Constructing intersubjectivity in representational design activities

Journal of Mathematical Behavior 19 (2000) 503 – 530

Constructing intersubjectivity in representational design activities Rafael Granados Graduate Group in Science and Mathematics Education, University of California-Berkeley, Berkeley, CA 94720, USA

Abstract What communication techniques do teacher and student use to achieve intersubjectivity when involved in design activities? In this article, I argue that people rely on an intuitive interactional method I call marking to define, constrain, and maintain their understanding of the task at hand and achieve common ground to solve collaborative design problems. To illustrate how this method works, I use data from video segments of middle and high school students and their teacher inventing representations of geometric figures as an executable set of instructions (as in a computer program). The analysis centers on the collaborative interaction — verbal communication, gestures, and representations — among teacher and students as they create and tune their understanding of the goal and nature of the activity. D 2001 Elsevier Science Inc. All rights reserved. Keywords: Collaborative activities; Collaborative learning; Discourse analysis; Design activities; Design space; Intersubjectivity; Meta-representational competence; Social interactions

1. Introduction Classroom design activities offer many interesting research possibilities for educational researchers. One topic of particular importance is how participants in design activities achieve intersubjectivity — that is, mutual understanding sufficient for effective collaboration. Consider the following activity description given by a teacher to a group of students. Teacher: . . . pretend that I’m a dumb turtle. . . . all I know how to do is walk and turn. I can follow very simple instructions. . . . I don’t know very much geometry. . . . I don’t know what a square vs. triangle vs. a circle is. I use simple instructions 0732-3123/00/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 2 - 3 1 2 3 ( 0 1 ) 0 0 0 5 5 - 4

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about how to walk and turn. So, here’s my question. How can you instruct me, the dumb turtle, to walk and turn in a square that is two steps long? This description raises a number of questions. In collaborative, open-ended design problems, how do the teacher and students build and maintain a common understanding of the task? From the teacher’s perspective, how can she or he position them in the appropriate domain? How can she or he guide and assist them as they invent and design an artifact? From the students’ standpoint, how do they know where to start? How do they know what to do and whether they have all the information to complete the task? How are they able to figure out which knowledge is useful or which is not? How do they decide on a goal? According to Vygotsky (1978) and others (Bretherton, 1991; Rogoff, 1990), an important aspect of the learning process within the social realm is to establish and maintain an understanding of ‘‘where the other person is coming from’’ or what is typically referred to as intersubjectivity. Bretherton (1991) refers to intersubjectivity as an ‘‘interfacing of minds’’ that involves an intuitive monitoring of one’s own perspective as well as the listener’s perspective. Clark and Brennan (1991) extend intersubjectivity further and propose that in order to establish and maintain a common understanding in communication, participants engage in ‘‘grounding.’’ They define grounding by using the metaphor of musicians playing a duet together, which involves ‘‘a coordination of process [that cannot be achieved] without a vast amount of shared information or common ground — that is mutual knowledge, mutual beliefs, and mutual assumptions’’ (p. 127). By establishing intersubjectivity, social partners build a joint level of understanding that allows them to achieve a common goal. Establishing and maintaining intersubjectivity is not necessarily a conscious process, but rather an intuitive process in which people naturally engage when interacting with one another. Furthermore, the processes through which intersubjectivity is achieved depend on the nature and goal of the activity in which participants are engaged. In order to investigate intersubjectivity, I make use of the term design space to refer to the ongoing understanding and actions that students and teacher perform to successfully accomplish a design activity. This term is inspired by, but somewhat different than, problem space as used in Information Processing Theory (IPT). A problem space includes the understanding of the problem, the goal, and the set of actions that lead to the solution (Newell & Simon, 1972). While there are apt parallels between these two constructs, in IPT research, the concept of problem space deals with prototypical problem-solving activities, which usually consist of a well-defined problem statement for which there is a set of actions that a problem solver can apply to get the ‘‘right’’ answer. Unlike problem spaces, design spaces stem from problems of a different nature. Design problems are more complicated than ‘‘well-defined’’ problems (e.g., a puzzle) because they do not have right or wrong answers. Instead, design problems have multiple solutions — which are only better or worse in various ways when compared to each other — and unbounded paths for reaching those solutions. The process for solving design problems requires that participants build criteria for considering what path or approach to take in finding solutions. Participants may go through cycles in which they introduce and reject proposals; they need to continuously redefine their understanding of the problem, the path they are taking, and the next steps they will take.

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Participants must not only coordinate their actions, they must also position themselves in a cumulative space of understanding as it changes. Under these conditions, the design space is more flexible, it is open to negotiation, and it is improvisational. One need not enumerate every possible move in order to reach the goal. Indeed, enumeration of moves is likely impossible. Consequently, the design space is defined as an open and negotiated conceptual structure that contains the ongoing collective specifications relevant to the design activity. It is made by participants for the purpose of defining their understanding of a design problem, coordinating their actions, and ultimately achieving the goal of the activity. This structure includes (1) the domain, or specific content area around which the design space is centered, (2) the goal of the activity, (3) specifications of information or knowledge to be used or excluded, and (4) the established processes of collaboration (e.g., how participants make requests or proposals to resolve uncertainties). Entering into and successfully traversing the design space requires establishing common understandings between participants. In this article, I investigate intersubjectivity in teacher– student collaborative design activities of geometric figures as an executable set of instructions (as in a computer program). I propose that an interactional method, marking, explains how a teacher and his students define, constrain, and maintain intersubjectivity. Marking, as defined in this article, is an intuitive interactional method in which participants consider how other participants ask questions, make statements, use gestures and representations for the purpose of defining the knowledge and or skills needed to solve the problem. I claim that this method is intuitive, not because it is innate, but because it is a process that is activated automatically in social interactions. The most apparent way by which people interact and share information in social interactions is through the use of language. Conversation analysis, which focuses on the ‘‘machinery, the rules, [and] the structures that produce and constitute’’ the orderliness of social action (Psathas, 1995), is one way to examine talk-in-interaction during everyday situations. According to Schegloff (1991), interactional structures that form the acts, messages, and utterances of participants are part of the ‘‘very composition, design, and structuring of conduct and is part and parcel of whatever processes — cognitive or otherwise — are germane to the conception and constitution of acts, messages or utterance in the first instance’’ (pp. 153–154). A case in point, turn-taking is fundamental to conversations since (1) it builds a motivation for listening, (2) it controls the understanding of utterance, and (3) it requires that participants display their understanding of the others turn’s in conversation (Sacks, Schegloff, & Jefferson, 1974). Turns of talk that fall into the practice of organization of repairs are interactional elements that contribute and sustain socially shared cognition by letting participants address speaking, hearing, and understanding problems1 (Schegloff, 1991, 1 Here is an example of a repair borrowed from Sacks et al. (1974, pp. 365 – 366). In line 3, F asks K to clarify whom they were talking about. K responds and in so doing accomplishes a repair (line 4). 1. F: This is nice, did you make this? 2. K: No, Samu made that. 3. F: Who? 4. K: Samu

