Can slope be negative in 3-space? Studying concept image of slope through collective definition construction

Can slope be negative in 3-space? Studying concept imageof slope through collective definition construction

Deborah Moore-Russo & AnnaMarie Conner &

Kristina I. Rugg

Published online: 23 October 2010# Springer Science+Business Media B.V. 2010

Abstract Developing deep conceptual understanding of what Ma (1999) calls fundamentalmathematics is a well-accepted goal of teacher education. This paper presents a microanalysis ofan intriguing episode within a course designed to encourage such understanding. An adaptationof Krummheuer’s (1995) elaboration of Toulmin’s (1958/2003) diagrams is used to examinevideo recordings and transcripts of a group of graduate students in secondary mathematicseducation grappling with the idea of a three-dimensional line having negative slope. The graduatestudents’ understandings of slope are examined using an expansion of Stump’s (1999, 2001b)categories of conceptions of slope. The episode ends in an interesting impasse, in which thegraduate students agree to pursue the idea no further, purposely ignoring the question of negativeslope, despite the clear intention of the task. The analysis explores the argumentation, factors ofthe learning environment, and conceptions of slope that may have contributed to this impasse.

Keywords Secondary mathematics teachers . Slope . Teacher education . Concept image .

Argumentation

1 Background

Slope is a universal topic in mathematics curricula that is usually introduced with lines orlinear functions. Its importance for describing the behavior of a curve and essential role inthe development of derivative are undeniable. Yet, as a secondary mathematics topic, it hasthe peculiar fate of being well-known but not well understood. Ironically this prominentconcept has received scant scrutiny; however, the research literature that does address slopemakes valuable contributions to understanding this complex concept.

Walter and Gerson (2007) suggest that the emphasis of the mnemonic phrase “rise-over-run” has contributed to an instrumental understanding of slope to the extent that studentsare poorly equipped to make connections between slope and line position or slope and rate

Educ Stud Math (2011) 76:3–21DOI 10.1007/s10649-010-9277-y

D. Moore-Russo (*) : K. I. RuggUniversity at Buffalo, The State University of New York, 566 Baldy Hall, Buffalo, NY, USAe-mail: [email protected]

A. ConnerUniversity of Georgia, 105 Aderhold Hall, Athens, GA 30602, USAe-mail: [email protected]

of change. Stump (1997, 2001a, b) has proposed representations for the concept in algebra,geometry, trigonometry, and calculus and in real-world situations as either static, physicalrepresentations or dynamic, functional representations. Her findings point to geometricratios as the dominating representation teachers use to think about slope. Stump (1999) alsonoted that teachers expressed concern with students’ understanding of slope, but the studentdifficulties they identified focused on procedures for determining slope rather thandeveloping conceptual notions of slope.

Based on her research results, Stump (1999, p. 142) suggested, “…both preservice andinservice mathematics teachers need opportunities to examine the concept of slope, toreflect on its definition, [and] to construct connections among its various representations.”One mechanism that allows teacher educators to provide such opportunities is definitionconstruction. De Villiers (1998) suggested that teachers should be involved in definingrather than presenting learners with finalized geometrical definitions. In professionaldevelopment, definition construction has been determined to be a means to enhance theconceptual understanding of mathematics teachers (Leikin & Winicki-Landman, 2001). Thegoal of the current study is to look at the collective argumentation surrounding graduatestudents’ engagement in a collaborative definition construction task to better understandhow the concept of slope in three dimensions is understood.

1.1 Conceptual framework

Following Voigt (1995), Yackel (2002), and others, we adopt an interactionist perspective.One of the primary tenets of interactionism is that “cultural and social dimensions are…intrinsic to the learning of mathematics” (Voigt, 1995, p. 164). This does not negate theindividual’s perspective, but instead asserts that mathematics learning is an individual and acollective process and as such allows a shift of focus to the social processes that existwithin the mathematics classroom community as influences on and evidence of the learningthat occurs. In accordance with the theory of co-constructivism (Valsiner, 1994), learning isan interplay between social and individual activities; it is the joint construction of anindividual’s ideational framework and “the ‘social others’ who influence the developmentof the individual psychological framework through attempts to communicate ideas”(Vidakovic & Martin, 2004, p. 468).

Meaningful mathematics learning occurs when learners use reflective abstraction as theyaccommodate perturbations, realizations that something is not as expected or does not fitwith past experiences (Cobb, 1994; von Glasersfeld, 1995). An important part of instructionis the introduction of appropriate perturbations to address the incorrect or underdevelopedideas held within learners’ concept images (Tall & Vinner, 1981; Vinner & Dreyfus, 1989).This study documents the perturbation–accommodation cycle as a group of four graduatestudents collectively defined the concept of slope in three dimensions. As illustrated both inthe particular episode and throughout the course of the semester, the discourse in the groupunder study was characterized by Burbules’ (1993) ideas of communicative relationship,participation, commitment, and reciprocity. That is, this discourse occurred betweenstudents who worked as equals to solve problems collaboratively by listening to eachother’s sometimes varied, occasionally conflicting, points of view. Moreover, it is thecollective discourse, when examined with written artifacts produced by the students, thathelps illuminate the individuals’ concept images of slope.

