is no group member who can help the others get unstuck (J2, LSI).</item> </ulist> <hd id="AN0187890701-36">4.3.1.4. Random selection theme #4: participants identified advantages and disadvantages of r...</hd> <p>Three issues were raised by three different participants regarding randomising groups <emph>each week</emph>.</p> <p></p> <ulist> <item> Due to the frequent changing of groups, one is never stuck with one 'crappy group' and 'you just have to stick it out' (C2, ESI).</item> <p></p> <item> When you are in a group you (do not) like, you (do not) look forward to coming to class for the rest of the week (K, LSI).</item> <p></p> <item> Whereas you get to know more people via frequent randomised grouping, you get to know people on a deeper level in fixed groups (E2, ESI).</item> </ulist> <hd id="AN0187890701-37">4.3.1.5. Random selection theme #5: new interactions and positions in new randomized groups</hd> <p>Different groups worked together in different ways (K, O, M, J2, C2, F2, N2, E2). This seemingly banal observation by many participants highlights an important aspect of frequently changing groups, particularly in contrast to fixed groups: Different groups will interact in different ways and positions will change. In E2's (ESI) words:</p> <p>Sometimes when groups change, you have to have that moment of okay, get used to working with these new people, since some people are more dominant over the group and some are more submissive and you need to figure out where do like I fit into the conversations, that part. And just, in general, how to work together without just saying, "Oh, you figured it out, I'll just get down what you wrote".</p> <p>(As we discuss under 'Student Selection', participants in fixed groups frequently developed fixed patterns of interaction and positions.) In addition to reporting differing group work experiences, several participants (M, J2, F2, E2) delineated how they acted differently in different groups (e.g. M and E2 reported taking a step back when grouped with more dominant personalities). Further, five participants (M, G, F2, N2, E2) reported how their groups had no designated writer (i.e. they rotated who documented the group's progress). Finally, five participants noted that they tried to help and teach others (G, J2, H2, N2, E2) with G and E2 also noting that they sometimes asked others for help. No participant who spoke to us about randomised groups reported being stuck (or seeing others stuck) in a position of help-giving or help-receiving.[<reflink idref="bib7" id="ref101">7</reflink>]</p> <hd id="AN0187890701-38">4.3.2. Student selection</hd> <p>For student selection, we focus on four themes based on the reported experiences of the 18 participants in the classes of instructors 2, 3, 4, 5, and 8. Similar to the previous section, we identified broad (student selection) themes since participants did not mention group formation constraints. (We presume that common constraints, such as 'work with your neighbors', are not perceived as [very] constraining.)</p> <hd id="AN0187890701-39">4.3.2.1. Student selection theme #1: participants offered a mixed picture of group work exper...</hd> <p>When instructors used student selection to form groups, some participants reported liking group work (L, C, H, F, G2, K2, I2, P2), some felt neutral or had mixed feelings (B, A, D, L2, A2, D2), and some disliked group work (J, E, I, N). All of the participants who disliked group work (as well as some of the participants with mixed feelings) were unhappy with the pace of their group, identifying the group's pace as too fast (E, I, N, D), too slow (J), or never right (D2). As we explore in Student Selection Theme #3 below, many students who disliked or had mixed feelings toward group work developed ways of mostly working alone. We offer this first theme in contrast to Random Selection Theme #1, which documented how participants felt positive or neutral about group work and where pace was not raised as an issue.</p> <hd id="AN0187890701-40">4.3.2.2. Student selection theme #2: student selection typically led to participants working...</hd> <p>None of the 18 participants reported having friends in the class when they entered it. Thus, many participants – especially from classes using student selection at the start – shared that they formed groups with their neighbours (I, A, L2, A2, D2, G2, K2, I2, P2). The only participant who discussed a different group formation approach was participant L, who noted that he formed a group with friends he had made during the early-semester instructor-made groups.</p> <p>After students were invited to form their own groups, those groups mostly remained intact, plus or minus losing (or gaining) a member from students dropping from (or into) the course. Switching one's group after groups had 'settled' was rare, and we only have two participants' stories to that effect (B, F). B switched groups several times at the start of the semester to be in groups with a smaller number of members. F, in the second week of classes, moved out of a group of Mandarin-speakers because they communicated in Mandarin, which she did not speak. She described this as a 'risky move that I felt stressed out about' (LSI) (see Hwang et al., [<reflink idref="bib16" id="ref102">16</reflink>] for more details). That said, even B and F eventually settled into permanent groups. Only two participants (sometimes) worked completely alone, isolating themselves from their peers (J, L2) (see Student Selection Theme #3).