Exploring mathematical exploration: how two college students formulated and solved their own mathematical problems.

Preparation of this paper is supported, in part, by a grant from the University of Delaware Research Foundation and a grant from the University of North Carolina at Charlotte. However, any opinions expressed herein are those of the authors. The first author is grateful for Ed Silver's guidance in the early stage of conducting this study. We are grateful for the editor and the three anonymous reviewers who made valuable suggestions concerning an earlier version of this manuscript, thereby contributing to its improvement.

Abstract

The study examined the mathematical explorations of two college students, with particular emphasis on how they formulated and solved novel problems that arose in the course of their on-going problem-solving activity. The results of this study suggest that mathematical exploration can be characterized as a recursive process in which solvers determine goals of action as they formulate their problems, solve the problems, and reflect upon their solution activities to formulate new problems. The results of this exploratory study contribute to the development of a conceptual framework and research tools to capture mathematical exploration processes.

Introduction

The mathematics education community has taken the position that observation, experiment, discovery, and conjecture are as much a part of the practice of mathematics as of any natural science and that our school mathematics curriculum needs to be representative of that position (NCTM, 1989,2000). For example, the National Council of Teachers of Mathematics has advocated that mathematics teaching should include the notion that "the very essence of studying mathematics is itself an exercise in exploring, conjecturing, examining and testing, all aspects of problem-solving. Students should be given opportunities to formulate problems from given situations and to create new problems by modifying the conditions given problems" (NCTM, 1989, p.95).

While various theoretical accounts of mathematical learning have focused on one or more of these processes (see for example, the work of Harel & Sowder 1998, that explains how conjectures evolve into formal arguments and proofs), the current paper will focus on how the solver incorporates these processes in mathematical problem-solving situations, as he/she develops novel solution activity. We refer to the solver's engagement in such sustained investigations, from his/her initial interpretive and sense-making of the problem situation through the development and carrying out of solution activity, as the solver's mathematical explorations.

The National Council of Teachers of Mathematics has been a strong advocate that U.S. students develop their problem-solving knowledge through mathematical explorations, as defined above. In addition, the international mathematics education community has also acknowledged the important roles played by mathematical exploration processes in the teaching and learning of mathematics. For example, the Chinese school mathematics curriculum stipulates that mathematics teaching should emphasize both the development of students' abilities to pose and solve mathematical problems, and the development of students' abilities to explore mathematics (Chinese Ministry of Education, 1998). This emphasis on mathematical exploration is also found in the mathematics curriculum of other Asian countries such as Japan and Singapore (Hashimoto, 1987), as well as in the Scandinavian countries, where open-ended mathematical tasks are used in the mathematics classroom to promote the development of students' mathematical problem solving processes (Pehkonen, 1995, 1997; Borgensen, 1994). In all of these countries, the activity of mathematical exploration in the classroom is viewed as an important focus of instruction that provides opportunities for students to enhance their mathematical thinking and reasoning abilities.

The current study investigated the problem-solving processes that help drive and sustain the mathematical explorations of problem solvers. The study focused on the solution activity of two college students working in a computer microworld, which simulated the path of a ball on a billiards table. The Billiard Ball task provided the solvers with an unusual degree of freedom to explore and develop novel mathematical relationships; hence, the results of the study build on existing research on problem solving in open-ended problem environments.

Theoretical Framework

Our focus on the solver's dynamic problem-solving actions and processes involved in mathematical exploration is compatible with the ideas of researchers who have examined mathematical problem-solving activity. Mathematicians and educators have long been interested in the processes that contribute to the mathematical explorations of learners (e.g., Hadamard, 1945; Poincare, 1908; Polya, 1962). For example, Polya (1945) focused on the important roles played by discovery processes in solving mathematical problems and proposed a four-phase model of heuristic processes to describe and explain the complexities of mathematical problem solving.

Drawing from Polya's ideas about problem solving, the research on mathematical problem solving during the past several decades has made great progress in developing an understanding of the affective, cognitive, and metacognitive aspects of mathematical problem solving (e.g., Borkowski, 1985; Cai, 1994; Lester, 1994; McLeod, 1989; Schoenfeld, 1985, 1992; Silver, 1985). For example, Garafalo and Lester (1985) developed a model of problem solving based on Polya's ideas (Garafalo & Lester, 1985; Lester, 1985). This model was useful because it both illustrated and clarified many of Polya's ideas, and thus provided some specificity to the sophisticated processes that underlie solvers' solution processes. For example, Polya believed that competent problem solvers engage in heuristical or provisional reasoning throughout the problem solving. Garofalo and Lester (1985) elaborated this view, by suggesting some of the ways competent solvers self-monitor their actions during problem solving, and thus highlighted the importance of the solver's meta cognitive actions throughout problem solving.