For this study, we use the tools of collective argumentation described by Krummheuer(1995) in order to efficiently represent the ideas expressed and debated by the individuals intheir engagement in mathematical processes as a collective definition for slope in three

4 D. Moore-Russo et al.

dimensions emerged and was negotiated. Our choice of argumentation as a lens for analysiswas influenced by Krummheuer’s (1995) description of student learning throughargumentation and by Yackel’s (2002) description of argumentation resulting in impasse.Krummheuer (1995, 2007), Yackel (2002), and others suggest that students learn frominteracting with each other as they work together in collective argumentation, whichKrummheuer (1995) describes as “a social phenomenon when cooperating individuals triedto adjust their intentions and interpretations by verbally presenting the rationale of theiractions” (p. 229). An examination of collective argumentation among a group of students isconsistent with our interactionist perspective, since such an examination highlights theinteractions of participants as they attempt to reach a collective consensus by verbalizingboth their claims and reasoning. In Krummheuer’s descriptions of collective argumentation,consensus is reached by the group(s) and individual(s) involved. However, Yackel describesa case in which the class “resolves” an impasse in a way in which at least one participant isnot satisfied with the outcome. Our episode of interest presents a slightly different way ofconcluding an episode of argumentation in which each individual was satisfied, for themoment, to end with an impasse, and the group created a collective definition for slope thatwas intentionally ambiguous (Zaslavsky & Shir, 2005).

1.2 Course environment and context of episode

The episode of argumentation that is presented took place among one group of fourstudents in a graduate course offered through a Graduate School of Education to facilitate adeeper understanding of key mathematical concepts and to foster mathematical insightthrough engagement in the authentic practice of mathematical inquiry. Mathematical habitsof mind do not come from teaching the same topic in the same manner year after year, butin exploring how mathematical ideas extend and connect across distinct fields ofmathematics. Teachers should be breaking ground that is new to them, just as they asktheir students to do, challenging themselves to explore mathematics in different ways.

The general goal of the instructor was that students in the class would acquire what Ma(1999) called a “profound understanding of fundamental mathematics” displayed byaccomplished teachers who understood key concepts deeply with a well-organized, richweb of connections among those concepts. To accomplish this goal adhering to the theoryof co-constructivism (Valsiner, 1994; Vidakovic & Martin, 2004) and the philosophy thatlearners need time to pass through the stages that are involved in understanding a complexconcept (Vinner & Dreyfus, 1989), the course instructor provided an environment in whichfundamental mathematics topics like bisection, intersection, and slope were defined and refinedin a three-dimensional environment through extensive interplay of individual activities andsmall group mathematical discussions, as defined by Pirie and Schwarzenberger (1988).Students in the course were given access to a three-dimensional manipulative kit (McGee,Moore-Russo, Lomen, Ebersole & Marin-Quintero, 2008) that allowed them to build x-, y-,and z-axes with points, lines, curves, and planes to aid them in their endeavors. Although itsuse was optional, this manipulative was often employed as students considered figures inthree dimensions.

The instructional design of the course fostered student–student interaction allowingextended involvement in small group work as a means to construct, negotiate, verify, andrefine their mathematical ideas. By philosophical design, instructor interventions werelimited and brief. The instructor circulated, “listening in” on the small group, rarelyattempting to refocus or redirect discussion. Each small group was video-recorded, allowingthe instructor access to all discussions after class.

Can slope be negative in 3-space? Studying concept image of slope 5

The particular goal related to the series of tasks reported was to help connect relatedalgebraic and multivariate calculus concepts. In the primary task under investigation, theinstructor tried to implement Borasi’s (1992) suggestion of beginning with a familiar,individually created, definition for slope and encouraging students to build on and refinetheir definitions to create a single collective definition that would apply in threedimensions. Through extensive small group discourse, students were able to use the actof defining to reflect on the concept of slope, going beyond making mathematical meaningfor slope as applied to lines in the traditional two-dimensional environment, to anexploration of what slope means in other contexts. The students started exploring andsubsequently re-forming their definitions in three dimensions and were then ready todevelop a more flexible and robust concept of slope in subsequent tasks.

During the prior class, the group was given two points in three-dimensional space andasked to find the slope of a segment between them. The students deftly applied thePythagorean theorem twice, once “horizontally” from the first point to a third that containedthe x- and y-coordinates of the second point and the z-coordinate of the first point, then asecond time “vertically” from the third point to the second. The group reported the ratio ofthe two distances was quite similar in three dimensions as it was in two dimensions beingthe “rise over the run”. None of the group members questioned the points’ ordering, whichwould have changed the sign they reported for slope.

The class session under consideration began with the instructor presenting a series oftasks related to slope. The first task was for the students to create individual definitions forslope that would hold in three dimensions; this took about 6 min. The second task askedindividuals to share their definitions with their groups and then come to a consensus andconstruct a single, collective definition of slope in three dimensions; this took about 14 min.In the third task, the group traced the concept of slope through the K-16 curriculum to seewhere it is found either implicitly or explicitly; this took about 11 min. After the groupreported their results to the circulating instructor, she presented a situation for the group toconsider in which intersecting lines had the same gradient as they revisited task 2 in anattempt to refine their collective definition of slope in three dimensions, a discussion thatlasted about 31 min.

While there is not a common definition for slope in three dimensions, the instructor wasexpecting the group to submit an algebraic equation for the slope between two points thatwould hold in three dimensions with at least some indication as to the critical nature that thedirection traversed, the points’ ordering, played in the equation. Without direction, there isno distinction between pairs of lines with the same gradient that intersect or are parallel.

