Journal of Mathematical Behavior 33 (2014) 168– 179

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The Journal of Mathematical Behavior

j ourna l h omepa ge: www.elsev ier .com/ locate / jmathb

Students’ sense-making frames in mathematics lectures

Aaron Weinberga,∗, Emilie Wiesnera, Timothy Fukawa-Connellyb,1

a Department of Mathematics, Ithaca College, 953 Danby Road, Ithaca, NY 14850, United Statesb Department of Mathematics & Statistics, University of New Hampshire, W383 Kingsbury Hall,33 Academic Way, Durham, NH 03824, United States

a r t i c l e i n f o

Article history:Available online 20 December 2013

Keywords:Sense-makingLecturesAbstract algebra

a b s t r a c t

The goal of this study is to describe the various ways students make sense of mathemat-ics lectures. Here, sense-making refers to a process by which people construct personalmeanings for phenomena they experience. This study introduces the idea of a sense-making frame and describes three different types of frames: content-, communication-, andsituating-oriented. We found that students in an abstract algebra class regularly engaged insense-making during lectures on equivalence relations, and this sense-making influencedtheir note-taking practices. We discuss the relationship between the choice of frame, thestudents’ sense-making practices, and the potential missed opportunities for learning fromthe lecture. These results show the importance of understanding the ways students makesense of aspects of mathematics lectures and how their sense-making practices influencewhat they might learn from the lecture.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Students typically spend approximately 80% of their time in class listening to lectures (Armbruster, 2000) and 63% ofprofessors in STEM fields use “extensive lecturing” in most or all classes (Berrett, 2012). Lecture listening is a difficultcognitive task for college students (Ryan, 2001), but it is important, since what students take away from lectures is closelylinked to what they learn (e.g., Titsworth & Kiewra, 1998).

The goal of this research project is to develop and use a framework to articulate the ways students make sense ofvarious discursive features of mathematics lectures, to use this framework to describe students’ sense-making practices,and to identify the various factors that influence and constrain this sense-making process. We apply the framework toanalyze students’ responses to a pair of abstract algebra lectures on equivalence relations. The lectures included definitions,propositions, proofs, examples, and exposition, meaning that the students’ experience was broadly representative of theirexperience of lectures in both algebra and other advanced mathematics courses. Thus we view this setting as providinginsight into students’ experience in upper-level, proof-based mathematics courses more generally. This work is part of abroader project with the overarching goal of identifying students’ opportunities to learn from mathematics lectures and howthose opportunities are affected by various aspects of the lecture and the demands that are placed on students’ knowledge

and understanding (see, e.g., Weinberg, Wiesner, & Fukawa-Connelly, 2012).

∗ Corresponding author. Tel.: +1 607 274 7081.E-mail address: aweinberg@ithaca.edu (A. Weinberg).

1 Present address: Korman Hall, School of Education, Drexel University, 3200 Market Street, Philadelphia, PA 19104, United States.

0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmathb.2013.11.005

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A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179 169

. Background

.1. Students’ experiences of lectures

Many students believe that the only way they can learn from lectures is to take verbatim notes (e.g., Bretzing & Kulhavy,981; Kiewra & Fletcher, 1984; Peper & Mayer, 1986); this perspective underlies the prior research on students’ participation

n lectures, which has focused on their note-taking habits. However, the existing research has been limited in the ways itas attempted to address students’ understanding of the lecture content, and we could find no prior research that describedhe ways that students engage with mathematics lectures. Some research (e.g., Barnett & Freud, 1985) has failed to addresstudents’ understanding of the concepts presented in the lecture. Other studies have relied on students’ self-reporting toescribe what they did during the lecture without determining whether students’ actual habits matched their descriptionse.g., Van Meter, Yokoi, & Pressley, 1994). Much of the research methodology has been to describe the number of “lectureoints” that students recorded and later recalled (e.g., Hartley & Marshall, 1974; Kiewra, 1991; King, 1992; Locke, 1977) oro analyze students’ learning by examining their performance on tests that only included factual questions (e.g., Peper &

ayer, 1978; Titsworth & Kiewra, 1998); this methodology implicitly frames “understanding” as simply recalling facts.A notable exception is the study by Ryan (2001), who generated metaphors that psychology students might use to

escribe and guide their lecture-observing habits and note-taking practices (e.g., “sponge,” “tape recorder,” “stenographer,”tc.). However, these descriptions were based on students’ self-reported beliefs (about their own habits) rather than onmpirical evidence about the students’ actual habits. In addition, the students in Ryan’s study were asked to describe theirote-taking approach over the entirety of the semester rather than in response to particular lectures and, more importantly,he descriptions failed to address the ways that these various practices might affect students’ opportunities for learning.

.2. Sense-making

When different students view the same lecture, they may perceive and understand the lecture in very different ways.e can understand this phenomenon by describing the students as having made sense of the lecture differently; the goal

f this project is to describe the various ways students make sense of discursive features of mathematics lectures.Although the terms “sense” and “sense-making” are widely used in mathematics education (e.g., NCTM, 2011; National

overnor’s Association, 2010; Schoenfeld, 1992), they do not have standard definitions in this literature. Grady (2013)escribed students as believing that mathematics is “sensible” when they view it as a “connected, coherent system in whichhere are reasons for such things as rules, procedures, and formulas, whether or not the individual yet knows or understandshe reasons” (p. 4). The focus of Grady’s framework is at a macro-level and facilitates the descriptions of students’ beliefsbout the process of doing mathematics; however, her framework does not facilitate analysis at a micro-level—that is, ofhe instances and ways students make sense of the mathematics they encounter as they experience a lecture.