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Schegloff, Jefferson, & Sacks, 1977). Clark and Shaeffer (1989) also analyze turn-in-talk and suggest that participants produce contributions to discourse for the purpose of achieving common ground. While conversation analysis research seems to be useful in suggesting how knowledge is socially shared in talk in everyday interactions, it does not offer any way to bind intersubjectivity to the conceptual context of activities. The constructs of design space and marking provide a link between the nature of the conceptual context and the processes of achieving intersubjectivity. In the following sections, I describe a specific setting from which the data was taken to illustrate how marking is used to achieve intersubjectivity in a design space. In this case, students were asked to invent representations of figures within the domain of turtle geometry. The data will be presented to define and illustrate the construct of marking, and to define specific types of marks that students and teachers make in the process of creating common understandings. Finally, I provide an analysis of interactions between participants across two activities to show how the domain and design space are directly linked to how marking is used to achieve intersubjectivity during a design problem.

2. Research design 2.1. Setting: MaRC course In the summer of 1997, Project MaRC (meta-representational competence) offered a 6week course entitled ‘‘Exploring the Symbols, Diagrams, and Images of Science,’’ to middle and high school students between grades 8 and 10 in the Academic Talent Development Program.2 The course met 2 days a week for 3 1/2 hours each day. The number of students who participated fluctuated initially between eleven and nine students. Ultimately, nine students completed the course. Practical and theoretical objectives motivated the design and implementation of this course. The practical objectives focused on the creation of activities that allow students to explore how to represent information in order to gain a general understanding of how symbols and diagrams are used in different scientific fields. These activities frequently required students to work in teams to invent representations. The theoretical objectives focused on understanding students’ meta-representational competence, that is, the students’ abilities to choose, critique, construct, invent, and productively use scientific representations (diSessa & Sherin, 2000; diSessa 1996; Hammer, Sherin, & Kolpakowsky 1991). Toward this end, classroom and computer lab activities were video taped, and the representations produced were collected.

2 The Academic Talent Development Program is a Summer Program administered by the Graduate School of Education at the University of California-Berkeley. The program has two divisions. The first division offers courses to students at the elementary level. The second division offers courses to middle and high school students. Courses offered are computer science, languages, mathematics, natural sciences, and social sciences. The MaRC course was offered to second division students.

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Students participated in activities such as representing motion, mapping specific model landscapes, turtle geometry (see next section), and the use of MaRC technology — scientific visualization tools — to represent and understand geographical (landscapes) and astronomical (moon, stars, and galaxies) information. For this study, the design space is made up of activities that are centered within the domain on turtle geometry. The following section provides a description of the domain of turtle geometry, the design spaces created by turtle geometry activities, and the types of marking that are used within these design spaces to achieve intersubjectivity. 2.2. Domain: turtle geometry Turtle geometry is the exploration of mathematics from a perspective that regards figures as dynamic. In this domain, figures are traced by a ‘‘turtle’’ (a turtle in a computational environment is a spatial creature that executes certain motion commands) whose movements can be described by suitable computational procedures (Abelson & diSessa, 1981). Thus, shapes can be constructed by telling a turtle how to move using simple terms such as forward (number of steps), left (degrees), or right (degrees). Fig. 1a illustrates a simple procedure that tells a turtle to draw an ‘‘L.’’ Fig. 1b shows how the ‘‘L’’ shape is represented in cartesian geometry. There are three important aspects to keep in mind concerning turtle geometry. First, turtle geometry describes figures in terms of procedures rather than equations. These procedures have to be executed to make the turtle draw the shape. Thus, the representation is dynamic. Second, descriptions of figures are local. In this system, figures are described a piece at a time or an instruction at a time (Fig. 1a) as opposed to defining the ‘‘L’’ shape based on a global coordinate system (Fig. 1b). Finally, turtles produce intrinsic descriptions of figures. In this perspective, figures do not depend on a frame of reference point. For example, the procedure above can tell the turtle to draw an ‘‘L’’ regardless of the direction it is pointing. An ‘‘L’’ is an ‘‘L,’’ regardless of how it looks from any reference. In a coordinate system, however, the description of an ‘‘L’’ requires coordinate pairs, which depend on a particular reference point and orientation. The next section describes the turtle geometry activities that we developed, why we used these activities, and how we implemented them in the course.

Fig. 1. (a) A procedure to tell a turtle to make an ‘‘L.’’ (b) A description of an ‘‘L’’ using coordinates.

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2.3. Design space: turtle geometry activities The two turtle geometry activities examined in this study we call Square and Triangle. These activities took place at the start of our course. In both activities, the students were asked to work in groups and invent a way to tell a turtle to produce a particular figure. For these activities, students did not use a computer. Rather, the teacher enacted the representations, pretending to be the turtle. The overall pattern of the activities was as follows: presentation of the design problem by the teacher; construction of representations for a specific figure (class and within group interactions); simulation of the representations (teacher executing the instructions provided by the students); and learning the Boxer language (proposing and introducing the technical notation for describing the figures in Boxer). At one level, these activities were intended to introduce students to a new way of making geometric figures and Boxer (Boxer is a programming language and software learning environment. See diSessa, Abelson, & Ploger, 1991 for a detailed description.). Learning Boxer was an important component of the course because scientific visualization activities depended on the students’ ability to use Boxer to examine astronomical images. At another level, we hoped that students would learn the Boxer language to be able to create their own tools to manipulate image data to pursue their own scientific investigations and goals. For the Square and Triangle activities, students were not told exactly what to do. They were given neither methods nor answers, nor were they expected to come up with one particular answer. Students had to deal with many uncertainties and unknowns. They had to make decisions about what information was important or needed and what path to follow for reaching their goal. They had to rely on knowledge and resources they had immediately available. Hence, the turtle geometry activities constitute the design space within which the teacher serves both as a guide and a codesigner with students. As a guide, the teacher presented the problem, clarified task ambiguities (e.g., provided specific answers to technical questions, defined abilities and limitations of the turtle), and modeled the actions of the turtle for students. The teacher’s actions gradually transformed a complex design problem into a simpler one to facilitate students’ understanding and engagement in the task. As a codesigner, the teacher needed to establish and maintain a common understanding of the different aspects of the design space as it was created with the students (i.e., domain, goal of activity, specific information needed, and participant collaboration patterns). In the next section, I will describe how analysis of the data led to the development of marking as a construct, and how different types of marks were used to establish and maintain intersubjectivity within the design space. 2.4. Preliminary data analysis: defining and categorizing marking In constructing an analysis of how participants built and maintained a design space, I transcribed the video segments of the two geometry activities. The transcripts included the gestures and representations, which highlighted or supplied information needed for the activity and thus were of importance to the interactional process. During this stage, I repeatedly examined both the video and the transcripts to identify patterns of behavior that