The transition from slope in two dimensions to three dimensions is not common but hasmerit. Parametric equations or vectors usually describe lines in three dimensions. Just asconnections are emphasized in single variable calculus between slope and derivative,connections can be made in multivariate calculus between the slope of the surface in aparticular direction and the directional derivative. The series of tasks helped demonstratethat in a three-dimensional environment both gradient and direction matter. Subsequentclass activities, not reported in this paper, required the group to consider the slope of a planein three dimensions in different directions, eventually connecting to the idea of directionalderivatives.

1.3 Analysis tools

The episode involving the second task was identified as an episode of interest by theresearch team. It began after the individuals had reported their definitions for slope as the


first step in constructing a group definition. The episode concluded with the collectivedecision purposefully to create an ambiguous definition. In contrast to task 1, which wascompleted in silence, and task 3, which involved shared ideas but no argumentation, abarrage of ideas and complex argumentation1 marked the group’s original attempt at task 2.The second task was revisited after instructor intervention, and the discourse then differedmarkedly with cooperative exploration rather than congenial argumentation. For thesereasons, the research team used a Toulmin diagram to analyze the 14-min episode duringwhich the small group initially worked on task 2.

Adaptations of Toulmin’s (1958/2003) diagrams have been used extensively inmathematics education to examine student learning and the role of teachers in collectiveargumentation (e.g., Forman, Larreamendy-Joerns, Stein & Brown, 1998; Krummheuer,1995, 2007; McClain, 2002; Yackel, 2002). Toulmin characterized arguments as consistingof claims, data, warrants, backings, qualifiers, and rebuttals, diagramming them as shown inFig. 1. Through a co-constructivist lens, we see data to be the taken-as-shared ideas thatgroup members accept as fact or do not challenge.

Expanding on traditional diagramming as a means to follow the conceptual developmentof the argument, we used color-coding both in transcripts and the diagram itself to examinethe contributions of the members of the small group engaged in this episode ofargumentation. The coloring allowed the researchers to determine the individualcontributions to the argument. The color-coded transcripts and color-coded diagramoutlining the argumentation in the second task were combined with written artifactsallowing the research team to form multimodal interpretations of video data in an attempt tounpack the individuals’ concept images of slope. We use the term concept image as definedby Tall and Vinner (1981, p. 152) “to describe the total cognitive structure that is associatedwith the concept, which includes all the mental pictures and associated properties andprocesses”, recognizing that an individual’s concept image is fluid and developed over timethrough experiences.

The research team then coded the individuals’ conceptualizations of slope using Stump’s(1999) original seven categories of representations for slope and an eighth category, real-world representations, addressed in her later work (Stump, 2001b). The research teamextended these eight categories, adding three that were identified during data analysis.The three additional categories reflect representations of slope that were apparent in theparticipants’ work but were not captured in the eight original categories. While datafrom a single group are highlighted in this study, the additional categories come fromconsideration of the interactions of all students in the course ensuring that theadditional categories were not unique to a particular participant or group. The ninthcategory addressed slope as a determining property. Students voiced that a unique line isdetermined in two dimensions if you are given point on a line and the slope of the line.They also discussed slope as the determining factor in related concepts such as parallellines and perpendicular lines. The tenth category dealt with slope as a behavior indicatorof a line. Students discussed how positive, zero, and negative slopes characterizedincreasing, constant, and decreasing lines, respectively. They also recognized that theabsolute value of slope denoted the severity of the line’s inclination. The 11th categoryrefers to students’ notion of a linear constant. Students talked about the preservation ofslope when a line is translated. They also recognized constant slope as being unique to

1 Note that defining is an activity in which meaning is assigned to a concept. When done collaboratively, theprocess often involves argumentation, the negotiation and discussion between collaborators with certain,preexisting ideas as they attempt to reach mutually acceptable conclusions.


“straight” or “flat” figures. The coding and categories, recorded in their two-dimensionalform in Table 1, were extended to include their three-dimensional counterparts for ouranalysis. The 11 categories are listed in the same order reported by Stump (1999, 2001b)with the three additional categories added in order of most to least observed.

1.4 Research questions

The episode of argumentation during the second task was interesting and unusual in that thegroup members ended in an impasse, which led to the creation of what the grouprecognized was an ambiguous definition for slope. This is the first and only time this groupreached an impasse during the semester in numerous hours of group tasks. In line with co-constructive theory, other tasks showed a negotiation among group members whereindividual comments or actions revealed changing individual mathematical ideas toaccommodate ideas that were expressed during group discussions. In turn, groupconceptions and collective responses to tasks often underwent a noted metamorphosisduring the course of group discourse to reflect a synthesis of the individual and collectiveideas. What about the act of defining slope to allow for a three-dimensional environmentled to this unusual outcome? Did the students’ understanding of slope in two dimensionscontribute to the impasse? The purpose of this paper is to better understand how and howdeeply the students understood the concept of slope and how their individual under-standings impacted their collective decision.

1.5 Methodology

The research reported here was part of a qualitative study that took place during the firstacademic semester in a research university in the Northeastern USA. The researchdescribed in this paper examined data from 64 consecutive minutes in a single class period,about midway through the semester, in an evening class that was 2 h and 40 min induration. The episode occurred in a small group setting, between secondary mathematics

DATA: Provide support

for the claim (facts or other

information)

Since,

WARRANT: Links the

data to claim by explaining

why the data are relevant

On account of

BACKING: The usually implicit but

commonly held reasons why the warrant

should be accepted by the participants

QUALIFIER: Gives the

strength of the claim (words

such as possibly or

certainly)

CLAIM: Statement

whose truth is

being established

REBUTTAL: Gives

circumstances under which the

claim would not be true

Unless

,

Fig. 1 Toulmin’s diagram (adapted from Toulmin, 1958/2003)


education graduate students who had worked together over substantial periods of time sincethe beginning of the semester. Group members included one inservice teacher (Abby,2

novice teacher with 2 years teaching experience) and three preservice teachers (Paul andDee, both had completed student teaching and were in their final semester of study andKeri, a former engineer in her first semester). All members of the group held bachelor’sdegrees in mathematics.