In Information Systems and Organizational Studies research literature, sense-making refers to a process by which peopleonstruct personal meanings for phenomena they experience (Dervin, 1983; Kari, 1998); Klein, Moon, and Hoffman (2006)escribe it as “a motivated, continuous effort to understand connections. . . in order to anticipate their trajectories and actffectively” (p. 71). In the context of a mathematics classroom, sense-making attempts to describe the process by whichtudents interpret and construct meaning for the activities, personal interactions, and discourse in which they participate;ense-making also focuses on the individual students’ mental schemas that that shape this process of meaning construction.his focus is in contrast with the way meaning-construction is typically viewed in the mathematics education literature,here “meaning” is typically thought of as ‘the meaning of X,’ where X is mathematical content to be taught” (Kilpatrick,oyles, & Skovsmose, 2005, p. 3). These perspectives of sense-making are in alignment with Grady’s (2013) focus on students’

ense-making orientation, but they also facilitate the description of students’ individual sense-making acts.A central organizing idea in Klein’s (e.g., Klein et al., 2006; Klein, Phillips, Rall, & Peluso, 2007) formulation of sense-

aking is a frame, which is based on Goffman (1974) and Minsky’s (1975) idea of conceptual frames as well as Piaget’se.g., 1952, 1954) idea of a conceptual schema. Beach (1997) describes a frame as “a [cognitive] structure for accountingor the data and guiding the search for more data.” Similarly, Klein, Phillips, Rall, and Peluso (2007) describe a frame as “acognitive] explanatory structure that defines entities by describing their relationship to other entities.” Frames are beginningerspectives or viewpoints that “define what count as data” and “shape [the interpretation of] the data” (Klein et al., 2006,. 88). Thus, we view a conceptual frame as an individual’s schema of interpretation. It is a mental structure that filters andtructures an individual’s perception of the world by causing aspects of a particular situation to be perceived and interpretedn a particular way. An individual uses a frame to organize his or her experience, to recognize patterns in new situations,nd to guide his or her activity.

From a sense-making perspective, students in a lecture initially experience a constant stream of phenomena, information,nd data that need to be organized (Chia, 2000). Based on their prior experience and contextual anchors in the situation,hey infer a frame in order to organize and interpret the data. As Klein et al. (2007) note, “The data identify the relevant

rame, and the frame determines which data are noticed. Neither of these comes first. The data elicit and help construct therame; the frame defines, connects, and filters the data” (p. 118).

Another basic concept used in the sense-making literature is that of an individual in a situation (that is situated in a socio-istorical context) who encounters a gap in his or her information or understanding, and gathers and interprets information

170 A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179

Fig. 1. An illustration of the sense-making process.

in order to bridge the gap (Dervin, 1983, 1989; Perttula, 1994; Savolainen, 1993; Solomon, 1997; Wilson, 1997). A gap is“an unclear aspect of a situation that a person feels the need to clarify in order to continue movement in a direction thatthe individual considers to be constructive or desirable” (Halpern & Nilan, 1988, 170). We characterize a gap in terms of aquestion that must be answered in order for the student to engage in or construct meaning for the mathematical situationor activity. Although a student may not consciously ask such a question or be explicitly aware of the existence of a gap,researchers may identify the existence and nature of a gap as it manifests itself in a student’s subsequent behavior. Viewinga gap in terms of a question provides a useful tool for researchers to summarize the hypothesized gap.

We view the use of a sense-making frame and the development and resolutions of gaps as complementary ideas. Sincethe sense-making frame supports and constrains the way a student interacts with data from the lecture, it influences thenature of any gaps that may arise. After encountering a gap, the student searches for relevant elements of the situation bynoticing particular pieces of speech, writing, and gestures and mentally grouping (i.e., bracketing) pieces together (Magala,1997; Weick, Sutcliffe, & Obstfeld, 2005). The student then constructs a bridge to address the gap by drawing upon his orher ideas, thoughts, conceptions, attitudes, beliefs, intuitions, and knowledge. In this process, the frame facilitates both thesearch for relevant elements and facilitates the construction of the bridge by highlighting significant elements, filtering outirrelevant elements, and structuring the creation of connections between elements. The process of bridging the gap resultsin the construction of meaning for the elements, and (potentially) a revision or reformulation of the frame to account forill-fitting data. Due to the fact that one first must experience phenomena, and notice and bracket the phenomena into events,this construction of meaning necessarily is a retrospective process (Paget, 1988).

The sense-making process is outlined in Fig. 1.

2.3. Other interpretations of frames

Although there has not been substantial work in the mathematics education literature on sense-making, Herbst (e.g.,2006) has adapted Goffman’s work (e.g., 1974) on the ways that the social structure of situations influence talk duringmathematical activity in classrooms. Specifically, Herbst described the effect that various instructional situations—such as“doing proofs,” “solving problems,” and “developing new knowledge”—have on the didactical contract and claims aboutknowledge in the classroom (Aaron & Herbst, 2012; Herbst, Nachlieli, & Chazan, 2011; Herbst, 2006). However, Herbst’sframes are used to make sense of socially accepted and normative ways of acting when engaged in different types ofmathematical activity; his frames are tied to well-established instructional situations and focus on the ways individualsinteract within these situations. In contrast, the work presented here focuses on the cognitive schemas individual studentsuse and the ways these schemas mediate the students’ personal sense-making process.

