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Teachersspatial literacy as visualization, reasoning, and communication Deborah Moore-Russo a, * , Janine M. Viglietti b , Ming Ming Chiu a , Susan M. Bateman a a University at Buffalo, State University of New York, Department of Learning and Instruction, 505 Baldy Hall, Buffalo, NY 14260-1000, USA b Buffalo State, State University of New York, USA highlights < Teaching experience and education have positive effects on spatial literacy. < Using multiple reasoning strategies had positive effects on spatial literacy. < Many teachers failed to make use of multiple reasoning strategies. < Teachers struggled with the vocabulary of spatial objects. < Many teachers do not possess the spatial literacy skills intrinsic to the curriculum. article info Article history: Received 16 March 2012 Received in revised form 28 August 2012 Accepted 30 August 2012 Keywords: Spatial literacy Spatial visualization Spatial reasoning Spatial communication abstract This paper conceptualizes spatial literacy as consisting of three overlapping domains: visualization, reasoning, and communication. By considering these domains, this study explores different aspects of spatial literacy to better understand how a group of mathematics teachers reasoned about spatial tasks. Seventy-ve preservice and inservice teachers worked on problems that involved spatial objects, their properties, and their relationships. Teachersresponses suggested that their spatial literacy skills were underdeveloped with deciencies most evident on problems that were solvable by dimensional reasoning. Poor vocabulary and misconceptions hindered teachersperformance. Teachers who used multiple reasoning strategies were more likely to solve a problem correctly. Ó 2012 Elsevier Ltd. All rights reserved. Spatial thinking can be learned, and it can and should be taught at all levels in the education systemdU.S. National Research Council (NRC) Geographical Science Committee (2006b, p. 3). Spatial literacy supports how we understand and interact with the physical world in which we live (Alsina, 2000; de Lange, 2003) and is needed in a variety of elds. Architects create two- dimensional renderings of three-dimensional structures that construction workers read and interpret. Athletes read two- dimensional diagrams that map out the spatial positioning of players on teams. Dancers enact spatial movements communicated through verbal commands. Researchers (McGraw, 2004) have shown that individuals differ in how they process and perceive two- and three-dimensional stimuli. Students with higher spatial abilities outperform other students in art courses (Haanstra, 1996), are more likely to solve problems and interpret graphs in physics (Kozhevnikov, Hegarty, & Mayer, 2002; Pallrand & Seeber, 1984), and learn more from computer animations of concepts in microbiology (Trindade, Fiolhais, & Almeida, 2002). Technological advances have brought attention to the role of spatial literacy. For example, digitization has reinvented the . spatial terms under which the arts are engaged(Brown, 2003, p. 287). Since GIS (Geographic Information System) technology has become more affordable and available, the role of spatial literacy in geography education has been examined in Australia (McInerney, Berg, Hutchinson, Maude, & Sorensen, 2009), Singapore (Yap, Tan, Zhu, & Wettasinghe, 2008), and Turkey (Incekara, 2010). While spatial literacy aids understanding in many disciplines, it is central in mathematics (NRC, 2006b; Presmeg, 2006). In order to prepare students to excel in numerous elds, teachers must possess spatial literacy skills and the ability to nurture these skills in their students. Since mathematics provides a common language to discuss and represent spatial relationships, it is particularly important that mathematics teachers develop a robust set of general spatial literacy skills (Newcombe, 2006). * Corresponding author. Tel.: þ1 716 645 4069; fax: þ1 716 645 3161. E-mail address: [email protected] (D. Moore-Russo). Contents lists available at SciVerse ScienceDirect Teaching and Teacher Education journal homepage: www.elsevier.com/locate/tate 0742-051X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tate.2012.08.012 Teaching and Teacher Education 29 (2013) 97e109

Teachers' spatial literacy as visualization, reasoning, and communication

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Teaching and Teacher Education 29 (2013) 97e109

Contents lists available

Teaching and Teacher Education

journal homepage: www.elsevier .com/locate/ tate

Teachers’ spatial literacy as visualization, reasoning, and communication

Deborah Moore-Russo a,*, Janine M. Viglietti b, Ming Ming Chiu a, Susan M. Bateman a

aUniversity at Buffalo, State University of New York, Department of Learning and Instruction, 505 Baldy Hall, Buffalo, NY 14260-1000, USAbBuffalo State, State University of New York, USA

h i g h l i g h t s

< Teaching experience and education have positive effects on spatial literacy.< Using multiple reasoning strategies had positive effects on spatial literacy.< Many teachers failed to make use of multiple reasoning strategies.< Teachers struggled with the vocabulary of spatial objects.< Many teachers do not possess the spatial literacy skills intrinsic to the curriculum.

a r t i c l e i n f o

Article history:Received 16 March 2012Received in revised form28 August 2012Accepted 30 August 2012

Keywords:Spatial literacySpatial visualizationSpatial reasoningSpatial communication

* Corresponding author. Tel.: þ1 716 645 4069; faxE-mail address: [email protected] (D. Moore-Ru

0742-051X/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.tate.2012.08.012

a b s t r a c t

This paper conceptualizes spatial literacy as consisting of three overlapping domains: visualization,reasoning, and communication. By considering these domains, this study explores different aspects ofspatial literacy to better understand how a group of mathematics teachers reasoned about spatial tasks.Seventy-five preservice and inservice teachers worked on problems that involved spatial objects, theirproperties, and their relationships. Teachers’ responses suggested that their spatial literacy skills wereunderdeveloped with deficiencies most evident on problems that were solvable by dimensionalreasoning. Poor vocabulary and misconceptions hindered teachers’ performance. Teachers who usedmultiple reasoning strategies were more likely to solve a problem correctly.

� 2012 Elsevier Ltd. All rights reserved.

“Spatial thinking can be learned, and it can and should be taughtat all levels in the education system”

dU.S. National Research Council (NRC) Geographical ScienceCommittee (2006b, p. 3).

Spatial literacy supports how we understand and interact withthe physical world in which we live (Alsina, 2000; de Lange, 2003)and is needed in a variety of fields. Architects create two-dimensional renderings of three-dimensional structures thatconstruction workers read and interpret. Athletes read two-dimensional diagrams that map out the spatial positioning ofplayers on teams. Dancers enact spatial movements communicatedthrough verbal commands.

Researchers (McGraw, 2004) have shown that individuals differin how they process and perceive two- and three-dimensionalstimuli. Students with higher spatial abilities outperform otherstudents in art courses (Haanstra, 1996), are more likely to solve

: þ1 716 645 3161.sso).

All rights reserved.

problems and interpret graphs in physics (Kozhevnikov, Hegarty, &Mayer, 2002; Pallrand & Seeber, 1984), and learn more fromcomputer animations of concepts in microbiology (Trindade,Fiolhais, & Almeida, 2002).

Technological advances have brought attention to the role ofspatial literacy. For example, “digitization has reinvented the. spatial terms under which the arts are engaged” (Brown, 2003, p.287). Since GIS (Geographic Information System) technology hasbecome more affordable and available, the role of spatial literacy ingeography education has been examined in Australia (McInerney,Berg, Hutchinson, Maude, & Sorensen, 2009), Singapore (Yap, Tan,Zhu, & Wettasinghe, 2008), and Turkey (Incekara, 2010).

While spatial literacy aids understanding in many disciplines, itis central in mathematics (NRC, 2006b; Presmeg, 2006). In order toprepare students to excel in numerous fields, teachers must possessspatial literacy skills and the ability to nurture these skills in theirstudents. Since mathematics provides a common language todiscuss and represent spatial relationships, it is particularlyimportant that mathematics teachers develop a robust set ofgeneral spatial literacy skills (Newcombe, 2006).

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Fig. 1. Spatial literacy domains.

D. Moore-Russo et al. / Teaching and Teacher Education 29 (2013) 97e10998

This paper has two purposes. First, it conceptualizes spatialliteracy as consisting of three interrelated domains. Next, it looksspecifically at the spatial literacy of secondary mathematicsteachers in the context of curricular recommendations (CommonCore State Standards Initiative, 2010; National Council forTeachers of Mathematics [NCTM], 2000a; Royal Society/JointMathematics Council [RSJMC], 2001; South African Department ofEducation [DoE], 2003).

1. Spatial literacy

While literacy typically refers to reading and writing, thebroader sense of being literate refers to being competent andknowledgeable in a certain area. de Lange (2003) described spatialliteracy as a person’s perception and understanding of spatialobjects and relationships. To be spatially literate, a person must (a)visualize spatial objects; (b) reason about properties of and rela-tionships between spatial objects; and (c) send (and receive)communication about spatial objects and relationships. Spatialliteracy is related to an individual’s cognitive processes and isindependent of a specific content area. (In contrast, geometry isa specific branch of mathematics where one often makes use ofspatial literacy skills while studying axioms, properties, and theo-rems related to points, curves, surfaces, and solids.)

Mathematics educators have acknowledged the interrelation-ship between spatial literacy and geometry (Battista, 2007; Bishop,1983; Clements & Battista, 1992; Del Grande, 1990; Usiskin, 1987).In the U.K., objectives for geometry education include the devel-opment of spatial awareness (RSJMC, 2001). In South Africa, thegeneral mathematics curriculum states that students shoulddevelop the ability to think spatially in order to analyze real worldsituations (DoE, 2003). Yet, the role of spatial literacy in somemathematics curricula is less than explicit. The NRC (2006b, p. 115)reported, “it is clear that spatial thinking pervades and permeatesthe detailed articulation of what is expected of students. . anunderstanding of spatial relations is presumed but never discussedexplicitly” regarding the key mathematics education document inthe U.S.1

2. Spatial literacy domains

For this study, we propose a framework suggesting that spatialliteracy comprises three domains (shown in Fig. 1): visualization,reasoning, and communication. We contend that spatial literacynot only involves a habit of mind that lends itself to spatial visu-alization and reasoning but also entails the ability to both processand communicate spatial information (Gorgorió, 1998). We nowdiscuss each of these three spatial literacy domains and how theyoverlap.