</p> <hd id="AN0187890701-41">4.3.2.3. Student selection theme #3: when given the option, some participants chose to work a...</hd> <p>Of the 18 participants, ten participants described working alone in some way: Five (J, E, I, L2, D2) described working alone (although three of them sat with a group), and a further five (A, D, A2, G2, P2) described their group work as sitting next to their peers working alone with an occasional check-in.</p> <p>Of the five participants who described working alone, three (J, I, D2) cited incompatible group work speeds. J noted in both interviews that he preferred to work alone so as not to be slowed down (e.g. by having to teach his peers). In a similar vein, he also shared that: (a) other people talking was distracting, and (b) he did not really get stuck and therefore did not need others' help. On the other hand, participants I and D2 cited experiences of being in overly quick groups as reasons for preferring to work alone. D2 also shared experiences of being in groups that were too slow and concluded that all her groups were either too slow or too fast, placing her in the 'leave-me-in-peace-I-want-to-work-alone camp' (LSI). Participant E chose to work by herself to use the group work time in a way she found more productive: rewriting and looking through her notes as well as working through problems. Finally, L2 was often observed working alone (sometimes while 'part of' a group) and shared that although he neither liked nor disliked group work, he often preferred working on his own. Participants E, I, and D2 (and sometimes L2) all sat with a group while working alone.</p> <p>G2's group collaborated in a style he described as 'three people, sitting in [<emph>sic</emph>] desks, who're working completely separately' (ESI) with the occasional check-in. A similar approach was taken by D's, A's, A2's, and P2's groups. But whereas D felt less inclined to show up to class on group work days and A found this group work unnecessary, A2 and P2 liked this form of group work. In short, even when 'with' a group, several participants worked mostly on their own.</p> <hd id="AN0187890701-42">4.3.2.4. Student Selection theme #4: fixed interactions and positions in student-selected gro...</hd> <p>As discussed in Student Selection Theme #3, of the 18 participants, ten described a very fixed way of interacting: working (mostly) alone – two sat alone, eight sat with a group and had the occasional check-in. Of these ten, several discussed fixed positions they saw for peers and themselves. In particular, three discussed being slow(er) and/or in a help-receiving position (I, D, P2), and four (J, A, L2, G2) described being fast(er) and/or in a help-giving position. The other three described more in-between positions, with E describing her understanding as being 'in the middle' (ESI), A2 describing that her group was not dominated by anyone (LSI), and D2 reporting that her groups were sometimes faster and sometimes slower than she was (LSI).</p> <p>Even when groups were more interactive, fixed patterns of interaction developed. Consider, for example, L's report that his group members would take turns solving problems with him always starting, followed by his male, and then his female group member. (The fourth person who later joined would go last.) Furthermore, even in more interactive groups, participants reported fixed positions of who was (and was not) smart and help-giving (B, F, I2). Consider, for example, B's words:</p> <p>For my group, I'm more just listening and learning [...] I have no idea what to do. I'm more or less confused, so my group members usually take it on. [...] For me, my group members are almost like my second teachers. (LSI)</p> <p>Of the 18 participants, only three indicated that to some extent group interactions and positions changed over time:</p> <p></p> <ulist> <item> H (LSI) reported that depending on the day, a group leader may (or may not) emerge.</item> <p></p> <item> F (LSI) noted that although she was the group's writer for some time, the group eventually began to distribute writing duties.</item> <p></p> <item> K2 (LSI) described that giving and receiving help from his group seemed balanced, with his role (as help-giver or -receiver) depending on the content.</item> </ulist> <hd id="AN0187890701-43">4.3.3. Instructor selection</hd> <p>We observed three forms of instructor selection: (a) instructors 2 and 4 grouped students with their neighbours (i.e. <emph>instructor selection (without student input)</emph>), (b) instructor 5's teaching assistant used his knowledge of the students to form groups (i.e. <emph>instructor selection (without student input)</emph>), and (c) instructors 6 and 7 sent out surveys to solicit students' input to create groups after students had gained experience working with several of their peers (i.e. <emph>instructor selection (with student input)</emph>).