Design of an Environment for Learning about Topology and Learning about Learning,” submitt

CAMBRIDGE RESEARCH CENTER

Design of an Environment forLearning about Topology andLearning about Learning

Carol StroheckerTR96-05 March 1996

Appears in Proceedings of the Second International Conference on the Learning Sciences,Evanston. Charlottesville: Association for the Advancement of Computing in Education, 1996.

Abstract

Studying people's thinking, particularly when both \"cognitive\" and \"affective\"aspects are of interest, requires a setting conducive to in-depth exploration of theparticipants’ dialogue and actions. This need has implications for the duration ofthe study and for the nature of the environment in which the inquiry is conducted.The researcher must be able to spend a good deal of time with the participants,encourage or design problem scenarios that will tend to surface certain kinds ofthinking, and develop a relationship with the participants that allows for theexchange of relevant information.

This paper describes how these requirements were addressed in an environment inwhich young people learned about knots and the topological relationships theyembody. The experimental design that I launched, and which the participantsappropriated and modified, consisted both of knots and of a social substrate thatencouraged the lively exchange of ideas about them.

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Copyright © Mitsubishi Electric Research Laboratories, 1994

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Imagine that you are in an inner-city elementary school – a gray buildingwith graffiti on the walls, long hallways populated by lines of children, distortedsounds coming from the public-address system, people chattering, computershumming. Amid the noise is the clutter of busy classrooms and senses ofurgency alternating with times of thoughtful quiet. One of the doors is

decorated with a sign that boldly announces the room as \"The KNOT LAB.\" Youenter, and find children playing with string, writing letters, watching videos,climbing to tack knots on display boards, arguing over the use of the videocamera, working together on stories about knots, and otherwise immersed inactivities related to their study of knots. They are surrounded by their ownconstructions: three of the four walls boast large, colorful displays showing knotsin various stages of formation, dangling from tree branches, tacked toaccompanying pieces of writing, sewn on cardboard, and drawn on variouspieces of paper. Books and other printed materials about knots are strewn about,and pieces of string are everywhere. You have entered a \"thinking environment\"dedicated to learning about knots and to reflecting on that learning.

What you do not see are aspects of the environment that are equallyimportant, but reside elsewhere. Miles away, an older child waits for the end ofhis school day so he can meet with me in a video editing room where we willanswer letters, in written and video form, which the younger knot-tyers haveentrusted me to give to him. I am the courier; he is the \"knot expert.\" A BoyScout and now, to some, a TV star, he takes his responsibility seriously and triesto answer the detailed questions that the other children pose. Eventually theywill all meet, when he comes to the \"Knot Fair\" that culminates the project.

This scenario evolved over five months’ time and enabled study ofdevelopments in twenty fifth-graders’ understandings of principles of topology.Knots are a class of objects well suited to this focus. They have inspired a branchof formal mathematics, \"knot theory,\" in which topologists seek algebraic meansof identifying the vast number of combinations of intertwinements similar towhat we call \"knots.\" But these are also among the most common of everydayobjects: pervasive through time and cultures, knots have become part of our arts,

mythologies, and symbol systems, in addition to our mathematics, physics, andpractical work.

Piaget and Inhelder (1967) recognized such advantages in their studies ofsmall children beginning to understand the relationships of proximity thatconstitute topology. Of interest were the homeomorphisms that can bediscovered as the forms are stretched or twisted: do the deformations retain theproximities, separations, or orderings? When and how do children recognizethese homeomorphisms?

In pilot studies preceding the research described here, participantsfocused on such deformations, but also learned to tie certain knots anddeveloped graphic and verbal representations of the tying processes. Althoughthese notations illuminate varying conceptions of the knots, it became clear thatthe depth of familiarity I was hoping to study would require a lengthy period ofimmersion, as well as an environment enabling a wide range of choices of knots,approaches to learning them, and media for expressing ideas about them. Thelonger duration also allowed time for relationships to grow among participantsand with the researcher.

The Piagetian research tradition is known for its emphasis on the selectionand design of settings and scenarios in which a certain line of exploration andquestioning will yield information about children's understandings of a specifictopic. More recently, \"post-Piagetian\" writers have acknowledged the

importance of social and cultural factors involved in learning, by extending thediscussion of design to entire environments that consist of people, projects, andplaces in which the activity happens (Ackermann 1987, 19, 1990; Papert 1980, 1984,

1987, 1990 \"Unified\"). Such environments grow through extended periods of time,

making it possible to examine cultural influences on the construction of ideas aswell as the subjects' personal involvement with the ideas.