Other researchers have extended our knowledge of problem solving by developing more contemporary methodologies to analyze problem-solving processes. In examining the problem-solving performance of college students, Schoenfeld (1985) focused on his students' problem-solving episodes at a more holistic or macroscopic level, describing how successful problem solvers evolve and "build-up" their mathematical knowledge in conceptual "chunks" and in an on-going fashion. In adopting a similar focus, Burton (1984) proposed a model of mathematical thinking that offered an explanation of the dynamic ways that cognitive and meta cognitive processes interact with each other as the problem-solving process commences.

Although these studies provided some insights into the processes of mathematical exploration, we do not yet have a comprehensive picture of the cognitive and meta cognitive processes that constitute and sustain the mathematical explorations of problem solvers. This has been noted by researchers (Lester, 1994; McGinn & Boote, 2003), suggesting that we still have some ways to go before we fully understand how the mathematical exploration processes of solvers contribute to their ongoing problem-solving performance. A central issue involves how the solver's initial sense-making actions within the problem situation develop into more formal ideas that can be implemented to solve the problem. Sumara (1996) suggested that solvers engaged in mathematical exploration benefit from becoming "lost within the problem". While they see no immediate ways to solve the problem, they are also free from any confining constraints (such as their prior experiences and biases) and can "see" the usefulness of pursing novel approaches that may eventually lead to a solution of the original problem. In trying to make progress towards a solution, the solver may reason provisionally as different courses of action are considered. Hence, the current study addressed the following questions: How do solvers structure and make sense of the problem situations they face? How do solvers develop their initial ideas and intuitions into conjectures and goals for action? How do solvers continually monitor and modify their actions throughout problem solving?

In framing these questions, the current study examined the research on mathematical problem solving. However, the problem-solving research is not clear on answers to these questions and provides only partial answers. It is difficult to draw many conclusions from the problem-solving research due to the many different (and sometimes incompatible) contexts researchers have used to examine problem-solving performance. For example, cognitive studies of problem solving have traditionally focused on the solver's performance on "well-structured" tasks (Schoenfeld, 1985). Specifically, in well-structured mathematical problem solving, both the goals and means are clearly specified within the task, and a solution consists of finding a path that connects the goal with the given state of the problem. The problem of Missionary-Cannibals crossing a river and The Tower of Hanoi problems are examples of well-structured tasks that have been used in several cognitive studies of problem solving (Gick & Holyoak, 1980). Within these kinds of puzzle tasks, problem-solving activity is limited to a "search and discovery" of acceptable moves, most of which can be found by employing trial-and-error strategies. Similarly, cognitive studies of algebra word problem-solving have limited their analyses to student performance on the prototypical algebra word problems (e.g., age problems, mixture problems, etc.) These problems are well structured in terms of the algebraic operations that lead to solutions; problems are viewed as having similar mathematical structures if they require the same algebraic operations for solution. Solvers of both kinds of tasks, puzzle and algebra word problems, have few opportunities to engage in the types of reasoning and problem-solving processes that we typically associate with mathematical explorations. Not surprisingly, the use of such well-structured tasks to examine problem-solving processes has been challenged by researchers who question their legitimacy as appropriate problem-solving opportunities, primarily because they give solvers few opportunities to develop their own goals and purposes for action (Lave, 1988, Sowder, 1985). According to Lave (1988), the assumption that puzzle-based tasks adequately simulate mathematics problems for subjects undermines most cognitive studies of problem solving because the "problems are assumed to be objective and factual ... yet they are constructed 'off-stage' by experimenters, for, not by problem solvers" (Lave. 1988, p.35). Similarly, Sowder (1985) criticized the research on algebra word problems (Mayer, 1985) because it adopted an overly simplistic view of problem solving, a view that characterizes expertise as the solver's ability to recognize similar problem types; rather, he suggested that a more useful characterization for mathematics educators is to view problem solving as involving "what do I do when I do not know what to do" (Sowder, 1985, p. 141). Hence, while the focus of puzzle-like and algebra word problem tasks undoubtedly brings a sense of objectivity to studying the problem-solving actions of a chosen population, it also tends to de-emphasize the subjective novelty that solvers engage in, which precedes the introduction of formal mathematical procedure into their solution activity.