</p> <p>We have little to report on <emph>instructor selection (without student input)</emph>. Mainly, the participants from instructor 2 and 4's classes confirmed that their instructors grouped students with their neighbours. Two participants shared how this led to unfortunate groupings. Participant I reported sometimes being grouped with people who always believe they are right and do not listen (and our observations suggest she was often in a group with a dominant male student). And F reported being stressed and worried about being grouped with students who communicated in Mandarin, which she did not speak.</p> <p>The participants from instructor 5's class (L2, A2, D2) also had little to say about <emph>instructor selection (without student input)</emph>. Only D2 noted in the LSI that it was 'kind of nice' how her instructor formed groups based on whom his teaching assistant thought might work well together because 'we were kind of carefully arranged with people who would be interested' and it was 'a good math experience'. That said, D2 acknowledged that two of the five group members did not participate. (Our observations show that the two international students in the group sat quietly at the edge of the group, and there was no communication between them and D2 or the other two domestic students.)</p> <p>Participants had more to say about instructors 6 and 7's grouping using students' preferences (i.e. <emph>instructor selection (with student input)</emph>), which led to the identification of the following two themes.</p> <hd id="AN0187890701-44">4.3.3.1. Instructor selection (with student input) theme #1: not all participants submitted g...</hd> <p>Of the seven participants from instructor 6 and 7's classes, J2, N2, and E2 spoke of submitting group work preferences to their instructor. J2 asked to work with fellow Mandarin-speakers with whom she harmonised, N2 asked to work with people he was productive and comfortable with, and E2 asked to work with her friends. B2 and F2 did not discuss whether they had submitted preferences, but F2's interview suggests she was grouped with friends. The only participant who discussed not submitting preferences was H2, who shared that he had no preferences and therefore had not submitted any. (C2 did not complete the LSI, and we therefore do not know whether [or which] group preferences she submitted.)</p> <hd id="AN0187890701-45">4.3.3.2. Instructor selection (with student input) theme #2: being in a permanent group creat...</hd> <p>Of the six participants from instructor 6 and 7's classes who completed the LSI, four shared positive feelings about their permanent group (J2, H2, F2, N2) and two shared mixed feelings (B2, E2). The former four explained that: (a) 'it's much better now' (J2, LSI) because she can now communicate effectively and work faster with her group in Mandarin, (b) despite not submitting preferences, 'randomly, each of the member [<emph>sic</emph>] in my group right now kind of works well together' (H2, LSI), (c) she is working with friends (F2), and (d) he is working with people of the 'general same knowledge level' (N2, LSI) with whom he is comfortable and productive.</p> <p>On the other hand, E2 shared that even though she was in a group with people she felt comfortable around, her friends positioned her as 'the smart one' (LSI), which she did not like: 'I don't mind helping out, but it's also like, I don't want to feel like I'm taking on the responsibility. I shouldn't be teaching them. It's kind of mixed feelings there' (LSI). Also offering some mixed feelings, B2 debated with himself in his LSI, 'Maybe it was a little bit more enjoyable before the permanent groups, but ... not really'.</p> <hd id="AN0187890701-46">5. Discussion</hd> <p>Below, we offer reflections in response to our research questions and results and then consider the relevance of the ITP and university context.</p> <hd id="AN0187890701-47">5.1. Instructor use of group formation methods</hd> <p>Even though the instructors we observed used a variety of group formation methods, they most frequently used student selection methods, allowing their students considerable say in whom they worked with. And although, as our literature review revealed, there is no research consensus on the best (combination of) group formation method(s), there is a consensus that student selection is rife with problems. Thus, the preponderance of student selection methods – particularly in the last third of the semester – is concerning. Anecdotally, we observed that in classes of instructors who leaned toward <emph>student selection (without constraints)</emph> – à la 'get into some groups' – students frequently got 'stuck' in groups of peers who sat close to them. These groups would often drift apart and stop working together, becoming groups only by virtue of their physical proximity – if that. This observation is in alignment with Student Selection Themes #2 and #3. The two departures from this pattern were the instructor who always used <emph>random (public) selection</emph> and two instructors who used <emph>random (private) selection</emph> before switching to <emph>instructor selection (with student input)</emph>. The former's practice aligns with Liljedahl's ([<reflink idref="bib21" id="ref103">21</reflink>]) recommendation, whereas the latter's is consistent with Reinholz's ([<reflink idref="bib26" id="ref104">26</reflink>]) suggestion. Since these recommendations come from different contexts – K–12 and undergraduate education, respectively – we reflect later in the Discussion on what the differences in context might mean for the suitability of different group formation methods.