Investigating Imagination as a Cognitive Space for Learning Mathematics

Educ Stud Math (2009) 70:259–274 DOI 10.1007/s10649-008-9154-0

Investigating imagination as a cognitive space for learning mathematics Donna Kotsopoulos & Michelle Cordy

Published online: 6 September 2008 # Springer Science + Business Media B.V. 2008

Abstract Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problemposing in learning mathematics, as well as imagination as a cognitive space for learning.

Keywords Cognitive . Imagination . Learning . Mathematics . Problem-posing .

Visual . Visualization

Yet, ultimately, mathematics reaches pinnacles as high as those attained by the imagination in its most daring reconnoiters. And this conceals, perhaps, the ultimate paradox of science. For in their prosaic plodding both logic and mathematics often outstrip their advance guard and show that the world of pure reason is stranger than the world of pure fancy. (Kasner & Newman 1940, p. 362)

D. Kotsopoulos (*) : M. Cordy Wilfrid Laurier University, Waterloo, ON, Canada e-mail: dkotsopo@wlu.ca

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1 Introduction Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor, Barry Mazur (2003). In his book, Mazur highlights, describes, and recreates the imaginative work of mathematicians. For example, he describes the work of 16th century mathematician, Girolamo Cardano, who is known to have invoked the earliest notions of imaginary numbers. In relation to Cardano and others throughout time, Mazur exemplifies that imagination has been shown to have been an integral part of the intellectual tool kits of both applied and theoretical mathematicians. These assertions about the imaginative work of mathematicians are not surprising given that mathematics consists of objects that, at best, can only be imagined (Lemke 2005; Moschkovich 2003; Sfard 2000).

Imagination and mathematics have often been coupled in research and scholarship, yet the relationship between the two is complex, often unexplainable, and elusive (Egan 1992; Gadanidis 2006; Hilbert & Cohn-Vossen 1952; Kasner & Newman 1940; Lakoff & Nunez 2000). Similar couplings have been made in mathematics education, where imagination is discussed as being important and useful (Davis & Simmt 2006; Hilbert & Cohn-Vossen 1952; Kasner & Newman 1940; Mazur 2003; Sfard 1997). Despite these connections, little is known about the ways in which imagination can be used as both a pedagogical and a mathematical tool for learning mathematics (Henderson 1995).

The work of Mazur (2003) led us to question whether teaching and learning mathematics can be thought of in terms of imagination. More specifically, we questioned whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Imagination as a means of learning mathematics has yet to be well defined, unlike this term as used in other areas such as language arts (see for example Egan 1992).

Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in a middle school setting. The research questions that guided our research were as follows: (1) How might teaching and learning mathematics be structured as a way of imagining? (2) What are we attending to pedagogically when we are teaching mathematics as a way of imagining?

2 Imagination and visualization: working definitions Mazur (2003) defines imagination1 as a new way of “providing a natural home for that something in the mind’s mind” (p. 138), that resists familiar “types of mental image-making activities” (p. 138). Commonly, imagination and visualization are taken as synonymous. As Mazur (2003) explains, “to imagine an . . . object and to visualize it can [our emphasis] be very different activities” (p. 138). He explains that visualization is an act of bringing a “picture into the mind’s screen” (p. 138) with already existing “mental equipment” (p. 138).

1

Mazur (2003) provides a comprehensive literature review of definitions of imagination (see pp. 15–16).

Our intention is to explore imagination in teaching and not to engage in a wider discussion on definitions of imagination beyond Mazur’s own explication, which has been adopted in our own analysis.

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In contrast, imagination has no such pre-existing mental equipment or picture, and the result is unfamiliar. The picture that Mazur identifies can be a single object (e.g., a triangle), a combination of objects (e.g., assorted polygons, symbols, etc.), or an experience (e.g., watching a teacher-led demonstration).

This framework distinguishes between asking students to engage in visualization and to imagine, based upon whether the picture or object is known to the student, or not. If a picture or object is unknown, students are engaging in imagination where a new picture or object is constructed on the minds’ screen. For example, asking middle-school students to “imagine” a square should generate a picture on their mind’s screen of a four-sided regular polygon with all sides equal and all internal angles 90°. However, asking middle-school students to “imagine” a rectangle that can also be a square may prove to be more challenging. Students may have no internal picture to bring into the mind’s screen. If students can visualize a rectangle that can also be a square, they are not by the definition advanced by Mazur imagining. Instead, they are visualizing. Our goal is to explore how new internal pictures or objects (e.g., a square that is also a rectangle) can be created through imagination.