In contrast, the current study addressed how solvers cope with problems that are more open and "ill-structured" (Kilpatrick, 1987; Simon, 1973). Ill-structured problems are those that "lack a clear formulation, or a specific procedure that will guarantee a solution, and criteria for determining when a solution has been achieved" (Kilpatrick, 1987, p. 134). An example of an ill-structured problem is provided by Kilpatrick (1987):

  An example is the problem of finding the paths from my house to my
  school. Structuring the problem would require specifying what is to
  count as a path. Solving it would require developing a procedure for
  identifying and enumerating the paths." (Kilpatrick, 1987, p. 135).

Unlike the aforementioned puzzle and algebra word problem tasks, the example of path finding given by Kilpatrick requires the solver to explore the problem and develop some foundational understanding within the problem environment before solutions can be constructed.

In addition to the feature that ill-structured problems may need to be reformulated by solvers before they can be solved, ill-structured problems also possess a conceptual "slipperiness" for solvers: the nature of what the solver interprets as problematic may change as the solver develops understanding and "gets a handle" on the problem and progresses to a solution.

While we have discussed the need to use tasks that give the solver opportunities for sustained investigation, we are also concerned about capturing in our analysis some of the informal reasoning actions of solvers. Contemporary studies of problem solving have seldom considered the important role played by the solver's subjective meaning-making activity that typically precedes the solver's introduction of formal algorithms and strategies into his/her solution activity. Mathematicians typically engage in informal reasoning such as provisional or hypothetical reasoning when they initiate work on a problem (Mason, 1995; Polya, 1962). However, theoretical accounts of problem solving seldom address the prominent role played by informal reasoning activity even though the informal reasoning actions of students in novel mathematics situations often pave the way for their development of formal algorithms (Cai, Moyer, and Laughlin, 1998).

The current study addressed the challenges discussed above in the following ways. First, we used an open-ended problem task to examine the problem-solving actions of students; this made possible a focus on the solver's solution activity on an ill-structured problem. Second, we presented the problem task to students via an interactive computer program called SNOOK; this mode of presentation afforded the students great freedom in their actions so that they could generate and explore a variety of problem situations. In the next section we provide rationale for the study's use of open-ended problem tasks and computer-based problem-solving environments.

Open-Ended Problem Solving: The Billiard Ball task. Open-ended problems arise for the solver when the goals for solving the problem are ill-structured in the sense that the solver's interpretation situation may give rise to several different problems and solutions. In contrast, many studies of problem solving employ experimental tasks that are well-structured and closed with regard to the guidelines that are placed on the solver's solution activity.

While open-ended problem solving has received surprisingly little attention by problem-solving researchers, several researchers have advocated the use of open-ended tasks in mathematics teaching, in order to stimulate the novel problem-solving processes of students (Borgensen, 1994; Pehkonen, 1995, 1997; Solvang, 1997). The current study used open-ended tasks with a goal of giving students ample opportunity to engage in novel problem solving. We will provide some examples to illustrate ways that open-ended problems enable students to demonstrate a level of novelty not possible with well-structured tasks.

In a similar way, the current study used a Billiard Ball task that provided opportunities for students to develop their intuitions and conjectures by generating and carrying out trials within a conceptually rich problem-solving environment. Specifically, students were asked to explore the path of a billiard ball once it is set in motion, until it ends up in a corner pocket. The students had an unusually high degree of freedom to explore their novel ideas and conceptions. They were able to explore various problem situations by manipulating the billiard table's dimensions as they wished, in order to see actual paths of the ball corresponding to specific length and width dimensions; using these self-generated simulation experiences, the students were able to explore their evolving ideas about relationships involving various table dimensions and the corresponding path of the ball. These experiences served as conceptual foundations from which the solvers developed a variety of conjectures about the relationships between the table's dimensions and the subsequent path taken by the ball.

The Billiard Ball task placed very few restrictive guidelines on the students' actions, and hence, can be viewed as an open-ended problem-solving task in the sense that it places very few restrictions on the students' actions, thus giving them a high degree of intellectual freedom to develop their mathematical knowledge. In addition, the task had the added feature that it could be presented within an interactive computer program that allowed students to easily generate new situations to explore.

