Moving Away from Lecture in Undergraduate Mathematics

Moving Away from Lecture in UndergraduateMathematics: Managing Tensionswithin a Coordinated Inquiry-Based Linear AlgebraCourse

Vilma Mesa1 & Mollee Shultz1 & Ashley Jackson1

# Springer Nature Switzerland AG 2019

AbstractThis study describes how nine university professors managed their teaching when ashift from lecturing to inquiry-based learning was mandated in a large enrollmentcourse, linear algebra for math majors. We describe the tensions that emerged andhow they were resolved, in part via the production of worksheets that were used inteaching. We describe the way in which professional obligations towards the disciplineand towards the institution shaped the tensions and the ways in which they wereresolved by this group of faculty. We offer a conceptualization of the enactment ofinstruction that was prompted by the study and implications for continued investigationof change strategies in undergraduate mathematics.

Keywords Linear algebra . Inquiry-based learning . Professional obligations

Promoting change in instructional practices in undergraduate mathematics educationhas been an important concern for over three decades, in part because of reports thatblamed math teaching as a culprit in students’ decisions to leave STEM fields(Seymour 1995, 2002; Seymour and Hewitt 1997). In a review of nearly 200 paperspublished over a 15-years period on how to promote change in instructional practices inundergraduate science, technology, engineering, and mathematics, Henderson et al.(2011) found that the typical approach of “developing and testing ‘best practice’curricular materials and then making these materials available to other faculty” (p.

https://doi.org/10.1007/s40753-019-00109-1

This work has been conducted with support from the Educational Advancement Foundation and theUniversity of Michigan, through a Transforming Learning for Third Century (TLTC) Quick Wins/Discoveryaward.

* Vilma [email protected]

1 School of Education University of Michigan, 3119, 610 East University, Ann Arbor,Michigan 48109-1259, USA

International Journal of Research in Undergraduate Mathematics Education (2020) 6:245–278

Published online: 26 December 2019

http://crossmark.crossref.org/dialog/?doi=10.1007/s40753-019-00109-1&domain=pdf

http://orcid.org/0000-0003-0460-1850

mailto:[email protected]

952) was ineffective in generating changes; the availability of materials was not byitself sufficient to spur all people into action. Likewise, they found that the secondtypical approach, “‘top-down’ policy-making meant to influence instructional prac-tices” (p. 952) was equally ineffective as it generates conflicts that threaten thecollegiality that is so necessary for departments to function well, when change is agoal. Instead, they found that the most effective change strategies were aligned with, orsought to modify, the beliefs of the individuals who were the target of the change byinvolving them over a long-term process that promoted an understanding of theircontext (e.g., department, university) as a complex system and that ensured the changeswere fully compatible with that context.

What does a change process look like? What challenges emerge and how do facultynavigate them when a radical instructional change is mandated in a department? Wehad the opportunity to witness this process when a department decided that the linearalgebra course intended for their math majors should be taught through an inquiryapproach instead of lecturing. After a year of preparation, the department assigned agroup of nine instructors, with various degrees of experience with the approach both asstudents and as instructors, to teach 12 sections of the course (~200 students) with anew textbook. We report on the tensions that instructors had implementing the newchange. Knowing what happens when a department seeks to change instructionalpractices of a foundational mathematics course such as linear algebra can shed lightinto what such a change entails. The study primarily contributes to this literature.

The paper is organized into five sections. We begin with the theoretical foundationsthat support the study followed by a review of relevant literature. After describing thecontext for the study and the methods for gathering and analyzing data, we present themajor findings. We conclude with a discussion and implications for future research.

Theoretical Foundations

We assume that when teaching, people enact either the role of instructor or the role ofstudent, and that they seek to fulfill specific goals, to teach a given content and toascertain that such teaching has happened (in the case of the teacher) and to learn thatcontent and to engage in activities that demonstrate that learning has indeed occurred(in the case of the students). The three-way interactions between teacher, students, andthe content (Cohen et al. 2003), are seen as a system in which the people in each rolerespond to obligations imposed by their specific roles, obligations that are not simplypersonal, but part of the social contexts inherent to the profession of mathematicsinstructor. Reinholz and Apkarian (2018) have noted that, for systemic change inSTEM departments to occur, the people enacting the change must be supported bythe structures (e.g., practices and norms) of the department, and be given the power toenact those changes. This system is bounded by the content to teach and learn and bythe institution where instruction takes place.

Whenever an instructional change is proposed, especially when it is advocated byprofessional organizations (e.g., Association of Mathematics Teacher Educators 2017;Australian Association of Mathematics Teachers 2006; Leinwand 2014), instructorsimplementing the change are put into what Herbst (2006) calls a “double bind” asituation in which instructors need to make a decision about a course of action by


choosing between two competing possibilities, each of which has potentially undesir-able outcome. We found Herbst’s (2006) sense of double bind useful to describetensions that emerge in a situation that requires choosing between two competingalternatives, and whose resolution is not necessarily desirable, no matter which alter-native is chosen.1 Herbst and Chazan (2011) proposed that in making decisionsregarding a course of action to resolve these tensions, instructors are continuouslyweighing four types of obligations: towards the knowledge developed by the discipline(the disciplinary obligation), towards the institution where teaching happens (theinstitutional obligation), towards the group of students who share common resourcesin the class (e.g., space and time; the interpersonal obligation), and towards theindividual student as a learner and human being who brings special learning andemotional needs (the individual obligation). Herbst and colleagues (Herbst andChazan 2011, 2015; Herbst et al. 2011) have argued that these obligations are tacit;they become evident only when instructors are confronted with situations in whichimplicit norms of the situation are breached (Herbst and Chazan 2015; Mesa and Herbst2011). This is because norms are regulated by those professional obligations. Wheneveran instructional change is promoted, breaches to norms of how things are done in anygiven department will occur.

There is evidence that professional obligations are central in the decision-makingprocess of post-secondary instructions when they are confronted with a situation in theclassroom that calls for enacting a particular change (Lande and Mesa 2016; Mesa2014; Shultz and Herbst 2017). Lande and Mesa (2016) studied differences in thediscourse of part-time versus full-time community college faculty while Shultz andHerbst (2017) investigated differences between the discourse of graduate studentinstructors versus that of full-time faculty at two universities. Both studies found thatprofessional obligations were relevant to the decisions they made and how agency wasexpressed, and that part-time instructors and graduate students expressed agency lessfrequently than full-time faculty. Mesa (2014), using video- and artifact-based inter-views with community college instructors teaching trigonometry, identified instructors’actions as being limited by a perceived responsibility to institutional requirements (e.g.,to cover the syllabus) rather than individual obligations (e.g., attending to student’sprior knowledge). Thus, whenever an instructional change is promoted we anticipatethat its implementation will be shaped by the instructors’ professional obligations.

For example, the disciplinary obligation, which, refers to the expectation thatteachers will represent the knowledge and practice of mathematics appropriately mayinclude the responsibility of checking the mathematical quality and appropriateness oftasks, textbooks, or other resources provided to students. In contrast, the institutionalobligation, which refers to the expectation that teachers will fulfill their role as a part oflarger organizations (e.g., the department, the university, the professional associations),involves considering regimes such as official pedagogies (e.g., lecturing or teachingwith IBL), policies (e.g., offering office hours or submitting timely grades), andassessments (e.g., number, duration, and frequency of examinations)—which exist

1 The notion of double bind was first proposed by Bateson (1973) to describe conflicting messages incommunication patterns between Bali mothers and children. Mellin-Olsen (1987, 1991) introduced the notionin mathematics education, also in the sense of conflicting messages tied to mathematical situations, specificallyregarding how explicit is the control over aspects of problem solving activity, specifically its goals and theavailability of resources. In this investigation we follow Herbst’s (2006) sense of double-bind.

International Journal of Research in Undergraduate Mathematics Education (2020) 6:245–278 247

regardless of teachers’ individual preferences (Chazan et al. 2016). Thus, the obligationto “oversee and question the quality of the representation of the discipline offered bythe curriculum” (p. 1967) is disciplinary, while the obligation to follow the givencurriculum of a course (with the assigned textbook, the pacing chart, and syllabus) isinstitutional. The individual obligation refers to an expectation that the instructor willattend to the needs of each student. When a student looks distressed when working on aproblem, the instructor’s obligation will require the instructor to seek to understand thenature of the distress (e.g., confusion, lack of knowledge, anxiety) and respondaccordingly. Assuming that a student may have experienced failure before in othermath classes, may lead an instructor into asking the student guiding questions that thestudent can answer to build that confidence, instead of pushing the student intostruggling with figuring out the solution because that may cause anxiety and affectthe student’s self-esteem. The interpersonal obligation refers that the expectation thatthe instructor attends to the need of the group of students as a whole: when individualsare in a classroom, they need to be aware that they are sharing resources: time andphysical “and symbolic space in socially and culturally appropriate ways” (Herbst andChazan 2012, p. 610). Thus, if students are supposed to be working in groups to workout solutions to problems, it may not be appropriate to blurt out the solutions whenother groups have not yet figured out a solution; the instructor will need to call on thestudent’s behavior in a way that both upholds the norms of sharing and keeps thestudent engaged in the work. For any given instructor, their individual interpretations ofwhat a particular obligation implies will be tied to their specific situation, and among agroup of instructors they could vary widely.