2.1. Visualization

Visualization is the process of generating cognitive representa-tions of spatial objects through visual images that may be facili-tated by external representations or physical actions (Arcavi, 2003;Gutiérrez, 1996; Presmeg, 1997; Zimmerman & Cunningham,1991).Presmeg (2006, p. 207) defined a visual image to be “a mentalconstruct depicting visual or spatial information” and, building onKrutetskii’s (1976) work, suggested that certain individuals are

1 Similarly, it was reported that, “A detailed reading of the [U.S.] science stan-dards suggests that there is spatial thinking and spatial reasoning content, butneither phrase appears in the text, nor are the concepts addressed explicitly andsystematically” (NRC, 2006a, p. 121).

more inclined to incorporate visual images when confronted witha problem. Visualization, without reasoning or communication,involves calling up a visual image without performing any opera-tions on the image or sharing information about the image withothers. For example, an individual can visualize two concentriccircles (i.e., circles sharing the same center) without communi-catingwith another or without reasoning about the circles; they aresimply seen in the mind’s eye.

Although often mentioned in the context of geometry(Gutiérrez, 1996), visualization also applies to other mathematicsareas like algebra, statistics, and calculus (Arcavi, 2003; Garfield &Ben-Zvi, 2007; Hughes Hallet, 1991; Mason, 1996; Rivera, 2007).Further, visualization has been found to impact science and reading(Phillips, Norris, & Macnab, 2010) as well as engineering (Contero,Naya, Company, Saorín, & Conesa, 2005).

When visualizing, some individuals hone in on the relationshipsbetween the parts of an object while others tend to focus on theoverall pictorial appearance of the object (Kozhevnikov, Kosslyn, &Shephard, 2005). An interior designer might mentally recall seeingtwo chairs in a room: one red and the other blue. This form ofvisualizing is not particularly helpful for spatial literacy. In contrast,the same interior designer would employ spatial literacy skillswhen visualizing the size, shape, and positioning of the two chairsin a room.

The difficulty of visualization in mathematics is in no small partdue to the paradoxical nature of mathematical knowledge. Incontrast to other scientific fields, there are no true mathematicalobjects (e.g., infinite lines) in the real world (Duval, 1999). Studentsare asked to work on material drawings but expected to considerthe figures that the drawings represent (Hollebrands, 2003). Anindividual’s visualization can be hindered by the inescapableduality between the abstract, ideal mathematical object and itsphysical representation (Laborde, 1993; Parzysz, 1988). Yet, it isalmost impossible to consider situations involving spatial figuresand relationships without first incorporating visualization(Goldenberg, Cuoco, & Mark, 1998).

2.2. Reasoning

The word “reasoning” generally describes the cognitiveprocesses that individuals use to form conclusions or make judg-ments from a given set of premises. We consider reasoning as theprocess of organizing, comparing, or analyzing spatial concepts and

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relationships (cf. Battista, 2007). Two types of reasoning that canapply to spatial situations include algebraic reasoning, whichinvolves the use of concepts, representations, and techniques thatare common in algebra (e.g., the use of equations to modelpatterns), and proportional reasoning, which involves the multi-plicative comparison of two quantities using ratios or proportions.Reasoning plays a key role in mathematics and other fields. Forexample, spatial reasoning has been found to be an essentialcomponent in carpentry training programs (Cuendet, Jermann, Arn,& Dillenbourg, 2011).

Reasoning may or may not include visualization or communi-cation. Individuals often engage in cognitive activity to developtheir personal understanding without either using visual images orexchanging information with others. A person can reason aboutspatial figures without visualization; for example, a person cancompare pizza sizes and costs using only the equation for the areaof a circle without ever visualizing the circular shapes involved.

2.3. Communication

Spatial literacy includes the ability to engage in spatial commu-nication (as seen in Gorgorió, 1998). For this study we considercommunication as the exchange of information through interactionswith others. As a spatial literacy domain, communication involvesindividuals using a variety of resources including language, writteninscriptions, gestures, etc. (O’Hallaran, 2008) to convey ideas toothers that relate to spatial objects or relationships (Moore-Russo,Conner, & Rugg, 2011; Moore-Russo & Viglietti, 2012).

The communication domain,whennot including visualization orreasoning, involves imparting and receiving existing information orknowledge through linguistic means and resources (e.g., algebraicformulas) that donotevoke visual images or reasoning. For example,a teacher need not visualize or reason when responding “lengthtimes width” to a student asking, “What is the area of a rectangle?”

When communicating, an individual’s understanding and use ofvocabulary is key (Adams, 2003; Orton, 1987) since flawed vocabu-lary use can hinder communication (e.g., Reehm & Long, 1996;Thompson & Chappell, 2007; Thompson & Rubenstein, 2000). It isnot enough for teachers to visualize and reason about spatialconcepts and relationships, they must also communicate and inter-pret others’ communication of spatial ideas. While teachers shouldmodel appropriate use of terminology, they also need to understandstudents’ colloquial expressions (e.g., “insidepart” rather than “area”to describe the two-dimensional space bounded by a circle).

2.4. Overlaps in visualization, reasoning, and communication

The visualization and reasoning domains overlap (VR) whena person invokes visual images to facilitate his or her reasoning.Many objects (e.g., cylinders) can be considered and manipulatedsolely with algorithms, but they also tend to evoke visual images(Fischbein & Nachlieli, 1998) especially when one reasons about thefigure. This echoes Godfrey’s (1910) claim that solving difficultproblems often requires more than logic e it requires seeing theproperties of a figure.

Dimensional reasoning typically involves both visualization andreasoning (Montiel,Wilhelmi, Vidakovic, & Elstak, 2009).We definedimensional reasoning as involving a person’s ability to fluidly movebetween working in one, two and three dimensions and tounderstand how the change between dimensions impactsmeasurement, the number of coordinates of points, the possiblevariables in an equation, etc. For example,2 if one shark weighs

2 Problem from NCTM’s Sixth World’s Largest Math Event (NCTM, 2000b).

6000 pounds and another shark is twice as long, approximatelywhat is the longer shark’s weight? To solve this, students mustrecognize that the shark’s body is a three-dimensional volume andthat different types of sharks tend to have similar structures. Sincethe sharks are proportional in length, then their widths and heightsshould also be proportional. Hence, all three dimensions should bedoubled, and the weight of the larger shark should be about2 � 2 � 2 ¼ 8 times the weight of the smaller shark or about8 � 6000 ¼ 4800 pounds.

The overlap between the reasoning and communicationdomains (RC) involves people using words, equations, etc. thatneither stimulate the use of visual images nor involve the use ofphysical representations (or iconic gestures) to share and pondermathematical ideas. For example, if students were collaborating todetermine how much the area of a circle changes if its diameter isdoubled, they may not invoke any visual images but theircommunication and reasoning might revolve solely around theformula for the area of a circle.

The overlap between the visualization and communicationdomains (VC) occurs when people use drawings, gestures, manip-ulatives, words, etc. to share the spatial images in their headswithout reasoning about them. For example a person mightdescribe a shooting target as a series of concentric circles with theintent not to promote reasoning but to help another visualize thenested arrangement of circles of graduated sizes.

The overlap of the three domains (VRC) occurs when peopleare engaged in reasoning, sharing their ideas, and using visualimages to assist in these processes. For example, a commoninstructional activity that incorporates visualization, reasoning,and communication is to provide students with an oatmealcontainer and have them disassemble it so that they cancollaboratively discover how to calculate the surface area ofa cylinder.

3. Research questions and methods

The data reported here are part of a study that exploredpreservice and inservice mathematics teachers’ spatial literacy. Aninstrument consisting of problem solving tasks was designed andadministered to a group of participants; the instrument and theparticipants are described in Section 3.1 through 3.3. First, quali-tative, content analysis was used to code each participant’s attemptto solve each problem, based on the relevant literature, expandingormodifying the codes as needed using emergent coding (Strauss &Corbin, 1990). Then, statistical methods were used to model thecharacteristics of each problem, each participant, and each of thesolution strategies.

To be consistent with the framework that underpins ourresearch, this paper focuses on reasoning and its intersection withvisualization and communication (VR, RC, VRC). While the visual-ization and communication domains outside of reasoning arecritically important, their investigation requires a different instru-ment than that employed in this study. We consider teachers’responses to items in two- and three-dimensional environment inthree categories: spatial objects, relationships between spatialobjects, andmeasurement of spatial objects. The following researchquestions guided the study:

1. What effect do the teachers’ academic or professional back-grounds have on their spatial literacy?

2. What effect do misconceptions related to teachers’ contentknowledge have on teachers’ spatial literacy?

3. What types of reasoning strategies do teachers use whenconsidering spatial objects and relationships?Which reasoningstrategies are most effective?

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4. What effect do visualization and communication have onteachers’ spatial literacy? Which reasoning strategies arerelated to teachers’ visualization and communication?

5. Do teachers possess the spatial literacy skills needed to facili-tate the types of tasks specified in national curricula?