To collect data, all group work was videotaped during the semester using one videocamera per group. The instructor-researcher and research assistant viewed the videotapesand observation data to identify salient episodes. Both independently identified thefollowing episode as being of interest. Then they shared the data with an “outsideresearcher” from a different university. In the tradition of multimodal analysis (Jewitt, 2008)and its inclusion of gesturing, tool use, tone of voice, and gaze, the research team usedvideotapes to jointly produce a thick, descriptive transcript of the episode. After numerousviewings of the episode, the team decided that a Toulmin diagram’s structure would afford

2 Pseudonyms are used.

Table 1 Conceptualizations of slope

Category Slope as… Coding

Geometric ratio (G) Rise over run G1

Vertical displacement (distance, change) over horizontaldisplacement (distance, change)

G2

Algebraic ratio (A) Change in y over x A1

Representation of ratio with algebraic expressions, y2�y1x2�x1

A2

Physical property (P) Property of line often described using expressions like “steepness”(“slant”, “pitch”, etc.); “how high up” or “it goes up”

P

Functional property (F) Constant rate of change between variables F

Parametric coefficient (PC) Coefficient m in equation y ¼ mxþ b PC

Trigonometric conception (T) Tangent of a line’s angle of inclination T1

Direction component of a vector T2

Calculus conception (C) Limit C1

Derivative C2

Tangent line to a curve at a point C3

Real-world situation (R) Static, physical situation (e.g., wheelchair ramp) R1

Dynamic, functional situation (e.g., distance vs. time) R2

Determining property (D) Property that determines parallel, perpendicular lines D1

Property with which a line can be determined, if you are alsogiven a point

D2

Behavior indicator (B) Real number with sign which indicates increasing, decreasing,horizontal trends of line

B1

Real number with magnitude which indicates amount ofincrease/decrease of line

B2

Real number that, if positive or negative, indicates line mustintersect the x-axis

B3

Linear constant (L) Property that is unaffected by translation L1

Constant property unique to “straight” figures L2

Constant property independent of representation L3


the best representation of the ideas that emerged during the small group discussion. In orderto contextualize the excerpt in the broader context of the series of course activities related toslope, the Toulmin diagram and descriptive transcript were used in conjunction with thecomplementary data from tasks 1, 3, and 2 (revisited), which were compiled in a matrixdenoting the individuals’ conceptualizations of slope.

To construct the Toulmin diagram, the instructor-researcher first considered the episodeof argumentation individually and diagrammed the basic structure of the arguments. Eachresearcher then reviewed the video and revised the diagram; the research team discusseddiscrepancies until agreement was reached. The instructor-researcher added color3 to denotethe individual contributors (and their order of contributions) of each part of the argumentand color-coded the linear transcript so it could be compared to the diagram (with theargument structure but no record of time). The diagram was examined for participationpatterns, argument structure, and key ideas development related to the concept of slope.Initial interpretations were filtered through multiple visits to the video and students’ writtenartifacts. This iterative process continued until the research team came to agreement as tohow the episode should be interpreted.

A slope conceptualization matrix helped the research team identify the evokedconcept images evidenced by the individuals’ comments and gestures on all tasks. Fora final analysis, the research team then compared results to the individuals’ homeworksubmissions of individually generated problems related to slope, group solutions (andthe discussion that led to those solutions) for other problems related to slope,individual reflections on the role of definition construction, and individual exitinterviews. The research team combed through the videos and transcripts repeatedly toidentify an exhaustive listing of the conceptualizations evident for each groupmember.

2 Findings

In this section, we present the data identified as crucial to answering the researchquestions. The Toulmin diagram that represents the argumentation and key conceptsraised in task 2 and the evoked conceptual image matrix based on evidence from theentire series of slope tasks are provided. In addition, excerpts from the transcriptrepresenting brief snippets of dialogue and student submissions of work are also sharedto further illuminate the individuals’ understandings about slope and the reason for theimpasse.

2.1 Toulmin diagram for episode of interest

The episode of interest provides a rare glimpse of graduate student interactions that isnot found in the published work on argumentation in mathematics classrooms. It is arecord of student–student interactions without instructor intervention (other than to posethe task). One thing that was immediately apparent from the diagram was the highincidence of rebuttals that occurred without qualifiers, which is a departure from mostreported research in mathematics education. Inglis, Mejia-Ramos and Simpson (2007)describe arguments by graduate students in mathematics who frequently used bothqualifiers and rebuttals in individually constructed mathematical arguments, which

3 References to color indicate work that was done in the analysis process.


occurred in one-on-one interviews. In contrast to this, task 2 discussion is a complexcollective argument, consisting of several connected claims and counter-claims alongwith an argument that is not treated as directly related to the others. The lack of qualifiersmay be due to an implicit understanding that statements are to be held as tentative (andthus implicitly qualified) until consensus is reached.

As is recorded in Fig. 2, the main claim in this episode is that slope can be negative in3-space. This claim is supported by four sets of data, most with at least one rebuttal. Thecounter-claim is that slope is always positive in 3-space. This claim is supported by twosets of data, and the primary argument contains a rebuttal that leads to a separate claimthat the definition of slope cannot rely entirely on distance.