</p> <hd id="AN0187890701-48">5.2. Student experience of group formation methods</hd> <p>Although we cannot (and do not) claim that the group formation method was the cause, it stood out to us that whereas participants who spoke about random selection reported positive or neutral group work experiences, participants who spoke about student selection offered a much more mixed picture, with several stating their dislike for group work. Less ambiguously, as advertised by the research literature, participants who spoke about random selection reported several positive consequences: getting to know their peers, learning from different perspectives, and understanding their position in the class. That said, participants also raised important problems with random selection that connect to its inflexibility and possible unfortunate group compositions: groups in which language is a barrier, groups with just one female member who feels uncomfortable, unproductive groups of friends, unproductive groups of people who do not work well together, and groups too homogeneous in ability.</p> <p>Furthermore, participants also discussed benefits and downsides of randomising groups <emph>each week</emph>, pointing out that: one is never stuck in a 'crappy group' for too long, one does (not) look forward to coming to class when one does (not) work with a group one likes, and although one does get to meet more people, one does not get to know them on a deeper level. The frequent changing of groups due to weekly randomising stands in contrast to what participants reported about student selection. Participants who spoke to us about student selection painted a picture of working mainly with their neighbours for the semester, subject only to minor adjustments. Further, a majority of participants who spoke about student selection methods described working either completely alone or alone with an occasional group check-in. In short, student selection appeared to lead to static groups of neighbours who often ended up looking but not acting like groups. Finally, we observed that whereas participants who spoke to us about random selection methods reported different group interactions and changing positions in their different groups, participants who spoke to us about student selection described more fixed ways of interacting and fixed positions in their fixed groups.</p> <p>Although only a few participants spoke to us about instructor selection, we learned that not all participants submitted group preferences to their instructor, but that those who did asked to work with peers with whom they were comfortable (e.g. due to productivity, due to a lack of language barriers, due to friendship). That said, working with the people one requested is not a guarantee for happiness, as E2's story suggests. Being in a fixed group with her friends, she experienced being in the fixed position of the group's 'smart one', a position in which she was not comfortable.</p> <hd id="AN0187890701-49">5.3. On the importance of the ITP context</hd> <p>Important differences between K–12 and undergraduate education may shape the suitability of different group formation methods. For example, one significant difference in context is the proportions of male and female students in the classroom.[<reflink idref="bib8" id="ref105">8</reflink>] Most university mathematics classrooms continue to contain a (large) majority of men as opposed to (non-single-gender) K–12 classrooms, in which male and female students typically exist in roughly equal numbers. Thus, in the K–12 context, <emph>random (public) selection</emph> – as advertised by Liljedahl ([<reflink idref="bib21" id="ref106">21</reflink>]) – is likely to lead to only a few groups with isolated female students. On the other hand, in the undergraduate context, <emph>random (public) selection</emph> is likely to lead to many groups of three or four with at most one female student. The latter is likely problematic: Heller and Hollabaugh ([<reflink idref="bib14" id="ref107">14</reflink>]), for instance, observed how female students who found themselves alone in a group with male students were often ignored.</p> <p>Second, the instructor–student and student–student relationships are typically different in the K–12 and the undergraduate context. Whereas in most K–12 classrooms, many students (and the instructor) are likely to have known each for years, most students (and the instructor) in a university classroom may not know each other prior to the first day of class. Thus, although Cohen and Lotan's ([<reflink idref="bib6" id="ref108">6</reflink>]) suggestion that teachers use their knowledge of the students to form groups (e.g. by separating fighting students) may be feasible in the K–12 context, this recommendation is likely infeasible [...] in an undergraduate classroom.