3. Participants and methods

The participants were six students at a large university in the northeastern United States; they were all enrolled in astandard junior-level abstract algebra course. The instructor had previously taught the course numerous times; he used

lecture as the primary didactical method and did not consult notes while lecturing. The lectures that form the basis for thiscase study included the presentation of definitions, propositions, proofs, and exposition; we view these lectures as broadlyrepresentative of proof-based mathematics classes in general.

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A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179 171

Four lectures were used as the basis for data collection; the lectures were selected from the beginning, middle, andear the end of the semester and included instances of presenting definitions, examples, theorems and proofs. Each lectureas video-recorded, and each student’s notes were collected shortly after the end of each class period. We transcribed the

nstructor’s speech verbatim from the recorded video and his board work was copied during observations and checked forccuracy with the video.

Within a week of each lecture, each student who had attended the lecture participated in a semi-structured interviewith stimulated recall. We adapted two interview protocols that are commonly used in sense-making research: message

/ing and abbreviated timeline (Dervin, 1983; Glazier & Powell, 1992; Spirek, Dervin, Nilan, & Martin, 1999). In message/ing, participants are asked to read a text and stop at places where they have a question (i.e., the gaps they encountered);t each stopping point, they engage in an in-depth analysis of their question. In the abbreviated timeline, participants aresked to focus on selected excerpts from a situation and to describe their perception of the events, the questions that aroseor them, and how they made sense of the situation. While we claim that students engaged in sense-making throughout theecture, in order to make the interviews a reasonable length, we did not ask the students to re-watch and comment on thentire lecture. Instead, to identify stopping points for the message q/ing technique, we used instances where there was aiscrepancy between the student’s notes and what the instructor had written on the board. We interpreted such instancess potentially having arisen from encountering a gap or an interesting perception of the lecture; we also hoped that studentsould be more likely to be able to describe their mental activities at these points. In addition, in order to allow for cross-case

omparison, we also identified and used instances where different students had made different decisions about includingext in their notes. Similarly, to identify excerpts for the abbreviated timeline technique, we selected instances where thenstructor described or instantiated examples of concepts that played significant roles in abstract algebra.

A necessary consequence of this study design is that participants were asked to talk about their sense-making after thenitial sense-making occurred. Consequently, the students’ reconstruction of their sense-making during the interview mayiffer from the meaning they constructed during the lecture. In addition, the students had additional opportunities to makeense of the lecture (via reviewing their notes) in the time between the lecture and the interview. Although the sense-makingrocess is necessarily retrospective, the methods used in this study cannot directly address these two issues. However, wexpect that using the students’ own notes and re-watching the lecture (via video) may have acted to anchor the students’ecollection, and we expect that the students’ descriptions during the interviews should be reasonably congruous with theirense-making that occurred during the lecture.

Although investigating students’ note-taking habits was not the primary focus of the interviews, we viewed their notess a “symbolic mediator between the content taught by the teacher and the knowledge constructed by students” (Castello

Monero, 2005, p. 268); thus, we used students’ note-taking strategies as a lens through which to view their experiencef the lecture and the ways they engaged in sense-making. Specifically, we used the students’ notes to identify points inhe lecture that might be the source of interesting data; to generate interview questions; and as a means to elicit students’escriptions of their own sense-making activity. In addition, we asked students to discuss their note-taking strategy to elicitescriptions of their sense-making practices.

The interview protocol included four main components:

Students were asked to describe their perception of their role as students in class, the aspects of the class the felt weredifficult to follow or record, their note-taking habits in class—including discussing what they write down and their purposefor taking notes, and what they hypothesized the instructor wanted them to include in their notes. They were also askedto identify which of Ryan’s (2001) metaphors for note-taking with which they most closely identified.Students were asked to discuss the discrepancies between their notes and the lecture, including how they decided whatto include in (or leave out of) their notes and the mathematical meaning or significance of the ideas related to thediscrepancies. In this component, we used the discrepancies as stopping points for the message q/ing technique.Students were asked about specific aspects of the mathematical content of each class period, interpreting the symbols,diagrams, terms, and theorems, as well as the relationship between the various representations of the mathematical ideas.In this component we used the abbreviated timeline technique by asking students to “talk through” excerpts of the lectureas they watched it via the recorded video. For example, we would show the student a brief clip of the instructor’s lectureand ask questions such as: “Why do you think he is presenting this example?” or “Why is this particular idea important?”Students were asked to describe the mathematical “big picture” of the lecture, explaining why particular ideas or theoremswere of mathematical significance, how the theorems or examples were related to the concepts, and how they woulddescribe the structure of the proofs that the instructor was presenting.

Although we had an initial framework in mind before beginning analysis, we had not previously identified particular typesf frames or gaps that we anticipated the students might have used. Consequently, we used techniques from grounded theoryn our analysis (Strauss & Corbin, 1994). First, we individually read the interview transcripts, making narrative commentsndicating where students appeared to describe aspects of the ways they made sense of the lecture, barriers to their sense-

aking, or their beliefs about their role or the instructor’s role in the lecture. Based on these comments, we identified annitial set of 30 different codes, which we categorized into five themes: (1) explicit beliefs about the student’s and teacher’soles; (2) modes of classroom engagement; (3) comments on the process and goals of notetaking; (4) modal challengeselated to speech and temporal aspects of the lecture; and (5) interpretations of mathematics content, the reasons for the

172 A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179

importance of the content, and the instructor’s motivation for presenting the content. Based on this fifth theme, we createddescriptions of possible frames and engaged in a second round of independent coding to make connections between theframes and the other codes and themes. Then, we compared codes, refined our descriptions of the frames, and arrived at aidentification of the frames that the students employed.

