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Factors Influencing ProspectiveTeachers’ Recommendations to Students:Horizons, Hexagons, and HeedAmi Mamolo a & Rebeka Pali ba University of Ontario Institute of Technologyb George Brown CollegePublished online: 05 Feb 2014.

To cite this article: Ami Mamolo & Rebeka Pali (2014) Factors Influencing Prospective Teachers’Recommendations to Students: Horizons, Hexagons, and Heed, Mathematical Thinking and Learning,16:1, 32-50, DOI: 10.1080/10986065.2014.857804

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Mathematical Thinking and Learning, 16: 32–50, 2014Copyright © Taylor & Francis Group, LLCISSN: 1098-6065 print / 1532-7833 onlineDOI: 10.1080/10986065.2014.857804

Factors Influencing Prospective Teachers’Recommendations to Students: Horizons,

Hexagons, and Heed

Ami MamoloUniversity of Ontario Institute of Technology

Rebeka PaliGeorge Brown College

This article examines pre-service secondary school teachers’ responses to a learning situation thatpresented a student’s struggle with determining the area of an irregular hexagon. Responses wereanalyzed in terms of participants’ evoked concept images as related to their knowledge at the mathe-matical horizon, with attention paid toward the influence of one on the other. Specifically, our analysisattends to common features in participants’ understanding of the mathematical task, and explores theinterplay between participants’ personal solving strategies and approaches and their identified pref-erences when advising a student. We conclude with implications for mathematics teacher educationresearch and pedagogy.

A current trend in research in mathematics education has sought to explore the specific facetsof pre-service teachers’ knowledge for teaching (e.g., Ball, Thames, & Phelps, 2008; Davis &Simmt, 2006) as well as how those facets interrelate to influence teaching (e.g., Watson, 2008;Zazkis & Mamolo, 2011). Informed by these works, this article examines pre-service secondaryschool teachers’ responses to a learning situation regarding a student’s struggle to determinethe area of an irregular hexagon. Our aim was to invite participants to consider a situation thatlay outside their repertoire in order to investigate what mathematical knowledge was evokedand “in focus” for participants and how this influenced the recommendations they gave. In thisarticle, we explore the broad research question: What influences pre-service secondary mathe-matics teachers’ recommendations when advising students on how to determine the area of anirregular hexagon? Specifically we attend to the interplay between pre-service teachers’ per-sonal solving strategies and preferences as related to shapes and areas, and their pedagogicalchoices in responding to a student’s struggle. Responses were analyzed in terms of participants’evoked concept images (Tall & Vinner, 1981) as they relate to their knowledge at the mathe-matical horizon (Ball & Bass, 2009; Zazkis & Mamolo, 2011), and how one might influence

Correspondence should be sent to Ami Mamolo, Assistant Professor, University of Ontario Institute of Technology,11 Simcoe Street North, P.O.Box 385, Oshawa, Ontario L1H-7L7, Canada. E-mail: ami.mamolo@uoit.ca

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FACTORS INFLUENCING PROSPECTIVE TEACHERS’ RECOMMENDATIONS TO STUDENTS 33

the other. We also examine the bases of these preferences, with the intent of shedding lighton how this knowledge may inform pre-service teachers’ choices and expectations for studentlearning.

SURVEY OF LITERATURE ON TEACHER KNOWLEDGE

Knowledge required for teaching mathematics has been widely discussed, from a variety ofperspectives. Attention has focused on what knowledge is required in teaching, for teaching,and of teachers (e.g., Adler & Ball, 2009; Davis & Simmt, 2006; Hill, Ball, & Schilling, 2008).A well-known analysis of teacher knowledge was provided by Shulman (1986) in his distinctionbetween subject matter knowledge (SMK) and pedagogical content knowledge (PCK).

In regard to teachers’ SMK, Shulman pointed out that “the teacher need not only understandthat something is so; the teacher must further understand why it is so, on what grounds its warrantcan be asserted, and under what circumstances our belief in its justification can be weakened andeven denied” (1986, p. 9). Ball and colleagues (2008) elaborated that SMK is a needed knowledgeby teachers for the tasks of teaching. Baturo and Nason (1996) mentioned that teachers’ SMKcan impact students’ opportunities to learn. They suggested a breadth in teachers’ SMK that“consists not only in ‘doing’ mathematics to generate ‘correct’ answers. Teachers should alsohave a sense of the mathematical meanings underlying the concepts and processes” (p. 236).Similarly, Ball (1990) noted that “teachers’ subject matter knowledge should not be merely acollection of disparate facts and procedures; instead it should be a collection of interconnectedconcepts and procedures” (as cited in Baturo & Nason, 1996).

In their extensive work on teacher knowledge, Ball and colleagues (2008) refined Shulman’scategorizations of PCK and SMK to include subdomains that offer a more specific windowinto teachers’ knowledge. As discussed in Ball and associates, the refinement of PCK separatedknowledge of the curriculum from knowledge of how to mobilize curriculum. They identifiedas necessary facets of teachers’ PCK a familiarity with students’ ways of thinking, includingcommon student errors (Knowledge of Content and Students), as well as knowledge of effectiveexamples or teaching sequences (Knowledge of Content and Teaching). Their refinement of SMKincluded distinguishing between knowledge for teaching that is common to knowledge required inother professions that use mathematics (Common Content Knowledge), mathematical knowledgespecific to teaching situations, such as how to explain rules and procedures (Specialized ContentKnowledge), and an awareness of how topics are connected across the curriculum (Knowledge atthe Mathematical Horizon).