</p> <p>Third, many university mathematics students are young adults, who are beginning to be independent and take on adult responsibilities, whereas K–12 students are not. Some researchers (e.g. Knowles et al., [<reflink idref="bib18" id="ref109">18</reflink>]) have posited that adult learners learn differently than child learners. 'Although the arguments [that adults and children are differently motivated to learn] no longer seem quite so clear' (Taylor & Hamdy, [<reflink idref="bib33" id="ref110">33</reflink>], p. 1563), we wonder if possible andragogic characteristics (e.g. greater autonomy, past experiences [with group work during school], intrinsic motivation to learn [Knowles et al., [<reflink idref="bib18" id="ref111">18</reflink>]]) change undergraduate students' attitude toward and behaviour in group work. Further, in school, where students spend several years with familiar classmates, there may be a greater incentive to contribute to a positive group work experience. In university, however, one's situation typically 'resets' each semester with new courses and often new people. Thus, we suspect that undergraduate students' andragogic characteristics and the comparatively short duration of undergraduate courses may increase the likelihood of students not participating in group work if given the choice (e.g. when the instructor uses a student selection method and does not enforce group membership). <emph>If</emph> instructors wanted to stop students from working alone, they could do so through purposeful use of group formation methods (e.g. by not using student selection methods or by taking into account students' preferences). Yet, instructors might also want to retain the possibility of students working alone, for example, for some students with autism, who may experience difficulties working in groups (Gurbuz et al., [<reflink idref="bib12" id="ref112">12</reflink>]).</p> <p>Finally, in addition to the three broad differences between K–12 and undergraduate education addressed above, we want to share three ways in which participants linked their positive group work experience to aspects of the ITP course. First, G, D, and I2 noted that group work was genuinely helpful in an ITP course (where the focus was on constructing arguments), whereas in previous calculation-heavy courses, group work was neither helpful nor necessary. Second, C2, L2, and G2 observed that the mindset of ITP students was different: They were more likely than Calculus and high school students to be motivated and serious about the work – and therefore willing to listen and discuss. Last, I2 shared that not receiving a grade for the group work in the class (and the group work's focus being on understanding the material) made a huge (positive) difference for her.</p> <hd id="AN0187890701-50">6. Conclusion</hd> <p>By observing eight instructors' use of group formation methods across a semester, we generated a refinement of Hagelgans et al.'s ([<reflink idref="bib13" id="ref113">13</reflink>]) classification of group formation methods. Using our refined classification, we reported findings about the group formation methods used across the semester and, like the CLUME survey (Baker & Hagelgans, [<reflink idref="bib2" id="ref114">2</reflink>]), found student selection methods to be most widely used. Further, we demonstrated that research on group formation methods is divided on (when) which method is most appropriate. As underscored by the participants' reported experiences, each group formation method has its advantages and disadvantages that vary across the semester and context. Nevertheless, the literature and our findings suggest that student selection is fraught with problems, whereas (frequent) random selection offers many benefits.</p> <p>If asked how we would form groups in our contexts, we would respond that we are currently most convinced by the arguments for letting students meet many peers at the start of the semester via <emph>pseudo-random (private) selection</emph> (pseudo-randomised to avoid known group composition issues) before moving to semi-stable groups via <emph>instructor selection (with student input)</emph>. Thus, at the start of the semester, students get to experience many of the benefits of random selection (e.g. meeting many of their peers) while some of its downsides are mitigated (e.g. avoiding one woman in a group with several men). Later, once students are familiar with their peers, students get to provide the instructor with their input and join more stable – and, ideally, harmonious and productive – groups. Since issues can still arise (see E2's reports), the groups should be thought of as semi-stable (Reinholz, [<reflink idref="bib25" id="ref115">25</reflink>]), that is, groups can be changed again every few weeks to respond to problems reported by students, to address issues observed by the instructor, and to mitigate students being fixed in positions. Anticipating, observing, and responding to group issues will require instructors to monitor how groups are working.