Imagination for the purpose of this paper is conceptualized as processing that takes place in a cognitive space for learning (Baars 1996). In this cognitive space, visuals are used to activate existing mental equipment. A visual is concrete and can be seen. Visuals can be pictures, objects, or experiences (i.e., watching a teacher-led demonstration). Visualization is the act of bringing a visual onto the mind’s screen (Mazur 2003); at which time the concrete picture or object may become redundant. Learning is seen as the shift between pictures and objects already existing on the mind’s screen, and new pictures or objects created by “the more difficult leaps of the imagination [that] force us to establish larger screens and, perhaps, new theaters of the mind” (Mazur 2003, p. 142).

3 The framework The framework is conceptualized as having three stages—“before,” “during,” and “after.”

Mazur (2003) points out, there is no such “temporal ‘before’ and ‘after’ in the logical structure of mathematics” (p. 150). However, as Mazur notes, distinctions of “before” and “after” are inherent in the epistemological development of imagination; consequently, these distinctions are also important in this framework.

Our identification of stages is not intended to suggest that imagination can be easily analyzed in a series of well defined and clearly distinguished stages. Nor is the framework intended to provide a prescriptive approach to teaching mathematics. Our articulation of the cognitive space of imagination is intended only as a guide for discussion and for the development of pedagogical practices.

3.1 Before imagination A key aspect of the before stage of imagination is the selection of a problem with which to actively engage students. Mazur (2003) describes such problems as follows: All the best mathematical problems are come-ons: there is a gentle irony behind them.

The problem setter usually presents to you a very precise task. Solve this! ... But if the problem is really good, a solution of it is nothing more than a letter of introduction to

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a level of interaction with material you hadn’t achieved before. Solving the problem gets you to a deeper level of question-asking. The problem itself is an invitation, a goad, to extend your imagination. (p. 23) Mazur emphasizes two important points. First, problems ought to have the potential to capture the attention of students. Capturing students’ attention does not imply that the problem is necessarily relevant to students’ lives, but more importantly that the problem is interesting. If children are to engage in imagination, then as teachers we need to select problems that are sufficiently stimulating. Second, the problem is one that leads to problemposing. The solution is not the conclusion of the problem, but rather the beginning of another.

Following the selection of a problem, teachers in preparation for imagination (e.g., determining prior knowledge, planning pedagogical approaches, etc.) will need to engage in some preliminary “corralling.” By definition, corralling means to surround and enclose.

Within this framework, corralling signifies the surrounding and enclosing of the pertinent prior, current, or future knowledge necessary for the particular mathematics in question to be fully explored by students (Mazur 2003, p. 63).

Corralling will also occur during learning. Any variety of pedagogical strategies can be used to corral, including exploration, experimentation, whole-group discussions, and so forth. Although some strategies used during corralling might be anticipated, others will occur reflexively in response to the negotiation of meaning occurring in the classroom.

Caution with corralling should be exercised. Students’ participation needs to be valued and encouraged, despite where they are in their own individual learning.

A key element of this framework is the use of a visual to introduce students to the problem (Mazur 2003, p. 132). Other researchers have also emphasized the importance of visuals as starting points for learning (see, for example, Pirie & Kieran 1994, growth of understanding framework). The visuals need not be initially mathematical. Through the approach proposed in this framework, students are actively involved in the negotiation of mathematical meaning in the classroom about objects, concepts, and experiences that may be initially ambiguous in terms of the ways in which they are mathematical or the intended mathematical meanings (Bauersfeld 1992; Voigt 1994). For example, a soccer ball (or football in some cultures) may not initially appear mathematical. Discussions or explorations related to spherical polyhedra or tiling of hexagons or pentagons on the soccer ball moves the initially non-mathematical object into the mathematical arena.

Bauersfeld (1992) explains that the visual, introduced initially by the teacher, “functions like an objectively given standard” (p. 479) from which to position learning. The visual (e.g., object, picture, or demonstration) becomes a “taken-as-shared” artifact in the classroom (Yackel & Cobb 1996). The taken-as-shared artifact permits students, alongside the teacher, to jointly and interactively negotiate mathematical meaning from a common and concrete reference point (Voigt 1994).

As previous experiences vary so much in the classroom, the common practice of starting with prior knowledge as a beginning point of learning can be problematic in that this starting point is often a case of starting with the teachers’ assumptions about students’ prior knowledge. By establishing a visual as a taken-as-shared artifact, more students can participate: The teacher can be more confident that all students are in some sense starting from the same point. This will later help students talk with one another, as well as explain their own thinking, since visuals are originating from the same starting point when developed from a taken-as-shared perspective.