Literature Review

By attending only to the etymology of the expression, inquiry-based learning could bedefined as learning that happens through inquiry. Below, we present definitions ofinquiry-based learning taken up in several contexts. Later, in the Discussion, we presentour conception of inquiry-based learning. The term inquiry has ontological, epistemo-logical, and methodological connotations that depend on the body of knowledge inwhich the expression is used. In other words, the discipline determines what is acceptedas objects of knowledge, what is the nature of those objects, and what kind of inquirywill lead to generate that knowledge. A literature review of the use of inquiry-basedlearning in science classrooms illustrates this point. Pedaste et al. (2015), after synthe-sizing nearly 60 articles that describe IBL as applied to educational contexts, proposedthe following definition for inquiry-based learning:

Inquiry-based learning is an educational strategy in which students followmethods and practices similar to those of professional scientists in order toconstruct knowledge. It can be defined as a process of discovering new causalrelations, with the learner formulating hypotheses and testing them by conductingexperiments and/or making observations (…) Inquiry-based learning emphasizesactive participation and learner’s responsibility for discovering knowledge that isnew to the learner. In this process, students often carry out a self-directed, partlyinductive and partly deductive learning process by doing experiments to


investigate the relations for at least one set of dependent and independentvariables. (p. 48).

Their synthesis led them to propose five phases as determining an IBL cycle: orienta-tion, conceptualization, investigation, conclusion, and discussion, each with several subphases. In this list, we notice specific elements that describe the inquiry that is to takeplace and the features that would allow one to recognize that such inquiry has happenedin a learning environment.

It is common to describe IBL in undergraduate mathematics as not lecturing(Yoshinobu et al. 2011; Yoshinobu and Jones 2012). In a lecture situation, the instructorpresents the content of the course using some combination of definitions, theorems andtheir proofs, and examples usually following a carefuly crafted script or set of lecturenotes. During a lecture, students are expected to listen, take notes of what is beingwritten on the board, and ask or respond to questions as needed (Bergsten 2007; Mesaand Herbst 2011; Movshovitz-Hadar and Hazzan 2004; Weber 2004). While somestudies have attempted to shine a positive side to lecturing (see e.g., Movshovitz-Hadarand Hazzan 2004; Weber 2004), the perceived lack of student engagement and strugglewith ideas during class has been seen as a major drawback (Leron and Dubinsky 1995).Mathematicians struggle with definitions, state conjectures, propositions, and theoremsthat can be derived from those definitions, and convince others, via a proof or anargument, that the propositions or theorems hold (Lakatos 1976; Phillips 2005;Yoshinobu and Jones 2012). In this context, IBL is seen as an alternative to lecturethat can give students the opportunity to experience mathematics in the same way thatmathematicians experience it, by producing proofs on their own and collaborating withpeers striving to understand mathematical principles during class time.

In undergraduate settings, IBL has been defined as a classroom instruction approachthat “invites students to work out ill-structured but meaningful problems (…); con-struct, analyze, and critique mathematical arguments; [and] present and discuss solu-tions at the board or via structured small-group work, while instructors guide andmonitor this process” (Laursen et al. 2014, p. 407). Laursen and Rasmussen (2019)defined Inquiry Based Mathematics Education (IBME) as the merging of IBL andinquiry-oriented instruction (IOI). They claim that while both strands of inquiry havedistinct origins, their implementation in the classroom similarly emphasize that “stu-dents engage deeply with coherent and meaningful mathematical tasks, studentscollaboratively process mathematical ideas, instructors inquire into student thinking,and instructors foster equity in their design and facilitation choices” (Laursen andRasmussen 2019, p. 10). In these classrooms students are expected to discover math-ematical principles from definitions, through problem solving or engaging with amathematical task, and to work collaboratively through ideas to generate new knowl-edge and connections, and in most cases, by answering questions or working problemsthat have been prepared in advance by their instructors. Students may be asked topresent their solutions to their classmates and to receive feedback on how theyconstruct their arguments (Gonzalez 2013; Hayward et al. 2016; Laursen and Hassi2010; Yoshinobu et al. 2011). There is a substantial variation regarding discovery,while the requirement for student involvement and collaboration is maintained. Theseapproaches to implement inquiry have several points of resemblance with Pedasteet al.’ (2015) definition of IBL, with cycles of orientation, conceptualization,


investigation, conclusion, and discussion, enacted as students grapple with problemsthat the instructors provide, present their work or discuss their thinking with each other,and reach a proof or a solution that is satisfactory.

Efforts to infuse inquiry in mathematics courses, such as differential equations(Rasmussen et al. 2009; Stephan and Rasmussen 2002; Wagner et al. 2007), linearalgebra (Wawro et al. 2012), and abstract algebra (Johnson 2013; Larsen et al. 2013),led by mathematics education researchers, have focused in the development of curric-ulum materials and in documenting their use in classrooms by individual instructors.Two exceptions include the Teaching Inquiry-oriented Mathematics: Establishing Sup-ports project (Johnson et al. 2018) and a survey-based quantitative study (Shultz 2019)that used responses from undergraduate mathematics instructors to show that variationexisted across many different instructional practices associated with IOI, organized byrelationships between students, content, and instructors on the instructional triangle(Shultz 2019). Her study, developed in conjunction with work on this project, showedthat the existence of inquiry is not dichotomous, but rather a multidimensionalcontinuum.

In spite of the strong theoretical and empirical support in favor of using inquiry(Laursen and Hassi 2010; Laursen et al. 2014), classroom implementation is not onlyrare (see e.g., Blair et al. 2013; Lutzer et al. 2007), it is also difficult. Instructors usinginquiry-oriented materials in mathematics courses report challenges understanding theirrole in the classroom, aligning the tasks in the inquiry-oriented curriculum with thecourse learning goals, and keeping the pacing required by the department (Johnson2013; Johnson et al. 2013; Mesa and Cawley 2016). The inquiry-oriented materials notonly have a very different content presentation, the explorations may lead to questionsthat the instructor might have not anticipated. Moreover, the organization of students ingroups, and the expectations to have large group discussion of ideas or have the classcritique a student presentation at the board, require a different skill set from thatrequired in a lecture setting. Doubts about whether students are learning the requiredcontent or reaching the learning goals set by the instructor can also prevent instructorsfrom using inquiry, in spite of reassurances that this should not be a worry (e.g.,Yoshinobu and Jones 2012). Analyses of questioning patterns among communitycollege instructors reveal reluctance in using challenging questions or activities duringlecturing. These instructors indicate a preference for questions students can answer inorder to build their self-confidence (Mesa 2010, 2012; Mesa et al. 2014).

As the group of faculty in our study engaged in the implementation of the linearalgebra course using an inquiry approach, we were privy to heated discussions amongthem regarding the use of the relatively unknown textbook, the different way ofteaching, and the mandate to keep a similar pace of content exposure across all thesections of a course with common examinations. Upon realizing that the discussionsspoke about threats to the implicit norms that shaped how things had been done incoordinated courses, we became interested in fully describing the nature of thosetensions. We posed the following research questions:

1. What tensions emerge as instructors of a coordinated linear algebra course teachwith IBL?

2. How were the tensions managed by this group of faculty?3. What role did the professional obligations play in these tensions?


Methods

This study took place in a coordinated2 Linear Algebra course taught at a researchuniversity. The university had created a grant program to support collaborative projectsthat would increase student exposure to active learning strategies; a small group ofmathematics faculty together with the first author, submitted a grant that was success-ful. The math department chose to implement IBL across all sections of a lowerdivision mathematics course, linear algebra for math majors, and to do so with acommercially available textbook that emphasized applications. The course had anaverage annual enrollment of about 400 students a year. The IBL course needed toinclude the same content that was included in previous semesters when it was taughtusing the lecture format.

The year prior to the study (Fall 2014-Winter 2015) the course was taught 4 days aweek in 50-min periods in sections of about 30 students. During that year, the coursematerials (the textbook and worksheets for students to work in groups in class) weredesigned and tested on two sections of the course in anticipation of a full implemen-tation in Fall 2015-Winter 2016. As a consequence of the testing, five major changeswere introduced during the year of the study. First, the course was taught three times aweek in 80-min periods, effectively adding 40 min of instruction per week (almost3 weeks more of classes with the old schedule); second, there was a class size reductionfrom 30 to 24 students; third, the department reserved rooms that had movable tablesand chairs and boards on all the walls to facilitate group work and student presenta-tions; fourth, there were weekly meetings of all faculty involved in teaching the course;and fifth, there were faculty development activities that included non-peer classroomobservations and student focus groups, a workshop illustrating the inquiry-basedlearning done in the department, peer-observations, and a monthly lunch that gatheredother faculty using inquiry-based learning to discuss their questions and offersuggestions.