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demonstrated participants’ construction and maintenance of intersubjectivity. Through this process, I identified marking as a generic means by which participants achieved intersubjectivity. Second, I developed a preliminary coding scheme that specified different ways in which marking was used as a tool to construct common ground. Each type of mark related to different aspects of the design space such as: (1) the goal of the activity; (2) specifications of information/knowledge to be used or excluded; and (3) the collaborative processes through which participants resolved uncertainties. Third, I coded the transcripts using the various types of marks. Fourth, to improve the validity of the coding scheme, on two separate occasions, a group of colleagues read the category definitions and reviewed the coded transcripts. Based on their comments and questions, the coding scheme was refined and the transcripts were recoded. Marking is an intuitive interactional method that allows participants to define, constrain, and maintain their ongoing understanding of a design space; the design space in these activities corresponds to the knowledge and skills that participants are building about not just what they need to accomplish for the task at hand, but also more generally, turtle geometry. Interactions among participants can involve language, gestures, and representations. Through the analysis, I identified five types of marks: Boundary Marks, Precision Marks, Target Marks, Attention Marks, and Directional Marks. This set of marks is a first attempt at characterizing marking based on a limited set of data. In order to make the framework generalizable to classroom design activities in different domains and to other forms of social interactions, these categories likely will need further development. Table 1 summarizes the marks. Below, I describe the various marks in more detail. 2.4.1. Boundary marks define and constrain the design space by including or excluding knowledge at the general level These types of marks set the frame of mind of the participants and provide general descriptions of the design space. They are usually found at the onset of the activity. They mark inclusions and exclusions of resources (knowledge and information) that narrow the design problem into a workable arena. If a design space is not delimited with boundary marks, unproductive effort can be spent exploring many possible solutions based on knowledge that was not excluded or specified. Table 1 List of marks Types

Function

Boundary Marks

These marks include or exclude knowledge from the design space. They appear mostly at the onset of the design activity to build the foundation of the design space. These marks further specify knowledge pieces for the production of a design. These marks set the goal of the activity. These marks serve three kinds of functions. They can reinforce previous information that was marked. They solicit resolution to ambiguity by proposing information. They can request specific clarification. These marks direct participants towards forthcoming marks or towards actions taken or to be taken.

Precision Marks Target Marks Attention Marks

Directional Marks

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Boundary marks are general in the sense that they do not supply detailed information that can be used in the design of the representation. For example, the Square activity description contains the utterance, ‘‘I know how to follow very simple instructions,’’ which informs participants that the commands to the turtle should be simple. It does not tell them what commands they should use nor even what ‘‘simple’’ means. Because these marks are general, there is room for multiple interpretations of the utterance. For example, the utterance above is not specific enough to eliminate the reasonable assumption that the turtle could understand graphical instructions (as opposed to words), which is another possible way of communicating actions. This may require another mark to clarify this utterance or to eliminate visual instructions as an alternative. 2.4.2. Target marks set the objective of the design activity In the Square activity, for example, the teacher asks students to make a set of instructions that will command a turtle ‘‘to walk in a square that is two steps long.’’ The teacher will usually set the goal of the activity when he or she specifies the activity. However, this does not mean that students cannot use target marks for themselves to achieve particular subgoals. The number of subgoals will depend on the complexity of the activity. If the design activity requires the design of a complex representation, both teacher and students may need to set achievable subgoals to reduce the complexity of the design. The activities examined in this analysis are challenging, but not overwhelming, and students are able to accomplish them in a short period of time and without setting elaborate subgoals. 2.4.3. Precision marks further specify knowledge or information necessary for the solution of the problem For example, in the Square activity, based on the statement that ‘‘OK, I can understand, like, turn left and turn right,’’ students acquire information that specifies some of the language the turtle can understand. Consequently, the students’ representations will have to include a command such as ‘‘turn left’’ or some variation. One could argue that these marks are just boundary marks since they also define and constraining the design space. However, the precise specification of pieces of information, at a detailed level distinguishes them from boundary marks. 2.4.4. Attention marks reinforce, propose, or request particular knowledge or information that might be of consequence to the production of the design space or the representation There are three functions that attention marks can perform. First, attention marks can reinforce particular information that was previously marked. For example, for the Square activity, the teacher makes the statement ‘‘Remember, I’m dumb . . . the simpler the instruction the better,’’ which reinforces the fact that the turtle should be given simple instructions. In this case, the mark emphasizes important aspects of the design space. Second, attention marks propose information that needs to be clarified. For example, later in the Square activity, a student asks the question ‘‘Can you understand steps?’’ This is a solicitation for confirmation or rejection of a proposed resource for design (that the turtle understands ‘‘steps’’). In performing these marks, participants solicit resolution to some ambiguity that

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was impeding the construction of the design space and or the solution. Third, attention marks can be specific requests for information. For example, the question, ‘‘Which way?’’ which was asked by the teacher immediately after he was given the instruction ‘‘Take two steps,’’ is a request for specific information. 2.4.5. Directional marks direct participants towards forthcoming marks or towards particular actions taken or that are about to be taken These marks add nothing to the understanding of the design problem, but they motivate and or engage the participants in the process of constructing the design space. Here are two examples from the activity descriptions of the Square and Triangle activity made by the teacher. First, the statement ‘‘Here’s my question’’ prepares participants for an important upcoming mark, which in this case happens to be the target mark (i.e., specifying the goal of the activity). The target mark, in turn, helps define the design space. Second, the statement ‘‘Hard one; this one has a little trick in it, so think carefully,’’ made by the teacher about the Triangle activity asks participants to be thoughtful and careful about their actions when finding a solution. Understanding each type of mark, and what each can achieve towards establishing and maintaining intersubjectivity, is an important first step in understanding the design space. However, it is also important to examine how different patterns of marking are used throughout the entire activity to generate the design space. The analysis below illustrates how patterns of different types of marks work together to achieve specific and global goals of the activity.