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3.2 During imagination The goal during imagination is the creation of a new picture or object on the mind’s screen (Mazur 2003, p. 40), which then becomes the “paradigm case” (Yackel & Cobb 1996, p. 469). Paradigm case, as used in this framework, refers to the new picture or object on the mind’s screen, which is established as the benchmark of what counts as mathematical or mathematically relevant within a particular context. The paradigm case is important in that it allows the teacher to negotiate meaning against the paradigm case to determine the extent to which learning is occurring. As Voigt (1994) explains, students may negotiate meaning of “empirical phenomena differently than expected by the teacher” (p. 280). Negotiation of meaning against the paradigm case is necessary during imagination, to clarify instances where difference may have occurred, and to move towards shared understandings.

One proposed strategy that can be used during this stage is “angling” (Mazur 2003, p. 5).

The term angling is developed metaphorically from physical construction of an angle consisting of two line segments meeting at a vertex, where the vertex can be conceived of as a pointer. Thus, angling within this framework is the cognitive pointing to essential aspects of knowledge drawing from disciplinary, academic, and other real world knowledge. For example, teachers may engage in angling when asking students to reflect upon certain aspects of a graphical representation to draw attention to specific features.

Alternatively, teachers may engage in angling during whole-class argumentation, drawing out more explicit responses and explanations from students as a means of isolating or highlighting pertinent information for the class.

Students may deviate from the intended direction of the learning by establishing alternative pictures or objects on the mind’s screen. It is not fruitful, in situations of time constraint, for students to deviate. Deviations can possibly lead to misconceptions for the students, while simultaneously creating additional challenges for teachers who struggle with time constraints and curricular pressures. For this reason, we propose that continued corralling is necessary to ensure that students remain within the cognitive spaces of the lesson and the discipline, thus keeping the imaginative space manageable for teachers and students. These two strategies, corralling and angling, may be teacher-lead or peer-initiated.

Teachers may encourage discussion amongst students about mathematics, in which case students are corralling and angling for one another.

3.3 After imagination The goal in the after stage is to determine the extent to which the student’s visualization in the prior stage is within the corralled spaces, or not. In simple terms, this stage involves some evaluation of the learning that has, or has not, occurred. We highlight three possible methods for evaluating learning: the aesthetic response, problem-posing, and hybridization of knowledge (Mazur 2003).

Mazur (2003) suggests that only after something has been imagined does an aesthetic response occur. Aesthetic, according to Mazur, signifies the affective (or emotional) judgment of the correctness/completeness/qualities concerning the mathematical knowledge, as well as judgment of the overall mathematical process or activity (cf. Egan 1992, 2005, see next section). This perspective is consistent with DeBellis and Goldin’s (2006) view “that the states of emotional feeling carry meanings for the individual” [emphasis in original] (p. 133). For example, students may reveal confusion (e.g., “I don’t understand”).

Conversely, a positive aesthetic response from a student (e.g., “I get it”) may suggest

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understanding, which may or may not be mathematically based (i.e., a social response following a peer’s lead). More complex are those aesthetic responses whereby students describe knowing in terms of “feel,” “hunch,” and so forth. As Mazur (2003) and others have pointed out, these affective responses have often been described by mathematicians at points of great clarity in their work (Liljedahl 2004).

Positioning of the aesthetic response here is not intended to suggest that an aesthetic response is strictly isolated to this stage. Indeed, aesthetic responses can and do occur throughout the other stages. However, Mazur (2003) positions aesthetic responses in this stage as carrying a particular meaning with respect to mathematics learning; namely, an identifying marker of a student’s progress in their learning.

Another important way for a teacher to determine whether learning has occurred within this framework is through an examination of the problems posed by students in response to the new picture or object (or paradigm case). As Mazur points out: [T]he great glory of mathematics is its durative nature; that it is one of humankind’s longest conversations; that it never finished by answering some questions and taking a bow. Rather, mathematics views its most cherished answers only as springboards to deeper questions. (p. 225) Problem-posing can and ought to be encouraged at any stage. However, we propose that problem-posing may be more effective for students when they have the necessary tools to construct questions most meaningful in moving them forward in further learning. Although there is a plethora of research concerning problem solving, relatively less has been done regarding children posing problems in relation to their own problem solving in mathematics. The springboard potential of problem solving is an area requiring more research attention.

Finally, Mazur (2003) introduces the powerful metaphor of “yoking,” drawn from the noun yoke, which means to join or connect together. As used in this framework, yoking describes the hybridization of knowledge into multiple and connected mathematical representations (i.e., algebraic, geometric, etc.) as evidence of deeper understanding (e.g., showing understanding in multiple ways by connecting graphical representations to real world problems, or algebraic solutions, etc.). This view of yoking is consistent with other research in mathematics education that describes deep understanding as the ability to make connections between mathematical representations (Goldin 1998; Goldin & Kaput 1996)

4 Alternative perspective Egan (1992) also differentiates between imagination and visualization in that visualization requires existing mental pictures, whereas imagination does not. In contrast to visualization, Egan proposed that imagination is the “capacity to think in a particular way” (p. 4). He suggests that imagination is the “crux where perception, memory, idea generation, emotion, metaphor, and no doubt other labeled features of our lives intersect and interact” (p. 3).