The audience of the course, as described in the syllabus, was “potential mathmajors” and “those interested in the theory behind the mathematics” (coursesyllabus, p. 1). During the semester we collected the data, there were nearly 270students enrolled in the course, across all of its 12 sections. The 14-weeks courseis described in the syllabus as “a rigorous introduction to linear algebra” coveringtopics such as “systems of linear equations; matrix algebra; vectors, vector spaces,and their subspaces; geometry of ℝn; linear dependence, bases, and dimension;linear transformations; eigenvalues and eigenvectors; diagonalization; and innerproducts” and presented as a study of these topics and their applications, in which“one proves the foundational results in linear algebra” (p. 1). The two major goalsfor the course are “to learn linear algebra and to learn how to write a rigorousmathematical proof. Students should leave this course prepared to use linearalgebra as well as to succeed in further theoretical courses in mathematics” (p.1). The syllabus also states that the course is difficult and suggests two alternative

2 By a coordinated course we mean a course with more than one section for which there are mechanisms put inplace to guarantee that all students are taught “consistent core material” (Rasmussen and Ellis 2015). Thisnotion does not imply that all sections of a course will be the same, as there are many elements to contribute tovariations across sections of a course.


courses that are offered to those interested in the “computational side of linearalgebra.”

The instructors used Linear Algebra with Applications (Bretscher 2013) as thecommon textbook, and were encouraged to use the worksheets that were developedthe prior year. The course covered Chapters 1 through 7 and section 1 in Chapter 8 inthe textbook.3 Every lesson was devoted roughly to one section of the book. The coursefollowed an internal day-to-day schedule which included “open” days without anyassigned section (See Fig. 1, top row center cell) that allowed everyone to address thesame content prior to exams. All the sections had the same day-to-day schedule, weeklyhomework and reading quizzes, and exams (two midterms and a final). The schedulewas for instructor use only and it was modified three times over the course of thesemester. Instructors were required to design and administer a quiz every week, onMondays.

Instructors modified existing, and created new, worksheets. The worksheetscontained between four and eight problems4 and were given to the students to workon during class. Some instructors chose to collaborate in the production and adaptationof the worksheets while some generated new sets of worksheets.

Data

We collected interviews from eight of the nine faculty members who taught the linearalgebra course in Fall 2015 at this university. There were four post-doctoral fellows,four tenured faculty, and one clinical, non-tenure-track lecturer with administrativeduties. Instructors conducted either applied or non-applied mathematics: Bethany,Henry, and Miles5 said they conducted research in applied math; Ed, Laura, Lewis,Thomas, and Ulrich said they conducted non-applied math research; Monica self-described as non-applied mathematician with research interests in applied fields. Threeof the instructors (Bethany, Ed, and Henry) had taught the course the year before thedata collection semester and tested and made modifications to the worksheets Henryhad developed; three instructors (Ed, Laura, and Miles) taught the course again in thesemester following the data collection term (See Table 1). Henry, in consultation withThomas, selected the textbook that would be used for the course. Three participants(Ed, Lewis, and Miles) each taught two sections of the course during the data collectionterm. Only three of the participants reported experience with some form of IBL, eitheras students, or as instructors.

We conducted the interviews in the semester following the Fall 2015 term. Theinterview protocol had 14 questions divided into three sections, background, resourcedesign and use, and comments on implementation (see Appendix A). In the backgroundsection, we asked about instructors’ research interests, their teaching, and their percep-tion of IBL and linear algebra in general and as it was taught at the campus using IBL.We sought this information to help us understand the characteristics that the participants

3 1. Linear Equations, 2. Linear transformations, 3. Subspaces of Rn and their dimensions, 4. Linear Spaces, 5.Orthogonality and Least Squares, 6. Determinants, 7. Eigenvalues and eigenvectors, 8. Symmetric Matricesand Quadratic forms, and 9. Linear Differential Equations.4 This number is difficult to establish, as some problems may have several parts; so individually a 4-problemworksheet may have 17 different sub-questions (e.g., Lewis’s worksheet for sections 1.2 and 1.3).5 Names are pseudonyms.


brought to IBL and to the course. In the resource design and use section,we asked participants toprovide two worksheets (one they really liked, one they really disliked) and a quiz, and askedquestions about purpose, origin, use and changes they would make.We focused on worksheetsbecause they were a primary topic of discussion during the planning meetings, and studentsspent their efforts working towards their completion during class. From our observations of theplanningmeetings,we noticed thatworksheets acted as a visible artifact for how instructorswereoperationalizing their goals for the course and their perceptions of IBL. By asking them for a

9th 26

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Open

WebHW 8

(5.2, 5.4)

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6 -8pm

Review

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WebHW 19

(6.3, 7.1, 7.2)

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(6.3, 7.1, 7.2)

Fig. 1 Excerpt from the third version of the day-to-day schedule, showing some changes to the originalschedule for the second third of the term. The schedule included the same material that was included prior tothe IBL implementation

Table 1 Characteristics of study participants

Participant Status Researchinterests

Experienceswith IBLa

Terms teaching Linear Algebra with IBL

Bethany Post-doc Applied No Winter 2015, Fall 2015

Ed Post-doc Non Applied No Winter 2015, Fall 2015, Winter 2016

Henry Clinical faculty Applied Yes Fall 2014, Winter 2015, Fall 2015

Laura Tenured faculty Non applied Yes Fall 2015, Winter 2016

Lewis Tenured faculty Non applied No Fall 2015

Miles Post-doc Applied No Fall 2015, Winter 2016

Monica Tenured faculty Both No Fall 2015

Thomas Tenured faculty Non applied Yes Fall 2015

Ulrichb Post-doc Non applied Unknown Fall 2015

aWhen asked about IBL experiences, instructors related being a student in an IBL class, or teaching one.b Ulrich did not participate in the interviews


worksheet they liked and one they didn’t, we sought to identify features of the worksheets thatthey found valuable. Because the interviews occurred after the course was over, and we hadinstructors who were also teaching the course again at the time of the interview, we addedquestions that sought to contrast the experiences in both semesters.

Analysis

We analyzed the interviews using a thematic analysis approach. Thematic analysis is useful forrepresenting themultiple layers thatmay be presentwithin a dataset and helps createmajor ideas(Creswell 2012; Guest et al. 2011). We listened to the interviews and, for each question, wegenerated statements that captured what we were hearing in the responses (e.g., ordering oftopics is an issue; the pacing of the course is unrealistic; the textbook is at the right level for thestudents; prior experience with IBL differs from what is done here). We were not originallylistening to the interviews to find tensions—but we heard multiple participants repeatedlymention some of the same frustrations. After listening to each interview, we created summarymemos that included these statements; these memos were then compared across all theinterviews. We then created what we call a thematic matrix: a table whose rows had specificelements of the summaries from each interview (a statement in support or against a point (e.g.,does not like the textbook; uses textbook as is because it ismandated by department) andwhosecolumns had each of the participants (see Table 2 for an abbreviated version). Its cells includedverbatim text and references to the sources for the evidence. The three authors participated infilling the matrix, clarifying the meaning of the statements, looking for disconfirming evidence,and shaping the final set of themes. To choose the themes developed in our findings section, wefocused on the rows that had multiple sources of evidence for at least five of the eightparticipants’ data. Once all the data sources had been assessed in this way, we sought additional

Table 2 Charting Evidence for Tensions

Bethany Ed Henry Lewis Laura Miles Monica Thomas

T: Negative views of textbook or itsdefinitions

* * * * * *

T: Emphasized ordering of problems andtopics

* * *

T: Kept using the textbook for the sake ofconsistency

* * *

I: Had little prior experience with IBLcourses

* * * * *

I: Had previous experience taking or IBLcourse

* * *

I: Said this is not true IBL * * * * *

I: Sacrificed discovery for the purposes ofthis course

* * * *

I: Defined IBL as not including lecture * * *

A: Had or wanted more applications onworksheets

* * * * * *

* indicates that there was evidence in support of the tension in the interview; T: Textbook, I: IBL, A:Applications


patterns by reordering the columns by disciplinary interest (i.e., applied versus non-applied) andby status (e.g., tenure track) to see whether specific patterns emerged across different groups.

To complement the thematic analysis,we also performed a content analysis of two textbooks,Bretscher (2013), the course textbook and a different linear algebra textbook that the director ofthe undergraduate programbelievedwould be appropriate for the intended goals and audience inthe campus (Friedberg et al. 2002, hereafter FIS; see Appendix B). Thus, the two textbookswould, in principle, be adequate for teaching the course. The purpose of this analysis was tounderstand the concerns about the textbook that were prominently discussed during planningmeetings. These analyses added depth to the way in which the instructors were using theworksheets to resolve the tensions generated in teaching the course.

We took three steps to establish the trustworthiness of the analysis and the interpretations.First, we followed a standardized protocol for conducting the observations and focus groups,for creating summary reports and giving feedback to the faculty, and for interviewing theparticipants. Second, to control for potential bias by the first and third author who participatedin the data collection as part of the evaluation project, the second author, who was new to theproject, led the analysis of the interviews. In this way she was able to bring fresh eyes to theanalysis and helped us to make our assumptions explicit. Third, we produced a 10-page,single-spaced summary of our findings and interpretations that we distributed to allinterviewed faculty seeking their feedback. We received comments back from four of themregarding clarifications on the textbook analysis and factual corrections to contextual infor-mation. In general, the faculty agreed with our interpretations, with some of them calling ourobservations “nice.” Their comments supported the findings, and helped us clarify our claimsand their relative importance.