3. Analysis of the square activity 3.1. Understanding marks within the design space In this section, an analysis of the Square activity transcript is used to illustrate how marks are applied to achieve intersubjectivity at specific key points in the process of developing the design space. Different types of marks serve specific purposes throughout the creation of the design space such that each type of mark may occur throughout the activity. However, some types of marks are more prevalent at the outset of the activity, other marks may be more frequently used at a time of confusion or ambiguity, still others may be used primarily to bring participants closer towards achieving a more global goal of the activity. For the transcripts we use the following notations: BM (Boundary Mark), TM (Target Mark), PM (Precision Mark), AM (Attention Mark), DM (Directional Mark), // (enclosed utterances that overlap), {} (description of physical action), . . . (a pause of 1 or 2 s), and Group number (specifies which group is involved). In addition, the interactions are separated into two levels: class level and group level. We do this to illustrate that marking happens at various levels. Class level refers to interactions happening publicly. Group level refers to interactions happening within groups. There were four groups of students: Group 1 (Ana,

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Tamara, and Edna), Group 2 (Angel and Natalie), Group 3 (Charlie, Peter, and Maria), and Group 4 (Dean and Luke).3

3.2. Bounding the design space: laying the foundation The initial goal of the activity is to bring the students and teachers together into the design space. In order to launch the students into the activity, the domain of the activity has to be understood, the goals of the activity need to be clear, and the specific parameters of the design problem need to be defined. In the example below, the teacher begins the Square activity using a series of boundary marks, with the apparent intention of laying the foundation of the design space.

(1) Teacher:

Class level In this game I want you to pretend that I am a dumb turtle. {some students laugh} Well, I’m not literally a {mimics a turtle} turtle. But, uhm, all I know how to do is, like, walk and turn. [BM] And, OK, I’m smarter then the average turtle. I know how to follow very simple instructions. [BM] But, I don’t, I don’t know very much geometry. And its . . . I don’t know what a square vs. a circle vs. a triangle is. [BM] I use simple instructions about how to walk and turn. [BM] So here’s my question. [DM] How can you instruct me the dumb turtle to walk in a square that is two steps long? [TM] {short pause} Think about that for a second. Discuss it with whomever you are seating next to. I don’t know what a square is. You just can’t tell me walk in a square. [BM] All I know how to do is the simplest instructions. [AM]

It is only natural that these statements are very dense with boundary marks since it is particularly important to establish the basic conformation of the design space at the beginning of this activity. Prior to the turtle geometry activities, students had been involved with the design of a stop sign for an international airport. In that design space, students were concerned with conveying the concept of stopping by drawing images. That activity is quite different from what the students should be doing here. In this case, the teacher does extensive work to set the appropriate design space for the students. The phrases ‘‘I am a dumb turtle,’’ ‘‘all I know how to do is . . . walk and turn,’’ ‘‘I know how to follow very simple instructions,’’ ‘‘and I don’t know very much geometry . . . I don’t know what a square vs. a circle vs. a triangle is,’’ are boundary marks. They are general in nature in that they give a general description of what a turtle can do in turtle geometry. Yet, they specify constraints and denote inclusions and exclusions of resources as part of the design space.

3

The names used in this study are pseudonyms.

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In addition to boundary marks, the teacher uses a directional mark, attention marks, and a target mark. The directional mark, ‘‘so here’s my question,’’ indicates that something important is coming. This mark energizes the students’ engagement with the process of constructing the design space since they know that something valuable is coming up. After the directional mark, the teacher uses a target mark (How can you instruct me, the dumb turtle, to walk in a square that is two steps long?) to begin setting a specific goal of the activity. The attention mark at the end of the description reminds students (i.e., reinforces particular information) that the instructions need to be simple. The teacher has done extensive work to set the foundation for the appropriate design space. Are the students now in the appropriate space? The answer is yes and no. As we shall see below, after the teacher finishes marking the design space and the goal for the activity, the students join the marking sequence, and contribute with efforts to further define the design space. However, as we shall see, the students’ initiation into the design space requires a distinct set of marks because they still need to know what kind of knowledge or information is going to be most useful to them. 3.3. The problem of residue After the teacher’s introduction, the students immediately begin to discuss the design problem. The following data provides a view of what is occurring at the group and class level. In this example, both the students and the teacher rely heavily on attention marks to determine what information or knowledge they will need to begin the activity.

(2) Edna (Group 1)

Group level

Class level

Step forward two feet turn left {Edna gestures with the right hand, using her pointing finger, the path of the turtle.}. You could do something with the angles, the angles from the square. [AM] We could make a 45-degree turn // walk two feet, then make {Edna becomes silent and shakes her head. (3) Teacher [AM] She is uncertain about the right angle.} // another . . . turn {Edna stops to listen to the teacher.}

{The teacher is pacing on one side of the room watching students break into groups and start to work.}

// Remember I’m dumb, so, so, the simpler the instruction the better. // [AM]

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(4) Tamara (Group 1)

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Why don’t we walk the square and have him tell us. {Everyone in group 1 laughs.} (5) Teacher

(6) Students Teacher

{Teacher is looking over students’ shoulders (Group 4) to see what they are doing.} Oh, I should clarify. [DM] {The teacher pauses a few seconds, then everybody becomes silent and turns towards him.} Unlike the last one, this one, we, we are // using words. // [BM] // Ooh. // Oooh. Yeah. After the last activity. You think, how can you draw some kind of turtle. I, I can understand very simple words. [BM]

Edna (line 2) immediately starts to present a solution to the problem to her group using attention marks. The way she presents her solution to her colleagues is as if the problem is rather trivial. Right from the start she verbalizes the procedure ‘‘Step forward two feet, turn left.’’ She does not finish the sentence, but she continues to describe the path of the turtle with her hand. As she performs the gesture, she realizes that there is an important piece of information that has to be considered as a part of the instructions, ‘‘angles,’’ the size of the turns. This is made evident by her utterance: ‘‘You could do something with the angles, the angles from the square,’’ which is then followed by a more specific statement, ‘‘We could make a 45-degree turn,’’ to her peers. Both of these statements are attention marks because they propose useful information to the participants. With respect to the terms ‘‘forward,’’ ‘‘two feet,’’ and ‘‘left,’’ these terms are not marked because she treats them as agreed upon terms. That is, she does not discuss the terms with the group, but rather, she takes these terms as givens and starts to formulate a solution immediately. We can speculate that because she is able to position herself in the appropriate design space immediately (based on the activity description), she assumes that her group is in the same space. Moreover, since her colleagues do not question or oppose these terms, we can infer that the group is working within a design space they have presumably agreed upon. For this reason, these utterances (‘‘forward,’’ ‘‘two feet,’’ or ‘‘left’’) cannot be classified as marks. Something else happens just as Edna is interacting with her colleagues. The teacher (line 3) performs two markings. The first is an attention mark to reinforce particular information, ‘‘Remember, I’m dumb . . . the simpler the instructions the better.’’ At this