Egan’s definition, similarly to Mazur’s (2003), emphasizes imagination as a process rather than an artifact.

Egan describes “imaginative teaching” (p. xii) as an alternative view that takes into consideration (1) cognitive tools (e.g., story, metaphor, binary opposites, rhyme, rhythm, pattern, jokes and humor, visualizations, gossip, play, mystery, and embryonic tools of

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literacy (pp. 2–6), (2) affordances associated with literacy, and (3) abstract theoretical thinking. Egan refers to the first two as “tricks” (xii) used to develop theoretical reasoning, which he describes as “the ability to develop theoretic abstractions and to use the other cognitive tools to that purpose” (p. 222). His framework has been most closely aligned with literacy and language arts.

One key difference between Egan’s (2005) and the proposed framework is the role of visualization. In the present framework, visualization is the central activity needed to develop imagination. Although Egan defines multiple tools, he appears to position story as central. Our view is that story may in fact support learning in some instances, whereas in other instances it may convolute learning through artificial lenses. The relationships between mathematical objects may be sufficient in their own right. Despite these differences, Egan acknowledges that “rarely, from [his] experience and from [his] reading of teacher-education methods texts, do we reflect on the vivid images that might be evoked by the content we wish to teach” (p. 27). Our proposed framework does just this: it requires teachers and students to reflect on visuals and engage in visualizations necessary for teaching specific mathematical concepts.

A secondary difference can be seen in Egan’s (2005) view of the role of the aesthetic. As he describes his framework, the aesthetic component is seen as a consideration that occurs on behalf of the teacher, in the “before” imagination. Egan’s view is more consistent with other scholarly work that explores the role of aesthetics in mathematics in terms of attentiveness (i.e., what to attend to, how to attend, and why to attend; Pimm 2006; Sinclair 2008). While this is an important and alternative perspective, we propose in this framework to use the construct of aesthetic response as one indication that learning has occurred. This aspect does not enter into Egan’s definition.

5 Methods This research was conducted in the classroom of the second author, who at the time was teaching seventh grade. Michelle had been a middle-school teacher for eight years. There were 20 students, nine females and eleven males, who consented to participate. The school had a diverse student population, and was situated in a large urban setting. The students in Michelle’s class engaged in four experiments developed collaboratively by both authors.

The development of the framework and the lesson planning occurred over several months and involved a concurrent reading of Mazur’s (2003) book, face-to-face meetings, as well as many electronic discussions. Wanting to ensure that we were developing ideas from Mazur’s (2003) text appropriately, we also solicited feedback about the framework from Dr. Mazur during the early planning stages, and on a preliminary manuscript.

The lessons were organized around four experiments that took place at the beginning of a geometry and measurement unit, where students would be analyzing properties of threedimensional objects (e.g., calculating and comparing volumes and surface areas of polyhedra, including minimal surface areas in relation to given volumes). The experiments were stimulated by Taylor’s (1993) research on soap bubble films. Taylor shows how bubbles always form the shape of a sphere, which has the smallest surface area for a given volume over other shapes. Taylor’s work contributes to the field of mathematics known as “Minimal Surface Theory.”

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The four experiments, all involving soap bubble film,2 occurred over a span of six mathematics periods, each of 40 min in duration. While developing the experiments we paid close attention to how learning might be corralled and angled to facilitate imagination.

Throughout the experiments, students engaged in whole-class discussions, discussions with their “elbow partners,” and small group discussions about their observations.

Data sources included: (1) transcribed videotape data of the six mathematical periods, (2) student artifacts, including worksheets and journal entries, and (3) researcher observational data. The first author of this paper attended each of the sessions as an observer/videographer, taking observational notes electronically when required.

Students were asked to write journal entries eight times at various points during the four experiments. Journal entries were semi-structured in that some angling by the teacher occurred through specific questions to guide students’ thinking and responses. These questions were frequently tailored to students’ emergent thinking often revealed through earlier journal entries.

The coding scheme, used to code both the student artifacts (e.g., journal entries and worksheets) and the transcribed video data, was developed from our research questions and from strategies identified through Mazur’s (2003) work. The coding scheme included: (1) angling (an), (2) corralling (c), (3) yoking (y), (4) in corralled spaces (in; i.e., moving in the intended direction of the exercise), (5) out of corralled spaces (off; i.e., learning offtrack), and (6) problem-posing (pp).