3.1. Participants

Data were collected from a convenience sample of 49 preserviceand 26 inservice teachers. All 75 teachers were enrolled in graduatemathematics education courses aimed at improving subject matterknowledge at a large research university in the U.S. In order tocomply with the university’s ethical clearance requirements,written consent for participation in the study was requested andobtained from each teacher.

While teacher-training programs in the U.S. vary greatly, at theuniversity where the study was conducted all teacher trainingtakes place in a Graduate School of Education, which offers bothchildhood (grades Ke6) and secondary (grades 7e12) mathe-matics education programs. Of the teachers, 91% were enrolled inthe secondary program, the rest were in the childhood program.Of the teachers in this study, 91% held Bachelor’s degrees inmathematics; the others held degrees in education with a math-ematics concentration. Most of the teachers were female (76%)and less than 30 years old (95%). The teachers had taken under-graduate mathematics courses at 29 different colleges anduniversities spanning 7 states and 3 countries (U.S., Canada, andEngland).

3.2. Data collection

Several explanatory variables were used to model spatialliteracy (see Table 1).

Data for the five demographic variables (listed in Table 1) wereobtained through school records and personal interviews. The

Table 1Variables used in the study.

Variable types Variables

Demographic GenderAgeBachelor’s degree in mathematicsPrevious non-teaching employment ina mathematics-related fieldTeaching experience

Item characteristics Content involving basic objectsContent involving relationships between objectsContent involving measurementAlgebraic reasoning possibleDimensional reasoning possibleProportional reasoning possible

Visualization Difficulty with visualizationConfusion between figure and representation

Reasoning Use of drawingUse of exampleUse of algebraic reasoningUse of dimensional reasoningIncorrectly forcing three-dimensional situationsto two dimensionsPresence of misconceptionMajor misconception

Communication Completeness of communicationVocabulary problems

remaining data were collected using a paper-and-pencil test con-sisting of true/false, multiple-choice, and open-answer items. Thedata collection instrument was administered over two years in fivegraduate mathematics education courses as a graded pretest toassess teachers’ spatial knowledge. To receive full credit, teachershad to give complete explanations for all responses (including thetrue/false and multiple-choice items).

There were two versions of the test, one with 21 items (usedduring the first five months of data collection) and one with 24items (used during the last 19 months of data collection). The datacollection instrument was initially developed by one member ofthe research team in consultation with another mathematicseducation professor and a veteran high school mathematicsteacher who was a mathematics education Ph.D. student. In jointmeetings, this group reviewed the NCTM documents thataddressed three-dimensional concepts until reaching consensuson an instrument to assess the teachers’ understanding of thethree-dimensional concepts in the Ke12 curriculum (Geddes et al.,1992; NCTM, 1989, 2000a, 2000b, 2006). This instrument consistsof 21 items, including 6 true/false, 5 multiple-choice, and 10 open-answer items. The instrument was modified once with the addi-tion of 3 open-answer items to assess common spatial miscon-ceptions during the courses in which the instrument wasadministered. As discussed below, we use item response models,which can analyze different tests with common questions (i.e.,anchor items). Of the full sample of 75 teachers, 35 teachersanswered 21 items, and 40 teachers answered all 24 items.Teacher responses to test items were the units of analysis(n ¼ 1695).

3.3. Item characteristics

Items on the instrument addressed three categories of contentinvolving: 1) basic objects, 2) relationships between objects, and 3)measurement as shown in Table 2. Items related to points, lines,planes, pyramids, cones, prisms, cylinders, and spheres wereclassified as having content that involved basic objects. Itemsinvolving bisecting, intersecting, parallel, collinear, or coplanarobjects were classified as having content that involved relation-ships between objects. Items involving length, surface area, andvolume were classified as having content that involved measure-ment. Note that some items were simultaneously classified intomultiple categories.

Specific forms of reasoning facilitated successfully respondingto each instrument item as shown in Table 2. While no onereasoning type was required for any given item, the use ofdimensional reasoning was possible for solving some items, theuse of algebraic reasoning was possible for solving other items,and a third group of items could have made use of both. Forcertain items proportional reasoning was possible, but

Table 2Frequency table for item characteristic variables on 21-item and 24-iteminstruments.

Variables Frequency

21-Iteminstrument

24-Iteminstrument

Content involving basic objects 14/21 17/24Content involving relationships between objects 8/21 9/24Content involving measurement 10/21 10/24Dimensional reasoning was possible 17/21 19/24Algebraic reasoning was possible 12/21 14/24Proportional reasoning was possible 5/21 5/24

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proportional reasoning alone would not have solved any itemcompletely.3

4. Measures

For all measures, two research team members independentlycoded all responses for a subset of 20 randomly selected partici-pants, using each of the general schemas described below The tworesearch team members discussed coding until consensus, afterwhich the item-specific rubrics were created. Interrater agreementexceeded 93% on all items.

4.1. Teachers’ spatial literacy

The following general schema was used to evaluate thecorrectness of the responses.

0: Missing, incorrect, or irrelevant response1: Some correct information with some incorrect or substantially

omitted, necessary information2: Correct but incomplete response e what is stated is correct but

with limited necessary information omitted3: Complete, correct response

To ensure consistency in evaluations for correctness, 24 of theitem-specifc rubrics were designed in accordance with this generalschema that addressed common responses for each item4 (asillustrated in Appendix A). Using the item-specific rubrics, a singleresearcher evaluated the teachers’ responses. In cases wherea response was not immediately apparent as having beenaddressed by its item-specific rubric, the research team evaluatedthe materials together until a consensus was reached. In thesecases, the general grading scale was used.

4.2. Coding for visualization

While visualization is critical to spatial literacy, the internalnature of visualization does not lend itself easily to measurement.The next two coding schemas are an initial attempt at consideringthe overlap of the visualization and reasoning domains; the firstconsidered difficulty with visualization and the second consideredthe confusion between ideal mathematical objects and theirrepresentations.

Without any prompting, several teachers expressed difficul-ties visualizing. Since the teachers were to provide completeanswers items, some wrote phrases like: “[I have] no idea how itwould look, so [I’m] guessing.” The research team wanted todetermine if these unprompted comments had an impact on theteachers’ spatial literacy scores, so the following coding schemafor the variable difficulty with visualization was included in theanalysis.

0: No explicit statement to denote difficulty conjuring up or usingmental images

1: Explicit statement that teacher had difficulty conjuring up orusing mental images

3 Since proportional reasoning had to be in conjunction with either algebraic ordimensional reasoning to successfully solve any item on the spatial literacy test, useof proportional reasoning was not included as one of reasoning variables listed inTable 1.

4 The instrument was administered to mathematics teachers. Hence, items usedmathematics vocabulary and were in a mathematical context. Since it is possible togeneralize the measures and coding schemes to other content areas the subject-specific items have been placed in appendices.

Due to the duality between abstract mathematical figures andtheir representations, certain teachers confused the representa-tion’s finite characteristics with the ideal figure. An example of thisis in Appendix B. A binary coding schema was used for the variableconfusion between figure and representation.

0: No obvious confusion between mathematical object and itsrepresentation

1: Confusion between mathematical object and its representationevident

4.3. Coding for reasoning

As reasoning was the primary focus of this study, it was also thedomain with the most measures. Included in this domain weremeasures in which it was impossible to determine which of theintersections of the spatial literacy domain diagramwere involved.What was clear was that all of the following measures involvedreasoning; what was not clear was whether the examples ordrawings provided by the teachers were used as a means ofcommunication or as a means to aid visualization. Thus thefollowing coding schemas measure responses that focused onreasoning and represented the following overlaps in domains: VR,RC, and VRC. Related examples are in Appendix C.

Five reasoning variables were coded as 0 for not evident and 1for present. These included use of drawing and use of example(which always involved specific quantities and often involveda drawing). Other variables included the use of algebraic reasoning,dimensional reasoning, and incorrectly forcing three-dimensional(3D) situations to two dimensions (2D).

During coding for these five reasoning variables, manyresponses were found to contain mathematical misconceptions.While such misconceptions impact teachers’ spatial literacy scores,the research team wanted to determine what effect, if any, thesemisconceptions had on other variables. So, the following schemawas used for presence of misconception (a score of 1 or 2 below) andmajor misconception (a score of 2 below).

0: No misconception evident1: Misconception present, however under certain conditions the

response is valid2: Major misconception present, response would not be valid

under any conditions

4.4. Coding for communication

Two variables were considered which represent the overlapbetween reasoning and communication. Since the teachers wererequired to write complete answers for each item, the first variablerelated to the thoroughness of the communication in the responses.The lack of a complete response might reflect vagueness, uncer-tainty, or imprecision in the teachers’ reasoning or communication.The second variable dealt with the effect of vocabulary issues onthe reasoning process.

Coding for the completeness of communication variable did notconsider the correctness of the response. While a teacher’sresponse could be assigned a code of 2 for its completeness ofcommunication, that explanation might contain false information.For example well-communicated reasoning could be based ona faulty premise. Therefore the following schema was used.

0: Missing response or “don’t know”

1: Incomplete response2: Complete, thorough response

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Without understanding the vocabulary used in an item, teacherswould certainly score lower on the spatial literacy test. Thevocabulary problems variable was coded to determine what effect, ifany, this aspect of communication had in conjunction with theother variables. Examples of vocabulary problems are given inAppendix D.