The episode begins just after the group has finished the first task of creating theirindividual definitions. The group starts the second task with Abby stating her individualdefinition with the intention that each group member will also do so.

2.2 Transcript excerpt preceding “episode of interest”

Abby [reading her paper, others attending to her]: Slope is the vertical changebetween two points divided by the horizontal change between two points. In twodimensions the vertical change is the change in y, and the horizontal change is thechange in x. In three dimensions the vertical change is the change in z, and thehorizontal change, and this is where I am getting stuck up, is the change in (pause)the distance between the two points [using pinched hands to denote two imaginary

Slope is rise over run, where rise is vertical change and run

is horizontal change

You just move from “higher to

lower” in 3-space.

Slope is always positive in 3-space

Counter Claim

Distance should be part of a definition

of slope

Data

Distance is always positive

Warrant

Distance on plane is related to Pythagorean

Theorem

Data

Pythagorean Theorem involves squaring

Warrant

Same slope for nonparallel lines

that “should” have different slopes

DataRebuttal

Warrant

Claim

Slope can be negative in 3-space

Formula for slope in 3-space should include z2-z1 in numerator and

distance on xy-plane in denominator

Data

This line appears to be something youcan “slide down”

Data

Rebuttal Ratio has positive

denominator and can have negative numerator

Warrant

Unless you switch z2-z1 in numerator to

make it positive

Rebuttal

“Negative coordinates,” line “under” xy-plane

Data

Can define line in 2-space with slope and a point, not so in 3-space

Data

Position vectors might be relevant

Claim

Unless you extend line to go “above”

xy-plane

RebuttalUnless distance can be negative in the xy-plane

Rebuttal

KEY Abby: green Dee: pink Keri: yellow Paul: aqua

Claim

Our definition can’t rely only on

distance

Slope can be negative in

2-space

Data

Slope is the change in z over the change in the xy-plane

Unlessformula uses

absolute value

Backing

Warrant

Rebuttal

Unless you realize that you can slide

down any nonvertical,

nonhorizontal line.

Unless distance alone is not enough to

determine slope

Tool represents 3-space accurately

You aren’t limited to just two dimensions.

Backing

Data

Fig. 2 Diagram of argumentation


points], but on the one plane [gestures with flat hand to indicate a horizontal plane,then looks up].

Paul: Yeah, yeah that’s good. [nods, looking at Abby] Same thing, except what I wantto know is can slope be negative in three space? [gaze shifts to paper, looks up whilefinishing question]

Keri: Yes. [looking at Paul]

Paul: How? [looking at Keri]

The contribution patterns and argument structure are presented in Fig. 2. UsingToulmin’s description of arguments allowed the parts of arguments to be identified and thestructure of the argumentation to be analyzed. This, in turn, provided insight into thecontributions of each participant and his or her conceptions of slope. The original claim isthat slope can be negative in three dimensions. Of note is the fact that each individual in thegroup contributed a warrant and backing for the data related to the algebraic formula forslope in three dimensions. These warrants and backings were all related to the idea of slopeas a geometric or an algebraic ratio. It is in this section of the diagram that we see everygroup member contributing variations of slope being “rise-over-run”. However, theargument contains three rebuttals, one collaboratively established by Paul, Keri, and Abbysuggesting that changing the order in which the z-coordinates were subtracted wouldchange the slope of the line from negative to positive. In other words, the expression couldalways be made positive. Hence, at least three members recognized the importance of thepoints’ ordering.

The counter-claim to the original claim is made by Abby, who claims the opposite ofKeri that slope is always positive in 3-space, referring to the use of squaring in thePythagorean theorem that the group had applied repeatedly to calculate the slopebetween two points during a previous task. The argument has data from Dee with abacking that in three dimensions a line is “just going from higher to lower”. Otherwise,this argument is primarily developed by Abby, who contributes data and part of thewarrants (with assistance from Paul). While Paul provided parts of the warrants forboth the main and sub-arguments, he began to rebut the argument that distance is notenough to determine slope. This led to a claim collaboratively constructed by Keri andPaul that the definition cannot rely only on distance with data that when only distanceis considered, nonparallel lines that should have different slopes have the same slope.In this particular claim, both Keri and Paul depended heavily on the accuracy of thetool they were using, the three-dimensional mathematics manipulative, for a warrantusing gestures and the tool to show nonparallel lines that would have the sameinclination. At times, students even contributed rebuttals for data that they contributed.Paul, Keri, and Abby did so for the data related to an algebraic representation thatsupported the original claim.

Keri stated a claim that position vectors might be relevant and provided data thatone can define a line in 2-space with a slope and a point, but not so in 3-space. This isthe first and only time during the episode that slope is hinted at having a representationrelated to trigonometry; most other comments in this episode refer to either thealgebraic or geometric depictions of slope. The reaction of the other group members toKeri’s claim was silence and then a comment by Abby that she had not seen positionvectors used in mathematics. The group continued to talk about multiple lines and


planes with the same slope, after which Keri gave her data by stating that you define aline with a point and slope in 2-space but not in 3-space. Paul responded, “Yeah” butwas focused on the three-dimensional model he was assembling. Abby acknowledgedKeri’s data about not being able to define a line with a point and a slope in 3-space,then Keri continued to give her data by directing the others’ attention to the fact thatshe tried to give a point and a slope (to determine a line) and realized that she couldnot because “it could go in any direction.” Even though Paul and Abby verballyrecognized Keri’s data, her claim (and thus her argument) was not pursued further bythe collective. In response to Keri introducing vectors to the discussion, Paul comparedit to the idea that multiple planes all have different dihedral angles, but none of theother group members asked what he meant or how this idea related to vectors, and hisstatement could be classified as an unrelated musing but not a claim, rebuttal, warrant,or backing. Since vectors are also commonly introduced to deal with three-dimensionalconcepts in multivariate calculus classes, it is notable that Keri’s claim is left“stranded” and does not result in further exploration of vectors.