</p> <p>We acknowledge that initial <emph>pseudo-random (private) selection</emph> followed by <emph>instructor selection (with student input)</emph> is not flawless, and it should not be misunderstood as our unequivocal recommendation – the arguments for and against other methods are context-dependent. Instead, we recommend that instructors carefully discuss with their (local) peers the logic and consequences of using different group formation methods in <emph>their</emph> context.</p> <hd id="AN0187890701-51">Acknowledgments</hd> <p>The authors gratefully acknowledge the data collection contributions of Sofía Abreu, Younggon Bae, Sarah Castle, and Robert Elmore, as well as the support offered by Mariana Levin and Shiv S. Karunakaran.</p> <hd id="AN0187890701-52">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <p>Correction Statement</p> <p>This article was originally published with errors, which have now been corrected in the online version. Please see Correction (<ulink href="http://dx.doi.org/10.1080/0020739X.2024.XXXXXX">http://dx.doi.org/10.1080/0020739X.2024.XXXXXX</ulink>)</p> <hd id="AN0187890701-53">Appendices</hd> <p></p> <hd id="AN0187890701-54">Appendix A</hd> <p></p> <hd id="AN0187890701-55">OBSERVATION TOOL – INTRODUCTION TO PROOF COURSE</hd> <p>Date of Observation: Section/Instructor: Observer:</p> <p> <bold>Objective:</bold> Early in the semester, we are particularly interested in how instructors describe the course as different from those experienced previously by students. Throughout, we are interested in how instructors present proof and proving work, generally (when and if that happens) and in the context of specific tasks. We are also interested in all questions posed in the classroom (from instructors to students, from students to instructors, and from students to students) and in how students participate tasks and classroom activities, especially group work.</p> <p> <bold>Instructions to Observers</bold>: Fill in the Lesson Summary using a new row for each 'new' activity in the lesson, adding space as added in the row as needed. You can and should judge what seems a 'new' instructional activity. While the lesson is unfolding, the two most important columns are 'time' and 'classroom activity and dialogue'. There may be times when it is hard to keep up. When instructors and students ask questions, do your best to capture the gist of each question; exact wording is great, but not expected. When students pose questions or make statements, indicate them with 'S:' (unless you know the identity of the student) to distinguish them from the likely more common instructor talk.</p> <p>At the end of the lesson (as soon as possible), examine the list of specific foci and add in description where it fits. Some foci are more relevant to 'lecture days' and some more relevant to 'small group work days'.</p> <hd id="AN0187890701-56">Lesson Summary</hd> <p></p> <p> <ephtml> <table><tbody><tr><td>Time (start; end)</td><td>Task/Activity</td><td>Classroom Activity and Dialogue (what the instructors and students say and do)</td></tr></tbody></table> </ephtml> </p> <p>Specific Foci:</p> <p></p> <ulist> <item> Did the instructor characterise the nature of the work in the course – what students had to do to be successful? If so, describe the gist of what he/she said.</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> Did the instructor describe the mathematical content of the course – either in terms of topics or argument? If so, how? The major course topics are represented diagrammatically on page 2 of the syllabus.</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> How did the instructor organise small group work in this session? What did she/he say (rationale for the organisation scheme) and do (to get students into groups)? Include any general statements about the purpose of group work that do not seem to fit above in 'nature of the work'.</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> Describe the instructor's pattern of questioning (students)? Were there sufficient questions to indicate a pattern? If not, describe the type of questions posed? Indicate if today's class was mostly lecture or mostly group work.</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> Describe the character of students' questions posed to the instructor. For example, were they pragmatic/logistical or mathematical in nature? Did students ever ask follow-up questions after the instructor responded to their initial question?</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> If substantial time was spent working in small groups, characterise the participation and powerdistribution in the groups you observed (without entering any group). Does one or more person appear to direct the group? Do some groups appear to work mostly independently (while sitting near each other)? When groups work collaboratively, how do they work (e.g. discuss the approach first, all work a bit and then confer about success)?