Limitations

The data for the study came from a grant whose general goals were to increase thenumber of undergraduate students in lower division courses experiencing IBL and toidentify efficient ways to train faculty to use IBL. The first and third authors partici-pated as evaluators of the grant, with the intent to document how well IBL trainingstrategies were implemented and assess which were perceived by faculty as beneficialin helping them understand how to teach with IBL. We positioned ourselves asimpartial regarding the use of IBL methods; we did not explicitly promote the use ofIBL with the instructors, but rather sought to understand what was working or notworking for them. However, as researchers and educators, we value constructivist-learning practices and are cognizant that these values could have potentially influencedthe interactions that they had with the instructors of the study when discussing their useof IBL in the classroom in the one-on-one conversations or occasionally during theplanning meetings or the monthly lunches. It is our impression that such influence isnot necessarily warranted, as we avoided presenting our own perceptions of how thingswere going or how things could be done better.

Findings

The three themes that we identified refer to three tensions that emerged as faculty werefaced with teaching linear algebra in the ways that the department demanded. The first


theme refers to the tension between faculty interpretations of the enactment of IBL andthe requirement to keep everyone teaching the course roughly at the same pacecovering the same content. The second theme concerns the textbook chosen and itrefers to the tension that arose from either dissatisfaction or agreement regarding theorganization and presentation of content in the mandated textbook. The third themeconcerns the role of applications and the difficulty faculty had to use them given thepacing of the course. We present each of these themes organized by our researchquestions: by describing the tensions, how faculty managed them, and the role thatprofessional obligations played in the management of those tensions.

“This is WYMCI:” The Meaning of “IBL”

This is the acronym that Henry, with a tongue-in-cheek attitude, used to describe whathe thought the department did under the name of inquiry-based learning. It stands for“Whatever You May Call It” and it means that “you just do not want to lecture aboutthings [students] can read, and you want to work on problems that reach beyond themechanics.” That what the department was mandating instructors to do was “not trueIBL6” was emphatically raised by the three participants (Laura, Henry, and Thomas)who have had prior experiences as students with other versions of “IBL.” These priorexperiences defined their somewhat stricter view of what using inquiry-based learningentailed, which in turn guided their worksheet design. According to this view, studentsshould be doing more discovery and the inquiry would be guided by their ownindividual interests, rather than following a prescribed path stated by the instructorthrough specifically crafted worksheets. Thomas had experienced a Modified MooreMethod (MMM) course taught by Paul Halmos and in that course, “there were just ahuge number of questions that allow[ed] you to explore on your own and createtheory.” Those questions were far more open-ended and did not presume that studentswould be following any particular order.

During the interviews Laura and Thomas said that what the department was doing,was not necessarily what they understood inquiry-based learning was. As Thomas putit, “[in this department] it [is] more interactive-based learning. Here we don’t requirestudents to figure out how it’s going to go. We require them to think about [the content]in small chunks… Certainly they [a]ren’t creating knowledge, which [would be] trueIBL.” For Laura, to actually do inquiry-based learning, there should not be a certainamount of material that needed to be “covered” on a certain day; there would need to bemore flexibility in the schedule and content of the course for “both students andteachers… but in [this implementation] there is not enough time.” Other instructorsagreed with Laura’s assessment, and shared how they attempted to resolve the issue intheir classrooms. For example, Monica attempted to allow time for the students toinvestigate themselves, but reported in a log that,

in order to get closure, and given the time limit, I have to wrap up the mostchallenging topics. Ideally, the students could figure it out by themselves andwith my help; but that would take maybe five times as much time. (emphasisadded)

6 We use “IBL” when participants are discussing their perceptions of inquiry-based learning.


In contrast, the other instructors had less strict characterizations of the inquiry-basedlearning enacted in the department. Bethany described the department’s inquiry-basedlearning as, “just not lecture” just as Yoshinobu and colleagues (2011; 2012) men-tioned. Miles stated, “it means an active learning environment where students are ableto learn on their own.” For Lewis “IBL” meant that.

the students are basically thinking in class… trying to figure things out on theirown. The class proceeds at a slower pace than a traditional class and that’s okay.You adjust the material based on how students are responding to it.

Monica, Ed, and Henry recognized the existence of a stricter version of inquiry-based learning but did not think students should or could discover everything.Monica, for example, said that while students should discover some things bythemselves, they couldn’t really discover the whole history of math on their own,and that guidance was needed. Ed said his “interpretation of IBL” was somewherebetween a complete hands-off MMM approach and “anything that makes thestudents think” approach.

In these descriptions of what the instructors perceived inquiry-based learning to beand what they said they were enacting in the classroom we see various interpretationsof the word inquiry, from student discovery of principles, to students thinking aboutideas, to students learning on their own. The descriptions also refer to time and to a “nolecture” stance on the part of the instructors. These interpretations were reflected in thechoices made for the course. For example, the textbook was not chosen because itwould support full discovery of linear algebra ideas but because it would be “the bestone for students to learn from” (Henry). The textbook was described as providing anintuitive approach to the ideas while having all the content the department demanded asgoals for the course. The constraints imposed by the coordination (teaching the samecontent, keeping similar pace in all sections, and using inquiry-based learning with atextbook that was not perceived as inquiry-based learning) contributed to these variedinterpretations.

With respect to the second research question, we found that the instructors used theworksheets to manage the double-bind of mandated inquiry but limited time and a non-IBL textbook. Laura, for example, infused inquiry in her worksheets trying to keep theschedule established by the department. She described her process of designing aworksheet as being guided by a key idea she wants her students to understand. Shesaid that she typically uses that key idea to create a sequence of problems that will leadthem to discovering and understanding that key idea. She said her worksheets start with“specific cases and then [move] to more abstract or general ideas. The students that cando the extra 3 x 3 case can also do the n x n case.” Her most successful worksheets werethose that walked “all the students” through a process of figuring something out, whilea worksheet in which many students were lost “was a disaster.” In one of her log reportsshe wrote,

The worksheet I’d prepared (an actual IBL discovery-type worksheet) seemed togo really well. The weakest groups were able to get through the main problemand really learn something, and the strongest groups were able to push throughhigher dimensional versions, too.


But another time she expressed concern that, “my attempts at IBL have causedmy section toget behind the others,” showing the challenge of implementing IBL in a coordinated course.

For Lewis, student understanding and being able to do something new was key inhis enactment of inquiry-based learning. He said that in order for students to be able todiscover something important the order of the problems in the worksheets and how thetopics were addressed needed to be carefully considered. A good worksheet would beone in which students would be able to prove a difficult theorem, “because it’s notsomething they would imagine being able to do before this class.” Lewis said thatteaching with daily worksheets gave him immediate feedback about student progress,which was seen as a benefit of using IBL. In a lecture class, he said, “your onlyfeedback is quizzes and exams, which students tend to do terribly on. [Those assess-ments] don’t tell you how much students are actually understanding.”

With respect to the third research question, the differences in how faculty defined themethod and designed the worksheets (which prescribed the content and the paths ofdiscovery for the students, in some cases subverting the order and content of the textbook,as we will see in the next section) uncovers a double-bind between complying with thedepartmental mandates of using IBL, keeping the pace, and teaching the content in thetextbook—all institutional obligations—and what mathematical inquiry should be: workingon undefined problems that lead to new mathematical ideas and insights—a disciplinaryobligation. Implicit in these participants’ interpretations of IBL is the assumption that duringlecturing the intellectual work is mainly done by the instructor whereas during the “IBL”lessons they led, they were able to help students think about, and wrestle with, the materialwith various degrees of discovery. Mathematical inquiry, in the broader sense of the term issacrificed, or rather re-interpreted as interactive work that seeks to make mathematicalconcepts, relationships among mathematical ideas and principles, and the mathematicalmethod of proving, transparent. The use of other instructional features in the classroom (e.g.,group work, student presentations, and very little lecturing) gave the impression that someform of inquiry-based learning was being implemented, and led to the consensus that thedepartment was not implementing “true inquiry-based learning.”While implicit in some ofthe instructors’ comments there were references to students, both as a group and asindividuals, we did not have enough evidence of ways in which the interpersonal orindividual obligations were contributing to the double-binds instructors experienced.

“The book is pretty awful”: The Role of the Textbook in Supporting MathematicalInquiry

We decided to name this theme using Lewis’s view of the textbook, because it wasshared by seven of the eight faculty we interviewed with some degree of vehemence.Definitions and sequencing of topics were two of the major points of contention.7

Although the instructors would agree that in mathematics more than one definitioncould be used and that some properties could be used to generate other properties, theyfelt that Bretscher’s choices were “wrong.” For Thomas, the “book is almost like nothaving a book. It has good problems and historical treatment, but doesn’t have thedefinitions that mathematicians use.” Ed, a non-applied mathematician, said that once

7 The analysis of the textbook in Appendix 2 expands on the textbook sections that were problematic to theinstructors.


the textbook was chosen, he followed it even if it was not ideal. He also affirmed thatBretscher’s definitions (e.g., linear transformations) were good for conceptual under-standing but not for theory building (i.e., for proving).