The structure of the episode of argumentation, then, is two opposing claims aboutwhether slope can be negative in three dimensions, both of which contain rebuttals, one ofwhich leads to a claim that the group’s definition cannot rely only on distance. Since theother constructive claim (the one Keri made) is effectively ignored, the group is left with aclaim about what the definition should not be rather than what it should include. This leftthe students in an impasse situation, one from which they felt they should not moveforward, as illustrated by the following excerpt toward end of the episode, just before thegroup started composing the definition.

2.3 Transcript excerpt (10:45–11:17 min into task 2)

Abby: I don't think we necessarily need to discuss in our definition if it could bepositive or negative... I think we should just go sort of with a combo... [Everyonelooking down, Paul just left room, Kate’s voice is tentative, Abby looks up then downas statement trails off.]

Silence for 2 s

Dee: I think we should just say what it is... and not include a formula, because thenthat way we don’t have to define whether there is a positive or negative slope[looking up, voice becomes louder]

Abby: negative slope *simultaneously*

Dee: We’re just defining

Abby: Right

Dee: What slope is. Okay...

Paul: What did I miss? [Paul enters, everyone turns]

Dee: Now...

Keri: We’re dodging the negative. [laughing, looking at Paul]


Dee: We’re dodging the positive/negative slope [looking at space in center of group]

Abby: Issue

Dee: Aspect *simultaneously*

[…]

Paul: Okay

Dee: And…

Paul: I mean, I’m sure it will come up. [smiling, pleasant tone]

Dee: I’m sure it will come up. [smiling]

Paul: So, lay it to rest.

2.4 Overview of tasks and student products

The group immediately discussed and refined their collective definition, which is groundedin a geometric conceptualization of slope, unanimously agreeing to the followingintentionally ambiguous definition.

SlopeIn 2-D: vertical change divided by the horizontal change, ie. [sic] rise over run, Δy/Δx.In 3-D: vertical change, change in z divided by the horizontal change, which is withrespect to the change in the xy-plane.

Once the group finished the collective definition, they moved to the third task andsketched an outline describing when students use ideas related to the concept of slope in theK-16 mathematics curriculum. The submitted outline below represents the basic ideasaddressed but does not do justice to the rich discussion involved.

Gr. K–2: NothingGr. 3–7: Used implicitly in line graphs and plotting pointsGr. 8: Define slope as m¼Δy Δx= ;m in y ¼ mxþ b; use of real life examples (like aslide)Gr. 9–12: Graph lines, slope of perpendicular and parallel lines, tangents to circles,learn other definitions of slope (rate of change), velocity (common example)Gr. 12–16: Determine instantaneous rate of change, differentiation

Upon completion, the group hailed the instructor who, aware of the struggle withnegative slope in three dimensions, asked about the collective definition. Abby volunteered,“We left out the whole positive, negative issue in 3-D.” Using the three-dimensionalmanipulative, Paul had repeatedly formed the segment with endpoints (0, 0, 0) and (4, 0, 4)then the segment with endpoints (0, 0, 4) and (4, 0, 0) as examples of lines with the sameslope if only rise and run were considered in task 2. The instructor adapted this and formedthe line segment with endpoints (0, 0, 0) and (4, 0, 4) then, keeping one endpoint at theorigin, swept the segment’s other endpoint around z-axis keeping the inclination of thesegment constant (with a rise of four units and a run of four units) stopping periodically to


ask the students what was the slope of the line segment represented. At each point theyresponded that the slope was one. When the instructor asked how many lines passingthrough the origin have a slope of one, Abby replied that infinitely many would. Theinstructor, still rotating the segment and looking from one member of the group to the next,stated, “Something about that doesn’t seem right. This doesn’t happen in the 2-D case,infinitely many lines with the same slope… Think about where slope leads. You’ve justoutlined the K-16 curriculum. Think about how you could define slope so that it doesn’tallow for infinitely many lines that pass through the same point to have the same slope.”

The group collaborated eventually using a polar coordinate system until they reached thefollowing “mathematical definition” (their adopted terminology for a definition withalgebraic equations). While the first definition was intentionally ambiguous, this definitionis unintentionally imprecise since the group failed to recognize that the inverse tangentfunction has a period of π.

Slope

Given points Aðx1; y1; z1Þ and Bðx2; y2; z2Þ such that z2 � z1;A0ðx1 � x1; y1 � y1; z1 � z1Þ ! A0ð0; 0; 0ÞB0ðx2 � x1; y2 � y1; z2 � z1Þ ! B0ðx0; y0; z0Þ

Then slope is

m ¼ ðz2 � z1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx2 � x1Þ2 þ ðy2 � y1Þ2q in the direction q;

where q¼tan�1ðy0=x0Þ; 0 � q � 360�:

2.5 Students’ conceptualization of slope

Even though individuals completed task 1 independently, all relied heavily on notions ofgeometric and algebraic ratios in their definitions. During task 2, all four referencedphysical properties of slope besides evoking geometric and algebraic ratio ideas. To a lesserextent, certain individuals remarked on functional properties and real-world situationsrelated to slope. There were no comments concerning slope as a parametric coefficient or asrelated to calculus. Only Keri alluded to the trigonometric conceptualization of slope in herstranded claim. It was during their analysis of this task that the research team first notedthree types of comments not identified in Stump’s work (the determining properties ofslope, slope as a behavior indicator, and the constant, linear nature of slope).