While their focus was on distinguishing facets of teacher knowledge, Ball and colleagues(2008) acknowledged their interconnected relationships. They suggested that “Teachers who donot themselves know a subject well are not likely to have the [pedagogical content] knowledgethey need to help students learn this content” (p. 404). Similarly, Chinnappan and Lawson (2005)considered teachers’ mathematical knowledge to play a crucial role in influencing the knowl-edge and understanding constructed by students. They suggested that “the lack of integrationbetween different branches of schema knowledge of teachers could impact on their teaching, inthat they and their students might not draw out important distinctions and similarities betweenkey knowledge schemas” (p. 217). Potari and colleagues (2007) observed that robust subjectmatter knowledge allowed teachers to interpret and develop student ideas with greater ease andeffectiveness. They also suggested that teachers’ ability to connect different mathematical areas

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34 MAMOLO AND PALI

and their awareness of the relevance of these connections were part and parcel to their ability toeffectively integrate SMK and PCK to create a rich mathematical learning environment. We seehere a connection to the construct of knowledge at the mathematical horizon (KMH) as it wasdeveloped by Ball and Bass (2009) and extended by Zazkis and Mamolo (2011).

Ball and Bass (2009, p. 6) described the construct of horizon knowledge as a teacher’sknowledge of students’ horizon, which consists of four elements:

1. A sense of the mathematical environment surrounding the current “location” in instruction2. Major disciplinary ideas and structures3. Key mathematical practices4. Core mathematical values and sensibilities

These elements speak to a structural, connected, and robust understanding of mathematics thatgoes beyond what is taught in school curricula. Ball and Bass (2009) described it as knowledgethat can “illuminate and confer a comprehensible sense of the larger significance of what may beonly partially revealed in the mathematics of the moment” (p. 6). They connected this knowledgeto teachers’ abilities to make judgments about mathematical importance, anticipate and makeconnections between ideas, and notice and evaluate mathematical opportunities, among otherabilities. Zazkis and Mamolo (2011) extended this interpretation of horizon knowledge, makingconnections to Husserl’s philosophical views of horizon (Follesdal, 2003), which allows a shiftin focus for the researcher whose attention is now toward the teacher’s horizon. We explore thisinterpretation in detail next. Zazkis and Mamolo presented several examples of teachers mobiliz-ing horizon knowledge in order to promote learning in their students. This KMH influenced theteachers’ understanding of students’ mathematical ideas and guided their subsequent pedagogicaldecisions. Talking about the impact of teachers’ content knowledge on classroom practices, Ball,Lubienski, and Mewborn (2001) highlighted that, “It is not only what mathematics teachers knowbut also how they know it, and what they are able to mobilize mathematically in the course ofteaching” (p. 451).

In this article we explore the horizon knowledge that pre-service secondary mathematics teach-ers were able to mobilize in a hypothetical teaching situation regarding the area of an irregularhexagon, with attention toward how this mobilization was triggered.

THEORETICAL UNDERPINNINGS

As mentioned, our interest is in pre-service teachers’ KMH, its facets, and its influences on teach-ers’ developing pedagogical moves. To shed light on this, we integrate two theoretical constructsthat will allow us to access and interpret pre-service teachers’ thinking. We appeal to Husserl’sphilosophical constructs of inner- and outer-horizons of an object, which were connected toteachers’ KMH by Zazkis and Mamolo (2011), to help interpret different aspects of pre-serviceteachers’ thinking. To gain access to this thinking, we make a connection to Tall and Vinner’s(1981) notion of evoked concept image.

Horizons

Husserl’s description of horizon relates to an individual’s focus of attention—in particular, whenan individual attends to an object (conceptual or physical), the focus of attention centers on the

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object itself, while the “rest of the world” lies in the periphery (Follesdal, 2003). The horizon ofan object thus includes all the features in the periphery. Some features in the horizon of an object,while not at the focus of attention, are specific attributes of the object in question—these relateto Husserl’s notion of the inner-horizon. That which exists in the inner-horizon of an object isdependent on the individual’s choice of focus as he or she attends to the object—what may lie atthe inner-horizon of an object in one instance may be at the center of attention in another, and viceversa. In contrast, the outer-horizon of an object refers not to specific attributes of the particularobject itself, but rather to the “greater world” in which the object exists. That is, the outer-horizonof an object is composed of features that are connected to the object and that embed it in a greaterstructure. It is independent of focus of attention and consists of generalities that are exemplifiedby the particular object.

As mentioned, Zazkis and Mamolo (2011) focused on teachers’ mathematical horizon, andconnected this interpretation with that of Ball and Bass (2009), who (in contrast) discussed teach-ers’ knowledge of students’ mathematical horizons. In particular, they connected inner-horizonwith Ball and Bass’s description of “a kind of peripheral vision” (p. 5) that provides teacherswith a “sense of the mathematical environment surrounding the current “location” in instruc-tion.” A further connection was made between Ball and Bass’s elements of KMH and Husserl’snotion of outer-horizon. For instance, Zazkis and Mamolo identified “major disciplinary ideasand structures as features of the world in which an object lives, yet which are not in and ofthemselves features of the object” (p. 10). This connection provides for us a window into inter-preting aspects of pre-service teachers’ SMK as they relate to what lies in and out of pre-serviceteachers’ focus of attention, and also what aspects of the “broader mathematical world” areaccessed.

At this point, examples are helpful to illustrate what is meant by inner- and outer-horizons.We first consider a physical object, and then turn to a mathematical one. For the physical object,take for instance, a table. At the focus of attention might be the height of the table, its wobbliness,and its color. The inner-horizon would then include all the other aspects of “table” that lie outsideour current focus, either because they are taken for granted or because they are not yet within ourawareness. Such features might include the fact that there is a sticky spot from spilt juice, or that itis a dinner table rather than a coffee table, or that there is room for six to dine. The outer-horizonwould thus include features that are not solely attributes of that specific table, but that are con-nected to the “world of tables.” This might include the class of objects made of wood, or with fourlegs, or with a surface parallel to the floor. In a similar vein, consider as a mathematical objectthe graph of the function y = 2x2 + 5. When attention is focused on, say, its shape (concave up,vertically stretched) and location on the plane (with vertex at (0,5)), the inner-horizon would theninclude all other aspects of that graph, such as the fact that y = 2x2 + 5 has no real roots, is sym-metric about the y-axis, has a minimum but no maximum point, and so on. In the outer-horizonlie aspects that include generalizations and abstractions of the specific graph, such as the setof all graphs of polynomials, differentiable functions, transformations of graphs, techniques forsketching graphs, and so on. These general aspects, which apply to the class of objects in whichour specific graph is contained, comprise only a part of the “greater world” in which an objectexists. Also included in the outer-horizon of a mathematical object are the connections betweendifferent disciplinary strands and contexts. For this parabola, that would include a connectionbetween the “worlds” of calculus and geometry via specific features of curves and surfaces thatexist in different contexts within these two “worlds” (such as differentiability, symmetry, andevenness).