We engaged in the coding process jointly and compared and discussed each of the codes applied. Our goal with the coding was to develop a deeper understanding of the data, while illustrating aspects of this framework for readers. We saw the coding process as a corralling of our own thinking, while at the same time it was a reliability check of our progression against our initial goals. We do not report examples of every code used in our results since the coding structure was intended only to guide our own thinking.

Following the coding, we utilized an interpretive framework for our joint analysis, which allows “researchers to treat social action and human activity as text ... expressing layers of meaning” (Berg 2004, p. 266). Our goal was to interpret the actions in relation to the framework. Consequently, our results are presented as a text of the human actions of the four-part experiment, including details of the experiments themselves. Some analysis occurred in “real-time,” during the data collection, in order to verify our strategies against the framework.

6 Results The before stage began with a “placemat” activity. Working in small groups, students were asked to record what they knew about bubbles on a section of a large piece of paper, laid flat on their desks (hence “placemat”). The purpose of this activity was to establish the existing outer limits of our corralling beyond our own assumptions about what students knew about bubbles. In addition, this activity provided an opportunity for students to share

2

The soap bubble film was created from a mixture of three parts dish washing liquid detergent, seven parts hot water, and one part sugar. We refer to this mixture throughout the paper as “bubble solution.”

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their individual understandings. As was evident throughout the results, learning in this framework “emphasizes the individual’s sense-making processes as well as the social processes” (Yackel & Cobb 1996, p. 460).

Discussions during the placemat activity were shared as a whole class. Students made general comments about bubbles such as, “fun to play with,” “they are round and they float,” “they’re shiny,” “they are sparkly, ” “they are like a mirror,” “light bounces off of them,” and “there are flavoured bubbles.” Two students seemed to identify the spherical property of bubbles. One student wrote, “They are round and they float,” and another says, “bubbles are round ... no matter what shape they are.” The properties of elasticity and tension of bubbles were not identified. Students did not appear to initially view bubbles as mathematical objects.

Each students’ contribution was validated by Michelle with a “thank you” (lines 7 and 9) whether or not it was in the corralled spaces. By accepting all answers, Michelle created a safe and expansive arena for participation where students’ thinking and participation are valued (Lave & Wenger 1991; Yackel & Cobb 1996). As Michelle explained, this approach “makes a classroom safe and opens up the possibilities of what students will say and share.”

Consequently, the classroom became a space where creative and imaginative work could take place more easily because she was open to pushing the boundaries and exploring even potentially trivial observations. Others have also emphasized the importance of listening and valuing student responses as a means of supporting and understanding student learning (Davis 1997; Yackel & Cobb 1996).

The placemat activity, as well as each experiment, concluded with a summary statement

by Michelle: ... so we already had to think about what you already know about bubbles. Let’s build on that knowledge. So, bubbles are made from bubble film.... Kind of feels like water and dish soap and we call that bubble film. Not film as in old camera but film as in a layer of something fluid and slippery.

Her intention in her summary statements was to draw out what counts as mathematically relevant (Yackel & Cobb 1996), which in this case was establishing a clear understanding amongst students about the type of bubbles that were going to be used in the experiments.

Two teacher-lead experiments, also part of the before imagination stage, followed the placement activity. The goal of these two experiments was to further corral students’ thinking about bubbles, with the explicit outcome of developing taken-as-shared understanding of tension and elasticity necessary for the third experiment. The spherical shape of all bubbles is directly related to the tension and elasticity properties of soap bubble films. Discovering these properties would later enable students to explore why spheres have the least amount of surface area in relation to a given volume over other polyhedron (Taylor 1993).

The first experiment involved dipping a large-sized bubble blower wand3 in bubble solution and waving the wand back and forth to show the elasticity of the soap bubble film.

3

The large-sized bubble blower wand was made of plastic and consisted of a handle, approximately 15 in. long, with a plastic circle (which is dipped in the bubble solution) having a diameter of nine inches. We refer to this instrument as a “wand” throughout the remainder of this paper.

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As Michelle dragged the wand slowly through the air, the soap bubble film gently pulled through to one side of the circular part of the wand and then the other, Michelle first asked the students to discuss in their groups of four what they had observed, and then to report to the class their deliberations.

In this activity, unlike the placement activity, students offered less peripheral information about bubbles. Throughout the discussion Michelle waved the wand through the air. The sustained visual of the wand and the bubble film, by Michelle, was a form of angling in reference to what they were seeing: Line 1 2 3 4 5 6 7 8 9 10 11 12 13

Transcript Lila: It was bendy.

Michelle: It was bent. It was bendy. What else? Tawnya?

Tawnya: It’s flexible.