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point, a lot of discussion is going on, much of which is undecipherable. Nevertheless, the video shows the teacher observing the work of Group 4, which apparently compels him to perform a directional mark (line 5), ‘‘I should clarify.’’ This mark engages the participants for forthcoming boundary marks that are of importance for the construction of the design space. He utters, ‘‘Unlike the last one [referring to the previous activity], this one, we, we are using words,’’ and ‘‘I can understand very simple words.’’ While it may seem that the teacher (at the beginning of the activity) is telling them to write words, ‘‘I can follow very simple instructions’’ (line 1), some students are not in this space. They assume that they need to draw the representation (i.e., a picture of a turtle making a square). Apparently, the marking ‘‘simple instructions’’ is not specific enough to nudge them into the appropriate space. Let us explore this further. Prior to the Square activity, students participated in the Stop Sign activity. In this activity, students designed a stop sign for a new international airport. The sign was used to communicate to car rental drivers from different parts of the world that they should stop as they exited the rental car parking area. Students in this activity mostly drew pictures to convey the message of ‘‘stop.’’ This design space is quite different from the design space of the square activity. The data above suggest that there is design space residue from the previous activity that needs to be acknowledged and then discarded to successfully establish the new design space for this task. The teacher makes this evident when he says: ‘‘Oooh. Yeah. After the last activity, you think, how can you draw some kind of turtle.’’ Another reason that students may be in a slightly different space is domain residue. That is, the word ‘‘geometry’’ (line 1) may conjure up the idea of drawing and measuring images. This may provide an explanation for Edna’s usage of the world ‘‘feet’’ (line 2) instead of steps. It may also reinforce the students’ presumption that the representation they are supposed to create for the Square activity should be similar to the one in the ‘‘stop sign’’ activity — it should be a drawing. When introduced to a new design problem, these students relied on knowledge and information gained in earlier activities. Certainly, one useful skill that we hope students develop is the ability to apply the same concept to different, but related problems. However, sometimes the residue from a previous design problem can impede rather than enhance engagement in a new design problem, and create misconceptions about the new task (Nesher, 1987, Resnick et al., 1989). Thus, it is important not to assume that students can easily shift between activities; these data show that shifting and nudging students into the appropriate design spaces is not a trivial process. Residue can impact not only students’ understanding of the domain within which they are working, but also the development of the design space. While it may be impossible to avoid design space residue and domain residue, it is important to understand that they may be evident in the development of a new design space. Teachers who can anticipate the residue can help students work through it as they establish new design spaces and activities where there is little or no overlap from previous design spaces and activities. Students are also active collaborators in the development of the new design space, and can help themselves work through both types of residues by asking informed questions.

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3.4. Smart marking: informed questions Recognizing that students were inappropriately utilizing information from a previous activity, the teacher provides additional boundary marks for the new design space within the Square activity. Students also recognize the confusion, and begin to ask ‘‘smart questions’’ using attention marks that result in gaining valuable new information from the teacher in the form of precision marks. In this process of ‘‘smart marking,’’ both the teacher and the students work together to refine the design space, and to understand the activity at a deeper level.

Group level

Class level (7) Natalie (8) Ana (9) Teacher

(10) Ana (11) Dean (12) Edna (13) Teacher

(15) Edna (Group 1) (16) Tamara (Group 1) (17) Edna (Group 1)

// Yeah, we could just . . . // {Pauses to let Tamara speak} It alternates . . . {in audible} // take 2 steps forward, make a 45-degree turn, 90-degree turn, two steps forward, . . . {Edna expresses the rest of the instructions by moving her hand on

(14) Natalie

(18) Dean

Can you understand steps? [AM] How about left and right? [AM] OK, I can understand, like, turn left and turn right. [PM] OK. Uhm, OK Can you understand like 45-degree angles? [AM] OK, let’s make it so that I understand angles. [PM] {As the teacher says the word angles his hand makes a gesture of a circle — clockwise.}. . . Just express them in degrees. [PM] 90 degrees. Please make a // 90-degree turn. Pretty cool. //

// Can you, like, uh, take steps? [AM]

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(20) Angel (Group 1)

top of the table and simulating the path of the turtle.} // Wouldn’t you say? Yeah, he said it was two steps. Soo. {Angel gestures the steps of the turtle. the turn, and more steps.} 90 degrees, two steps, yeah. {Tamara gets a clean piece of paper from her binder.}

(19) Teacher

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You can uh, you can tell me how many, tell me how many steps to take. [PM] Sure. //

In this segment, we find the students and teacher working towards polishing the design space. They want to be more specific about the turtle’s instructions. It seems reasonable to presume that the students realized that there are many different ways for defining these instructions. Thus, they feel that it is important to agree on the specific set of instructions. They want resolution to particular issues. Consequently, they ask informed questions: ‘‘Can you understand steps?’’ ‘‘How about left and right?’’ and ‘‘Can you understand like 45 degree angles?’’ The teacher replies with precision marks that help them with the design of the procedure. Evidence that students are forming and inhabiting a similar if not the same space can be found in the questions and later in the statements in lines 14, 16, 17, and 20. For example, after accepting the idea of using degrees to specify the amount of turning, Natalie publicly utters to the class the appropriate angle the turtle needs to turn (‘‘90 degrees’’: line 14). She apparently also feels pleased with their progress (‘‘pretty cool’’). Etna builds on the contributions of the class, uses both ‘‘degrees’’ and ‘‘steps’’ in her proposal to her partners (line 17). Students are not only making progress and building on each others’ contributions, they are also reflecting on the joint construction of the solution as is evident when Tamara, asserts that the parts of the solution alternate (e.g., forward  steps, turn 90°) before Edna offers a solution. It seems that the entire class is making progress refining and negotiating the design space. However, not all students may be in the same space. Domain residue may still be influencing some students’ thinking. This is made evident when they propose different turtle instructions in following interaction between a student and teacher.

Group level (21) Luke (Group 4) (22) Teacher

Do you know length? [AM] OK, if, uhm . . . I don’t know what length is. [BM] But I do know one step, two steps, vs. three steps. [PM]

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Luke’s question of length suggests that he is still influenced by the geometry domain. In geometry, the term length specifies the distance from a starting point to an ending point. Thus, it seems that Luke is still trying to use geometry to define the size of each side. Perhaps he considers the instruction ‘‘make a segment of length 2’’ to be a reasonable command specification for a turtle. Yet, the informed question made by Luke prompts the teacher to provide a boundary mark and a precision mark that will eliminate the residue. In the next part of this activity, students are given one more minute to work on the procedure. The teacher then decides to move to the simulation of the procedure the students have generated. He asks Dean and Luke (Group 4) to give him instructions (line 23). Class level (23) Teacher

(24) Dean (25) Teacher (26) Dean (27)Another student (28) Teacher (29) Dean

(30) Dean (31) (32) (33) (34)

Dean Teacher Dean Teacher

(35) (36) (37) (38) (39)

Dean Teacher Natalie Teacher Angel

OK. Uhm. Dean and Luke. You want, uh, you want to tell me the instructions. I’m going to go into dumb turtle mode. So, it better be simple. [AM] Just tell me what to do. Take two steps Which way? [AM] // FORWARD // [PM] // Straight // OK. {The teacher physically walks two steps — thump, thump.} Now make a 90-degree angle to the right. {Teacher extends the left and right hand out in front of him. He then rotates the right hand 90 degrees. The teacher then physically turns to the right — squeaky sound is made from turning.} Take two more steps forward. {The teacher takes two steps — thump, thump.} 90-degree angle to the right. {The teacher turns 90 degrees.} {The teacher walks two steps — thump, thump.} Oh, I anticipated. Two more steps forward. {Dean laughs.} Forward two. Right? Then you are going to tell me the same thing until I’m done. Right? Yeeeah. So. That’s what we did. OK That’s what we did.