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Concept Images

As we reflected on our own experiences determining the area of an irregular hexagon, we realizeda connection between the evoked aspects of our concept images (Tall & Vinner, 1981) and ourknowledge at the inner- and outer-horizons. The well-known construct of concept image wasdescribed by Tall and Vinner as “the total cognitive structure that is associated with the concept,which includes all the mental pictures and associated properties and processes. It is built upover the years through experiences of all kinds, changing as the individual meets new stimuli andmatures” (1981, p. 152). An evoked concept image is that portion of an individual’s concept imagewhich is activated in a given situation. A concept image need not be coherent or self-consistent,and indeed could contain contradictory features, which may (or may not) remain tacit.

Vinner (1992) suggested that a concept image is “shaped by common experience, typicalexamples, class prototypes, etc.” (p. 200). As such, one may predict features of an individ-ual’s concept image, which are likely to be developed or evoked in a given teaching situation.Bingolbali and Monoghan (2008) elaborated on this idea, noting that classroom experience,teaching practice, and departmental affiliation of undergraduate students each had influence onstudents’ developing concept images. In relation to our research, studies that have attended toindividuals’ concept images in geometry have noted that children classify shapes by using a com-bination of visual prototypes and specific attributes (Clements & Battista, 1992). These attributesdo not necessarily align with the formal properties of the shape, and instead include aspects suchas relative sizes of shapes (Lehrer, Jenkins, & Osana, 1998). Walcott, Mohr, and Kastberg (2009)observed a similar trend, noting that when classifying shapes, children “attended to attributesand properties of the shape including the size (area) of the shapes or the number of sides and/orangles of the shape” (pp. 38–39). They also observed that students who had developed a “static”figural concept of shape held inflexible prototypes of shapes, which were largely determinedby these attributes. These mental images were often in conflict with the formal definition ofthose shapes, and Walcott and colleagues concluded that students’ understanding was “disjointedas students focus on individual parts of the object” (p. 39) rather than viewing it holistically.These conflicts may not be surprising. Ward (2004) similarly observed such conflicts or “gaps”between “a student’s concept image and the concept’s definition as taught by the teacher” (p. 40).She explored prospective K-8 teachers’ concept images of shapes and noted that “their con-cept image of a hexagon was that of a regular, convex, gravity-based hexagon” (p. 48), while“non-traditional” shapes were not likely to be recognized as hexagons. Taking these studies intoaccount, we wondered if individuals with more robust mathematics education would also beinfluenced by factors such as size of shapes, or if they were likely to reason with a shape’s formalproperties.

In this study, we use the construct of evoked concept image as a lens to view what mathe-matics is “in focus” for an individual, and what might lie on the periphery. As such, our analysisof participant responses attends to common features that we identify as aspects of the evokedconcept images of participants, and draw a connection between these features and knowledgeat the inner- and outer-horizons. We then attend to how this horizon knowledge manifested inparticipants’ advice and preferences regarding area computations. While we are aware of manystudies that utilize the construct of evoked concept image to explore learners’ understanding ofspecific concepts in mathematics and science (e.g., Bingolbali & Monoghan, 2008; Moore-Russo,

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Conner, & Rugg, 2011; Saglam, Karaaslan, & Ayas, 2010), we have not to date found studies thatapply this construct to teachers’ developing understanding of SMK and PCK. As such, we offerthe integration of these constructs as a novel lens with which to explore and connect facets ofteachers’ knowledge.

METHODOLOGY

Participants for this study were 20 prospective secondary school mathematics teachers enrolledin a teacher education program. Each participant had the equivalent of either a major or minordegree in mathematics, and professed a high level of confidence with secondary mathematicscontent. Participants were invited to respond to two written questionnaires, each taking approx-imately 30 minutes, and administered one week apart. Participants were informed of the scopeof the questionnaires, which sought to explore their mathematical and pedagogical knowledge,given a hypothetical teaching situation, but they were not made aware of the specific content inadvance, aside from being told that the second questionnaire would follow-up on ideas raisedin the first session. Both questionnaires were administered with the instructions to answer hon-estly and reflectively, that it was appropriate to provide an answer of “I don’t know.” It was alsoemphasized that there were no right or wrong answers.

The first questionnaire, depicted in Figure 1, was designed to uncover participants’ instinc-tive strategies and approaches when advising a student, Delia, on how to compute the area ofan irregular hexagon. Delia was described as a “good student working on an extra-curricularproblem,” and our intent in doing so was two-fold. First, we hoped it would encourage par-ticipants to reason freely with the problem, drawing on intuitive and formal understandings,and perhaps using strategies and ideas that are not specific to the secondary school curriculum(but may be part of elementary or tertiary curricula, for example). Second, we wanted partici-pants to feel comfortable engaging with the mathematics without a perception that we might be“testing” their knowledge of specific curricular content (e.g., if Delia were in this class or thatclass) or their awareness of pedagogical sensitivities (e.g., if Delia had special needs or mathanxiety).

There were several factors that motivated our interest in “Delia’s hexagon.” First, we expectedthat while participants would have substantial experience determining areas of irregular trianglesand quadrilaterals from school, determining the area of an irregular hexagon would not have beenas strongly emphasized and as such would provide a novel context for mathematical engagement.

Imagine you are a teacher in the following situation: Delia, a high school student with good grades, is working on an extra-curricular math problem and approaches you for help. Here is the problem:

You are given a hexagon ABCDEF, where the lengths of the sides are equal to AB = CD = EF = 1 and BC = DE = FA = , and AB is parallel to DE, BC parallel to EF, and CD parallel to FA.

1. What is the measure of each interior angle?2. What is the area of the hexagon?

Delia has found that all of the interior angles are of equal measure, but is unsure how to find the area. How do you recommend Delia go about finding the area?