4. Results

The excerpts reported in this paper are taken from two lectures that focused on aspects of equivalence relations: in thefirst, the instructor discussed the notion of equivalence classes and in the second, he constructed the rational numbers asequivalence classes of pairs of integers; we refer to these, respectively, as Lecture 1 and Lecture 2.

4.1. Sense-making frames

Students employed three types of sense-making frames when they observed the mathematics lecture.

• The first type of frame is content-oriented sense-making. With this frame, students notice mathematical aspects of thesituation (e.g., symbols, definitions, facts, and concepts) and encounter gaps about the meaning of the mathematicalcontent or how to use it in an example that is being presented. They seek to determine whether a mathematical statementis true, to follow each step in a formal proof, or to understand how to apply a definition or determine properties ofmathematical objects. For example, if an instructor creates a specific equivalence relation, the student’s response to thelecture may be interpreted in terms of questions such as “How do the properties of an equivalence relation appear in thisexample?” or “How is this equivalence relation being used in this example?”

• The second type of frame is communication-oriented sense-making. With this frame, students notice the instructor’s spoken,written, and gestural actions for organizing and presenting mathematical ideas. They seek to understand the ways theinstructor is categorizing or connecting ideas, the ideas communicated by board layout, and the instructor’s organizationalcues. For example, if an instructor divides the board into multiple sections (with the intent of presenting a general theoryor argument on one side and specific examples or an informal interpretation on the other (cf. Weber, 2004)), the student’sresponse to the lecture may be interpreted in terms of the question, “What does the separation of the board tell me aboutthe type of content that is in each section?”

• The third type of frame is situating-oriented sense-making. In contrast to content- and communication-oriented frames,which may be characterized by questions of the form “what?” or “how?”, a student uses a situating-oriented frame whentheir engagement with the lecture can be understood in terms of “why?” questions. A student uses a situating-orientedframe when they attempt to construct meaning for events or concepts in terms of the purpose they serve in the lecture.There are two types of situating-oriented frames: mathematical purpose and pedagogical purpose.

◦ A student uses a mathematical purpose frame when they notice mathematical aspects of the situation but, instead of ask-ing about the meaning of the particular concept or implementation of the procedure, they seek to determine why theconcept is useful or why it is mathematically significant. For example, when they see a particular equivalence relation,their response may be interpreted through questions such as, “Why are equivalence relations important?” or “Why areequivalence relations typically used in this way when constructing arguments for proofs?” Similarly, a collection of exam-ples may prompt students to think about the activity that the examples enable us to accomplish and the properties theyshare, reflecting questions like, “Why is the instructor presenting these examples in this collection or ordering?” Althoughconstructing a bridge with this frame will likely involve content- and communication-related data, the gap is related tothe usefulness or significance of the data.

◦ A student uses a pedagogical purpose frame when they notice communicational aspects of the situation and seek to under-stand how the instructor’s pedagogical actions and decisions—such as choosing and ordering lecture content—are relatedto the meaning or significance of the mathematical ideas. For example, if a lecturer presents specific examples beforestating a general rule or definition, the student’s response may be understood through the question, “Why is the instructorpresenting examples before a definition?” Similarly, if the instructor presents a general rule in a new section of the board,the student may think about the aspects of the situation or example that the rule might be highlighting (implicitly) guidedby the question, “Why is the instructor including this rule at this point in the lecture?”. Although constructing a bridgewith this frame may involve content-, communication-, and mathematical-purpose-related data, the gap is related to thepedagogical significance of the action or concept.

In the interview excerpt in Table 1, one student, Jocelyn, demonstrated the use of all three types of frames when discussingLecture 2. During the lecture, when the instructor began talking about the idea that equivalence classes partition sets, hedrew the following diagram on the board, far to the right of the specific examples he had previously been talking about (see

Figs. 2 and 3). Prior to the excerpt, Jocelyn had been discussing the instructor’s example of the equivalence class of (3,4) (i.e.,the fraction ¾ expressed as an ordered pair):

In this excerpt, Jocelyn has already noticed and bracketed numerous elements of the instructor’s speech, writing, andgestures, labeling these “example,” “side-note,” “picture of equivalence classes,” “answers,” “three-fourths” and “equivalence

A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179 173

Table 1Jocelyn’s explanation of the instructor’s diagram.

Interviewer: So what kind of things do you think the diagram is trying to communicate?Jocelyn: Well, he’s trying to show. . . like, that’s not part of this example. It’s like, a side-note thing. He always has a lot of side notes and I don’t have

room on my paper for side notes. . .. So, anyway, that’s a side-note of the. . .he drew a picture of all the equivalence classes. But. . .those weren’t theanswers he was looking for so they were just other examples. So if I just added them in here, I would think that it’s part of this when it’s really notpart of that.

Interviewer: So going back, that might actually cause. . .Jocelyn: Cause confusion, yeah.