The textbook was less bothersome for instructors with applied backgrounds, whoindicated that the textbook “was good for keeping an intuitive feel” (Henry) andthat using the textbook’s definitions was important to maintain consistency with therest of the group of instructors and not jeopardize the carefully designed pacing ofthe course. Henry said that a pure mathematician would want to define linearindependence in terms of vectors (as opposed to redundancy) in order to have themost helpful definition for proofs instead of using a more intuitive approach, whileBethany said,

A lot of [the discussions about Bretscher] came down to how much we shouldrely on the book versus how much you should rely on what you think isconvention. From my perspective, this is the book that students read, and weshould try our best to provide them a consistent picture. I’ll sacrifice nonstandarddefinitions for the sake of that consistency.

Henry, Bethany, Ed, and occasionally Miles, worked together on the available set ofworksheets to generate the ones that they each used. Henry, Bethany, and Ed had testedthis set of worksheets previously, so they were familiar with them, and were lesstroubled by the difficulties raised by faculty who were teaching the course withBretscher for the first time.

Instructors were required to follow a set timeline to address the content in specificweeks of the term. Instructors found it very difficult to do just that. The textbookpresentation of the content did not match how instructors felt the content should betaught; but it was the textbook required by the department, and that meant they had touse it. Below, we discuss how instructors managed this bind through their use ofworksheets.

With respect to the second research question, the tenured faculty8 either createdworksheets that contained the definitions they preferred, introduced topics earlier thanwhat the textbook had, or told their students not to look at the definitions in thetextbook. Thomas used Lewis’s worksheets “without any changes [because of] lackof time [to modify them]” while Laura and Monica created their own worksheets afterconsidering the available set. Laura introduced general vector spaces in the worksheetsfor Chapters 2 and 3, instead of waiting until Chapter 4. Once this order change wasdone, she introduced concepts in terms of vector spaces before showing them in ℝn.Monica, who declared “certain things [vector spaces] were in the wrong order,” did notdeviate from it. While she liked that the textbook used examples to assist with students’intuition she also noted “but then the book stays with the example ofℝn and if that’s thefirst time you hear about a vector space, which is really an abstract concept, …it doesharm [if] that’s all we know.” To overcome this perceived weakness, she supplementedthe textbook with examples of vector spaces and general structures in the worksheetsand during class discussions, maintaining allegiance to her notion of the disciplinaryknowledge needed.

8 Ulrich, the post-doc who did not participate of the interview, also created his own worksheets.


Figure 2 helps to illustrate how the two groups of faculty managed this tension in theirworksheets. Figure 2a shows three problems included in the worksheet that Henry used towork on linear independence (Section 3.2 in Bretscher). Figure 2b shows the first and lastproblems of a three-problem worksheet that Lewis used to work on the same content.

Problem 2 in Henry’s worksheet gives a context for using Bretscher’s definitions ofspan and redundant vectors, using vectors that make computations simple. Students wouldpresumably identify the vectors that are redundant by looking for linear combinations ofpairs of vectors that would give the third one. In this way, Henry abides with the textbook’sdefinitions and gives students tasks to work out their meaning. Problem 3 gives anotheropportunity for students to apply the definition of redundant vectors. Problem 4a isdevoted to linear independence, asking questions that emphasize why the presence of aredundant vector results in vectors that are linearly dependent, as in the normativedefinition. Parts b and c of Problem 4 would constitute a proof for the equivalence ofthe definitions of redundancy and linear dependence, thus creating an intuitive context forlinear independence: linear independence occurs when there are no redundant vectors.Lewis’s worksheet, in contrast, begins with the normative definitions of linear indepen-dence, span, and basis, and then gives four sets of vectors for students to identify the onesthat can be a basis for ℝ3. The vectors are chosen to keep the computational burden lowand the directions point to verifying whether conditions in the definition are satisfied. Thesecond problem (not shown) is a direct application of the notions of span and basis; it asksstudents to show that the set of three vectors satisfying a given relationship (x1 + 2 x2 - 4x3 = 0) is a subspace of ℝ3 and to find a basis for it. To conclude his worksheet, Lewis, inhis regular “Something to think about” section, asks a question to the reader about aredundant vector that complains saying that it is not fair that “others were ahead” of it “inthe line.” The task, indirectly seeks to make a connection between the definitions he hasprovided in the worksheet and those offered by Bretscher. By doing this, Lewis showscompliance to the institutional obligation of using the textbook, while maintaining hispreference for using the normative definitions he believes would facilitate proving, thusabiding to his obligation towards the discipline.

To respond to our third research question, these examples illustrate the double-bindbetween the need to comply with the institutional requirement of using the mandatedtextbook across all sections of the course and the need to follow the normativepresentation of the mathematical content, which instructors perceived as more condu-cive to accomplish the goal of helping students learn how to prove. The participants,both those who liked the textbook and those who did not, used their worksheets tomanage this tension, either using them to subvert the textbook order, or by includingnew definitions and problems with explicit connections to the content in the textbook,or by using the textbook definitions that are then re-interpreted with the more normativelanguage. In our analysis we saw some evidence for concerns for students as individ-uals (e.g., maintaining the textbook notation) but it was not spread across the faculty.Concerns that would signal a professional obligation towards students in the space werenot identified.

“I don’t have time for applications”: Managing Time to Support Inquiry Experiences

Though time management is an issue for any instructor, the participants constantlydescribed needing, and spending, more time in the work associated with their


(a)

(b)

Fig. 2 Excerpts from worksheets for linear independence by a Henry, October 2, 2015 and b Lewis, October5, 2015


inquiry-based learning course as compared to what they would usually do forother courses. Monica, referring to class preparation, reported in a log that,“Teaching IBL style requires much more time than a regularly taught class forthe same amount of material,” while Miles said “I feel that with IBL teaching it ismore difficult for me to catch up [to] the class compared to traditional lectures. Istill don’t know how to overcome this problem.” Laura also expressed frustration,writing that, “The pace of this course was *ridiculous* … I think it is unfair tomake the course move so quickly.”

To cope with the need to make sure students explored the assigned material in thecourse, faculty mentioned needing to make difficult decisions about the inclusion ofapplications in their worksheets.

Laura said that she liked the applications in Henry’s worksheets “[but] I don’t havetime for applications.” She preferred to prioritize abstraction and the development ofmathematical theory. The worksheet she produced and liked, for example, guidedstudents to generate a proof for the area interpretation of the determinant. She wouldhave liked using applications, if she had had more time.

In contrast, Monica said that in her worksheets she liked to pair concepts andapplications about a central idea, and that she tended to put more concrete or moreabstract supplementary material on the board depending on how her students wereresponding to it. Though she was aware of time constraints (“on the worksheet, the ideais to cover the topics for that day [because] we know by a certain date there is anexam”), she emphasized that because this course was the students’ first experience withabstract worlds, they needed support [through applications]. She contrasted students’experiences to those in calculus, saying that “in calculus, on a more basic level, you canstill picture things. Linear algebra you can do that, but it’s a step up with vector spaces.”For this reason, her vector spaces worksheet (see Fig. 3) contained an applied probleminvolving springs and differential equations, followed by questions dealing with thedefinition of vector spaces. By including a reference to the modeling of the motion of amass on a spring, she wanted to make the abstractness of linear algebra manageable. Inthis process, however, the emphasis on abstraction is in the background in favor of theapplications.

To respond to the third research question, instructors’ needed to abide by theinstitutional obligation of making sure to expose students to the prescribed content,while also deciding whether or not to use applications, a decision related to thedisciplinary content. Applications help illustrate abstract concepts, but using them

Fig. 3 Excerpt of Monica’s 14th worksheet on Section 4.1, Oct. 2015


could reduce the time needed to keep pace with other sections of the courses, which inturn is needed given that the examinations were common.

Discussion

In the context of a coordinated, university linear algebra course, we sought to answerthe following three questions:

1. What tensions emerge as instructors of a coordinated linear algebra course teachwith IBL?

2. How are the tensions managed by this group of faculty? and3. What role did the professional obligations play in these tensions?

Our analysis identified three tensions, first, between the instructors’ perception of whatIBL should be for the students and what the department was asking them to implementas IBL; second, between the instructors’ perceptions of the correct mathematicalknowledge and the one presented in the textbook mandated by the department; andthird, between the need to illustrate notions via applications and the pacing of thecourse. We found that instructors managed these tensions through the production ofworksheets, which allowed them to make room for their views of linear algebra, ofinquiry, and applications. Underlying these tensions are obligations towards the disci-pline and the institution, which to some extent led to a re-interpretation of inquirylearning itself. There might have been tensions that were motivated by the interpersonalor individual obligations, but those were not salient in our data.

Our impression is that the disciplinary obligation provoked and guided the mostcomplex decisions for the faculty because of the boundaries set by the department,which are part of the institutional obligation. First, instructors had to consider the scopeof inquiry they wanted to use because the requirement to cover a pre-establishedcontent that would be tested in a common final exam did not lend itself to what theyfelt inquiry was about. Second, instructors had to select between using normativedefinitions from the start and therefore changing the order of presentation of ideas orusing the definitions in the textbook and follow the day-to-day schedule. Third,instructors felt they had to be mindful about the applications they could include becauseof the time constraints. In some ways, faculty wanted to make sure students couldexperience inquiry, learn normative definitions, and have the opportunity to see themathematical ideas exemplified in applications.