FIGURE 1 The first questionnaire: Participant recommendations.

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Second, Delia’s hexagon has some symmetry to it, but not too much—visually, it appears “dif-ferent” without being “strange.” In addition, the side lengths—alternating between 1 unit and

√3

units—provide a numerical challenge that requires some attention, and, finally, the area can becomputed with a variety of strategies common to secondary school curricula, some of which aremore easily realized given the specified side lengths. We further saw Delia’s hexagon as an inter-esting context in which to elicit participants’ horizon knowledge. Connecting to Ball and Bass’s(2009) four elements, in the tasks related to Delia’s hexagon (see Figures 1 and 2), requires:(1) a sense of the mathematical environment (definitions, properties, techniques, applicable to theproblem and within the repertoire of a “good” secondary student); (2) major disciplinary ideasand structures (as embedded in the symmetry and irregularity of the hexagon); (3) key mathe-matical practices (relating to ways a shape’s measurements may be determined indirectly); and(4) core mathematical values and sensibilities (as related to the connections between elements2 and 3, namely knowledge of appropriate key practices given the specific structure embedded inthe problem).

At this stage in the data collection, we deliberately omitted a diagram of Delia’s hexagon.We were interested to see how (and whether) participants would introduce their own diagrams,what these might look like, and how they influenced strategies and recommendations. Figure 2presents an accurate depiction of Delia’s hexagon. We anticipated that the process of creatingand using a diagram would be influential in participants’ reasoning, as well as in focusing theirattention. We speculated that there might be a connection between drawing and using a diagram,and the “horizon of the problem”—what is kept in focus or at the periphery of attention couldbe impacted by an individual’s interpretation of what Delia’s hexagon should look like. As such,our analysis tends in part to the possibility of such a connection.

The second questionnaire was, as mentioned, designed to follow up on trends in participants’recommendations to Delia. While we reserve our analysis of these trends for the following sec-tions, we mention briefly some of our observations now, as they relate to the subsequent stage indata collection. Of the 20 participants, 18 drew diagrams and all 20 described “deconstructing”the hexagon into smaller “easier” shapes; these are summarized in Appendix A. One participantalso drew a diagram that extended the hexagon, although he identified this approach as “underused.” Briefly, of the 18 respondents who included diagrams, 15 drew regular hexagons; the twomost commonly observed diagrams are depicted in Figure 3.

In our follow-up questionnaire, we aimed to further probe reasons and preferences behind par-ticipants’ recommendations. Since the majority of recommendations went rather briefly into themathematics (and none verified if their recommendations would actually work, although we notethey were not explicitly asked to do so), we decided to follow up with a questionnaire that invitedparticipants to consider two different recommendations for finding the area of Delia’s hexagon,

A B

C

DE

F

FIGURE 2 A scaled depiction of Delia’s hexagon.

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a) b)

FIGURE 3 Decomposing Delia’s hexagon into (a) two triangles and arectangle, and (b) six equilateral triangles meeting at the center point.

and then to discuss their preferences. Our initial analysis suggested that participants relied tooheavily on the regularity of the depicted hexagon—the reasoning accompanying the diagram waseither inappropriate (e.g., Figure 3a) or incomplete (e.g., Figure 3b) to generalize to an irreg-ular hexagon. We found several of the approaches rather complicated, requiring trigonometry,approximated square roots of square roots, and generally messy computations. These emergedin contrast to our own approach (which involved extending the hexagon, as in Figure 4), andwe wondered what participants would make of our strategy. Based on these observations andcuriosities, we decided that the follow-up should include recommendations: (i) with a diagramof a regular hexagon, and one with an irregular hexagon, (ii) that could apply to both regularand irregular hexagons without introducing any additional mathematics, and (iii) that reflectedand contrasted participants’ inclination to decompose the hexagon. The second questionnaireadministered to participants is presented in Figure 4.

We turn now to a more detailed analysis of participants’ responses to both questionnaires.As the discussion section will illustrate, the trends that emerged from our analysis reflect topicssuch as diagrammatic reasoning, reliance on formulas, and structural features of mathematicalconcepts. In our systematic examination of the data, we attended to cues in language, visual

Recall Delia’s hexagon ABCDEF, with sides lengths AB = CD = EF = 1 and BC = DE = FA =

To determine the area, Delia was given a variety of different recommendations. Here are two of them:

Recommendation A: Extend the hexagon into an equilateral triangle as in the figure below. Then use the areas of the large triangle, and small outer triangles, to determine the area of the inscribed hexagon.

Recommendation B:Decompose the hexagon into three triangles (1, 2, 3, which are all equal), and an equilateral triangle 4, as in the figure below. Then sum the areas of the inscribedtriangles to determine the area of the hexagon.

Which approach do you prefer, and why?

A B

C

DE

F

A B

C

DE

F

FIGURE 4 The second questionnaire: Participant preferences.

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40 MAMOLO AND PALI

representations, and symbolic notations. Our focus is on exploring the connection between pre-service secondary teachers’ horizon knowledge and their strategies and recommendations, as wellas on identifying the bases for why they prefer some recommendations over others.

RESULTS AND ANALYSIS

We begin by reporting on participant responses to the first questionnaire (Figure 1). We firstidentify general trends in their recommendations, and then zoom in to focus on more specificaspects. Following that, we turn our attention to responses to the second questionnaire.

Recommendations for Delia (The First Questionnaire)

We note two prominent trends in how participants interpreted the request to provide a recommen-dation to the hypothetical student Delia:

1. Responses based on broad ideas; and2. Responses giving step-by-step procedures of how to solve.

For example, exemplifying the first trend, Sophia advised that math problems “can always bebroken down into smaller, simpler parts” and that “Delia should be able to put in lines to breakup the hexagon into shapes which we have established rules and laws to work with.” Sophia alsoidentified the importance of “building and developing on prior knowledge and understanding” andnoted that “By high school, Delia should have learned about all the basic shapes and know howto solve for the areas.” In her response, Sophia seems to take the solving strategies of “breakingdown a problem” literally as “breaking up the hexagon” to “smaller, simpler parts.” As such,we identify a connection between her horizon knowledge regarding solving strategies and herfocus of attention regarding the specific mathematical content of the problem. In particular, itwas her knowledge of “key mathematical practices” (Ball and Bass, 2009, p. 5) related to solvingstrategies, which seemed to evoke a particular interpretation of how to work with Delia’s hexagon.This solving strategy was intended to be “simpler,” and took for granted the idea that it would beeasier to work with smaller shapes. This latter idea emerged in several responses, and we discussit in more detail next.