Interviewer: Confusion instead of making it clear what’s going on. That’s interesting.Jocelyn: Like here, like this I know three-fourths is the equivalence class like, he was writing like, different things that could be equivalence classes

and not saying if they were or anything, so. . .like, if he was more definite about something, like if he said, “These are the equivalence classes,” then Iwould have wrote them down probably.

. . .Jocelyn: But I don’t, yeah, I don’t copy a lot of the side notes, because that just confuses me in the end.

Interviewer: I see. So you try to stay straight to the. . .Jocelyn: Yeah

Interviewer: . . .what’s actually important.Jocelyn: Yeah, what I think is important.

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Fig. 2. An illustration of the board and the instructor’s diagram.

lass.” These various elements appear to have led to her using multiple frames to make sense of aspects of the lecture inifferent ways.

Several of Jocelyn’s comments suggest she was employing a content-oriented frame. When Jocelyn talked about a “picturef all the equivalence classes,” she was identifying the mathematical concept that the diagram might be related to. Similarly,hen she talked about “three-fourths is the equivalence class” she was identifying an “equivalence class” and connecting

t with her knowledge of the fraction ¾. In other utterances, Jocelyn appeared to be making sense using a communication-riented frame. When she said, “It’s like, a side-note thing,” she was interpreting the instructor’s use of a previously unusedection of the board to draw the diagram as indicating a new aspect of the lecture. Finally, when Jocelyn said, “those weren’the answers he was looking for,” she was trying to understand the instructor’s purpose for including the diagram; this is ineeping with a pedagogical situating-oriented frame. Similarly, when she described identifying “what I think is important,”he was most likely alluding to using a situating-oriented frame to identify the reasons for including examples; she may

ave been using a mathematical frame to try to determine how the examples were related to the “big ideas” of the lesson,r potentially a pedagogical frame to determine the importance (or lack thereof) of something being a “side note.”

Fig. 3. The instructor’s diagram.

174 A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179

Table 2Meredith and Petra describe their note-taking.

Interviewer: He just asked, will a be related to c? I’m trying to understand—how did you decide that that wasn’t worth writing in your notes?Meredith: I guess because I didn’t know what the answer was yet.

Meredith: He just goes on big rants and it’s like okay, I don’t see the connection I don’t see why this would be relevant so, I don’t, I just listen insteadof writing it down.

Interviewer: Are you able to tell me, or at least able to describe. . .you said that there’s stuff he says out loud that sometimes just doesn’t make sense.Can you give me some rule, some way to think about why it doesn’t make sense?

Petra: Hmmm. . . If he’ll just relate it to something else, and I’m not. . . sometimes I just won’t get the connection. But if I do, then I’ll try and write itdown

Table 3Landon explaining his understanding of the situation.

Interviewer: So, looking at what you’ve got, you’ve got almost everything that was on the board, but I do want to point out something that you havethat he didn’t. You actually added a comma and a “but.” It seems really small, but I’m still going to ask you: Why’d you do it?

Landon: Oh, because. . . um, it seemed to flow better.

Interviewer: Excellent. So what’s the tension here? Why is he contrasting these two things?Landon: Because he defined, well he said, a b, he’s writing it as a b but you have to realize that when you write it like that you can’t, you still say

3/4ths doesn’t equal 9/12ths, you can’t assume that—not 3/4ths; 3 comma 4 doesn’t equal 9 comma 12.

Interviewer: Why not?Landon: Because they’re different ordered pairs.

Interviewer: OK. So that would be a bad thing. But what is it that gets the “but” about it?Landon: Because it’s just looking at that (points) and it’s defining it as a over b, you would just looking at you would define a b equals the same. Like 3

over 4 doesn’t equal 9 over 12 or it, 3 comma 4 equals 9 over 12.

Interviewer: Well, we want it to.Landon: We want it to, but. . .

Interviewer: It doesn’t.Landon: It doesn’t.

4.2. Students’ sense-making practices and connections to note-taking

Of the six students in the study, five claimed to copy what the instructor wrote on the board verbatim, suggesting that theyclaimed the same approach as previously described in literature (Bretzing & Kulhavy, 1981; Kiewra & Fletcher, 1984; Peper& Mayer, 1986). However, the students’ notes differed from the instructor’s writing in numerous places. In reviewing theirnotes and re-watching excerpts from the lectures, the students’ explanations of these discrepancies frequently indicated actsof sense-making. For example, in the interview excerpts in Table 2, Meredith and Petra described reasons some of their notesfrom Lecture 1 differed from the instructor’s writing; both of these students described engaging in sense-making practicesthat included looking for connections between concepts. In particular, they reported not writing ideas in their notes whenthey could not see how the ideas were connected or when they identified a gap for which (they believed) they would notbe able to construct a bridge.

Rather than unthinkingly copying the instructor’s writing, all of the students in the study described note-taking practicesthat indicated they were regularly engaged in making sense of the lecture and the course content.

In addition to omitting aspects of the instructor’s board-work from their notes, there were places in each student’s notesin which they added to what was written on the board; many of these discrepancies occurred when the student appearedto have been engaged in sense-making. For example, in the interview excerpt in Table 3 and Fig. 4, Landon was watching

a segment of Lecture 2 in which the instructor focused on the equivalence of 3/4 and 9/12. In general, Landon tended toomit much of the instructor’s board work from his notes (and, on some days, did not take any notes). Prior to this excerpt,Landon had struggled to accurately describe the mathematical components in the lecture; in particular, he was only able

Fig. 4. Landon’s notes about the equivalence relation.