These environments have come to be called \"learning environments\" or,acknowledging the researcher's participation and the double purpose of thework, \"environments for learning and research\" (Ackermann 1987). In a furtherattempt to emphasize the participants' examination of their own learning, I havedubbed the Knot Lab a \"thinking environment.\" Characteristic of my approach

were a certain wariness about the potential influence of my interventions,willingness to change hypotheses or courses of action as the project developed

(Berg and Smith), and dedication to the task of developing a \"thick description\"(Geertz 1973) of the children's work and thinking (Strohecker 1991). The setting had

to be flexible enough for the project to evolve in response to ideas and events thatoccurred during its course. I presented an initial context for learning aboutknots, which we gradually modified as the children became immersed in theproject.

To begin, I described a Boy Scout who wanted to be their \"pen pal\"through exchanges of videotapes. This Boy Scout would demonstrate how to tievarious knots and respond to questions from the other participants as the projectprogressed. He was an older child who was himself in the process of learning totie certain knots. I assumed the roles of facilitator and \"courier,\" the personcommon to each end of the communication, who videotaped the sessions andarranged for the children and the Boy Scout to see each others' video mail.

Videotaping the exchanges between the children provided a means ofrecording visual and aural data, and also stimulated the children's excitementabout the project. They enjoyed \"seeing themselves on TV.\" The inevitable self-consciousness that resulted among the participants supported a theme ofbecoming aware of one's own thinking processes so that they can be madeavailable to others – in this sense, the children were also researchers, and severalcame to think of themselves in this way.

This \"video correspondence\" served a kick-off purpose but did not definethe scope of the project, which quickly took on a life of its own. Most dramatic inits evolution were gradual shifts from an emphasis on video as an instructionand communication medium, to the children's initiation of their own activitiesand greater use of paper correspondence. The children worked individually orin teams, initially within four separate working groups. As the end of the projectapproached, distinct boundaries between these groups relaxed and increasinglyfrequent but casual merging of the groups occurred. With this change cameincreases in the incidence of borrowing of ideas and of collaboration betweenmembers of initially different working groups. The children were building not

only understandings of certain knots, but a culture dedicated to learning aboutknots and thinking.

Three important elements of the research occurred through the workingsessions: they formed a period of culture-building and of immersion in thinkingabout the knots, so that discussions in the form of the \"final interviews\" fit withina context that all the children shared; the working sessions, in their own right,generated data on thinking about knots; and in the course of these sessions,many of the children built up a relationship with me that came to involvecomfort and trust. Gradually through the course of the project, and in the finalinterviews, many of the participants were willing to show and explain to mewhat they thought. We had taken an approach that assumes that people willhave different ways of thinking about aspects of knots, and which values thesedifferences. Although many of the children seemed hesitant at first, they came toaccept that this approach was genuine – that they weren't going to be told theywere wrong or stupid if they risked articulating what they thought.

The Knot Lab, as a center of this approach, became a place where theparticipants began having dialogues and debates about different ways to thinkabout knots (and eventually, other issues in life, too). Many children engaged in\"dialogues\" with themselves – that is, they developed a form of critical thinkingin which they would launch an interpretation of a knot and then retract ormodify it as they continued the exploration. It was not unusual for a child toarrive at an understanding very different from the one she had started with, andto describe the initial interpretation as being \"wrong.\" This form of self-critiquewas refreshing for its lack of the punitive overtones that can stem fromincorporation of voices of authority who emphasize mistakes as beingproblematic.

The environment became particularly supportive of conducting a studythrough participant observation. There was so much going on that my presencecould not help but blend in with the activity. Of course, as an older personwhose involvement with the project extended beyond the Knot Lab, my role wasdifferent from that of most of the participants. Still, they understood and sharedmy interest in looking at learning as well as at knots, and we exchanged ideas

about knots, school, our personal lives, practical considerations in the use ofvarious media for communicating about knots, and so on.