Michelle: Thank you. I like your word choice.

Codes in c in c an Kain: It can stretch. in Michelle: It can stretch. Thank you. c Michelle: It holds together well. Nice. Raise your hand if you have other things to say. c Kain: The movement in your wrist can make the bubble go each way. off Michelle: Definitely. There has to be some sort of movement. c Perry: It displays a high level of elasticity of the film. in Michelle: Can you repeat that? c an Perry: It displays a high elasticity of the bubble film. in Michelle: Excellent word. I like that word elasticity. I heard the words flexible, bouncy, c bendy...umm ... and ...and what other words did we hear? Flexy [sic], bouncy, elasticity...so an something about how this film moves and how it sort of stretches. That was really good thinking. So, some of the properties that we’ve learned...alright...and these are “student” words we have come up with. ...So know that bubble film is elastic. Thank you ... Thank you, Perry for talking about elasticity. We know that bubble film is elastic and is something that we can stretch like a rubber band. Umm, it stretches. It allows the bubbles. This is what allows us to make bubbles: this stretchy property allows us to eventually make bubbles.

In our planning of this first experiment, we had established that our objective was to encourage students to build on one another’s ideas with the distinct goal of establishing the property of elasticity as a taken-as-shared property of bubbles. We see this in the excerpts from Kain and Perry. Kain begins with the idea that bubbles stretch (line 5). The idea is then taken up by Perry (line 10).

We observed that some of the ideas seemed unrelated, and perhaps off task (line 1 and 8).

As Voigt (1994) points out, students may characterize empirical phenomena differently than intended. The experiment concluded with the property of elasticity being named and strongly reinforced by Michelle, thus leaving no uncertainty among students about what was relevant (line 13).

The second experiment, also part of the before imagination stage, involved tying a string across the wand, creating two sections within the wand. The wand was then immersed in the bubble solution, and the soap bubble film on one side of the string was popped. The students observed that when one side was popped, the string levitated to a taunt position enclosing the remaining soap bubble film. A student volunteer repeatedly dipped the wand into the bubble solution and popped one of the sides of the string, while Michelle facilitated a whole-class discussion about what they were observing.

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Her use of why/what-type questions in the next excerpt are examples of angling.

Michelle angled students’ attention to particular physical changes that were occurring (line 14). We see that a student, Kahla, also takes up angling for other students (line 17): Line Transcript 14 Michelle: Why is the ...that string in the middle jumping like that. ... How we are able to get this string that naturally hangs down to kind of levitate up. Why is that? Why are we making it levitate? How is that happening? Could you talk with the people at your group and take a guess why that is happening? How do we explain the levitating? RJ? 15 RJ: The...ah...when you pop the bottom bubble, the ahh...the air that when you pop the bottom the air pushes the rope up.

16 Michelle: Thank you very much for that explanation. What do you think, Kahla?

17

Kahla: Well, it has like both bubbles but when you pop the bottom bubble it stops the stretching of the string and then the other one like expands and goes smaller.

18 19

Michelle: The other one expands and goes smaller?

Kahla: Yeah, so it takes less space.

20

Michelle: Thank you for that thinking. Mohammed, what do you think is happening?

Why is that happening?

Mohammed: Cause, like there is just one bubble left and that bubble pulls the string.

Michelle: One bubble left and it pulls. Some other ideas...Alim?

21 22 23

Alim: I think that the air is springy cause of the force. The force is pushing the string so the air is tightening.

24

Michelle: Okay, so you’re sort of in agreement with RJ that there is something to do with air that is happening here.

Codes an

c in c an c an in an an in an in c an an in y c y

Participating in such discussions reinforced the notion that the student’s thinking was valued by Michelle. Michelle routinely responded to a student response with a question (line 18, 20). She routinely asked students to explain their thinking, give examples to illustrate their ideas, and in instances where they were unable to provide their own clarifications, she invited other students to engage in negotiating understanding. Meaning was an ongoing negotiation occurring reflexively through sharing.

Two important ideas emerged from this last whole-class discussion, which were consistent with the intended goals. Firstly, Kahla began to identify the conservation of surface area of soap bubble films (Taylor 1993) in her contribution (lines 17 and 19).

Secondly, the remainder of the conversation converged on the corollary idea of force, or to be specific, tension of soap bubble films.

Alim’s use of the word “force” (line 23), to describe what he was witnessing, can be seen as a student initiated form of yoking where prior knowledge, in this case from science, was drawn upon. Michelle made further connections to a contribution made earlier in the conversation by RJ (line 15). In doing this she was building consensus amongst students about important aspects of the experiment, while creating additional learning opportunities for students to compare and contrast shared perspectives.