In this segment, we find that the students are now well positioned in the design space. They are giving simple commands to the turtle. However, the teacher still wants them to have a better grasp of turtle geometry by asking them to be more specific. They cannot just say, ‘‘take two steps.’’ They have to specify the direction of the steps. The teacher marks this with an attention mark, ‘‘Which way?’’ (line 26). Dean replies with a precision mark ‘‘FORWARD’’ (line 26).

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Although another student offers an alternative (‘‘straight’’, line 27), the teacher adopts forward. Had Dean said ‘‘straight,’’ the teacher would likely have offered ‘‘forward’’ to be consistent with the Boxer language. In this instance, the marking of ‘‘FORWARD’’ helps define more precisely the design space since this information can later be used to make instructions for other shapes. Continuing with the solution, Dean provides simple instructions that the teacher can execute in his role as the turtle, and, thus, no other marks appear. Let me emphasize that no other statements or actions are marked in the transcription after line 26 because participants are working with information that they already have. In other words, at this point in time, they are sharing a common understanding of turtle geometry that contains all the information that they need to create a representation for the Square activity.

4. Learning turtle language After the execution of the procedure, the teacher stands by the board and writes the procedure for the square (Fig. 2) that the students created.4 This part of the activity centers on learning the technical language that the turtle understands — Boxer. Notice that the teacher filters the instructions made by the students into a more precise language. This representation, made by the teacher, is a precision mark. The marking, however, does not stop here. He then writes the same representation as a more efficient Boxer program (Fig. 3). Again, this representation constitutes a mark. Both marks serve the purpose of specifying the forms of possible solutions, and are, of course, incorporated into the design space of the students. In sum, throughout the activity, the teacher plays a prominent role in helping students define and work within the same space of understanding. He does this very generally, ‘‘I can understand simple instructions,’’ and very specifically, ‘‘I can understand, like, turn left and turn right.’’ When he is specific, he allows students to commit to particular information. This, in turn, allows students to proceed more quickly towards solving the problem. Both the immediate feedback (a quick response to their questions) and the specification of important pieces of information reduced the complexity of the design problem. As a result, they were able to produce three representations: (1) the procedure that the students invented; (2) the execution of the square done by the teacher; and (3) the Boxer representation. In the next activity, we explore the marking of the Triangle design space.

5. Analysis of the triangle activity 5.1. Similar marking patterns with a new twist The Triangle activity directly followed the Square activity. One possible goal for the teacher is for students to apply their newly developed knowledge of turtle geometry and 4

The text in the images was digitally enhanced to make readable the writing on the board.

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Fig. 2. The teacher writing the procedure for making a square. This is a precision mark. [PM]

turtle language to a new, but similar design problem. However, as the teacher and students work through this new design problem, important differences between the Square activity and the Triangle activity emerge, leading to a deeper level of understanding of turtle geometry and turtle language. Because they are in the same design space in the Triangle activity as they were in the Square activity, we shall see a similar pattern of marking as the student and teacher embark in this new design problem. However, there are a few important differences in marking that are important for setting up opportunities for a heightened level of conceptual understanding.

Fig. 3. The teacher writing the procedure in the Boxer language using the repeat command and a box. This is a precision mark. [PM]

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5.2. Bounding the space: laying the foundation While the Square activity had a high number of boundary marks, the teachers’ introduction of the Triangle activity has virtually no boundary marks. Instead, as can be seen below, the teacher utilizes marks with more specificity such as target marks, articulating new goals for this task. Conceivably, this is due the fact that students are not shifting to a new design space or defining a new domain, but rather adding depth to already established intersubjectivity. They are still in the same design space and most, if not all, of the design space and domain residue from the stop sign task has been eliminated.

Class level (40) Teacher

All right. Hard one. This one has a little trick in it. So think carefully. [DM] Using uh . . . You know how I was turning. Right? [AM] I was like measuring off the angle, between my arms and then turning my body? {The teacher shows students how he was measuring the angle with his arms, Fig. 4.} [PM] How can I make a triangle? [TM] Write a little {The teacher points to square procedure located on the board, Fig. 5.} instruction set along these lines for the triangle that is easy. [TM]

In this activity statement, the teacher performs a directional mark at the start of the activity, trying to influence students’ forthcoming actions by asking them to be thoughtful and careful since the design of the triangle is not going to be as easy as the design of the square. In other words, he is asking students not to take the design space they already have from the previous

Fig. 4. The teacher measuring an angle with his arms. This is a precision mark. [PM]

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Fig. 5. The teacher pointing to the procedures on the board. This is a precision mark. [TM]

activity as sufficient for producing the solution for this problem. He says to the students, ‘‘Hard one. This one has a little trick in it. So think carefully.’’ Besides cautioning students to think carefully as they embark on this new activity, the teacher identifies a possible tool they might want to use for this problem. Using an attention mark, he communicates the importance of the upcoming information, ‘‘You know how I was turning. Right,’’ and using a precision mark, he reminds students of a tool he used to measure a ninety angle for the square, ‘‘I was like measuring off the angle, between my arms and then turning my body?’’ (Fig. 4). After marking a tool for possible use, the teacher provides students with two key target marks, ‘‘Write a little . . . instruction set along these lines for the triangle,’’ adding ‘‘along these lines,’’ and pointing to the representations on the board (Fig. 5), that set the goal for the new activity. This gesture is a target mark since it points to examples of an appropriate representation and indicates a specific structure for the representation of the triangle. As these examples illustrate, gestures and physical representations can also be used to mark the design space. In this case, the teacher used gestures to mark a tool and to augment the target mark. 5.3. Smart marking: informed questions Once again, both students and teacher utilize attention and precision marks to clarify the design problem. For example, following the introduction of the new design problem, Luke asks a smart question by using an attention mark (line 41) to find out what type of triangle they should be building. This attention mark is very important since triangles can have sides of different lengths and different angles. If this information were not specified, students would have to do more work (the procedure would require more instructions since one could not use a repeat command). The teacher provides important information in response to the question with a precision mark, specifying that the triangle should be an ‘‘equilateral triangle’’ (line 42).