For each participant, the number of working sessions ranged from ten totwenty during the course of the project, and these sessions ranged from one tothree hours in duration. At the end of the study, I had an individual meetingwith each participant. We conversed while the child tied various knots,compared two similar knots, and arranged a set of knots into groups accordingto perceived similarities. These comparative techniques were useful in elicitingunderstandings of relationships among parts of the configurations. Importantly,the immersion in Knot Lab projects that preceded the interviews had the effect ofpreparing the children for these detailed discussions.

In comparing the Square and Thief knots, several children acted on thesuggestion that they imagine themselves to be a small ant crawling along thesurface of the knot. This is a technique that Piaget and Inhelder had used in theirstudies of younger children working with simpler knots. It proved helpful inassisting these older children to imagine a change of scale (which was bothspatial and temporal) in which portions of the knot could be consideredseparately, making more manageable the problem of finding one's way throughthe complicated configurations. Without this change of context, many of thechildren would have found difficulty in describing some of the differences.

I collected data through note-taking, audiotaping, videotaping, andtangible projects that the children produced. Sources of information includedconversations with the children (and with their teachers); the children's videoand paper correspondence with the Boy Scout; their written descriptions ofknots, illustrated stories, instructions, etc.; and the bulletin-board displays thatthey constructed.

Video was a key element of the Knot Lab. The idea of the \"video pen pal\"was one that the children accepted and played with immediately. The \"videocorrespondence\" launched the project and provided a way for the kids to getinvolved quickly. It also encouraged communication about knots and a degreeof self-consciousness not just about one's own appearance, but about how todescribe knots and issues related to them. This aspect of self-reflection should

not be underestimated. The video correspondence also established the presenceof the camera as an everyday element of the research situation, so that its use asdata-collecting device became relatively unobtrusive.

Several important questions emerged as I looked at the data: How did thechildren describe knots and tying maneuvers? What difficulties did theyencounter? What strategies did they use for getting out of difficult situations?What knots did they perceive as being similar or related, and why?1

Thinking about knots tends to elicit a wide range of diversity; the methodsand materials used in this study enabled understanding of particular ways inwhich general patterns of thought may become mobilized within the mind of anindividual.References

Ackermann, E. 1987. \"Helping Children Become Epistemologists.\" Epistemology

and Learning Group, MIT.

Ackermann, E. 19. \"Circular Reactions and Sensori-Motor Intelligence: A

Discussion on Piaget's Theory of Early Cognitive Growth.\" Epistemologyand Learning Group, MIT.

Ackermann, E. 1990. \"From Decontextualized to Situated Knowledge: Revisiting

Piaget's Water-Level Experiment.\" Constructionist Learning. Epistemologyand Learning Group, MIT.

Berg, D. N., and K. K. Smith. 1985. Exploring Clinical Methods for Social Research.

Beverly Hills: Sage Publ.

Geertz, Clifford. 1973. The Interpretation of Cultures. New York: Basic Books.Papert, S. 1980. Mindstorms: Children, Computers, and Powerful Ideas. New York:

Basic Books.

Papert, S. 1984. \"New Theories for New Learnings.\" School Psychology Review.Papert, S. 1987. \"About Project Headlight.\" Cambridge, MA: Epistemology and

Learning Group, M.I.T.

1

Discussions of the major findings are in preparation for publication elsewhere, but the most completereport is in (Strohecker 1991).

Papert, S. 1990. \"A Unified Computer Environment for Schools: A

Cultural/Constructionist Approach.\" Proposal to National ScienceFoundation. Cambridge, MA: Epistemology and Learning Group, M.I.T.Piaget, J., and B. Inhelder. [1948, 1956] 1967. The Child's Conception of Space.

Trans. F. J. Langdon and J. L. Lunzer. New York: W. W. Norton.Strohecker, C. 1991. Why Knot? Ph.D. diss., Massachusetts Institute of

Technology.Acknowledgments

The research described here was conducted in 19 with the supervision ofSeymour Papert and the support of: Apple Computer, Inc.; Fukutake PublishingCo., Ltd.; IBM Corporation (Grant #OSP95952); the LEGO Group; MacArthurFoundation (Grant #874304); National Science Foundation (Grants #851031-0195, #MDR-8751190); Nintendo Co., Ltd.; and the MIT Media Laboratory. The ideas expressed

here do not necessarily reflect the positions of the supporting agencies.