Michelle continued with yoking by reminding students about prior learning about tension and compression during a science unit on structures and mechanisms. Michelle had asked students to find a partner, face one another, link hands, and pull. Her reminder about tension exemplified corralling, while simultaneously angling to highlight specific ideas and

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concepts. At the conclusion of the first and second experiments, the students were able to identify elasticity and tension as important bubble film properties.

The students’ journal entries could be organized into three groups during the analysis following the first two experiments. Some students suggested that the first two experiments confirmed what they knew about bubbles (i.e., tension and elasticity). However, as described earlier, neither tension nor elasticity was mentioned in the placemat activity.

Other students reported that the experiments expanded their knowledge about bubbles. One student said that she “already knew that bubbles had a bouncy property to them, but I had no idea they had the tension ... I always thought they were WAY too light, and airy.” Others reported that the experiments represented new knowledge for them. For example, one student reported that “when we popped the film on the bottom of the string, the string rose up. We learned therefore that bubble film is like elastic. It shrinks.”

The third experiment was seen as the during imagination stage. Students were asked individually to construct bubble blowers in the form of regular (e.g., circle, square, etc.), and irregular (i.e., heart-shaped, etc.) shapes. Students were directed to predict in their journal entries the anticipated shape of the bubbles that would be blown from their blowers.

The goal of this experiment was to establish the sphere as the paradigm case and to begin to relate the paradigm case to ideas of tension and elasticity from experiments one and two.

In the school court yard, students proceeded to blow bubbles over multiple trials, and recorded the shapes of the bubbles in a table format. Each student discovered that all bubbles were spherical, regardless of the shape of their blower. Students were asked, following their trials, to contemplate in their journal entries how their predictions shifted or changed.

In their journal entries prior to the third experiment, nine students had initially predicted that their blower would produce a bubble related to the shape of their blower (i.e., nonspherical), while the other 11 predicted the bubble would be spherical. As one student, Nazeem, said about one of her blowers, “[It] is just a plain rectangle and I think it will make a more rectangular bubble.” Three of these nine students predicted that the bubble would morph into a sphere after the initial bubble formation based upon the random blower.

We saw an interesting change in students’ perspectives in their journal entries following the third experiment. Explanations from students, both those who did not predict spherical shaped bubbles and those that did, discussed bubble properties as though they had known this information all along; yet, earlier brainstorming sessions and journal entries revealed the contrary. We attribute this sense relayed by the students of having this knowledge all along to the continual process of establishing taken-as-shared norms through the highly visual nature of the framework.

This sense of “I know that,” demonstrated by students in their final journal entries suggests a consolidation of knowledge between what they had known and what they had learned through the three experiments completed thus far. As researchers and teachers, we were both looking for an “aha moment” (Liljedahl 2004) as the aesthetic response.

However, students actually presented a more muted “it makes sense” response that related their understanding back to the properties of tension and elasticity from the first two experiments. This should have been anticipated given the explicit intent to build take-asshared knowledge, beginning with a visual, throughout the experiments. This muted aesthetic response highlighted to us the importance of building taken-as-shared understandings.

The summation of this third experiment involved a whole-class discussion. In this discussion, Michelle affirmed that the paradigm case was indeed a sphere. She also engaged in angling to draw students towards additional problem-posing about why all bubbles formed spheres, despite the shape of the blower.

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The final experiment, which was also teacher-led, and seen as the after imagination stage, examined the shapes and angles of soap bubble film inside a cubical-shaped bubble blower. We had pre-constructed a cube using bendable wire for this experiment. The cube was then dipped in bubble solution to examine how soap bubble film formed. The goal of this final experiment allowed us to gauge the extent to which students were successful in the during imagination stage. That is, we looked for instances when students referred to the paradigm case of bubbles as “spheres” and posed additional problems related to their prior learning and visualizations.

Prior to the experiment, Michelle engaged in a whole class discussion on what students anticipated they might see: Line 25 26 27 28 29

Transcript Michelle: . . . Crystal, what do you think is going to happen?

Crystal: A sphere.

Michelle: Tawyna, what do you think it will look like?

Tawyna: The air inside, it will look like a sphere. I don’t think it would form a square.

Michelle: You don’t think the shape inside would be a form of a square?

30 31 32

Tawnya: I can’t really explain it. None of the other bubbles were squares.

Michelle: Okay. Caleb?

Caleb: I think it’ll all be circles.... It will look like normal soap ...

Codes c in c in c an c c in

Students had the ideas in place that the bubbles formed on the cube would have a spherical shape (lines 29, 28, and 32). Tawyna related her understanding back to the paradigm case from experiment three (line 30). Caleb, even though he used “circle” also related back to the paradigm case with his use of “normal” to describe the soap film (line 32).