These tensions were more salient because of the constraints generated by thedepartment, and the instructors’ clear commitment to work within them (a manifesta-tion of their professional obligation towards the institution). If a different textbook hadbeen available or if there had not been a requirement to keep the sections covering thesame content in the same weeks, and if they had not been required to use IBL, it islikely that these tensions would not have emerged as strongly. In the absence of theseconstraints, it is likely that some sections of the course would have had more discoveryand less exposure to course content than other sections. The departmental mandate ofmaking sure that all the students experienced the “same” course across the multiplesections of the course, was translated into a shared commitment across the instructors;


their professional obligation manifested in how they strived to make sure they compliedwith the institutional requirement despite their personal preferences.

At the same time, each instructor created a slightly different course because theydesigned their worksheets and taught the course in ways that accommodated theirviews about inquiry-based learning, linear algebra, and applications. We sawdifferences in how groups of instructors organized themselves to work out theirviews. These differences seemed to fall along two intertwined dimensions, thestatus of the instructors in the department (tenured vs. non-tenured) and theirindividual research interests (applied or non-applied). Of the five instructors whotended to go along with the textbook presentation (Bethany, Ed, Henry, Miles, andMonica) a presentation that ostensibly favored an intuitive approach to the material,three had applied research interests (Bethany, Henry, Miles) and four were post-doctoral fellows or on special renewable contracts (Bethany, Ed, Henry, Miles). Thefour instructors who strongly opposed the textbook presentation and changed theirworksheets to accommodate their personal view of the content (Laura, Lewis,Thomas, and Ulrich) had non-applied research interests, and all, except for Ulrichwere full-time tenured faculty. They also were the ones who indicated not havingtime for applications.

We believe that their own temporary contractual obligation towards the department canmake faculty on short-term contracts more inclined to follow the institutional demandssimilarly to what Lande and Mesa (2016) documented in their study of community collegemathematics faculty. Theremight be two possible incentives for faculty on short-term contractsto abide to the institutional obligation. First, these instructors may presume that a supportiverecommendation letter for a future faculty position or for re-appointment might be more likelyto be secured if they play along with the rest of the group and maintain congeniality.Alternatively, the amount of work that would be needed to create a new set ofworksheets—a set that would follow the normative view of linear algebra—would take timeaway from research, and this would be detrimental for their future bids at faculty positions.Thus, they would be more likely to use the available worksheets and follow the textbook.

Faculty with declared applied research interests appreciated the concrete way inwhich the textbook presented basic notions. They saw its potential for visualizationand for giving students easy access towards formalization. While they worriedabout their eventual leap towards abstraction, they did not have as much fear astheir colleagues with non-applied research interests that students would not be ableto manage the leap.

In the end, all students were exposed to roughly the same material, completed thesame assignments, and took the same final exam. Evidence from student focus groups,in which all students took part, indicated that they acknowledged that how the coursewas being conducted was helping them understand the material better. In spite of theheavy load of the course, they appreciated its emphasis on proofs and the collaborativenature of class work. All students recognized the inquiry nature of the course—using asreference other mathematics courses they had taken prior to this class—and found it tobe quite appealing (Jackson and Mesa 2016). Thus, in practical terms, whether what thestudents experience is IBL or not, what textbook is used, or whether they need to domore applications, might be less important than how students engage with the materialin the day-to-day interactions in the classroom. Of course, further work needs to bedone to investigate this claim.


Conceptualizing the Enactment of Inquiry-Based Learning

Using participants’ descriptions of the enactment of inquiry-based learning, we cate-gorized the variety of practices along three distinct dimensions: (1) the students’relationship with the mathematical content, (2) students’ relations with each other,and (3) instructors’ interpretation of the department’s demand for supporting inquiry.First, is the level of discovery that the instructors felt the word inquiry entailed; itappears that on one extreme, the discovery meant proposing definitions that were usedto investigate the object being defined, leading to conjectures or propositions that couldbe explored to determine whether they were true or not, similar to what is done in theMoore Method.9 In this process, there would not be constraints on what could bediscovered. It would be up to the person working on the definitions to determine thepath of discovery and to generate theory. On the other extreme, the discovery starts withexisting definitions and theorems, and rather than focusing on creating new knowledgederived from those, the discovery is about making sense of the given definitions andtheorems, relating concrete and abstract notions, and using them proficiently. The pathof discovery is, for the most part, prescribed. This level of discovery, lower than fulldiscovery, was reflected by instructors who said that the department was notimplementing IBL. Though we only represent the two extremes of discovery, inpractice the amount of discovery would range over these in a continuum: for example,the idea of reinvention from the realistic mathematics education program suggestsvarious paths of re-discovery of mathematical notions (see, e.g., Artigue andBlomhøj 2013; Gravemeijer and Doorman 1999; Johnson et al. 2018; Rasmussenand Kwon 2007; Wawro et al. 2012).

The second dimension, peer engagement, refers to the students’ engagement witheach other while working on mathematical ideas. On one extreme, the students wouldbe required to discuss their thinking with their peers in organized groups or presenta-tions, with other students acting as a sounding board and offering feedback for thethinking process. On the other extreme, the students would be working individually.The instructor may or may not contribute to shape the discussion. High peer engage-ment with group work was mentioned by our participants (both faculty and students) asbeing enacted regularly, while student presentations at the board was emphasized morein the literature as being a typical feature of the Modified Moore Method than in whatwe heard from the participants.

The third dimension, instructor-provided information, refers to expectation that theinstructors impart the needed information in the classroom. On one extreme, theinstructor withholds all the information from the student (or students) letting themfollow any path, including incorrect ones, and avoiding assessments of value or qualityof their productions. On the other extreme, the instructor explicitly supplies all theinformation such as definitions, propositions, theorems, proofs, examples and counter-examples through lectures or other resources. The instructors in our study said theyprovided various levels of information to their students; some giving “mini-lectures”

9 The Moore Method is named after R. L. Moore, a mathematician who believed that students should createmathematics by working ideas on their own starting with a few key definitions, proposing conjectures, andexploring them in ways that would allow further discoveries of mathematics (Jones 1977; Mahavier 1999). Ina class using Moore’s method, the instructor writes a definition on the board and lets the students ponder it andcome up with propositions based on that definition without direct instruction.


occasionally and some abstaining from giving direct answers to students’ requests forinformation.

Radar representations with three axes—Discovery, Peer engagement, and Instructor-provided information—allow us to represent the various methods that instructorsmentioned during the interviews, lecture, the Moore Method, and the implementationof inquiry-based learning in this department. Each axis is used to represent what couldbe the average percentage of time that each of these dimensions might be enacted in theclassroom over, say, a semester. The purpose of these diagrams is not to give precisepercentage measurements, but instead to help visualize a conceptualization of IBLenactment. Lecture (see Fig. 4a) illustrates a possible high percentage on the instructor-provided information axis, to suggest that the source of information in this setting isexpected to be the instructor; in addition, there are low percentages (in this hypotheticalexample, 0%) for the other two axes, assuming that students do not engage indiscussing ideas with each other and that no discovery takes place. The Moore Method(see Fig. 4b), according to what the faculty described, would have a higher percentageon the discovery axis, and a lower percentage on the other two dimensions; in thisenactment, the instructors are not expected to provide much information to the studentsand students are not expected to engage with each other in the process of discovery,except during presentations when students are expected to give feedback. Finally, theinquiry-based learning that was described in this study (see Fig. 4c), would have ahigher proportion of student engagement with each other via group work or presenta-tions at the board, some level of discovery of preexisting mathematical ideas, and anexpectation that instructors would provide some, but not much, information to thestudents (e.g., presenting problems that can be used to illustrate notions, giving hints).

We argue that these variations can be seen as the result of instructors’ decision-making based on how they manage double-binds with their institutional and disciplin-ary obligations. For example, the need to expose student to a prescribed content in14 weeks shaped decisions that resulted in lowering the level of discovery in the course.In addition, the institutional requirement of this department of organizing students ingroups to work together in class can be seen as increasing the time that such activitywas done. Moreover, the institutional mandate of giving students the space to figure outthe worksheets on their own limited the amount of information instructor couldprovide; instructors, however, from time to time gave “mini-lectures” and createdworksheets that guided the discovery. Because the process of discovery and engage-ment with new mathematical ideas is essential to mathematical research and is part ofinquiry, we see this as a force that would move the proportion of time higher; however,the inquiry-based learning taught by the instructors in our study contained less discov-ery than they would have liked because of the constraints imposed by the department(see Fig. 5).

The evidence from this investigation suggests that the tensions that can emerge asmathematics departments seek to increase the number of students experiencing adifferent teaching approach can be navigated via a system that allows faculty to exerttheir professional commitments even in a fully coordinated system. The instructors whoparticipated in the study included faculty who had experienced IBL before as well asinstructors for whom IBL was new. The department’s decision of engaging them withthe course through teaching with IBL, together with the systems that it put in place sothe course could be viable with IBL, can be seen as a strategy that Henderson et al.