0: No obvious difficulty with vocabulary1: Presence of difficulty in teachers’ use of or interpretation of

vocabulary

5. Analysis

The statistical strategies used to address the varying analyticaldifficulties posed by the study are shown in Table 3.

Teacher’s responses to each itemwere measured and convertedinto an index before an analysis identifiedmajor factors linked withthe spatial literacy scores. Compared to traditional methods, itemresponse models (a) create more precise estimates of participantcompetence, (b) allow comparisons of participants who have takendifferent tests with overlapping items, (c) model items that allowpartial credit, (d) model items of varying levels of difficulty, and (e)model items of different quality with respect to distinguishinghigh- vs. low-competence participants (Baker & Kim, 2004;Joreskog & Sorbom, 2004). Furthermore, the item response modelconverts the discrete, ordinal measures on each item into contin-uous interval measures for analysis.

A regression models how the teachers’ demographic infor-mation; the item characteristic variables; and visualization,reasoning, and communication characteristics evidenced in theteachers’ responses were linked to teacher performance on eachitem. Unlike a traditional, ordinary least squares regression,a multilevel cross-classification regression correctly models thestructure of a set of teachers’ responses to each test item to yieldunbiased standard errors (Goldstein, 1995). To test for indirecteffects through mediators (X / M / Y) at different levels(teacher-level vs. test item-level), a multilevel mediation test isneeded (Krull & MacKinnon, 2001). Lastly, testing manyhypotheses increases the likelihood of a false positive, which isaddressed by a two-stage linear step-up procedure, which out-performed 13 other methods (e.g., adaptive Hochberg procedure,Holm’s procedure, median adaptive linear step-up procedure)

Table 3Statistics strategies to address each analytical difficulty.

Statistics strategies used Analytical difficulties addressed

Factor analysis on polychoriccorrelation matrices; itemresponse models

Participants took different, overlapping testsTest items allowed partial creditDifficulties of test items variedConverted ordinal measures into intervalmeasuresQuality of each test item to distinguishhigh- and low-competence participants(discrimination) varied

Multilevel cross-classification Allowed for modeling participantcompetence on each test itemScores on test items done by same personare more similar than those done by differentpeople (can cause biased standard errors)

Multilevel mediation testsand path analysis

Indirect, mediation effects at different levels(teacher or test item)

Two-stage linear step-upprocedure

False positives

according to computer simulations (Benjamini, Krieger, &Yekutieli, 2006).

5.1. Measurement

To model the underlying competencies of the tests, a factoranalysis of the polychoric correlations was conducted. As separateitem response models could be needed for each type of spatialliteracy competence, an exploratory factor analysis (principal factor)identified whether the items reflected one underlying competenceor multiple competencies. As the partial credit on the items resultedin ordinal values, an exploratory factor analysis on unbiased poly-choric correlations was needed (Joreskog & Sorbom, 2004).

For each student, a competence score were computed for eachset of underlying factors that accounted for differences in itemdifficulty and quality. Evidence for a single dominant factor include:(a) the largest factor’s explained variance exceeds 20%, (b) a largeratio of the eigenvalue of the largest factor over that of the secondlargest factor, and (c) small eigenvalues aside from that of thelargest factor (Tabachnick & Fidell, 2006). The eigenvalue of a factorindicates how much of the variation in the test scores is accountedfor by the factor. The larger the eigenvalue, the greater the varianceexplained by the factor and the less variance explained by otherfactors. Hence, a factor with a large eigenvalue and small eigen-values for the other factors indicates that there is one underlyingcompetence that accounts for the difference in test scores (as is thecase for this data set).

The factor analysis results (in Table 4) showed that the largestfactor’s explained variance was 30% (exceeding 20%). Furthermore,the largest factor’s eigenvalue was much larger than those of allother factors. Together, these results indicate a single dominantfactor, suggesting a single underlying competence for the test items(Tabachnick & Fidell, 2006). It is this competence that we are callingspatial literacy.

To model item responses that allow partial credit, two-itemresponse models are used to identify the best fitting model. Apartial credit item response model (PCM) captures differences in thedifficulty of each item. A generalized partial credit model (GPCM) notonly captures difficulty differences but also allows the quality of anitem in distinguishing higher and lower competence test takers todiffer (discrimination). Higher discrimination items provide moreprecise information about each teacher’s ability, and the GPCM-IRmodel weights these items more than lower discrimination items.

The GPCM model Pi(rjq):

PiðrjqÞ ¼ ePr

j¼1aiðq�bijÞ

1þPmi�1k¼1 e

Pk

j¼1aiðq�bijÞ

(1)

is the probability that teacher jwith competence qwill have a scoreof r (where 0 � r � 3) or better on item i, accounting for each item’s

Table 4Results of factor analysis indicating a single factor.

Factor (i) Eigenvalue % Varianceexplained

Ratio to nexteigenvalue (Fi/Fiþ1)

1st 7.20 30% 2.322nd 3.10 13% 1.163rd 2.68 11% 1.314th 2.05 9% 1.065th 1.93 8% 1.166th 1.66 7% 1.207th 1.38 6% 1.118th 1.25 5% 1.199th 1.05 4%

93%

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discrimination strength (ai) and difficulty for each level of partialcredit (bij). The simpler PCM assumes that all items have the samediscrimination (same as equation (1), except ai is replaced witha constant). The models were fit to the data via Bayesian expecteda posteriori (EAP) estimation, and a log-likelihood difference (DLL)c2 test indicated which model fit the data better (Baker & Kim,2004; Bock & Mislevy, 1982).

The discrimination strengths of the items differed (data avail-able upon request). Results of the DLL c2 tests providedevidence that the GPCM fit the data better than the PCM(c 2¼ 60 ¼ 1793 � 1733; p < 0.001).

As the best item response model produces more precise, cor-responding partial credit levels for each item than the arbitrary,uniform original coding (0, 1, 2 or 3), each teacher’s raw score wasreplaced with the corresponding GPCM score. The GPCM scoreswere standardized so that the maximum total score was 100%. Thestandard errors at any level of teacher spatial knowledge weresmall (all less than 10%), showing precise measures of eachteacher’s spatial knowledge (data available upon request). TheseGPCM scores were then analyzed with an explanatory model toidentify key factors related to the teachers’ spatial literacy.

Table 5Relative frequency chart showing percentage of teachers demonstrating between0 and 11 errors for the three spatial literacy domains.

Types of errors Percentage of teachers with ___ error(s)

0 1 2 3 4 5 6 7 8 9 10 11

VisualizationDifficulty with visualization 92 8Confusion between figure and

representation82 13 5

ReasoningIncorrectly forcing 3D

situations to 2D38 36 17 9

Presence of misconception 5 9 13 15 16 11 8 7 5 4 4 3Major misconception 29 31 19 13 5 1 1 0 1

5.2. Explanatory model

To identify key factors associated with each teacher’sperformance on each item, a multilevel, cross-classificationregression model was used (Goldstein, 1995). This explanatorymodel tests whether teachers’ demographic characteristics, testitem characteristics, or teacher actions (visualization, reasoning,or communication variables), all listed in Table 1, were linked toteacher performance on each item. As higher discriminationitems are more informative and more precise than lowerdiscrimination items, each teacher’s response to each item wasweighted by the item’s discrimination from the GPCM (to ensuretheir total weight was equal to the total number of test items,the discrimination weights were standardized). As the dataconsist of two levels (teacher and item), a 2-level cross-classification model was used.

Test itemðijÞ ¼ b0 þ ei þ ej (2)

b0 is the grandmean intercept of the test score Test_item(ij) on itemi by teacher j. The item residuals and teacher residuals are ei and ejrespectively.

Next, sets of explanatory variables were entered into the modelaccording to temporal order (thereby restricting the possible orderof causation): demographic variables, item variables, visualizationvariables, reasoning variables, and communication variables.

Test itemðijÞ ¼ b0 þ ei þ ej þ bdDemographicsð0jÞþ btItem Characteristicsði0Þ þ bvVisualizationðijÞ

þ brReasoningðijÞ þ bcCommunicationðijÞþ bxInteractionsðijÞ ð3Þ

The five Demographic variables were entered first.5 Theseexplanatory variables were tested for significance with a nestedhypothesis test (DLL c2, Kennedy, 2008). Non-significant variableswere removed. This procedure was applied to all subsequent vari-ables. Next, the six Item_Characteristics variables were entered intothe model. As visualizing and reasoning processes can influenceeach other, it is not clear which of these two processes occur first.

5 Note that all variables are listed in Table 1.

Hence, the Visualization and Reasoning variables were enteredtogether. Next, the two Communication variables were entered intothe model. Finally, to test for moderation effects, Interactionsamong the above variables were added.

To identify indirect, mediation effects at multiple levels, multi-level mediation tests and path analyses were used (Krull &MacKinnon, 2001). All significant variables were tested forpossible significant mediation effects on significant variablesentered in earlier sets. For significant mediating variables, thepercentage change was 1 � (b0/b), where b0 and b were theregression coefficients of the explanatory variable, with andwithout the mediator in the model, respectively. Path analyseswere performed on significant mediating variables (Kennedy,2008).

Testing many hypotheses increases the likelihood that at leastone of them incorrectly rejects a null hypothesis (a false positive).To control for the false discovery rate, the two-stage linear step-upprocedure was used (Benjamini et al., 2006). An alpha level of 0.05was used.