A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179 175

Table 4Meredith and Petra’s explanations of their note-taking decision.

Meredith: Because I remember him writing that and saying like, yeah that should be true, and him saying it wasn’t. And then I understood that thepoints weren’t the same, but then I didn’t understand, I guess the bigger concept of why the whole thing didn’t, so I didn’t write it. Because I think itwould just confuse me looking back at it.

Petra: I didn’t draw the dotted line because. . .I don’t know why. I didn’t write symmetric and reflexive because he didn’t say what they meant, andthen I didn’t want to just write them and look at my notes and be like, “Wait, what were those?”

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Fig. 5. Petra’s notes about the equivalence relation.

o provide vague descriptions about properties of equivalence classes and was unable to connect the lecturer’s generaliagram (see Fig. 2) with the specific example of the rational numbers. However, at the place in his notes where he haddded the word “but” (see Fig. 4)—a word that was neither written nor spoken by the instructor—he was able to describe theelationship between the specific fractions (i.e., ¾ and 9/12) and the ordered pairs (i.e., (3,4) and (9,12)), and, consequently,he mathematical motivation for using this relationship.

Not only did students engage in in-the-moment sense-making, but they also prepared for future sense-making. Forxample, in the interview excerpt in Table 4, Meredith and Petra described how their goals for using their notes to makeense of the course material outside of class mediated the way they engaged in class. In response to Lecture 2, Meredithndicated that she understood that the fractions ¾ and 9/12 should be equal and that the two points (3, 4) and (9, 12) wereifferent but she could not make sense of the instructor’s reason for discussing these examples. In response to Lecture 1,etra indicated that she could not make sense of terms related to equivalence relations. Both students expressed a beliefhat they would face a gap for which they would not be able to construct a bridge and, as a result, decided to not record thenformation in their notes.

.3. The importance of communication- and situating-oriented frames

Although most students regularly used content-oriented frames to make sense of elements of the lecture, some students’pparent failure to use communication- and situating-oriented frames limited their ability to construct a robust under-tanding of the mathematical concepts and their relationship with other mathematical ideas and processes. For example,hen defining the rational numbers as a set of equivalence classes on ordered pairs of integers in Lecture 2, the instructorsed the example of 3/4 and 9/12 to motivate the use of equivalence classes: the fractions are equivalent, but the orderedairs (3,4) and (9,12) are not equal, so their equivalence can be identified by showing that 3 × 12 = 4 × 9. Fig. 5 shows anxcerpt from Petra’s notes in which she copied down most of what the instructor wrote on the board, but did not includehe equation 3 × 12 = 4 × 9. In the excerpt in Table 5, she indicated that she did not include this equation because it was atatement she knew was true. This suggests that Petra had experienced (and bridged) a gap that was only related to theruth of the “multiplication fact” represented by the equation. She did not attempt to connect this “fact” back to the exampler the concept of equivalence classes, potentially reflecting that she did not encounter a gap related to the mathematicalignificance of equivalence relations (in particular, the need to specify the equivalence relation to determine whether twoet elements are equivalent) or the instructor’s choice to include this particular example. This suggests that Petra was not

pplying a situating-oriented frame in this situation.

In contrast to Petra, Tod appeared to use a situating-oriented frame during Lecture 1. In the excerpt in Table 6, Todescribed “something paralinguistic” that told him not to be concerned with the placement of the numbers in the diagramshown in Fig. 6). This suggests that Tod had experienced a gap related to the instructor’s purpose for including the specific

able 5etra’s explanation of the situation.

Interviewer: One of the things that he wrote down is that he wrote right under this that three-fourths equals nine-twelfths and three times twelveequals four times nine, and you didn’t write it down. So, how come?

Petra: It was easy? I don’t know. I know why those are equal, I guess.

Interviewer: And so, it wasn’t going to help you out?Petra: No.

176 A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179

Table 6Tod’s explanation of his modifications to the diagram illustrating equivalence classes.

Interviewer: The first line that he drew was here and this is 2 and negative 2 and then 3 and negative 3 were in the middle. . . the picture is slightlydifferent than what is on the board. How come that’s okay?

Tod: I guess I understood that there was no direction to the circle. That there was no coordinate system to the circle so I, I knew that, I thinksomething about the way that he drew it indicated to me that he wasn’t concerned with where numbers were, just how they related to each otherso that I could put two—I must have, maybe I heard him say 3 and negative 3 first and then when there was a break in the action I saw that therewas 2 and negative 2 so I decided to add those on there.

Interviewer: Because he definitely never said that there was anything about the placement in the circle, but he never said that there wasn’t either. . .Tod: So something paralinguistic indicated to me that he didn’t care what the coordinates were in the circle—yeah, the layout of the circle

Fig. 6. Tod’s modified copy of an illustration of equivalence classes defined by the absolute value.

Table 7Tod’s explanation of his omission of a diagram from his notes.

Interviewer: Cool, and then I actually stopped it here because this triangle is going on and you’ve recorded that there are no triangles, but you haven’tactually drawn the triangle. Help me out.