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Class level (41) (42) (43) (44) (45) (46) (47) (48)

Luke Teacher Tamara Dean Natalie Several Students Ana Tamara

You have to make it, uh, an equilateral triangle? [AM] Yeah, lets make it, uh, let’s make it an equilateral triangle. [PM] OK. Oh that makes it a lot harder. [AM] It’s easier. [AM] // Yeah it’s easier. // // Because of its sides // [PM] // It makes it a lot easier. //

The teacher’s precision mark (42) brings common understanding of the design problem to most students, except one, Dean, who thinks it may make the task harder (line 44). It turns out that his difficulty is that he does not know the definition of an equilateral triangle. However, Dean uses an attention mark to solicit assistance to help bring him to same level of understanding as his peers. Other students point out that this triangle should be simple, but without enough information to fully understand why it should be easy (line 45, line 46, and line 48). Although Dean does not get a response publicly, he finds an answer to his question from his partner Luke who explains what equilateral means. The video shows Luke leaning and saying something to Dean. Later, we hear ‘‘OK’’ from Dean suggesting that he now understands. This segment illustrates that achieving a common understanding is challenging, particularly in a setting in which there is a large group discussion occurring in conjunction with several small group discussions. Some participants resort to the smaller group to get them into the appropriate design space. Even though the domain and design space are quite similar to the Square activity, these examples illustrate how smart marking, in both large and small group discussions, is a necessary component of successfully engaging in this new design problem. The new design problem requires slightly different goals, and some new knowledge. 5.4. Residue revisited Although we might expect little to no residue as students engaged in this new design problem, there are unresolved issues from the Square activity that now surface as students move to a deeper level of understanding the domain and design space. While students have been told to think carefully about this activity, they immediately rely on their geometry knowledge and what they learned about turtle turning in the Square activity. We shall again find evidence of domain residue; relying on what they know about regular geometry will not always work well for turtle geometry. In the design space achieved in the Square activity, students did not develop a full understanding of what turning means in turtle geometry. They did not understand that to a turtle the command ‘‘90-degree angle to the right’’ means sweeping around its’ own position. In other words, to a turtle, turning does not signify an angle between two lines that are connected end to end. In the Square activity, the internal angle of a square happens to

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coincide with the appropriate turning that allows a turtle to make a square. Hence, the students did not have to face this issue, an issue they now face in the Triangle activity. An erroneous assumption that was not problematic in the Square activity became problematic in the Triangle activity. However, the teacher and students successfully utilize different types of marks to work through the residue problems. The discussion below shows that students were thinking about equilateral triangles as having 60° angles. Some may have been thinking about the fact that the sum of the internal angles in a triangle is 180, and all they need to do is divide by 3 (since sides are equal) ‘‘180 divided by 3 is 60.’’ Hence, it seems that students continue to have domain residue because their general knowledge of geometry with respect to equilateral triangles is involved in their thinking. Group level (52) Dean (Group 4) (53) Tamara (Group 1) (54) Tamara (Group 1)

I want 60 degrees. 60 degrees. 180 divided by 3 is 60.

Following the above segment, students spent a couple of minutes discussing the problem. Later, the teacher asks Tamara, Angel, and Edna (Group 1) to provide the instructions to the turtle. Group level (55) Tamara (Group 1) (56) Tamara (Group 1) (57) Tamara (Group 1) (58) Tamara (Group 1) (59) Teacher (60) Students

Forward 3 steps. {Teacher walks three steps — thump, thump, thump.} Left turn 60 degrees. {Teacher measures 60 degrees with his hands.} Take 3 steps forward {Teacher walks three steps — thump, thump, thump.} (58) Tamara: Turn 60 degrees to the left. I’m measuring {Teacher measures 60 degrees with his hands} // off about 60 degrees // [AM] //Oh, no! {Students notice that the procedure is not going well because the sides are not going to connect, and the path of the turtle is not making the equilateral triangle.} //

The instructions begin with ‘‘Forward 3 steps,’’ and ‘‘Left turn 60 degrees.’’ Fig. 6 illustrates the path that is being constructed by the teacher, acting as a turtle, as he executes the interactions. After the teacher’s statement, ‘‘I’m measuring off about 60 degrees,’’ and his physical action (line 59), students notice a problem (line 60). They realize that the triangle is not going to connect based on their instructions. The teacher’s utterance and physical move-

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Fig. 6. The path being constructed by the teacher as he executes the instructions.

ment, which measures the angle for the turtle, is an attention mark. Notice that students did not react to the teacher when he was measuring the 60° angle in line 56, or after the second walk made by the teacher (line 57). Since students were not attuned to their problem, the teacher raises the possibility that their angle might be wrong by specifically bringing attention to his action of measuring the angle. As this marking is done, students react and notice that the next instruction, if it is executed (forward 3 steps), will not make a triangle. Without this mark students might not have noticed that the angle they deduced was wrong. Moreover, they would not have started to notice that to a turtle ‘‘turning’’ left or right a number of degrees means turning around its own position. The fact that students notice that ‘‘turning’’ signifies something important at this point does not necessarily mean they know how to deal with it. Once the execution is completed the teacher asks students what happened. In performing this directional mark, the teacher is asking the students to review their actions (how they came up with the solution and where they made their mistake). Realizing that the students know that something is wrong, but that they are not sure of the nature of the problem, the teacher goes to the board to help them.

Group level

Class level (61) Teacher (62) Angel

(63) Teacher

What, uh, what happened? [DM] We’ve, uh, what did you . . . The angle, the angle wasn’t. . . [PM] Let’s see what just happened here. [AM] So let’s say this is like we are looking at me from the ceiling. So I took, 3 steps forward. Right? {Teacher draws a straight line representing the three steps the turtle takes and then it extends it by using a dotted line.}

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(64) Dean (65) Teacher

(68) Angel (Group 1)

// 60 from 180. // [PM]

(66) Tamara and Ana (67) Tamara (69) Angel

(70) Tamara (Group 1)

Yeah. Now, now I’m currently facing this way. And then // I turn . . . 60. right {Teacher estimates a 60-degree turn and makes another line for the next walking steps (Fig. 7)} // [PM] // The angle is off. // [PM] Oh, shoot. The angle is supposed to be // 120. // [PM] Yeah is 120. [PM]

We were picking the internal angle. [PM]