6. Results

6.1. Summary statistics

The most common reasoning type employed by the teacherswas use of drawing (evidenced in 47% of all responses) while use ofexample occurred in 21% of the responses. The use of drawing or useof example was possible for any item. Dimensional reasoning waspossible for 17 of the 21 items, and algebraic reasoning was possiblefor 12 of the 21 items. Despite the fact that 81% of the items couldhave made use of dimensional reasoning, this reasoning strategyoccurred in only 4% of the teachers’ responses while the use ofalgebraic reasoning occurred in 22% of their responses.

Despite their prior mathematics coursework and similar careertrajectories, the teachers’ spatial literacy scores were low andvaried widely. Even though the items were developed usingsuggestions for secondary mathematics tasks, the teachers’ meanscore on the test after adjusting for the difficulty of each item was39% (standard deviation ¼ 0.29; minimum ¼ 0; maximum ¼ 1;skew ¼ 0.06; kurtosis ¼ 2.5). The high standard deviation denotesconsiderable spread among the scores; only 20% of the teachersscored 50% or higher on the test.

Next, we consider the summary statistics for the visualization,reasoning, and communication variables. The relative frequenciesof errors evident in teachers’ responses are in Table 5.

CommunicationVocabulary problems 16 24 28 17 4 3 1 4 3

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6.2. Visualization

Without prompting, 8% of the teachers expressed difficulty withvisualization in their written responses. While 82% of the teachersshowed no evidence of confusion between figure and representation,several teachers (18%) did (see Table 5).

6.3. Reasoning

As seen in Table 5, 62% of the teachers tried incorrectly forcing 3Dsituations to 2D at least once. Ninety-five percent of the teachershad at least one presence of misconception in their responses with71% demonstrating a major misconception. In fact, over fortypercent of the teachers showed evidence of 5 or moremisconceptions.

6.4. Communication

The teachers received the following scores for completeness ofcommunication: 14% of the responses received a score of 0; 39%received a score of 1 (incomplete response), and 47% of theresponses received a score of 2 (complete and thorough response).In terms of communication errors, 84% of the teachers had at leastone vocabulary problem (as seen in Table 5). Vocabulary problemswere visible in 10% of all responses, largely by one third of theteachers.

6.5. Explanatory model

The multilevel, cross-classification identified key factors relatedto teacher performance on the items. The results showed thatteacher performance differed mostly across items (87% of thevariance) rather than across teachers (13%). Many teachers foundsome items very easy and others very difficult. The path analysisresults are shown in Fig. 2.

Use of Example

AlgebraiReasonin

Presence ofMisconceptio

Algebraic ReasoningPossible

+0.03 **

-0.31 ***

+0.05 **

+0.18 ***

Demographics Item characteristics Reasoning

Dimensional Reasoning

Dimensional ReasoningPossible

Use of Drawi

Bachelor’s Degree in Math

Teaching Experience

Fig. 2. Path diagram of the final multilevel, cross-classification model predicting the score odashed lines represent negative effects. Dashed boxes indicate negative total effects. Thicke

The columns of the path diagram indicate categories ofexplanatory variables that were significantly linked to spatialliteracy (demographics, characteristics of items, reasoning, andcommunication). The total effect of an explanatory variable onspatial literacy is indicated by the box around its name in thediagram (dashed for a negative effect, solid for a positive effect,thicker line for a larger effect). An explanatory variable’s total effectis the sum of its direct effect and all of its indirect effects. A directeffect is represented by a single arrow linking an explanatoryvariable (X) to spatial literacy (S), namely X/ S, and its value is thatof the regression coefficient on the arrow. An indirect effect occursthrough a mediating variable (M), namely X / M / S, and iscomputed by the product of the regression coefficients on thearrows (see computations below). Note that if the mediating vari-ableM is linked to spatial literacy through indirect effects, all ofM’sdirect and indirect effects must also be included in the computationof X’s indirect effect. Only significant variables and links appear inthe diagram.

Demographic variables were linked to spatial literacy (see farleft column in Fig. 2). Teachers with past teaching experience hadhigher spatial literacy scores on each item (þ0.34 points onaverage) compared to those without teaching experience. Likewise,teachers with a Bachelor’s degree in mathematics had higher spatialliteracy scores on each item (þ0.50) than those without (its thickerline shows its effect exceeds that of teaching experience). As theseexplanatory variables only had direct effects and no indirect effects,their total effects are equal to their direct effects.

Like the demographics variables, the significant communicationvariables only had direct effects on spatial literacy (see fourthcolumn in Fig. 2). Teachers who showed completeness of commu-nication on an item often had higher scores on it (regression coef-ficient b ¼ 1.91; total effect ¼ direct effect), and those who hadvocabulary problems averaged lower scores (�1.53).

Reasoning variables (presence of misconception, dimensionalreasoning, use of drawing, use of example, and algebraic reasoning)

c g

Vocabulary Problems

n

Spatial Literacy

Score on

Each Test Item

+0.34 *

+0.50 *

-0.61 ***

-0.27 *

-0.36 **

+1.42 ***

+0.52 ***

+0.52 ***

+1.91 ***

-1.11 ***

-1.53 ***

-0.18 ***

+0.42 ***

+0.36 ***

+0.40 ***

-0.09 ***

-0.04 *

+0.16 ***

Communication

ng

Completeness of Communication

f each spatial literacy item (n ¼ 1695 responses). Solid lines represent positive effects;r lines indicate stronger links (larger regression coefficients).

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were also linked to spatial literacy (see third column in Fig. 2). Asexpected, teachers who had a misconception on an item scoredlower on it (�1.11 ¼ total effect).

Meanwhile, dimension reasoning showed a positive direct effect(þ1.42). Furthermore, teachers who used dimensional reasoningshowed more completeness of communication (þ0.36), resulting inan indirect effect of þ0.69 (¼[0.36][1.91]); as shown above,completeness of communication’s total effect on spatialliteracy¼ 1.91. Altogether, teachers who used dimensional reasoninghad a total effect of þ2.11 extra points on each item(þ2.11 ¼ 1.42 þ 0.69; direct effect plus indirect effect viacompleteness of communication).

Likewise, the use of a drawing, use of an example, or use ofalgebraic reasoning also had positive total effects on spatial literacy.When teachers used a drawing (direct effect ¼ �0.36), they showedmore completeness of communication (þ0.42), resulting in an overallpositive total resulting in an indirect effect ofþ0.69 (¼[0.36][1.91]);as shown above, completeness of communication’s total effect onspatial literacy ¼ 1.91. Altogether, teachers who used dimensionalreasoning had a total effect of þ2.11 extra points on each item(þ2.11 ¼ 1.42 þ 0.69; direct effect plus indirect effect viacompleteness of communication).

Likewise, the use of a drawing, use of an example, or use ofalgebraic reasoning also had positive total effects on spatial literacy.When teachers used a drawing (direct effect ¼ �0.36), they showedmore completeness of communication (þ0.42), resulting in an overallpositive total effect of þ0.44 extra points on each item(þ0.44 ¼ �0.36 þ [0.42][1.91]; direct effect plus its indirect effectvia completeness of communication’s total effect of 1.91). Whenteachers used an example (direct effect¼þ0.52), they showedmorecompleteness of communication (þ0.16), resulting in an overallpositive total effect of þ0.83 on each item (þ0.83 ¼ 0.52 þ [0.16][1.91]; direct effect plus indirect effect via completeness ofcommunication). Teachers who used algebraic reasoning on an itemshowed more completeness of communication (þ0.40) and fewervocabulary problems (�0.04), resulting in a total effect of þ1.35points higher on each item (þ1.35 ¼ 0.52 þ [0.40][1.91] þ [�0.04][�1.53]; direct effect plus indirect effects via completeness ofcommunication and via vocabulary problems).

Lastly, items in which algebraic reasoning was possible ordimensional reasoning was possible were more difficult than otheritems, resulting in lower scores (see second column in Fig. 2). Onitems in which algebraic reasoning was possible, teachers oftenshowed less completeness of communication (�0.18) and slightlyfewer vocabulary problems (�0.09), resulting in an overall totaleffect of �0.82 (¼�0.061 þ [�0.18][1.91] þ [�0.09][�1.53]; directeffect plus indirect effects via completeness of communication’s totaleffect of 1.91 and via vocabulary problems’ total effect of �1.53). Onitems inwhich dimensional reasoning was possible, teachers showedmore presence of misconceptions (þ0.18), slightly more dimensionalreasoning (þ0.03), less use of drawing (�0.31), and slightly more useof example (þ0.05), resulting in an overall total effect of �0.50(¼�0.27 þ [0.18][�1.11] þ [0.03][2.11] þ [�0.31][0.44] þ [0.05][0.83]; direct effect plus indirect effects via presence of misconcep-tions’ total effect [-1.11], dimensional reasoning’s total effect [þ2.11],use of drawing’s total effect [þ0.44], and use of examples’ total effect[þ0.83]; see above for total effect computations). The remainingvariables were not significantly linked to spatial literacy.

7. Discussion and conclusions

This study used a spatial literacy lens to investigate teachers’solutions on related 21-item and 24-item spatial literacy tests. Byconsidering three overlapping domains, visualization, reasoning,and communication, we explored different aspects of spatial

literacy to better understand how a group of mathematics teachersperformed on tasks that involved spatial objects and relationships.