Tod: Yeah, I think he draws this a little bit later. Yeah, so, I uh suspect, which I didn’t get, I suspect that something was going too quickly for me and Ieither was writing something else then, maybe I was filling out this, or I decided that it was going quickly and I should hang back for a second andjust listen until the main point came up. Because he’s kind of went off to the side, he said, well hold on, let’s think about this for a second. Often thatindicates for a me that I can listen for a little bit and then pick up the main point and put that in there so I suspect that there is some time in theblank space here during which I listened then I realized that I could sum up what he was saying, that’s something that I do on a regular basis thatkind of figure out a quick sentence, what sentence summarizes this whole thing regardless of whether he says it or not. I figure that, I decided that Icould sum in this sentence and at least I would mention the triangle and when I went back I would be able to figure out pretty quickly that thiscould also be drawn as a triangle and it would have the same effect, it would get the same point across.

Interviewer: What does, what’s the triangle representing in this case?Tod: The triangle is representing the fact that there can be no distinct integers A, B and C that are, such that A is, in equivalence right? Yeah, since A is

equal to B and B is equal to C then A is equal to C because if A is equal to B and B is equal to C then A must be equal to C so that other line issuperfluous.

Interviewer: okay, so, there’s a particular property of equivalence relations that he was using this diagram to illustrate. . ., which one is that?Tod: Oh—transitivity, yes, yeah, yeah.

numerical examples (i.e., that they were simply illustrating a type of relationship between numbers) and was using apedagogical situating-oriented frame to make sense of the data.

Although communication-oriented frames do not have the same direct connection to mathematical concepts and pro-cesses as content- and situating-oriented frames, they are nonetheless important for supporting mathematical sense-making.For example, in the excerpt in Table 7, Tod described the reasons for why a triangle-shaped diagram (shown in Fig. 7) that the

instructor wrote on the board did not appear in his notes. He indicated that he focused on the instructor’s use of the boardspace—specifically, writing on the board “off to the side.” Tod used a communication-oriented frame to make sense of theinstructor’s action as indicating that an illustrative example or additional bit of detail was being presented; this information

Fig. 7. The instructor’s triangle-shaped diagram that Tod omitted from his notes.

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A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179 177

asn’t part of the “main point,” so it was something that he could “summarize” retrospectively and could return to his notesater to fill in the connections to the main mathematical ideas.

By using these communication- and situating-oriented frames, Tod was able to connect the diagram and examples tohe broader concept of an equivalence relation, to identify abstract properties of the equivalence relation (i.e., transitivity)nd connect these to the specific examples, to recognize and reflect on his own sense-making process and make proactiveecisions about which activities were most important for him to encapsulate the result of this process.

Both Tod (in Table 7) and Jocelyn (in Table 1) made sense of the instructor’s use of board space, with Tod describing how thenstructor “went off to the side” and Jocelyn discussing “a side-note thing.” They both appear to have used communication-riented frames to identify the instructor’s work in this new section of the board as distinct from what he had previouslyeen discussing. However, their use of other frames led to them making sense of the lecture elements in different ways.od appeared to focus on the instructor’s motivation (i.e., thinking about “something paralinguistic” and the instructor’smplied comment to “hold on, let’s think about this for a second”), suggesting he had been using a situating-oriented frameor pedagogy; he had also made sense of the “big idea” (i.e., transitivity) that was being suggested by a triangle in the diagramnd thinking about how that property could be used to structure the diagram in Fig. 5, suggesting he had also been using

situating-oriented frame for mathematics. In contrast, Jocelyn described the instructor’s board-use as indicating that theontent it contained simply showed additional details. This conclusion may be the product of a communication-orientedrame, with Jocelyn believing that any “side note” was superfluous to the main ideas of the lecture. Alternatively, she mayave encountered a gap related to the instructor’s motivation for including the “side note” but failed to use an appropriateontent-oriented frame to make sense of the mathematical connections between the “side note” and the main ideas. Althoughoth Tod and Jocelyn experienced the same presentation, Tod appeared to have made significant mathematical sense of thelements of the lecture, while Jocelyn’s use of frames led to missed opportunities for learning.

. Discussion

The perspective of sense-making adds to our understanding of the ways students engage with and come to understandathematics lectures. In contrast to prior research, which has focused on students’ ability to take verbatim notes or record and

nternalize particular “lecture points,” a sense-making perspective focuses on the ways students notice and frame elementsf the lecture, the questions that they generate based on the data and the frame, the ideas they draw upon to answer theseuestions, and the type of meaning they construct as a result of this process. In particular, this perspective highlights theeflexive relationship between the data and the frame—how the students’ questions influence what they notice, which inurn influences the way they make sense of the situation.

A main result of this study was to identify three types of sense-making frames that students appeared to employ in ordero make sense of various aspects of the lecture. Specifically, students used a content-oriented frame to notice and bridge gapselated to the mathematical content of the lecture; a communication-oriented frame to notice and bridge gaps related to thenstructor’s presentation of the content; and a situating-oriented frame to notice and bridge gaps related to the purpose thathe content and communicational aspects serve in the class. The lectures upon which the interviews were based includedeatures that are common to most proof-based mathematics classes (i.e., propositions, theorems, proofs, examples, andxposition), so we anticipate that students in other upper-level classes might also use these frames. However, it is possiblehat students in other classes—particularly in classes that rely on pedagogical techniques other than lecture—might employther types of frames as well.