During the experiment, students gathered around the tub with the bubble solution to observe more closely what was happening to the cube. Surprising for students was the creation of an additional cube-like structure of soap bubble film in the center of the cube, with bubble film vertices, extending from the each bubble blower vertex to the structure in the center of the cube. This was an important aspect of this fourth experiment. Students made observations about the two structures in terms of ratios, angles, proportions, volumes.

In the final journal entries we saw important evidence of yoking (i.e., connecting knowledge) in students’ problem-posing. The intention in the final journal entry was to check if (a) additional questions developed about the relationship between minimal surface area for a given volume of a sphere and other three dimensional shapes that would lead students into the upcoming unit on three-dimensional geometry (e.g., volume, surface area, optimization, etc.), and (b) student thinking was consolidated.

One student asked whether transformations could be used to describe symmetries between bubble films. This question was interesting and unanticipated. We had not contemplated thinking about soap bubble films in terms of symmetries with middle-school students. Olga asked, “is there at all ANY [her emphasis] possible way to get different shaped bubbles?? ” Two students asked a chemistry-related question about the role of sugar and humidity in sustaining bubble films. Others raised questions about testing minimal surface area of spheres against another polyhedron (i.e., cube), which was one of the desired outcomes of the four experiments. In all of these questions, we saw evidence of yoking. The questions raised about testing spheres against other polyhedra were a useful in segue to the forthcoming geometry unit. Michelle’s beginning point, consequently, for the next unit was testing the minimal surface theory of spheres against other polyhedra.

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7 Educational significance and conclusions Following our reading of Mazur’s (2003) text, we wondered how teaching and learning mathematics might be structured as a way of imagining. We also wondered what might be attended to pedagogically, within such a framework, that might be different from other frameworks we had experienced in our classrooms. Thinking about how to occasion learning via an application of the framework was very useful.

Imagination, as a cognitive space for learning, was thought of in terms of three stages: before, during, and after. In the before stage, we focused on selecting a problem, establishing a common visual (i.e., the two teacher-led experiments with the bubble film) as a taken-as-shared artifact, and corralling requisite knowledge. In the during stage we attended to the task of negotiating the meaning of the common visual in order to establish the paradigm case (i.e., spherical bubbles). In the after stage, we focused on affirming the paradigm case, deepening understanding by making connections to other knowledge (i.e., yoking), while simultaneously encouraging problem-posing. Throughout the process, angling was used to point students cognitively to what was mathematically relevant. The framework was not linear, but rather recursive, in that learning began and ended with a mathematical problem situated in a visual context.

One important aspect of this work was the focus on problem-posing by students. This allowed us to see if were on-track with our learning objectives for the four experiments.

Furthermore, it gave the students the sense that they were mapping their own learning paths. Using problem-posing, as a means of checking for understanding and for directing future learning, was a new focus for Michelle.

Michelle, as the classroom teacher, identified the emphasis on visualization within the framework to be particularly transformative, both in her teaching and in the students’ ability to learn. Students appeared to have benefited from the continuous use of familiar visuals, as springboards, to facilitate visualization and imagination. This emphasis on visuals and visualization is a deviation from the common and dominant emphasis of real world or childrelevant applications (National Council of Mathematics Teachers/NCTM 2000).

An important lingering question in relation to this framework remains, namely, Can all mathematical learning be initiated with a visual? Mazur (2003) easily and, in fairly layman’s terms, does just this with imaginary numbers. This being said, we envision cases where an argument can be made that a visual approach is not possible or pedagogically appropriate. In a cursory review of our own elementary curriculum (Ontario Ministry of Education and Training/OMET 2005), our view is that mathematics at this level can be approached from a visual perspective. Indeed, we propose that a visual approach may be more readily taken-as-shared than hypothetical real-world simulations that beg the question, whose real-world?

In our endeavor to engage in the development and testing in a middle school setting, the teaching mathematics as a way of imagining framework was not without challenges. As Egan (1992) points out, imagination does not readily “lend itself to practical methods and techniques that any teacher can employ in classroom instruction” (p. 1). We had concerns that imagination, from a pedagogical point of view, may be without boundaries, and not seen as mathematical. This was unsettling for Michelle, and may be unsettling for other teachers, who may not necessarily be mathematics specialists, as is the case for Michelle.

Nevertheless, the focus on visuals, visualization, and problem-posing within this model, while attending to the pedagogical strategies of corralling, angling, and yoking, and the three imagination stages, proved to be both important strategies for structuring learning, and novel approaches to thinking about teaching and learning mathematics.

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Acknowledgment Thank you to Dr. Barry Mazur, Harvard University, for his thoughtful feedback.

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