</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> How does the instructor characterise or evaluate students' work in class, either to praise or critique, individually or collectively as a class, or to comment on specific solutions to tasks, from homework or presentations in class?</item> </ulist> <p>[Response]</p> <p></p> <ulist> <item> If there are audible interactions between instructor and students before or after class, what is the character of those interactions? Are they pragmatic/logistical or more mathematical in nature?</item> </ulist> <p>[Response]</p> <p>Seating chart (optional):</p> <hd id="AN0187890701-57">Appendix B</hd> <p>Table B1. Observed instructors' group formation method use.</p> <p> <ephtml> <table><tbody><tr><td><graphic href="tmes_a_2369651_ilg0001.gif" content-type="Graph" /></td></tr></tbody></table> </ephtml> </p> <p>4 Note<emph>.</emph> This table shows the group formation methods used by the eight observed instructors over the course of a semester. The seven observed group formation methods are abbreviated as: R (Pb) = Random (PuBlic); R (Pr) = Random (PRivate); PR (Pr) = Pseudo-Random (PRivate); I (I) = Instructor selection (with student Input); I (woI) = Instructor selection (WithOut student Input); S (C) = Student selection (with Constraints); and S (woC) = Student selection (WithOut Constraints). Days without group work are indicated by 'n/a'. To distinguish between instructors who, for example, randomised groups every class and those who randomised groups once at the start of the semester and then continually told groups to get into the same group as last time, we underlined table entries if groups were newly formed on that day. We also underlined table entries if groups had been newly formed since our previous observation. To capture the uneven rate of our observations (i.e. observing instructors more at the start of the semester), the thicker vertical bars represent the ends of the first and second thirds of the semester. Random selection methods are presented in darkest gray, pseudo-random selection methods in dark gray, instructor selection methods in middle gray, and student selection methods in light gray.</p> <ref id="AN0187890701-58"> <title> Notes </title> <blist> <bibl id="bib1" idref="ref9" type="bt">1</bibl> <bibtext> One benefit of pseudo-random selection with private modifications is students' ability to privately offer grouping suggestions to the instructor, for instance, if they do not feel safe working with a particular student. The instructor can then make these modifications before class (or during class with, for example, a rigged Excel sheet). Despite being pseudo-random, the instructor may present the selection method as random.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref63" type="bt">2</bibl> <bibtext> Liljedahl ([19]) posited that some students who wish to work with their friends may cease collaborating when placed in a group of non-friends by their teacher.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref43" type="bt">3</bibl> <bibtext> All necessary approvals were obtained from the Human Research Protection Program at the institution at which the research was conducted.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref5" type="bt">4</bibl> <bibtext> In most classes, some students arrived late, and instructors would tell them to join a particular group. Thus, we were confronted with the question of whether to, for example, change a coding from <emph>random (public) selection</emph> to <emph>pseudo-random (public) selection</emph>. We chose not to make this change and, instead, coded for instructors' intended group formation method. Further, sometimes students worked together even though the instructor had not asked students to work together or had asked students not to work together. We did not code this as group work.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref10" type="bt">5</bibl> <bibtext> Note that we are not using 'random' and 'pseudo-random' in the way statisticians use these terms. For instance, whereas Excel's random number generating functions are considered pseudo-random by statisticians, we would consider using Excel's randomisation function to compose groups a random selection method. For us, what makes a group formation method pseudo-random is if the method has some semblance of randomness with the instructor still conceivably manipulating groups (e.g. grouping students by height which is more likely to create homogeneously gendered groups).</bibtext> </blist> <blist> <bibl id="bib6" idref="ref33" type="bt">6</bibl> <bibtext> In different countries, the birth rate peaks at different points of the year. This is also true for the two countries that most of the students originated from: the U.S.A. and China. Thus, the likelihood – however small – was increased that a U.S. student would work with another U.S. student and a Chinese student would work with another Chinese student. Although this increase in probability is likely negligible, we choose to include selection by birthday as a pseudo-random method to highlight how seemingly random methods may not actually be random.