The first research question considered the teachers’ academicand professional backgrounds. Teaching experience or a Bachelorsdegree in mathematics had positive direct effects on the teachers’spatial literacy scores. However, the low spatial literacy scores oneach item indicate that the teachers performed poorly overall.Despite what should have been adequate preparation, manyteachers’ responses were incomplete or incorrect with numerousmisconceptions regardless of their teaching experience oracademic backgrounds. The second research question consideredthe effect of misconceptions on the teachers’ spatial literacy. Thepresence of such a large number of misconceptions suggests thatmany teachers did not have deep, flexible knowledge of the subjectmatter; while not encouraging, this is not surprising consideringfindings from other countries (Fujita & Jones, 2006; Jones, Mooney,& Harries, 2002; de Villiers, 1997).

The teachers’ low scores and numerous misconceptions havetwo implications. First, the content knowledge needed by teachersrequires more than just a solid grasp of the subject (Ball, 1990;Fennema & Franke, 1992). The teachers’ lack of robust subjectmatter knowledge could impact how they are able to “mediatestudents’ ideas, make choices about representations of content,modify curriculummaterials, and the like” (Ball & Bass, 2000, p. 97).Second, misconceptions were prevalent in the teachers’ responses.Such misconceptions may not only carry over to their students butalso lead to a host of relatedmisconceptions stemming from the oldones.

It is interesting to note that academic background and profes-sional experience had an impact on the teachers’ spatial literacy butno statistical effect on their content knowledge. One possiblereason for the increased spatial literacy scores corresponds withfindings that free-hand sketching of figures, a common task inmathematics courses and in the teaching of mathematics, has beenshown to be a significant factor in the development of spatial skills(Leopold, Górska, & Sorby, 2001; Sorby, 2009). It is possible thatbackground and experience failed to impact the teachers’ contentknowledge due to the resilient nature of misconceptions (Chi,2005; Fraser, Tobin, & McRobbie, 2012) or since many of thetopics in upper division mathematics classes do not address thesecondary topics on the instrument in this study.

The third research question focused on the reasoning domainof spatial literacy by considering teachers’ reasoning strategies.Teachers used a variety of strategies in their responses to theitems. While each of the five strategies employed by the teachershad a positive overall effect on their spatial literacy scores, thesingle most effective reasoning strategy was the use of dimen-sional reasoning. Even though use of dimensional reasoning wasthe highest contributor to spatial literacy scores, the teachersemployed dimensional reasoning in only 4% of their responses.As the results showed that dimensional reasoning was effectivebut underutilized, it seems that teachers were either unfamiliaror uncomfortable using such a potentially beneficial reasoningstrategy. Past research (Webb & Farivar, 1999; Webb, Nemer, &Ing, 2006), shows that if a teacher is not aware of a strategy,his or her students are unlikely to experience that strategy in theclassroom. The responses suggest that it is unlikely that theteachers in this study would use dimensional reasoning in theirclassrooms.

Each strategy contributed to the teachers’ spatial literacy scores.Teachers who demonstrated a more robust spatial literacy (asmeasured by their use of multiple reasoning strategies and use ofcomplete communication, among other things) performed better.Since many teachers did not make effective use of multiple strat-egies to solve problems, our findings extend past the teachers’

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content-specific performances to suggest that they may need to“shift their perspectives about teaching from that of a process ofdelivering information to that of a process of facilitating students’sense making about mathematics” (Martin, 2007, p. 5). Besidesa deep knowledge of their subject matter, these teachers could alsobenefit by developingmore general spatial literacy skills that can beapplied across content areas. Increasing the number and types ofstrategies available to these teachers might help them, and theirstudents, successfully solve more problems involving spatialconcepts.

If students are not provided opportunities to work with physicalmanipulatives, sketch diagrams by hand, and engage in otherpractices that have been found to develop spatial literacy skills(Wright, Thompson, Ganis, Newcombe, & Kosslyn, 2008) in theirmathematics classes, where will they be provided such opportu-nities?Moreover, these opportunities could have a greater impact ifthey were accompanied by teaching specific spatial reasoningstrategies.

It should not be assumed that students automatically acquirespatial literacy skills; Schonborn and Anderson (2006) assertedit is “essential to explicitly teach and assess this type ofknowledge” (p. 101). Others suggest that spatial literacyinstruction has a positive impact (Terlecki, Newcombe, & Little,2008) that is both lasting and transferable (Mohler & Miller,2008; Sorby, 2009). Moreover, Rovet (1983, p. 171) claimed, “itappears that 12 min of instruction was roughly equivalent tothree years of untutored development.” Hence changes, evensmall changes, that acknowledge and explicitly teach spatialreasoning skills might have significant impact without the needto substantially reduce time spent on typical instruction topics.Since spatial reasoning plays a critical role in many fields, spatialeducation should not only be embedded in mathematics coursesbut also interwoven across different academic areas. This alignswith suggestions (Braukmann & Pedras, 1993) that context,including discipline-specific applications, is important in devel-oping spatial skills.

The fourth research question dealt with the impact of visuali-zation and communication on teachers’ spatial literacy and theeffect of reasoning strategies on these domains. Few teachersexplicitly mentioned difficulties with visualization; as such, thevisualization variables did not have a statistically significant effecton the teachers’ spatial literacy scores. The communication vari-ables, however, did have an impact on teachers’ spatial literacyscores. Total effects for completeness of communication had a strongpositive effect whereas vocabulary problems had a strong negativeeffect. Despite their mathematics training, many of the responsessuggested that the teachers did not grasp the vocabulary of basicspatial concepts. Vocabulary, which falls in the communicationdomain of spatial literacy, should be fostered in the mathematicsclassroom. If teachers have not mastered the most fundamentalaspect of the contente vocabulary, then they cannot be expected touse “spatial language in sensitive and precise ways [which] can beinstrumental in maximizing spatial learning” (Newcombe & Stieff,2012, p. 965).

Communication of spatial ideas is an integral part of the worldaround us. We live and work in a three-dimensional world, and astechnology advances, individuals are more likely to be asked tonavigate virtual spatial environments. A person’s success in fieldsranging from athletics to engineering depends on successfulcommunication of spatial ideas.

While no one reasoning strategy made the teachers moresusceptible to lower spatial literacy scores caused by vocabularyproblems, reasoning strategies had an effect on the teachers’completeness of communication. The use of drawing, use of example,use of algebraic reasoning, and use of dimensional reasoning

cumulatively increased the teachers’ completeness of communica-tion and, hence, the teachers’ likelihoods of achieving higher spatialliteracy scores. Although the teachers’ spatial literacy scores werepoor, in all cases the reasoning strategies had a positive impact ontheir completeness of communication. Communicating ideas isintegral to the practice of teaching; furthermore, the communica-tion of spatial concepts and relationships extends past academiaand impacts people in a wide variety of fields. It would seem thatthe availability of multiple spatial reasoning strategies wouldbenefit numerous individuals.

The fifth research question considered the teachers’ spatialliteracy in terms of the requirements of the curricula discussedpreviously. The results of this study are consistent with deLange’s (2003) assertion that spatial literacy has been neglec-ted and should encourage the education community to considerhow spatial literacy is addressed, especially in teacher education.In light of the findings of Pittalis and Christou (2010), increasedspatial literacy should positively impact teachers’ instructionaldecisions and expectations, especially as related to teaching thatinvolves three-dimensional situations. Therefore, we contendthat since spatial literacy applies to fields beyond mathematics,teachers (and their students) need experiences that support andencourage visualization, reasoning, and communication ofspatial objects, their properties, and their relationships acrosscurricula.

8. Limitations and future work

Although spatial literacy involves all three domains, additionalresearch is needed that specifically addresses the visualizationdomain. Since enough teachers’ responses contained explicitmention of difficulty visualizing, the decision was made to includethis variable in the study. However, too few responses mentionedthese difficulties to demonstrate an impact on the path diagram.Likewise, too few teachers’ responses contained evidence ofconfusion between mathematical objects and their representationsto demonstrate an impact on the path diagram, although it ismentioned in the literature.

Visualization, as measured in this study, did not impact theteachers’ spatial literacy scores; however, its analysis included onlytwo measures: difficulty with visualization and confusion betweenfigures and representations. The paper-and-pencil instrumentemployed did not lend itself to probing teachers about visualizationor its overlap with reasoning, rather the instrument allowed theresearch team to record only what was evidenced in the teachers’responses. Future work can develop alternate means to assessteachers’ visualization.

A recent study of literature teachers’ use of textbook materials(Poyas & Eilam, 2012) suggested a three-phase navigation processto (a) find commonalties in the visual elements and text, (b) cyclerepeatedly between the visual elements and the text, and (c)construct of a holistic view of a “textevisual” interpretive space.Applying this three-phase process to both visual and textualcomponents of mathematics (including words and algebraicequations) might shed further light on the spatial literacy skills thatmathematics teachers possess.

Because a non-random convenience sample was used for thisstudy, its results are not intended to be generalized. Nevertheless,this research provides valuable insight into spatial literacy asa construct comprised of three intersecting domains and can serveas a catalyst for others to explore spatial literacy. Future studiescould use randomized, representative samples and extend toteachers in other content areas or in other countries. If the results ofthis study are replicated with a larger sample, it would providestrong evidence that spatial literacy goals are not being addressed.

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Acknowledgment

We gratefully acknowledge Yik Ting Choi for her assistance.

Fig. 5. Use of example to solve problem.

Appendix A. Sample rubric

The item-specific rubric forWhat can bisect a plane? is displayedin Table 6.

Table 6Sample rubric for specific item.