During task 3, eight conceptualizations of slope were evidenced. This was the only taskin which slope was conceptualized as a parametric constant. Slope as a linear constant andthe trigonometric notion of slope were not mentioned. Surprisingly, since it had dominatedthe discussion in tasks 1 and 2, nobody alluded to slope as a geometric ratio.

When the group revisited task 2, each individual used a trigonometric conceptualizationof slope but only so far as using the tangent of a line’s angle of inclination. Nobody alludedto vectors or their direction components. The group relied heavily on the algebraic andgeometric ratios of slope. They also discussed the physical properties of slope and exploredslope as a determining property, a behavior indicator, and a linear constant.

The group discussions during tasks 2, 3, and 2 (revisited) extended past the initial,individual definitions with their algebraic and geometric ratio emphases. The one


commonality among all members found in warrants and backings during task 2 dealt withthe connections between slope as a geometric and algebraic ratio. Until the instructorintervention, Paul, Dee, and Abby had not discussed any trigonometric conceptualizationsof slope. After the intervention, all four recognized the need to use angles in the definingtask. The minimal intervention coupled with the group’s prior reflection about slope beingnegative seemed to help trigger new aspects of the concept image and to connect these newaspects to ones that had already been raised. Table 2 shows data regarding the evokedconcept images during the series of tasks.

The research team noted that all group members seemed to bestow superiority on algebraicformulas, regarding slope as a geometric ratio as inferior to slope as an algebraic ratio. Paul isthe first to refer to Keri’s initial definition with its algebraic formula as a “mathematicaldefinition”. This terminology was echoed by both Abby and Dee. During task 3, no commentswere made concerning slope as a geometric ratio in the K-16 curriculum.

3 Discussion

While this study was limited to four students, its findings reverberate and expand otherresearch to give the following interpretations credibility. In this section, we revisit thefindings in light of the research questions that drove the study. We also reflect on additionalfindings, placing them in context within the existing literature base.

3.1 Concept images of slope impact argumentation

Yackel (2002) described a situation of impasse in a second-grade class where two studentsdisagreed over the answer to a question. Without teacher intervention, the disagreement wassettled by use of a calculator rather than by argumentation, an outcome that was neithersatisfactory to the observers nor to the dissenting student. In our situation with graduatestudents, the outcome was impasse by decision of the participants. Since they could think ofevidence for and against both the claim and the counter-claim, they decided not to decide(until later intervention by the teacher). We find this outcome to be interesting given thetask in which the students were engaged and their acknowledgement that this outcomecould not be the final one. However, they seemed satisfied and moved forward constructingan ambiguous definition for slope in 3-space. The impasse stands as evidence that the groupmembers had trouble accommodating the perturbation that was experienced when thefamiliar topic of slope was considering in three dimensions.

When considering the factors that led to the impasse, two stand out. First, the group’sconclusion from the claim and counter-claim was a negative conclusion—definition cannotrely only on distance. Rather than concluding that something was needed in the definition,the group decided simply that the definition is not dependent on a single factor. This factor(distance) is crucial in the algebraic and geometric conceptualizations of slope in twodimensions with which they were familiar. The second contributor to the impasse was thestranded claim that position vectors might be relevant. We cannot know why Keri did notrestate this claim later in the argument to address the need for direction or to suggest the useof angles of inclination, although this may suggest that her trigonometric notion of slopewas ill-formed or disconnected from her other conceptualizations of slope. The groupmembers seemed to hold default assumptions related to slope as a geometric ratio. Eventhough these notions were very much a part of each individual’s personal conceptdefinition, they were not regarded as highly as algebraic conceptualizations. In fact task 3’s


Tab

le2

Evo

kedconceptual

imagematrix

Geometry

ratio

Algebraic

ratio

Physical

properties

Fun

ctional

properties

Param

etric

coefficient

Trigonometric

conceptualization

Calculus

conceptualization

Real

world

Determining

property

Behavior

indicator

Linear

constant

Paul

XX

XX

–b–a

XX

XX

Dee

XX

X–a

–b–a

–aX

X

Abb

yX

XX

–a–a

–b–a

–aX

–a–b

Keri

XX

XX

XX

X–b

–b

aNot

evidencedun

tiltask

3bNot

evidenceduntil

task

2(revisited)


discussion was entirely devoid of mention of geometric ratios and had very few commentsrelated to slope as a physical property. This was in sharp contrast to the visual perspectiveemphasis of task 2. Of all the tasks in the series, it was only during task 3 that the groupwas primarily focused on formulas in which slope was an algebraic ratio, a parametriccoefficient, the determining factor for parallel and perpendicular lines, or the definition of aderivative involving limits.