Other responses that were based on broad ideas alluded to solving procedures but did not gointo details. For instance, Aiden responded that, “You need to cut the shape into shapes (triangles).The area of a triangle is 1/2bh. Then you would add all the shapes together.” Such responses tendedto avoid specifics of how to break up the shape (e.g., how many triangles, which triangles) or howto determine the unknown dimensions of the triangles. More significantly, there was no indicationfrom participants that determining these unknowns could be problematic for students. Indeed,regardless of how the hexagon is broken up, determining the base or height of the compositetriangles is not immediate, nor is it a particularly “nice” computation to execute. We may interpretAiden’s avoidance to consider the details behind his recommendation as an indication of limitedor restricted KMH. That is, his attention to the problem stayed focused on the hexagon and itsdecomposition—the knowledge that one can decompose the hexagon to composite shapes maybe viewed as an aspect of the outer horizon, as it connects to the “broader world” of reasoningwith shapes. In his response, Aiden drew a regular hexagon, decomposed into two triangles and

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a rectangle, as in Figure 3a (also Figure c of Appendix A), and, as such, the specific dimensionsof the hexagon seemed to fade into the periphery. We return to this idea later on as we analyze inmore depth participants’ use of regular hexagons in their responses.

Regarding our second observed trend, responses giving step-by-step procedures to solve, wenoted a range of provided details in the solving procedures described by participants. We alsoidentified two main themes that influenced participants’ recommendations:

1. The specific diagram evoked by the question; and2. Knowledge of specific formulas for finding dimensions of composite shapes.

Of these themes, we observed that the first had a more direct influence on how participantsapproached the problem and what were their recommendations, while the second impacted howfar participants were able to go with the recommendation. For the present article, we restrict ourattention to the first theme as participants referred only to one particular formula for calculatingtriangle area: 1/2 × base × height.

By and large, the evoked diagram which was elicited from participants in response to thefirst questionnaire included a regular hexagon decomposed into composite shapes. The specificdiagrams produced by participants, and their frequency, are summarized in Appendix A. Thisobservation is in accordance with prior research on prospective elementary teachers’ conceptimages of polygons (e.g., Ward, 2004). We note that the evoked image of a regular hexagonhad a direct impact on the recommendations given for Delia, particularly with respect to howthe hexagon was decomposed into shapes. Responses that decomposed the hexagon into twosimilar triangles and a rectangle (e.g., Abigail), or six equilateral triangles (e.g., Miles), or twosimilar trapezoids (e.g., Lucas) do not generalize to an irregular hexagon and nor do the associatedsolving strategies proposed by participants. For instance, Abigail gave explicit steps on how Deliacould determine the area—all of which relied on a conceptualization of a regular hexagon. Abigailwrote:

Since Delia already knows two lengths of the triangles, she can easily find the third length of thetriangle which is also the longer side of the rectangle. Then she can solve the area of the two trianglesand the rectangle in the middle using formulas for the areas. Once this is calculated, she can just addthe area of the rectangle and the two triangles and this total will be equal to the area of the entirehexagon.

In her response, Abigail’s focus stays resolutely on her diagram—she draws it in two differentplaces, at the beginning and end of her explanation, and identifies a line of symmetry from onevertex to its antipodal vertex. Through maintaining this focus, the specific side lengths of Delia’shexagon wind up in the periphery of Abigail’s attention, and thus in the inner-horizon. Shiftingthe side lengths to the periphery also seemed to influence Abigail’s assessment of the level ofdifficulty of the problem, which we connect to evaluating mathematical opportunities and makingjudgments about mathematical importance—features Ball and Bass (2009) attributed to teachers’KMH. For example, Abigail takes for granted the ease with which a student could find the areaof the triangle she imagines. In fact, given participants’ exclusive interpretation of the area of atriangle as 1/2 × base × height, determining the unknown length of the triangle would involvea rather messy calculation with cosine law and radicals of radicals, which then lead to furthercalculations to determine the triangle’s height.

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Similarly, participants who decomposed Delia’s hexagon into six equilateral triangles alsotended to focus on their diagrammatic construction, with the specific side lengths fading to theperiphery. Participants who drew six equilateral triangles in a regular hexagon seemed to have anexpectation that this construction would naturally extend to any irregular hexagon. For example,Miles, who offered two possible diagrams—one that decomposed into six triangles and one thatextended to a rectangle (depicted in Figure e of Appendix A)—preferred the decomposition as “itworks for all polygons.” With respect to horizon knowledge, we interpret Miles’ preference for ageneralizable approach as part of the outer horizon, as it suggests an awareness of the disciplinaryideas and structures as well as core values and sensibilities (Ball & Bass, 2009). He noted thathis latter method of extending the hexagon into a rectangle would “allow the student to beginthinking differently and creatively about figures” but that it was a method “often under used insituations where it could be helpful.” Miles appeared to have a more robust KMH than mostparticipants in this study, as he was able to articulate his judgments on mathematical importanceand connections to broader mathematical ideas, both elements of outer horizon. While we foundevidence of Miles’ outer horizon knowledge, we did not find evidence of his knowledge at theinner horizon—for example, the specific concepts necessary to execute the broad ideas he wassuggesting.

An interesting interpretation to Delia’s hexagon came from Lucas. His evoked diagramis pictured in Figure 5, and very clearly shows a regular hexagon. Indeed, he even cre-ated an unconventional (and inappropriate) labeling in order to “force” regularity on Delia’shexagon. We offer this example as an illustration of how robust the prototype of a regularhexagon is in the concept image of prospective teachers, even those with advanced mathematicalbackgrounds.