Score

0 1 2 3

Blank; incorrectobject (e.g.,point, segment,ray, angle)

Correct object, butincorrect condition(e.g., another planeintersects the first ata right angle); correctconception of planebut incorrect notionof bisection beinglimited to finite objects(e.g., nothing can bisecta plane since it extendsinfinitely in twodirections)

Mention of onlyone of the twoobjects that canbisect a plane(e.g., any linethat lies on theplane cuts itinto two halfplanes)

Mention of bothobjects that canbisect a plane(e.g., any line onthe plane andany plane thatis unique from butintersects thefirst, such thattheir intersectionis a line)

Fig. 6. Incorrectly forcing 3D situations to 2D.

Appendix B. Examples of teachers’ misconceptions related tovisualization measures

One of the true/false items stated: Planes A and C must intersect ifthe intersection of planes A and B is nonempty and the intersection ofplanes B and C is nonempty. One response that exemplified confu-sion between a mathematical object and its representation is pre-sented in Fig. 3. In this response, the teacher attributed the

Fig. 3. Confusion between planes and their representations.

5

46

Fig. 7. Correct response to solid of revolution problem.

properties of finite representations (i.e., drawings with definiteboundaries for the planes) to the figure. In fact, planes extendinfinitely, and, as drawn, planes A, B, and C would actuallycompletely overlap rather than overlap only where shaded.

Appendix C. Examples of teachers’ responses andmisconceptions related to reasoning measures

One item asked the teachers to determine the relation betweenthe radii of two balls if the volume of the larger was double that ofthe smaller. Fig. 4 displays a response that combined two reasoningstrategies, use of drawing and algebraic reasoning.

Fig. 4. Combined use of drawing and algebraic reasoning to solve problem.

The teacher whose response is seen in Fig. 5 employed use ofexample to solve the same problem. This teacher arrived at the

solution by assigning a particular numeric value for the figure’sradius.

Dimensional reasoning was evidenced when teachers movedbetween dimensions. When asked about the equation ax þ by ¼ din three dimensions, one teacher responded, “This is the graph ofa line on a two-dimensional plane, but in three dimensions you alsohave to think about the z variable, even when it is not in theequation. So, this equation would be a graph of a plane perpen-dicular to the xy-plane that passes through that line since z can beany real number.”

An example of incorrectly forcing 3D situations to 2D is shown inFig. 6. Even though the instrument gave instructions that problems

were to be considered in three dimensions, one teacher’s responseto the following item (a problem commonly found in calculuscourses): If the right triangle shown below is revolved about its 4inch side, describe, draw, and label the lengths of the resulting solidof revolution is an example of the persistence to limit three-dimensional situations to a two-dimensional environment. Thecorrect response, a cone, is shown in Fig. 7.

For an example of a presence of a misconception, consider onetrue/false item that asked if any three distinct points in three-dimensional space would lie on a single line. One teacher claimedthat two distinct points would lie on a single line and a thirddistinct point would define a plane. This response was assigneda code of 1 (presence of misconception) because it can be, but is notalways, true. Three points, like the three feet of a tripod, can lie ona flat two-dimensional surface hence defining a plane. However,this is not the only possibility since three distinct points can all lieon the same line (i.e., be collinear). If youmake the feet of the tripodline up with each other, it will not stand but will fall over so noplane is defined.

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Appendix D. Example of teachers’ misconceptions related tovocabulary measures

Each content area has its own specific terminology. Thefollowing figures provide evidence of the mathematics teachers’vocabulary struggles with the wording “regular tetrahedron” fromone item. Even though the term “regular” (i.e., all edges and angleshave the same measure) is commonly used in both two- and three-dimensional geometry curricula, the responses shown in Fig. 8

Fig. 8. Teachers’ vocabulary problems with the word “regular.”

demonstrated two teachers’ difficulties with the term, sinceneither struggled to produce edges of equal lengths in otherdrawings when needed.

The response seen in Fig. 9 illustrates another teacher’s difficultywith the term “tetrahedron”, a solid with four faces.

Fig. 9. Teacher’s vocabulary problem with the word “tetrahedron.”

References

Adams, T. L. (2003). Reading mathematics: more than words can say. The ReadingTeacher, 56(8), 786e795.

Alsina, C. (2000). Gaudi’s ideas for your classroom: geometry for three-dimensionalcitizens. In Abstracts of plenary lectures and regular lectures (pp. 22e23). 9thInternational congress on mathematics education. Tokyo, Japan: ICME 9.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics.Educational Studies in Mathematics, 52(3), 215e241.

Baker, F. B., & Kim, S. H. (2004). Item response theory. Boca Raton, FL: CRC Press.Ball, D. L. (1990). The mathematical understandings that prospective teachers bring

to teacher education. Elementary School Journal, 90, 449e466.Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and

learning to teach: knowing and using mathematics. In J. Boaler (Ed.), Multipleperspectives on the teaching and learning of mathematics (pp. 83e104). Westport,CT: Ablex.

Battista, M. T. (2007). The development of geometric and spatial thinking. InF. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching andlearning (pp. 843e908). Charlotte, NC: National Council of Teachers of Mathe-matics/Information Age Publishing.

Benjamini, Y., Krieger, A. M., & Yekutieli, D. (2006). Adaptive linear step-up proce-dures that control the false discovery rate. Biometrika, 93(3), 491e507.

Bishop, A. (1983). Space and geometry. In R. Lesh, & M. Landau (Eds.), Acquisition ofmathematics concepts and processes (pp. 175e203). Orlando, FL: Academic Press.

Bock, R. D., & Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a micro-computer environment. Applied Psychological Measurement, 6(4), 431e444.

Braukmann, J., & Pedras, M. J. (1993). A comparison of two methods of teachingvisualization skills to college students. National Association of Industrial andTechnology Teacher Educators, 30(2), 65e80.

Brown, N. C. M. (2003). Are we entering a post-critical age in visual arts education?Studies in Art Education, 44(3), 283e289.

Chi, M. T. H. (2005). Commonsense conceptions of emergent processes: why somemisconceptions are robust. The Journal of the Learning Sciences, 14(92), 161e199.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. InD. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning(pp. 420e464). New York, NY: NCTM/Macmillan Publishing Co.

Common Core State Standards Initiative. (2010). Common core standards for math-ematics. Washington, DC: National Governors Association Center for BestPractices and the Council of Chief State School Officers.

Contero, M., Naya, F., Company, P., Saorín, J. L., & Conesa, J. (2005). Improvingvisualization skills in engineering education. IEEE Computer Graphics andApplications, 25(5), 24e31.

Cuendet, S., Jermann, P., Arn, C., & Dillenbourg, P. (2011). A study of spatialreasoning skills in carpenters’ training. Retrieved from the École PolytechniqueFédérale de Lausanne website. http://infoscience.epfl.ch/record/164668/files/zollikofen.pdf.

Del Grande, J. (1990). Spatial sense. Arithmetic Teacher, 37(6), 14e20.Duval, R. (1999). Representation, vision and visualization: cognitive functions

in mathematical thinking. Basic issues for learning. In F. Hitt, & M. Santos(Eds.). Proceedings of the 21st North American PME conference, Vol. 1 (pp.3e26).

Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. InD. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning(pp. 420e464). New York, NY: National Council of Teachers of Mathematics/Macmillan Publishing Co.

Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning.International Journal of Science Education, 20(10), 1193e1211.

Fraser, B. J., Tobin, K., & McRobbie, C. (Eds.), (2012). Second international handbook ofscience education, Vol. 1. Dordrecht, The Netherlands: Springer.

Fujita, T., & Jones, K. (2006). Primary trainee teachers’ understanding of basicgeometric figures in Scotland. In J. Novotná, H. Moraová, M. Krátká, &N. Stehlíková (Eds.). Proceedings of the 30th conference of the International Groupfor the Psychology of Mathematics Education, Vol. 3 (pp. 129e136). Prague, CzechRepublic: PME.

Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: a currentreview of research on teaching and learning statistics. International StatisticsReview, 75(3), 372e392.

Geddes, D., Bove, J., Fortunato, I., Fuys, D. J., Morgenstern, J., & Welchman-Tischler, R.(1992). Geometry in the middle grades. Curriculum and evaluation standards forschool mathematicsIn Addenda series: Grades 5e8, . Reston, VA: NCTM.

Godfrey, C. (1910). The board of education circular on the teaching of geometry.Mathematical Gazette, 5(84), 195e200.

Goldenberg, E. P., Cuoco, A. A., & Mark, J. (1998). A role for geometry in generaleducation? In R. Lehrer, & D. Chazan (Eds.), Designing learning environments fordeveloping understanding of geometry and space (pp. 3e44) Hillsdale, MJ: Law-rence Erlbaum.

Goldstein, H. (1995). Multilevel statistical models. Sydney, Australia: Edward Arnold.Gorgorió, N. (1998). Exploring the functionality of visual and non-visual strategies in

solving rotation problems. Educational Studies in Mathematics, 35(3), 207e231.Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: in search of

a framework. In L. Puig, & A. Gutiérrez (Eds.). Proceedings of the 20th annualmeeting of the International Group for the Psychology of Mathematics Education,Vol. 1 (pp. 3e19). Valencia, Spain: Program Committee.

Haanstra, F. (1996). Effects of art education on visualespatial ability and aestheticperception: a quantitative review. Studies in Art Education, 37(4), 197e209.

Hollebrands, K. F. (2003). High school students’ understandings of geometrictransformations in the context of a technological environment. Journal ofMathematical Behavior, 22(1), 55e72.

Hughes Hallet, D. (1991). Visualization and calculus reform. In W. Zimmerman, &S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp.121e126). Washington, DC: Mathematical Association of America.