Fig. 4 Enactment of instruction along the Discovery, Peer Engagement, and Instructor-provided information,for a Lecture; b Moore Method; c the IBL described in the study


(2011) would qualify as more successful for instilling change: it was sustained over along period of time (14 weeks, the duration of the course with the expectation that allinstructors would teach again at least another term) and it sought to make the depart-mental beliefs explicit and understood by the participants via the weekly coursemeetings, which served as the prime mechanism for negotiating those beliefs withtheir own views of the discipline, the nature of IBL, and the role of applications. Fromour vantage point, we see that it was useful for instructors to have the autonomy tomanage these tensions through the worksheets. Having the flexibility of adaptingexisting—or creating new—worksheets was fundamental for the system to work.Likewise, the mandate of minimizing lecture and increasing group work and studentpresentations was well coupled with the idea of giving students in-class work, againthrough the worksheets, that gave instructors flexibility in exposing students to themandated content in ways that allowed them to fulfill their disciplinary obligations.

We believe also that having a textbook that supported some of the inquiry by helpingstudents develop intuitive understanding of the mathematical notions, was a strategicdecision that facilitated the change. On the one hand, the textbook provided studentswith a resource to draw from whenever needed (many problems for parts of thehomework were directly taken from the textbook) and on the other, it gave faculty adifferent way to think about presenting the material. Having a textbook that wouldallow students develop intuition made the need for developing worksheets that empha-sized proving more salient for the faculty. The label of being an “awful” textbook wasapplied to the difficulty some faculty saw in the textbook’s ability to support proving.But it was precisely this weakness that pushed faculty into developing worksheets thatwould guide students’ development of concepts and ideas for proving.

Finally, we would be remiss not to mention how important was that as part of thechange strategy, the department invested in developing materials a year before of the

Institutional Obligation:Give students time to figure things out by themselves

Institutional Obligation:Include all the prescribed content

Institutional Obligation:Use group work

Fig. 5 Examples of impact of various institutional obligations that would move instructors away from lectureand into IBL in this study


full implementation so they became available during the full implementation. Acting inthis way, the department contributed to a set of “default options” (Rasmussen and Ellis2015, p. 112) that, in tandem with the commercially available textbook, were meant tomake instructors’ work easier. As we saw, instructors did take advantage of theavailable worksheets: even the faculty who decided to create their own, used theavailable set as a starting point.

Implications

This study describes what happened inside a sustained change strategy that targetedteaching in undergraduate settings by unpacking the obligations that faculty respond towhen they went through that process. The lens of practical rationality helps interpretthese difficulties in terms of the binds in which instructors are put when they have toalter practice.

More needs to be done to understand the role that obligations towards studentsas individuals and as participants in a collective class work. Our lack of detailedclassroom enactment does not allow us to trace the role of these obligations onhow faculty chose to teach or planned their lessons, although we have someevidence that they had students in mind when they were designing some of theiractivities and when they were changing gears in the middle of a lesson. A differentdesign would be needed to pursue such investigation and to untangle the ways inwhich the four obligations contribute to how IBL is enacted at scale with acommercially available curriculum.

Other studies document that student resistance is an issue for IBL implemen-tation (e.g., Hayward et al. 2016). In our study, we did not come across studentvocal resistance to IBL, on the contrary, students as a whole, appreciated its role inhelping them learn the material via discussion, collaboration, and the one-on-oneand large group conversations with the instructor in class. We think that the lackof choices in how instruction was to be delivered (all sections were taught withIBL) might have a role in such lack of opposition to the method. We observed andheard about some forms of student resistance such as skipping class, not using thetextbook, working individually, or refusing to go to the board. The flexible way inwhich IBL was implemented, however, allowed such forms of resistance to occurwithout being detrimental to the experience of the rest of the students. In someways, resistance can be seen as a natural aspect of the implementation and wesuspect that studying how instructors manage such resistance will shed light onhow they deal with their professional obligations towards individual students andthe students as a whole; we speculate that these obligations will be in directtension with institutional mandates to use IBL.

Some questions of a practical nature remain in light of these findings. First, it iswidely accepted now that any form of active learning is strongly associated with largerknowledge gains and course performance in STEM fields (Freeman et al. 2014) andthere is evidence that using IBL has a positive association with student gains incognitive and non-cognitive outcomes in undergraduate math courses (Kogan andLaursen 2013; Laursen et al. 2014). It would be important for making a case foruniversity administrators that implementing IBL in this course resulted in greater


benefits for the students, at least in terms of standard measures of course performance(e.g., grades, performance in subsequent math courses, selecting mathematics as amajor). Because this department had only one year of transition, such investigationcan be accomplished with an analysis of historical data comparing pre-implementationto post-implementation; the institutional character of the implementation allows foreffectively controlling for student characteristics.

Second, it would be important to investigate the extent to which instructors havetransferred some of the strategies they learned while teaching the linear algebra IBLcourse to their non-IBL courses. Anecdotally, we have learned that faculty implement-ed some aspects of IBL in their non-IBL courses, by designating one full session forinquiry work per week, in which they gave students more difficult problems to work ingroups and then present at the board. It would be important to know how widespreadthese practices are and when are instructors more likely to use them (e.g., terminalcourses, advanced courses).

Finally, this implementation occurred in a department that had several mechanismsto support the use of IBL, specifically, an initial set of worksheets, small class size,rooms with movable chairs and boards on all the walls, observations of teaching andfocus groups, pre-semester workshop, and weekly and monthly discussions led by acoordinator. Such implementation is costly. An important question to address iswhether the benefits of the implementation (e.g., sense that students are understandingideas better, more students are majoring in mathematics, fewer students fail the course)outweigh the cost of these support mechanisms (e.g., having a coordinator, needingmore faculty for more sections that have fewer students, observing all instructors,talking to all students, etc.). Such cost-benefit analysis needs to be performed andappropriate data collected so departments can make an informed choice about imple-mentation such as the one described here.

IBL has been described as an instructional approach that has the potential to helpstudents understand mathematical ideas and become more proficient in the practicesrequired for doing mathematics: proving, communicating ideas, following arguments,etc. The findings from this study help build knowledge about ways to manage tensionsthat inevitably emerge in a departmental implementation. Investigations of similarnature with different departments and institutions would contribute to this knowledgebase.

Appendix 1: Interview Protocol10

Part 1: Background, Linear Algebra, Inquiry-Based Learning

1. Please tell us a little about yourself: what is your PhD in and what teachingexperiences have you had?

2. Prior to coming to this campus what did you know about I-B-L? What experiencedid you have teaching with I-B-L?

3. I-B-L can have different meanings to different people. What does inquiry-basedlearning mean to you?

10 For full protocol please contact the corresponding author.


4. In your opinion what should be the goals and objectives of any Linear Algebracourse? How are they different from the goals and objectives of the Linear Algebracourse in this campus?

5. Fall 2015: Tell us your impression of how well students mastered the goals for thecourse. (Winter 2016: Tell us your impression of how well students have masteredthe goals for the course so far)

6. Who did/do you collaborate with most during the Fall semester/this semester?What was the nature of the collaboration?

7. Please describe the role of <your role as> the course coordinator. What kinds ofsupport did you receive from the coordination?

8. What was most useful to you to learn about I-B-L?9. Please describe the supports that were useful to you to learn to teach with I-B-L.

How were they helpful?

<there were probes for the various elements that were designed to support thecourse, including textbook weekly meetings, discussion with colleagues, andobservations>.

Part 2: Worksheets and Quiz

10. Please tell us what are key features you seek to ensure your worksheets have (e.g.,variety, order, level of inquiry, types of problems, etc.)

11. Tell us about your (Worksheet they like-WL, Worksheet they dislike-WD, Quiz-Q)

(a) How did you come up with this (WL, WD, Q)?(b) Why do you like it?(c) What objectives/goals does this (WL, WD, Q) target?(d) What resources did you use in creating it?(e) How did the students work on the worksheet?(f) What changes, if any, would you make to it?

12. Bi-weekly log question: What was your strategy for adapting the worksheets andat the same time managing your time constraints?

13. For instructors teaching for a second semester:

(a) What is different about teaching this course for a second semester?(b) What did you learn from last semester?(c) What are you using now to create the worksheets for your class?

Appendix 2: Content Contrast of Two Textbooks

We provide a contrast between two textbooks, Bretscher (2013) and Friedberg et al.(2002), hereafter FIS, in order to provide a context for the discussions about contentthat framed much of the tensions we describe in this study.


Bretscher’s Presentation of Content

According to Henry, Bretscher was chosen as the textbook for the course because itprovided students with a resource that helped them develop intuition about linearalgebra concepts and procedures. Henry chose the textbook after reviewing variousoptions and, after having given it “a try,” found it sufficiently readable and usable bythe students, one from which “students could learn best.” His perception was that theproblems were “decent” and sufficient for helping students develop the needed intuitiveunderstanding of linear algebra ideas, despite its shortcomings regarding the definitionsused for linear independence and linear transformations, and the particular ordering ofthe chapters. During the first semester in which all the sections used IBL withBretscher, the faculty expressed significant concerns with these shortcomings duringthe weekly course planning meetings.