Of the 20 participants, only one indicated there would be different reasoning involved whendealing with a regular versus irregular shape. We identify the ability to distinguish betweenwhat holds for a regular versus irregular shape as an understanding of major disciplinary ideasand structures (outer horizon), as well as key mathematical practices (outer horizon) regard-ing the use of definitions and resulting properties. As such, participants who omitted or failedto realize such distinctions were interpreted as having limited or restricted KMH. In con-trast, participants who demonstrated a shift in focus of attention toward such distinctions andtheir consequences were interpreted as having more robust KMH. One such example wasOlivia.

A

BE

F

D

C

1

1

1

11

1

√3

√3

FIGURE 5 Lucas’ decomposition of Delia’s hexagon.

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FACTORS INFLUENCING PROSPECTIVE TEACHERS’ RECOMMENDATIONS TO STUDENTS 43

Olivia at first sketched several regular hexagons decomposed as in Figures c and f of AppendixA. The shifts in her focus of attention were well documented by the progression of figures, eachof which refined the earlier attempts. From her early attempts, it was clear that her initial evokedconcept image was of a regular hexagon, and while she meticulously labeled each side of thehexagon, she did not at first connect the numeric values with the side lengths. Alongside herdiagrams she included the letters “SOHCAH” (a mnemonic which stands for Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse) as well as a formula for sine of an angle. As herevoked concept image of a regular hexagon triggered specific solving strategies related to decom-posing the shape, Olivia drew on her knowledge of trigonometric relationships to apply themdeliberately to one of her hexagon constructions (Figure c, Appendix A). The fact that trig couldbe used to gain more information about a shape can be seen as part of the outer horizon (sincethis is part of a “broader world” in which shapes exist), while anticipation of how those cal-culations would pan out with the specific hexagon in question can be seen as part of the innerhorizon (since this is a specific feature of the object in question). Notably, Olivia did not exe-cute these calculations. Rather, she identified a relationship between her sine formula and anarea formula, which again we interpret as applying knowledge from the inner-horizon. She sub-sequently abandoned this construction in favor of one with six equilateral triangles joined at acenter point of the hexagon (such as Abigail’s construction). The abrupt change in focus sug-gests that some part of her KMH motivated a new approach—perhaps she anticipated challengesin the computations, perhaps she felt this line of reasoning was too cumbersome. It is not clearfrom her work exactly what motivated the change; however we can infer that the change wasnot triggered by a realization that her construction of a regular hexagon was incorrect—at leastnot yet.

The new hexagon decomposition into equilateral triangles became the focus of Olivia’s atten-tion, and the formulae that were previously accessed seemed to fade back to the periphery andwere not referred to again. After a rough sketch of this construction, Olivia turned to a new page,drew the same decomposition (very neatly this time), labeled it precisely, and then crossed it outto sketch a messy irregular hexagon with exaggerated side lengths decomposed into six scalenetriangles. For Olivia, it seems as if the precise attention required to draw and label an accurateregular hexagon brought back into focus the specific side lengths of Delia’s hexagon, and thusevoked a new, revised, diagram. While she did not elaborate on how this construction could beused to determine the area of the hexagon, we did find evidence of her KMH when she observedthat this strategy would be “difficult since we have to find the point of interception [sic] of twodiagonals and then connect other two vertices to that point.” Here we interpret Olivia’s innerhorizon knowledge emerging to influence her pedagogical sensitivities—namely her assessmentof the appropriateness of the approach for Delia.

Participants’ Preferences (The Second Questionnaire)

Recall the second questionnaire (depicted in Figure 4) gave participants two choices of rec-ommendations for Delia and asked them to explain their preference. Of the 20 participants,11 preferred Recommendation A (which extended the irregular hexagon into an equilateraltriangle), and 9 preferred Recommendation B (which decomposed a regular hexagon intofour triangles). A common feature in participants’ justification of their preferences for both

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Recommendations A and B was the ease and clarity of the approach. We identified the followingmore specific trends:

1. Participants who preferred Recommendation A attended to structural features andconsequences of the provided diagram; and

2. Participants who preferred Recommendation B attended to surface features of the solvingprocess and their prior personal experiences.

These different considerations led participants to different reasons for why one approach waseasier than the other. The focus on structural rather than surface features has a direct con-nection to KMH—both in terms of Husserl’s notion of outer horizon as well as Ball andBass’s (2009) second and third elements and their reflexive relationship. For example, Sarahexplained:

I prefer recommendation A because the image makes the idea very clear, where recommenda-tion B visually seems more complicated. Also with A, you’re only using equilateral triangles.. . . I know the smaller triangles are equilateral because the larger triangle is equilateral, mak-ing its interior angles 60 degrees by definition, which means the remaining two angles [of thesmall triangles] must be 60 degrees because they are all similar and the sum of all angles is180 degrees.

Accompanying her explanation, Sarah drew the following diagram (Figure 6), noting lines ofsymmetry (a major disciplinary idea) and illustrating her conclusion that the small and largetriangles were similar.

Sarah’s evoked concept image of equilateral triangles included properties of the formal defini-tion (a key mathematical practice) and structural features, which allowed her to assess the level ofdifficulty of this recommendation as opposed to the other. She identified the area of the hexagonas the difference between the areas of the large triangle and three smaller ones, and contrastedthis “easier” approach with recommendation B, for which she wrote: “is unclear and I don’t thinkI could find the side lengths [of the triangles] in this recommendation B. It seems as though thereis a lot more work to finding the areas of the triangles in B.” Sarah’s judgment that “there is alot more work” may be seen as an awareness that some of the important structural features ofrecommendation A were not present in B. Thus, it seems as though her evoked concept image ofthe equilateral triangle, and the associated horizon knowledge, influenced what was in focus forSarah as she considered both recommendations.

The specific diagram and the related mathematical thinking skills required to reason with itwere important considerations for Miles in his selection of recommendation A. Recall that Mileshad previously considered both decomposing and extending the hexagon in order to determine its

A B

C

DE

F

60

6060

1

FIGURE 6 Sarah’s diagrammatic reasoning.