The impasse might have occurred because of the group’s emphasis on algebraicrepresentations over other representations. It could have also occurred due to the group’spreference (as noted over the course of the semester) to work with the three-dimensionalmodel, with gestures, or with other tools in order to aid visualization in three dimensions.However, since no mention of the trigonometric conceptualization of slope surfaced in task3, it seems individuals’ concept images for slope did not trigger trigonometric notions. Thisseems to be the key factor for the impasse. Even during the revisit of task 2, nobody voicedany connection to vectors, although the entire group recognized the need to involve thetangent of the angle of inclination. In interviews with the individual group members aboutthis incident after the end of the semester, Paul and Keri did not recall the incident. Whenasked about vectors, Paul associated them with physics and did not recall seeing them inany undergraduate mathematics. Keri referred only to normal vectors being used todetermine the slope of a plane. Dee did remember the event but did not remember studyingvectors, confessing to know “nothing” about them. Abby also recalled the situation but didnot see the relevance of vectors to it.

The prevalence of geometric ratio has been documented by others (Azcarate, 1992;Stump, 1999), and the lack of a trigonometric conceptualization of slope is consistent withStump’s (1999) findings in which it was the least recorded representation for slope for bothpreservice and inservice teachers. The group’s final collective definition for slope involveda direction angle θ, the angle between the x-axis and the projection of the line on the xy-plane, which is related to Rasslan and Vinner’s (1995) angle for slope. In their concludingremarks after studying how a change in scale affects students understanding of slope,Zaslavsky, Sela and Leron (2002) suggested that “[t]eaching students to ignore [the anglefor slope] or treat it as irrelevant does not contribute to understanding slope and may in factlead to … confusion” (p. 139).

3.2 Importance of pedagogical and social structures of learning environment

Similar to Zaslavsky, Sela and Leron’s (2002) findings, the disequilibrium experienced bythe students in this study “provoked much discussion and reflection and led to re-examination and refinement of the assumptions concerning such basic notions as slope…considerations [that] can be regarded as the kind of mathematical understanding that Ma(1999) discusses with respect to teachers’ knowledge of mathematics” (p. 138). However,had the defining been only an individual assignment, the definition creation process wouldnot have been of much value to the students (as evidenced in task 1). The collective processof defining within an environment that fostered and required student–student interactionwhen joined with the perturbation of working in three dimensions combined for ameaningful mathematical experience. Even though the personal concept definitionsexpressed by the individuals were similar, the instructional design that demanded studentdiscourse demonstrated that the collective moved beyond to explore the plurality ofconceptualizations for slope held by the group members.

It was not just the collective act of defining that was important but also the group’sengagement in creating a minimal outline of slope in the K-16 landscape. The coordination


of all three tasks allowed the individuals to use and connect multiple conceptualizations ofslope. Moreover, each of the tasks provided the instructor critical information about howthe individuals regarded slope, including which conceptualizations of slope were evokedmost frequently and which were most valued in formal educational settings, allowing foreffective intervention.

Zaslavsky and Shir (2005) found that when a defining task was given to a group of fourIsraeli high school seniors (all with sound mathematical backgrounds) in a setting withoutinstructor interference, this created “a rich and stimulating learning environment, in whichthe learners were motivated to interact meaningfully” (p. 339). Our analysis of this episodesuggests that an intervention-free environment may lead to productive argumentation, butthe students may eventually reach an impasse, one that they are either unwilling or unableto navigate (without scaffolding from the instructor). However, critical intervention by theinstructor can provoke additional argumentation and discussion, resulting in interesting andnovel solutions. This suggests that even when the students involved are mature andmathematically advanced, the task is clear and mathematically rich, and the environment isdeliberately structured for productive discourse, appropriate intervention by the teacher maybe crucial to mathematical progress.

It was the minimal, yet directed, instructor intervention that facilitated the groupconsidering the role of direction and angle in how slope was defined. Yackel’s (2002)case and our described situation of impasse illustrate the importance of appropriateintervention by a teacher. By considering slope in a three-dimensional environment, theseries of tasks in this study provided just enough disequilibrium to promote reflection sothat the individuals were able to expand the representations in their personal conceptimages of slope until they were collectively able to connect personal concept definitionswith a formal concept definition. However, even though the series of tasks involvedcollective participation from four individuals, who might be expected to be equipped toask each other appropriate questions to provoke progress on mathematical problems,instructor intervention was required. This might imply that teacher educators need toprovide and model learning environments that allow reflection on and connection ofmultiple fundamental mathematics concepts as well as various conceptualizations andrepresentations of a single concept.

3.3 Unpacking students’ conceptual images of slope

This work builds on Stump’s (1999, 2001a, b) findings to further study how studentsconsider slope. The series of tasks help shed light on the fact that individuals evokedifferent conceptualizations of slope depending on the task at hand. Even though thestudents evoked more visual perspectives when working with the concept using geometricintuition or informal physical properties, they valued formula-driven conceptions of thenotion, especially in the K-16 curriculum.

Our investigation also yielded three additional conceptualizations of slope. The first wasthat slope can be a determining property for a line when taken with a given point, or it canbe the determining property for parallel or perpendicular lines. The second was that slopeindicates the behavior of the line. It can denote the increasing, decreasing, or horizontaltrends of a line and the extent or severity of those trends. The third conceptualization is thatslope serves as a linear constant property. Slope is the fixed property that remainsunchanged under translation. It is the one unique property that gives a figure its“straightness”, and it is a constant property that is present in all representations of linearfunctions.


This study puts forward 11 conceptualizations of slope; however, future research is neededto determine if there are other ways that students consider the concept. Further work is alsoneeded to better explain how the identified conceptualizations of slope are connected.

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Documents

Can slope be negative in 3-space? Studying concept image of slope through collective definition construction