On the board, the teacher repeats the instructions that were given to him (line 63). At the same time, he begins to draw the path that the turtle makes (Fig. 7). He begins by tracing the first instruction, ‘‘3 steps forward.’’ At this point, he does something interesting. He extends the line path of the turtle (dotted line in Fig. 7) and then estimates a 60° turn and makes another line three steps long. He then marks the 60° angle with an arc. At this point, students notice their mistake (lines 66 and 67). In this case, the representation being drawn by the teacher (and specifically the marking of the 60° angle on the drawing) becomes a precision mark since it is identifying quite specifically the angle that the turtle is turning in relation to it’s position. This, in turn, shows students that they were focusing on the wrong angle. The segment shows evidence of students picking up the mark, ‘‘the angle is off’’ (line 64), and finding a solution. Angel verbalizes the equation needed to find the angle, ‘‘60 from 180.’’ Tamara describes why they made the mistake, ‘‘We were picking the internal angle’’ (line 68). Despite the fact that the domain and design space are similar if not identical in the Triangle activity as in the Square activity, domain residue and design space residue could not be avoided. However, working through the residue in the square activity resulted in achieving enough common understanding to solve the design problem. Working through residue again, this time using mostly precision marks rather than both attention and precision marks,

Fig. 7. Diagram that the teacher draws on the board. This is a precision mark. [PM]

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students arrived at a more accurate and deeper understanding of turtle geometry than they achieved the first time through the domain and design space. After a solution is found, the teacher executes the procedure again with the correct angle. He then goes to the board and asks students how he would write the instructions in the Boxer language. Given their new understanding of turtle geometry, students give the teacher the appropriate instructions with little difficulty. After this activity, the students successfully carried out one more turtle geometry activity, designing a program that built a five-point star. Later, working in a computer lab, they entered their programs into the Boxer environment, and watched their designs in action.

6. Discussion In the process of examining marking and the construction of the design space, discoveries emerge regarding activities, the nature of students’ understandings, and questions. With respect to activities, a new activity that lays the foundation for a new design space requires a higher number of boundary marks. This is only natural since making students shift across different domains will require more effort than just setting different targets. For instance (Fig. 8), the Square activity requires more boundary marks than Triangle activity because it was a new activity to students. Prior to this activity, students had designed a stop sign. After the square activity, establishing a design space for the Triangle activity was easier because students were using information and knowledge that was established in first activity. Hence, achieving intersubjectivity for the second activity was easier. The lesson is that the guidance that teachers provide to transition to a new activity is critical for successfully establishing an appropriate design space. If they do a reasonable job

Fig. 8. Design space of the Square activity showing a higher density of boundary marks than the design space of the Triangle activity.

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in describing the new activity, students will be positioned in the appropriate design space and activities that require collaboration will be easier to coordinate. However, if the description does not sufficiently specify a design space, achieving intersubjectivity will be more laborious. Naturally, no activity description will put all students in exactly the appropriate design space. Nevertheless, attention and preparation is needed to help transition students from one design space to another. With respect to students’ understandings of the design task, two interesting findings emerge. First, when students construct a new design space, there will be design space residue from previous activities. This will have an affect on how students interpret the boundary marks laid down by the teacher and other participants. There was evidence of this phenomenon in the square activity when students interpreted the objective as a picture rather than a textual procedure. The residue was due to the prior stop sign activity. Second, when students construct a new design space there may be domain residue. Particular markings will always be interpreted in the context of prior experience and knowledge — not all of which may be relevant or productive. For example, prior experience with ‘‘geometry’’ encouraged students to think in typical geometric terms (length, feet), and may also explain why students felt compelled to draw in the square activity. While design space residue may typically be easier to discharge in a set of activities, domain residue may linger and resurface. For example, in the Triangle activity, students automatically used their geometry class knowledge of equilateral triangles to find the angle the turtle needed to turn. Some students directly used their knowledge that equilateral triangles had 60° angles, while others calculated the angle by dividing the sum of the internal angles (180) by three. In either case, the students were not looking at the problem from the perspective of a turtle, and this resulted in their some difficulty in solving this problem. However, with guidance from the teacher, and by asking their own informed questions students achieved a strong understanding of turtle geometry and turtle language. These discoveries suggest that activity descriptions are open to varying interpretations. Students’ interpretations may come from recent experiences or from information and knowledge that they have learned in the past. Using information from what we know and have experienced is a natural and effective strategy when facing a new activity. However, the findings here suggest that, in spite of a good activity introduction, particular knowledge may be cued by words or information provided in the activity description itself. Being aware that some terms or descriptions may conger up conflicting information may help teachers understand the conceptual space students are utilizing to approach a new design problem. This, in turn, will help teachers guide students towards the appropriate design space. Teachers need to be attuned to the form of particular marks that they make and the affect that they may have on the kind of resources that students utilize. Finally, from a marking perspective students frequently ask informed questions. In both activities, students and teacher worked together to be on the ‘‘same page’’ in order to successfully accomplish the task. While the questions students asked may have seemed simplistic and obvious, the questions served to clarify uncertainties and synchronize information and knowledge. The fact that questions arose from the need to achieve intersubjectivity and solve the given design problem attests to the fact that students’ questions

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were smart and informed. Accordingly, the questions students’ ask are important informational resources for teachers. Some students’ questions indicate places where students need further clarification and information to fully understand the design space. Questions by students about the very nature of the task may seem off-target to a teacher who knows what he or she intends. However, we have seen how much work is done (much of it through marking) to establish a frame for coherent action, and teachers need to be alert to questions of this type.

7. Conclusion In this special issue, the various articles have added to the understanding of metarepresentational competence by looking at several issues from different points of view. This study contributes to this understanding by examining how students and a teacher in collaborative design activities coordinate their actions to design representations. More specifically, this study explored the discourse of the participants as they build their understanding of the nature of the activity. By making use of design space as a construct to bind intersubjectivity to the conceptual context, we discover that the participants made use of marking to feed the ongoing interaction in the design of the representations. Identifying marking in discourse is a first attempt at shedding light on the process through which participants achieve intersubjectivity in collaborative design activities. Some questions remain: Is marking a method that is autonomous (i.e., independent of context)? Can we use marking as a method for exploring intersubjectivity in other types of interactions such as (1) scientific class discussions, (2) two people meeting for the first time trying to find out what they have in common, and (3) a couple seating in a coffee shop interacting and trying to find out if they love each other? Do we need different conceptual structures similar to a design space to bind intersubjectivity to different contexts and thus be able to examine marking in different kinds of activity? Based on the data presented here, I propose that marking is present whenever we are trying to achieve intersubjectivity. However, research is needed to explore these questions in detail. A valuable goal for future work is to continue to examine marking in other forms of collaborative activities. We may find that marking is a systematic model by which people coordinate their understanding and problem solving in all social interactions.

Acknowledgments This paper is based on work done by the Project MaRC Research Group: Flavio Azevedo, Andrea A. diSessa, Andrew Elby, Noel Enyedy, Jefferey S. Friedman, Rodrigo Madanes, and Nathaniel Titterton. This work was funded by the National Science Foundation under grant RED-9553902, Andrea A. diSessa, principal investigator. The opinions expressed in this paper are those of the author and do not necessarily represent those of the Foundation.

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