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref11" type="bt">7</bibl> <bibtext> We wish to acknowledge that although changing groups each week may have led to changing interactions and positions, dominant personalities may have still found a way to dominate their groups. Group work norms such as determining the group's writer via 'first come, first served' (e.g. N2) may still benefit dominant personalities.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref1" type="bt">8</bibl> <bibtext> As many surveys of university mathematics (e.g. the College Board of the Mathematical Sciences [CBMS] surveys, the Characteristics of Successful Programs in College Calculus [CSPCC] survey) have presented gender as a binary category, we do not know the proportion of non-binary students in U.S. university mathematics classrooms. Further, we know very little about the experience of non-binary students during group work. We do have reports, however, of binary gender being used as a group forming criterion, which devalues the experiences of non-binary students (Francis & Monakali, [10]).</bibtext> </blist> </ref> <ref id="AN0187890701-59"> <title> References </title> <blist> <bibtext> Armstrong, N., Chang, S.-M., & Brickman, M. (2007). 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  Data: How University Mathematics Instructors Form Groups and How Students Experience Them
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  Data: <searchLink fieldCode="AR" term="%22Valentin+A%2E+B%2E+Küchle%22">Valentin A. B. Küchle</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-2997-7128">0000-0002-2997-7128</externalLink>)<br /><searchLink fieldCode="AR" term="%22John+P%2E+Smith+III%22">John P. Smith III</searchLink><br /><searchLink fieldCode="AR" term="%22Jihye+Hwang%22">Jihye Hwang</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-7075-4508">0000-0002-7075-4508</externalLink>)<br /><searchLink fieldCode="AR" term="%22Reshma+Menon%22">Reshma Menon</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2025 56(9):1804-1831.
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  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Data: 28
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  Data: 2025
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  Data: National Science Foundation (NSF)
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  Data: <searchLink fieldCode="DE" term="%22College+Faculty%22">College Faculty</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Teachers%22">Mathematics Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Grouping+%28Instructional+Purposes%29%22">Grouping (Instructional Purposes)</searchLink><br /><searchLink fieldCode="DE" term="%22Ability+Grouping%22">Ability Grouping</searchLink><br /><searchLink fieldCode="DE" term="%22Homogeneous+Grouping%22">Homogeneous Grouping</searchLink><br /><searchLink fieldCode="DE" term="%22Heterogeneous+Grouping%22">Heterogeneous Grouping</searchLink><br /><searchLink fieldCode="DE" term="%22Decision+Making%22">Decision Making</searchLink><br /><searchLink fieldCode="DE" term="%22Introductory+Courses%22">Introductory Courses</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Active+Learning%22">Active Learning</searchLink><br /><searchLink fieldCode="DE" term="%22Classification%22">Classification</searchLink>
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  Data: 10.1080/0020739X.2024.2369651
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  Data: 0020-739X<br />1464-5211
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  Data: 'Group work' is a vague description of an instructional activity, because many factors shape its character and effect on students. One important factor is group formation, that is, how groups are formed by the instructor. In this paper, we sought to better understand the variation of group work with respect to group formation by addressing: How do university mathematics instructors form groups over the course of a semester? To this end, we examined eight instructors' methods of forming groups in one multi-section introduction to proof course. Our findings include a classification of group formation methods and descriptions of how instructors varied their formation methods across the semester. Further, we sought to understand: How do students experience different group formation methods? We analysed interviews with 29 students from the eight instructors' classes and identified central themes among students' experiences of different group formation methods. Finally, we discuss the sometimes conflicting research on (when) which group formation method is most appropriate and offer our thoughts on how the differences between typical undergraduate and K-12 mathematics classrooms may contribute to different recommendations.
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  Data: 2025
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        StartPage: 1804
    Subjects:
      – SubjectFull: College Faculty
        Type: general
      – SubjectFull: Mathematics Teachers
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