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area, and had noted that extending offered students opportunities to think creatively with skillsthat he deemed as “under used.” In explaining his preference for recommendation A, he againemphasized that:

Negative space thinking is something that is rarely cultivated and can prove very useful in fieldsextending beyond pure mathematics. While method B is applicable for most polygons (dividing thepolygon into various triangles), method A offers a different approach that will get the student to beginthinking of alternative methods to exact the same end.

In this excerpt, we see an element of Miles’ outer horizon knowledge, which situates the specificconcepts and strategies in a world “beyond pure mathematics.” His acknowledgement that thestrategy of recommendation B is broadly applicable is also indication of horizon knowledge,showing an awareness of key mathematical practices (Ball & Bass, 2009) that apply beyond thespecific question (outer horizon). There is also evidence of Miles’ knowledge at the inner horizonwhen he attends to the specifics of the provided diagrams:

It must also be noted that both figures shown are only one possible configuration. In fact, fig-ure B is further from an accurate scale representation than figure A (the sides are not accuratelyproportioned).

His focus on the diagrams attended specifically to their structure and shape—in fact, Miles wasone of only four participants who alluded to diagram B as an inaccurate representation. His needto note that the figures showed only one configuration suggests that, while alternate configurationswere not in focus, they were certainly in his periphery. This led him to observe that even “if theinternal angles aren’t equal, figure A’s approach can still be used. The triangle form though,may not be equilateral, but it will be isosceles.” Here again, we see that Miles is considering thesolving strategies broadly, as they would apply to other irregular hexagons—suggesting that hisconcept image of hexagons included more than the regular prototype, and that he was able toreason with these shapes without having them directly in view (inner-horizon).

Throughout his response, Miles kept his student in mind, and our analysis suggests a possibleconnection between a teacher’s KMH and his or her knowledge of content and students (KCS).We note such a connection in his previously mentioned statement that recommendation A is a“different approach that will get the student to begin thinking of alternative methods,” as well asin his concluding statement:

In any case, both methods should be shown and Delia should be encouraged to question bothapproaches. There is never one way to approach a question—some work better for some than others.Finding these different approaches will allow a student to choose a method that works well for him,and will deepen their understanding of the concept being taught/explored.

While we note limitations in our data collection instruments in terms of elucidating such a con-nection, we suggest that Miles’ KMH seemed to enable him to keep in mind his future students’horizons. His attention to the broader thinking skills cultivated by the strategies from recommen-dation A, as well as the importance of choice and comparison to student thinking, illustrates away that KMH may influence KCS, and vice versa (since the mathematical knowledge evokedand in focus may depend on what one sees as important for student learning).

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Miles’ response also caught our attention as it contrasted sharply with a common responsegiven by participants that preferred recommendation B. Namely, the argument that recommen-dation B is more familiar and comfortable, and more closely connected to solving strategiesparticipants used as students or during their practice teaching. For instance, Lucas (who had pre-viously tried to “force fit” Delia’s hexagon into a regular shape) remarked that: “RecommendationB is easier for me to visualize and comprehend, and therefore I would feel more comfortable rec-ommending this approach.” Lucas showed consideration of students’ ways of thinking when henoted that: “A would be preferred over B for some because A allows students to ‘think out-side of the box’ . . . which in turn could help them solve even more complicated problems inthe future.” However, he concluded that “all-in-all, I prefer Recommendation B, as it makesmore sense to me. It would be what I feel more comfortable suggesting.” Lucas did not get intothe specifics of the recommendations or what they would entail. He focused his preference onthe difference between extending and decomposing and based his decision on his abilities toexplain one over the other. While Lucas was aware of key mathematical practices of extend-ing and decomposing, he did not consider these in light of the major disciplinary ideas andstructures related to generalizing from a regular to irregular shape, and as such, we interpret alimited KMH.

Personal preference and comfort were also important factors for Abigail, who noted that “rec-ommendation B is the approach I would take because of the way I learned geometry. The hexagondivided into triangles is the approach I learned in school.” For Ella who claimed to “like the sec-ond approach, because it is showing Delia that a hexagon is a composite shape” and becauseit “reminds me of a grade 9 class . . . where he [the teacher] asked me to show that a trape-zoid is a composite shape and asked me to guide them [students] to finding the area of it. Themethod I used was similar to Recommendation B.” We see from Ella a sense of the mathemati-cal environment (Ball and Bass, 2009, element one) situating the problem within the secondaryschool curriculum, but note that this horizon knowledge appears limited in its focus on makingconnections to previous experiences, without acknowledging relevant distinctions.

Similar observations were made for Lexi, who chose B because “automatically I can recallthe Pythagorem Theory [sic] to solve for the individual triangles 1, 2, 3, which are all equal”and “I prefer formulas in mathematics, and checking that my work is correct, as opposed toguessing what number relates to what side, which is probably how I would go about solvingapproach A.” It should be noted that the Pythagorean theorem is not applicable to the trian-gles in recommendation B, although it can be useful in determining the area of the equilateraltriangle from recommendation A. Lexi’s strong preference for formulas, and her desire to staywithin the comfort zone of how she would solve, limited her sense of the problem and steered hertoward an inappropriate approach. While we acknowledge that personal comfort and familiarityare certainly valid reasons for choosing one approach over another, we argue that for pre-serviceteachers they should not be the only reason, and further suggest that robust KMH allows pre-service teachers to see beyond their comfort zone to better meet the learning needs of theirstudents.

In general, we found participants’ preferences for B to be limited in scope and to focus onsurface features of the recommendation without taking into consideration any of the details of itsimplementation, such as focusing on decomposing versus extending, formulas versus guessing,and what was learnt in school versus what is most appropriate for this question. Another surfacefeature we noticed is illustrated by Victor’s response:

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I prefer recommendation B. When calculating something like area (or volume), I find it easier to con-ceptualize adding small portions to get the new total portion instead of calculating a larger portionand subtracting from it to arrive at the right answer. Also, using the latter option, only one func-tion is required (addition) vs. using the former option which requires two functions (addition andsubtraction).