Incekara, S. (2010). The place of geographic information systems (GIS) in the newgeography curriculum of Turkey and relevant textbooks: is GIS contributing tothe geography education in secondary schools? Scientific Research and Essays,5(6), 551e559.

Jones, K., Mooney, C., & Harries, T. (2002). Trainee primary teachers’ knowledge ofgeometry for teaching. Proceedings of the British Society for Research IntoLearning Mathematics, 22(2), 95e100.

Joreskog, K., & Sorbom, D. (2004). LISREL 8.7. New York: Scientific SoftwareInternational.

Kennedy, P. (2008). A guide to econometrics. Cambridge, UK: Blackwell.Kozhevnikov, M., Hegarty, M., & Mayer, R. E. (2002). Visual/spatial abilities in

problem solving in physics. In M. Anderson, B. Meyer, & P. Olivier (Eds.), Dia-grammatic representation and reasoning. Berlin: Springer.

Kozhevnikov,M., Kosslyn, S., &Shephard, J. (2005). Spatial versusobjectvisualizers: anewcharacterization of visual cognitive style.Memory & Cognition, 33(4), 710e726.

Krull, J. L., & MacKinnon, D. P. (2001). Multilevel modeling of individual and grouplevel mediated effects. Multivariate Behavioral Research, 36, 249e277.

Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren.Chicago: University of Chicago Press.

Laborde, C. (1993). The computer as part of the learning environment: the case ofgeometry. In C. Keitel, & K. Ruthven (Eds.), Learning from computers: Mathe-matics education and technology (pp. 48e67). Berlin: Springer.

de Lange, J. (2003). Mathematics for literacy. In B. L. Madison, & L. A. Steen (Eds.),Quantitative literacy: Why numeracy matters for schools and colleges (pp. 75e89).Princeton, NJ: The National Council on Education and the Disciplines.

Leopold, C., Górska, R. A., & Sorby, S. A. (2001). International experiences indeveloping the spatial visualization abilities of engineering students. Journal forGeometry and Graphics, 5(1), 81e91.

Martin, T. S. (Ed.), (2007). Mathematics teaching today: Improving practice, improvingstudent learning! Reston, VA: National Council of Teachers of Mathematics.

Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran,& L. Lee (Eds.), Approaches to algebra: Perspective for research and teaching (pp.65e86). Dordrecht, The Netherlands: Kluwer.

McGraw, T. M. (2004). The effects of two-dimensional stimuli and three-dimensional stereoptic stimuli on spatial representation in drawings. Studiesin Art Education, 45(2), 153e169.

McInerney, M., Berg, K., Hutchinson, N., Maude, A., & Sorensen, L. (2009). Towardsa national geography curriculum for Australia. Queensland, Australia: AustralianGeography Teachers Association Ltd., Royal Geographical Society of QueenslandInc., and the Institute of Australian Geographers Inc.

Page 13: Teachers' spatial literacy as visualization, reasoning, and communication

D. Moore-Russo et al. / Teaching and Teacher Education 29 (2013) 97e109 109

Mohler, J. L., & Miller, C. L. (2008). Improving spatial ability with mentoredsketching. Engineering Design Graphics Journal, 72(1), 19e27.

Montiel, M., Wilhelmi, M. R., Vidakovic, D., & Elstak, I. (2009). Using the onto-semiotic approach to identify and analyze mathematical meaning when tran-siting between different coordinate systems in a multivariate context. Educa-tional Studies in Mathematics, 72(2), 139e160.

Moore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space?Studying concept image of slope through collective definition construction.Educational Studies in Mathematics, 76(1), 3e21.

Moore-Russo, D., & Viglietti, J. M. (2012). Teachers’ communication and under-standing of axes in 3-dimensional space: An introduction to the K5 ConnectedCognition Diagram. Journal of Mathematical Behavior, 31(2), 235e251.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluationstandards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000a). Principles and standards forschool mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000b). Animals as our companions:World’s largest math event 6. Reston, VA: Author.

National Council of Teachers of Mathematics. (2006). Curriculum focal points forprekindergarten through grade 8 mathematics. Reston, VA: Author.

National Research Council. (2006a). National science education standards. Wash-ington, DC: National Academies Press.

National Research Council. (2006b). Learning to think spatially: GIS as a supportsystem in the Ke12 curriculum. Washington, DC: National Academies Press.

Newcombe, N. S. (2006). A plea for spatial literacy. The Chronicle of Higher Education,52(26), B20.

Newcombe, N. S., & Stieff, M. (2012). Six myths about spatial thinking. InternationalJournal of Science Education, 34(6), 955e971.

O’Hallaran, K. L. (2008). Mathematical discourse: Language, symbolism and visualimages. London: Continuum.

Orton, A. (1987). Learning mathematics: Issues, theory, and classroom practice. Lon-don: Cassell Education.

Pallrand, G., & Seeber, F. (1984). Spatial ability and achievement in introductoryphysics. Journal of Research in Science Teaching, 21(5), 507e516.

Parzysz, B. (1988). Knowing versus seeing: problems of the plane representation ofspace geometry figures. Educational Studies in Mathematics, 19(1), 79e92.

Phillips, L. M., Norris, S. P., & Macnab, J. S. (2010). Visualization in mathematics,reading and science education. Dordrecht, The Netherlands: Springer.

Pittalis,M.,&Christou,C. (2010). Typesof reasoning in3Dgeometry thinkingand theirrelationwith spatial ability. Educational Studies in Mathematics, 75(2), 191e212.

Poyas, Y., & Eilam, B. (2012). Construction of common interpretive spaces throughintertextual loops e how teachers interpret multimodal leaning materials.Teaching and Teacher Education, 23, 89e100.

Presmeg, N. C. (1997). Generalization using imagery in mathematics. In L. D English(Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 299e312).Mahwah, NJ: Erlbaum.

Presmeg, N. C. (2006). Research on visualization in learning and teaching mathe-matics: emergence from psychology. In A. Gutiérrez, & P. Boero (Eds.), Handbookof research on the psychology of mathematics education (pp. 205e235). Rotter-dam: Sense Publishers.

Reehm, S. P., & Long, S. A. (1996). Reading in the mathematics classroom. MiddleSchool Journal, 27(5), 35e41.

Rivera, F. (2007). Visualizing as a mathematical way of knowing: understandingfigural generalization. Mathematics Teacher, 101(1), 69e75.

Rovet, J. (1983). The education of spatial transformations. In D. R. Olson (Ed.), Spatialcognition: The structure and development of mental representations of spatialrelations (pp. 164e181). Hillsdale, NJ: Lawrence Erlbaum Associates.

Royal Society/Joint Mathematics Council. (2001). Teaching and learning geometry11e19. London: Royal Society.

Schonborn, K. J., & Anderson, T. R. (2006). The importance of visual literacy in theeducation of biochemists. Biochemistry and Molecular Biology Education, 34(2),94e102.

Sorby, S. (2009). Educational research in developing 3-D spatial skills forengineering students. International Journal of Science Education, 31(3),459e480.

South African Department of Education. (2003). National curriculum statementgrades 10e12 (general). Mathematical literacy. Pretoria: Department ofEducation.

Strauss, A. L., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theoryprocedures and techniques. Newbury Park, CA: Sage.

Tabachnick, B. G., & Fidell, L. S. (2006). Using multivariate statistics. Boston: Allyn &Bacon.

Terlecki, M. S., Newcombe, N. S., & Little, M. (2008). Durable and generalized effectsof spatial experience on mental rotation: gender differences in growth patterns.Applied Cognitive Psychology, 22(7), 996e1013.

Thompson, D. R., & Chappell, M. F. (2007). Communication and representation aselements in mathematics literacy. Reading & Writing Quarterly, 23(2), 179e196.

Thompson, D. R., & Rubenstein, R. N. (2000). Learning mathematics vocabulary:potential pitfalls and instructional strategies. Mathematics Teacher, 93(7),568e574.

Trindade, J., Fiolhais, C., & Almeida, L. (2002). Science learning in virtual envi-ronments: a descriptive study. British Journal of Educational Technology, 33(4),471e488.

Usiskin, Z. (1987). Resolving the continuing dilemmas in school geometry. InM. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, Ke12 (pp.17e31). Reston, VA: NCTM.

de Villiers, M. (1997). The future of secondary school geometry. Pythagoras, 44,37e54.

Webb, N. W., & Farivar, S. (1999). Developing productive group interaction in middleschool mathematics. In A. M. O’Donnell, & A. King’s (Eds.), Cognitive perspectiveson peer learning (pp. 117e149). Mahwah, NJ: Erlbaum.

Webb, N. W., Nemer, K. M., & Ing, M. (2006). Small-group reflections: parallelsbetween teacher discourse and student behavior in peer-directed groups.Journal of the Learning Sciences, 15(1), 63e119.

Wright, R., Thompson, W. L., Ganis, G., Newcombe, N. S., & Kosslyn, S. M. (2008).Training generalized spatial skills. Psychonomic Bulletin & Review, 15(4), 763e771.

Yap, L. Y., Tan, G. C. I., Zhu, X., & Wettasinghe, M. C. (2008). An assessment of the useof geographical information systems (GIS) in teaching geography in Singaporeschools. Journal of Geography, 107(2), 52e60.

Zimmerman, W., & Cunningham, C. (1991). Editor’s introduction: what is mathe-matical visualization? In W. Zimmerman, & S. Cunningham (Eds.), Visualizationin teaching and learning mathematics (pp. 1e8) Washington, DC: MathematicalAssociation of America.