Linear Independence

In Section 1.5, FIS presents the following definitions:Definition. A subset S of a vector space V is called linearly dependent if there

exists a finite number of distinct vectors u1, …, un in S and scalars a1, …, an, not allzero, such that

a1u1 þ a2u2 þ⋯þ anun ¼ 0

In this case we also say that the vectors of S are linearly dependent.Definition. A subset S of a vector space V that is not linearly dependent is called

linearly independent. As before, we also say that the vectors of S are linearlyindependent. (Friedberg et al. 2002, p. 36-37)

In contrast, Bretscher defines linear dependence in a logically equivalent way butusing the non-normative11 term redundant:

Redundant vectors;4 linear independence; basisConsider vectors v!1;…; v!m in ℝn.

a. We say that a vector v!i in the list v!1;…; v!m is redundant if v!i is a linearcombination of the preceding vectors v!1;…; v!i−1:

5

b. The vectors v!1;…; v!m are called linearly independent if none of them isredundant. Otherwise, the vectors are called linearly dependent (meaning that atleast one of them is redundant).6

c. We say that vectors v!1;…; v!m in a subspace Vofℝn form a basis of V if they spanV and are linearly independent.7 (p. 125)

In the call for the fourth footnote in the title, “Redundant vectors;” Bretscher states:“The notion of a redundant vector is not part of the established vocabulary of linearalgebra. However, we will find this concept quite useful in discussing linear

11 By non-normative we mean that mathematicians would not recognize the term. A search in Wolfram alphagives the dictionary (non-mathematical) definition of redundant as “repetition of same sense in differentwords.”


independence” (p. 125). With this statement, Bretscher explicitly recognizes that thatredundancy is not a normative definition in linear algebra.

Bretscher justifies the inclusion of a different definition because it is “useful.”Exactlywhat hemeant by this is not explicitly stated. One affordance of this definition, which he might bereferring to, is that it helps demonstrate how each vector of a linearly independent set cannot beformed by linear combinations of the other vectors in the set. The normative definition of havingno nontrivial solution to an equation does not reveal this relationship. Bretscher’s definitionmakes the relationship between the vectors in the setmore intuitive.One could imagine that oncea first vector is pictured in a vector space, the next vector in the linearly independent set cannotbe formed by transforming the first vector.A third vector that needs to be linearly independent inturn cannot be formed by transforming or combining the first two vectors, and so on for themvectors in the linearly independent set. This presentation is consistent with Bretscher’s intent forhis textbook, described in the preface as emphasizing visualization.

Laura summed up the definition’s main affordance and limitation saying: “it isbasically a formalization of our intuition of linear independence but it is hard to usein proofs.” Consider the problem of showing that vectors in a given set are linearlydependent. With FIS’s normative definition one needs to find the scalars that satisfy theequation a1u1 þ a2u2 þ⋯þ anun ¼ 0 (see Friedberg et al. 2002, p. 36–37). But inorder to use this equation in proof-writing, Bretscher needs to do additional work. First,he defines an intermediary definition, linear relation, as follows:

Definition 3.2.6: Consider the vectors v!1;…; v!m in ℝn. An equation of the form

c1 v!1 þ⋯þ cm v!m ¼ 0!

is called a (linear) relation among the vectors v!1;…; v!m”(p. 127).

Next he introduces a theorem that relates this definition his own definition of lineardependence:

Theorem 3.2.7… The vectors v!1;…; v!m in ℝn are linearly dependent if (and onlyif) there are nontrivial relations among them (p. 128).

The inclusion of the extra definition 3.2.6 and theorem 3.2.7, which resemble thenormative FIS definition, show that Bretscher presents both the normative and non-normative definitions to students rather than only one, on account of facilitatingstudents’ “imagin[ing] the computations” (p. ix).

Linear Transformations

FIS defines a linear transformation as follows:Definition. Let V and W be vector spaces (over [a field] F). We call a function T :

V→W a linear transformation from V to W if, for all x, y ∈ V and c ∈ F, we haveT(x + y) = T(x) + T(y) (and)T(cx) = cT(x) (FIS, 2002, p. 65)Bretscher, in contrast, proposes two definitions for linear transformations; one in

Chapter 2 and one in Chapter 4. In Chapter 2, he defines linear transformations asfollows:

Linear transformations2

A function T from ℝm to ℝn is called a linear transformation if there exists an n x mmatrix A such that T x!� � ¼ A x!,

For all x! in the vector space ℝm (p. 45).


and states Theorem 2.1.3 as follows:Theorem 2.1.3: Linear transformationsA transformation T from ℝm to ℝn is linear if (and only if)

a. T v!þ w!� � ¼ T v!� �þ T w!� �; for all vectors v! and w! in ℝm; and

b. T k v!� � ¼ kT v!� �; for all vectors v! in ℝm and all scalars k:(p. 45)

which corresponds to the normative definition of linear transformations. In the proof forTheorem 2.1.3 Bretscher constructs the needed matrix, justifying his non-normativedefinition by arguing that “correct” definitions do not exist. In the call for footnote 2Bretscher writes,

This is one of several possible definitions of a linear transformation; we could justas well have chosen the statement of theorem 2.1.3 as the definition (as manytexts do). This will be a recurring theme in this text: Most of the central conceptsof linear algebra can be characterized in two or more ways. Each of thesecharacterizations can serve as a possible definition; the other characterizationswill then be stated as theorems, since we need to prove that they are equivalent tothe chosen definition. Among these multiple characterizations, there is no “cor-rect” definition (although mathematicians may have their favorite). Each charac-terization will be best suited for certain purposes and problems, while it isinadequate for others (p. 45).

This is of course correct. It might be a matter of preference, and in a course in whichstudents are expected to understand how mathematics is generated, such work might beuseful, even if the definitions proposed do not conform to normative, accepted prac-tices. An advantage of Bretscher’s definition is that it is useful for the purpose it isbeing used for. The first three chapters of the textbook focused on matrices in Euclideanspaces (i.e., n-dimensional spaces of real numbers). Proposing a definition of lineartransformation that requires a matrix furthers his goal of helping students gain intuitionabout what the linear transformations do. Though it is slightly more limited than thedefinition in FIS (it reduces vector spaces to those in ℝn), it gives students a context formanipulation and recognition of linear independence. By defining lineartransformations as those involving matrices, Bretscher delays the use of typical lineartransformations such as differentiation, integration rotations, reflections, or projectionsuntil Chapter 4.

In Chapter 4, Bretscher introduces the normative definition of a lineartransformation:

Definition 4.2.1. Linear transformations, image, kernel, rank, nullityConsider two linear spaces12 V and W. A function T from V to W is called a linear

transformation if

T f þ gð Þ ¼ T fð Þ þ T gð Þ and T k fð Þ ¼ kT fð Þ

12 Bretscher uses the term linear spaces (as opposed to vector spaces) here.


for all elements f and g of V and for all scalars k. These two rules are referred to as thesum rule and the constant-multiple rule, respectively. (…) (p. 178).

By waiting until Chapter 4 to introduce this definition, Bretscher delays the use oftypical linear transformations such as differentiation, integration rotations, reflections,or projections. This aligns with Bretscher’s interest in “keep[ing] abstract exposition toa minimum… The examples always precede the theorems in this book” (2013, p. x).The tension between contextualization and abstraction is not new (Tall 2004). Choosingthe contextualization can be rationally justified on grounds of accessibility, while thenormative definition (e.g., FIS’s) is useful for proof-writing, but will not give studentsthe contextualization needed to make the material more accessible.

Order of Material

The order in which content is presented in Bretscher differs from the order in FIS. FISintroduces linear independence before linear transformations. Bretscher presents lineartransformations before linear independence, and as mentioned earlier, Bretscher intro-duces the two definitions a second time when he defines general vector spaces inChapter 4. The first three chapters in Bretscher are dedicated to showing students “thelanguage of linear algebra in ℝn” (2013, p. 166). The two definitions are presentedagain in terms of general vector spaces in the fourth chapter, and then in the same orderas in FIS. In FIS and in Chapter 4 of Bretscher, the definition of linear transformationsis introduced after linear independence because transformations are about preservingthe structure of the vector spaces, which we conjecture is motivated by wanting topreserve the relationships between the vectors in those spaces. Emphasizing thepreservation of linearity with linear independence in the early chapters of Bretscherwould not have been appropriate, because the reader would not yet know why therewas anything important to preserve.

The decision to introduce linear transformations before linear independence also fitswith Bretscher’s goals for intuition and visualization of the subject. Chapter 1, “LinearEquations” (p. vii), includes topics such as linear systems of equations and matrixalgebra, so defining linear transformation functions gives a formal label to the opera-tions Bretscher wants the student to become familiar with. In Section 2.2, immediatelyafter the section that gave the definition of a linear transformation, Bretscher showslinear transformations visually with scaling, projections, reflections, rotations, andsheers. In doing so, Bretscher prioritizes giving students a concrete, graphic intuitionof these transformations before stepping into the more abstract concept of linearindependence, where visualization beyond three dimensions is more difficult.

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