This idea that “only adding” makes recommendation B easier or more straightforward was echoedby the majority of participants who chose recommendation B, such as Abigail, who found sub-traction “confusing to me,” Nimah who claimed that “the process of adding comes more naturallyto me than subtraction,” Sebastian who found adding “more straight forward” than “having tofigure out how to subtract,” and Zack who felt it “makes the whole question ‘fit’ really well. . . while ‘A’ doesn’t even attempt to calculate the area directly.” The evoked concept imagesof these participants included broad considerations of the level of difficulty of subtraction ver-sus addition, but avoided any specific consideration of what was being added or subtracted andhow difficult it would be to determine those addends, minuends, or subtrahends. As Nimahspeculated:

I think this has to do with the fact that as children, we are always taught addition before subtrac-tion. Our simple additions come naturally to us and we can automatically recall the answers in ourheads at a young age. But, for some reason, we have a more difficult time recalling our memory forsubtraction.

Attention to the operations of addition versus subtraction, and the evoked ideas connected to par-ticipants’ concept images of these operations, shifted their focus away from the substance of thequestion, and as such, obscured their horizon knowledge of the related concepts. Victor’s insis-tence that smaller shapes rather than larger, and one operation rather than two, are easier to workwith, avoids considering how many and (more importantly) which operations were needed beforeone is ready to add or subtract areas. Nimah’s connection to the simple arithmetic learned aschildren also shifts focus away from the problem, and creates a skewed interpretation of whatis really involved in the solution. We view this focus on surface features as limited horizonknowledge, as well as a resistance to consider deeply an approach that was outside their reper-toire. The two are connected, and in turn connect to how teachers view and guide their students’learning.

CONCLUSION

What influences pre-service secondary mathematics teachers’ recommendations when advis-ing students on how to determine the area of an irregular hexagon? We found several factors.In accord with research done with children (e.g., Clements & Battista, 1992; Lehrer et al.,1998; Walcott et al., 2009), participants with strong mathematics backgrounds also relied onprototypes of regular hexagons to reason about Delia’s hexagon. Visual information from theseprototypes, such as the relative sizes of shapes, as well as superficial features such as the numberof operations, influenced judgments concerning level of difficulty and appropriateness of differ-ent recommendations. Specific features of recommendations were largely ignored by participants,in favor of general observations and personal preferences (e.g., Aiden’s response, questionnaire1; Lexi’s response, questionnaire 2). We highlight the prominent and influential role of regular

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prototypes and personal comforts in individuals with strong mathematics backgrounds as one ofthe contributions of this study.

Ball and Bass (2009) considered that “some horizon knowledge is about topics, some is aboutpractices, and some is about values” (p. 10) and noted that “knowledge of the horizon does notcreate an imperative to act in any particular mathematical direction” (p. 10). To some degree, ourresearch substantiates this claim—as illustrated by participants such as Lucas, who showed evi-dence of KMH but still acted based on personal comfort level. However, for others, knowledge ofthe horizon provided—perhaps not an imperative, but an invitation to act in a particular direction.This was illustrated in our analysis of the first questionnaire, in which participants’ evoked con-cept images influenced what lay in focus of attention and also directed shifts in attention towardan inner horizon, such as with Abigail. We noted a reflexive relationship—knowledge at the hori-zon also influenced what was evoked for participants. This in turn influenced personal solvingstrategies, which then influenced participants’ recommendations and preferences. By integratingthese two constructs—evoked concept image and KMH—we offer a novel lens through which toanalyze and interpret the connections between teachers’ personal mathematical knowledge andthe knowledge they see fit to mobilize in teaching situations.

Our analysis also indicates an important possible connection between a pre-service teacher’shorizon knowledge and his or her knowledge of content and students. We propose that a robustKMH may help individuals see beyond their personal comforts and experiences as students tothink more broadly about the student learning experience. For instance, participants such as Mileswere able to view different aspects of the problem and the recommendations in light of differentcurrent and future experiences for their students. For Miles, horizon knowledge brought to viewways to foster important (for him) student thinking skills—KMH influencing KCS—and whatknowledge was evoked seemed to rely on what he saw as important for student learning—KCSinfluencing KMH. In contrast, we suggest that limited horizon knowledge may be connected to alimited knowledge of student thinking, and vice versa. For instance, participants who focused onsurface features (limited KMH) were unable to accurately predict what would be challenging oreasy about the recommendation for students (suggesting a limited KCS), while participants whofocused on their personal thinking and comforts as students seemed to generalize this broadly tostudents (limited KCS), and this in turn may have influenced how deeply they were willing toconsider the mathematics of an unfamiliar approach (limited KMH). This study identifies a needfor further research regarding the reflexive relationship between an individual’s KMH and KCS.Our ongoing research explores this relationship, with particular attention toward how the formermay influence understanding of common student errors and students’ difficulties with content orways of thinking.

While further research is needed to expound the connections among a pre-service teacher’sevoked concept image, his or her knowledge of student thinking, and his or her knowledge at theinner and outer horizons, our results suggest some pedagogical implications. Namely that keepinga flexible view of what is in focus and what may lie in the horizon is important and should befostered in teacher education programs. As teacher educators, we cannot expect that pre-service(or in-service) teachers have a robust concept image for every concept in the school curriculum.However, we suggest that developing a robust KMH may help pre-service teachers negotiatethrough different aspects of their evoked concept image, which in turn could lead to “better”recommendations for students—ones that are more focused on the mathematical possibilities andstudent thinking rather than on comfortable strategies or surface features.

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APPENDIX A—DISTRIBUTION OF RESPONSES TO QUESTIONNAIRE 1

First QuestionnaireNumber ofResponses Participants’ Diagrammatic Representations

Responses with a single diagram 15/ 20 5 Fig. a

Decomposing the hexagon into composite shapes 2 Fig. b

6 Fig. c

2 Fig. d

Responses with two diagrams 3/20 1 Fig. e

Extending and decomposing the hexagon

Decomposing the hexagon in two different ways 2 Fig. f

Responses with no diagram 2/20

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