The idea of a sense-making frame is particularly useful because it enables us to specify the aspects of the lecture to whichhe students attend, to anticipate the types of meaning the students might construct, and to identify missed opportunities for

aking sense of the lecture. The students in this study appeared to draw on the three types of frames in different ways and atifferent times. In some of the excerpts, it appeared that students were able to construct a richer mathematical meaning forspects of the lecture when they used multiple frames—in particular, the communication and situating-oriented frames—toake sense of a particular collection of events in the lecture.The students in this study described routinely attempting to make sense of the lecture while they observed it. This is in

ontrast to results of prior research as well as some of the students’ professed beliefs that their goal in observing a lectures to take verbatim notes. Related to this, one of our most important findings is that the students in this study often omittedspects of the instructor’s writing from their notes when they did not understand the mathematical content, the purposeor including it in the lecture, or how the content was related to other mathematical ideas. Similarly, students who didot use particular frames may not have constructed meaning for the mathematical significance of various concepts. Thisesult presents a dilemma to instructors who view lecturing as “an efficient way for the professor to dictate the pace andonvey his vision to the students, on the condition that students would do their share of groping and staggering towardhe goal [of learning the mathematical content] on their own” (Wu, 1999, p. 5). Specifically, if students construct a differentnderstanding of the mathematical and pedagogical content of the lecture than the instructor, then they will have difficultyeeting the lecturer’s learning goals in such a pedagogical model.

The students’ note-taking practices appeared to be connected to their sense-making practices. They reported taking notes

o support out-of-class sense-making, and we found that many of the discrepancies between the instructor’s written worknd students’ notes indicated instances of sense-making. In addition, the students also appeared to make conscious decisionsbout note-taking that reflected a belief that they had encountered a gap that they felt they could not bridge outside of class.

178 A. Weinberg et al. / Journal of Mathematical Behavior 33 (2014) 168– 179

Although compiling accurate notes is not necessarily the primary goal of observing a lecture, these notes play an importantrole in the students’ learning by mediating the construction of knowledge, so it is important to understand the relationshipbetween note-taking and sense-making.

We hypothesize that students’ sense-making practices are a key component of their opportunity to learn from observinga lecture, that is, the “circumstances that allow students to engage in and spend time on academic tasks” (National ResearchCouncil, 2001, p. 333). The data collected in this study show how the type of meaning that students may construct from alecture can be understood in terms of the sense-making frame that they bring to a particular set of phenomena. For example,Tod used a situating-oriented frame when making connections between specific examples and broader concepts aboutequivalence relations; this frame supported his making sense of particular aspects of the lecture and led to his engagingwith important mathematical aspects of the situation. Although all of the students in this study were successful in theirmathematics classes (e.g., all had, at least, a 3.2 cumulative GPA in both their mathematics and non-mathematics classes),we hypothesize that the students who are most successful may be the ones who are more likely to flexibly select and switchbetween various sense-making frames.

It also appears that it is possible for two students to apply similar sense-making frames but construct different kindsof meaning; this meaning may be poorly aligned with the meaning that would be constructed by an expert observer. Wehypothesize that these differences may result from the demands that the lecture itself creates in order for an observer toconstruct certain types of meaning, and individual students’ abilities to meet these demands. Specifically, a student may needto possess certain knowledge or to know how to interpret symbols, words, and gestures in specific ways to fully understandthe ideas in the lecture. The idea of the implied observer (outlined in Fukawa-Connelly, Weinberg, Wiesner, Berube, & Gray,2012) provides a framework for describing these demands in terms of codes, competencies and behaviors required of anobserver.

The work of Herbst on instructional situations (e.g., Aaron & Herbst, 2012; Herbst et al., 2011; 2006) further suggests thatthe frames that students employ in a lecture may be influenced by aspects of the lecture itself. For example, Aaron and Herbst(2012) showed that when the instructor is presenting a definition, students might have different expectations about the typeof mathematical thinking in which they will engage than if the instructor were motivating a proposition or presenting a proof.An area for future research is to explore whether students have multiple content-oriented frames that they invoke dependingupon the particular piece of mathematics that the professor is presenting. Moreover, this study expanded upon the workof Aaron and Herbst by showing that students also use pedagogical and communicative frames to make sense of the class.The professor in this study used different parts of the board for different types of mathematical content, likely promptinggreat use of a communication-oriented frame due to the clear and consistent use of board organization. For example, Jocelynused the placement of text on the far right side of the board as a cue to consistently invoke a communication-orientedframe to make sense of that content as being unrelated to the main themes of the instructor’s presentation. A larger-scaleinvestigation of such behavior is could have broad implications for instructors’ pedagogical practices in lectures.

The framework presented in this paper represents an initial effort to describe the ways students make sense of mathe-matics lectures in terms of the sense-making frames they employ. The exploratory nature of the study limited our abilityto investigate how and why students employed specific frames in specific instances. In addition, since we asked studentsto describe their sense-making at researcher-selected instances and we limited our analysis to two lectures, we may nothave captured exhaustive or completely representative descriptions of the students’ sense-making practices. Thus, there area number of open questions about students’ use of sense-making frames in mathematics lectures, such as: are students ingeneral more likely to use some sense-making frames than others? Under what circumstances do individual students usedifferent sense-making frames? What relationships are there between a student’s mathematical knowledge and their framechoice? How are students’ choices influenced by their own beliefs about mathematics? Is there a relationship between thecontent of the lecture and the students’ choice of frame? Finally, in what ways might an instructor’s method of presentationinfluence the frames students draw upon?

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