Mathematical Exploration with a Computer Microworld.

The use of interactive computer problem tasks or computer microworlds has given researchers an effective means to capture and analyze mathematical novelty in problem-solving settings. Computer microworlds can serve as conceptually rich environments for mathematical exploration because they allow the solvers the necessary autonomy and freedom to explore and develop their mathematical knowledge (Steffe and Olive, 1996).

Subjects of the study worked within a computer microworld call SNOOK to explore mathematical relationships. SNOOK is a computer program that simulates the path of a cue ball on a billiard table having certain dimensions. The ball starts from the bottom left-hand corner, and initially is set to travel at an angle of 45 degrees and at a gradient of 1.

After SNOOK has been loaded, a prompt appears on the screen, asking the subject to enter the dimensions of the particular table they wish to examine. Once numbers for the HEIGHT and WIDTH of the table are entered, the screen will show the path of a ball on the table. The bottom of the screen displays the dimensions of height and width, number of hits, and the corner in which the ball finished. The case of a table having dimensions 2 feet by 8 feet is shown in Figure 1.

Working within SNOOK provided the subjects with opportunities to explore a great variety of mathematical relationships. Specifically, in SNOOK, there are several distinctive correlation variables, including the height of the tables (HT), the width of the tables (WT), the number of hits (NH), the location of the final pocket (FP), the number of squares passed through by the ball on its path (NS), and the number of regions formed by the trace of the path of the ball (NR). The correlation structure of SNOOK allowed subjects to explore the relationship between or among these variables. For example, subjects could explore the relationships between table dimensions and the number of hits. Hence, subjects could easily generate many trials to explore and analyze. In addition, the subjects' results are immediately available to them to aid their reflections as they contemplate new explorations. Finally, and most importantly, SNOOK provided the subjects total freedom and control over their problems-solving actions, ensuring that they were able to explore the viability of their evolving conjectures.

[FIGURE 1 OMITTED]

In completing the Billiard Ball task, the solvers were able to physically manipulate the problem variables to generate images and simulations corresponding to specific table dimensions; the solvers were able to pose new problems to explore, reflect on potential solution activity, and draw inferences from the results of their actions. In these ways, the solvers could develop structure within their solution activity.

Since our purpose was to describe and explain the mathematical processes that constitute the solvers' mathematical explorations, we were especially interested in the student's interpretations and evolving conceptualizations. Hence, we make a distinction between the open-ended nature of the Billiard Ball task and the open-end-ness of the solvers' solution activity, with the latter designation encompassing the ways that the solvers evolved and extended their problem-solving goals based on their interpretations of the situation.

We believe that mathematical exploration is a complex set of problem-solving processes that commences whenever solvers find themselves facing unexpected problematic situations (Pask, 1985), or situations in which they see no immediate way to achieve their goals and purposes. By having students solve an open-ended task within a computer microword, we were able to observe solvers truly engaged in the dynamic activity of mathematical problem solving. This made possible a focus on the crucial ways that problem solvers make sense of the mathematical situations they encounter, and how they generate and develop ideas and conjectures to resolve dilemmas and questions that arise for them in the course of their ongoing activity (Lave, 1988).

Methodology

Subjects

The subjects of the study were two mathematics majors, one a junior (SI) and the other a sophomore (S2), from a university located in the United States. The problem solving of undergraduate college students has become a popular area of study among mathematics education researchers (Schoenfeld, 1992). From our standpoint, the advantage of studying the problem-solving actions of college students is that we believe that it enables us to observe and explain a broad range of reasoning and problem-solving actions.

Administration Procedure

A written description of SNOOK was provided to the subjects, who were then given approximately 10 minutes to familiarize themselves with the operation of the program. After this preliminary orientation, each subject had 45 minutes to work with the program to explore mathematical relationships. Subjects were given the following instructions: "There are interesting mathematical relationships in this microworld. I would like you to think about this microworld, as a mathematician, and explore and see what you can discover. You may generate as many examples as you like to help your mathematical questions or problems, or make any interesting mathematical conjectures that occur to you. Remember, it's very important for you to talk aloud during your exploration." All subjects were videotaped as they worked in SNOOK.

Data Coding and Analysis

The data used in the analysis consisted of the videotaped protocols, the experimenters' field notes, and the subjects' written work. Written transcriptions of the videotapes were generated and verbal protocol analytic techniques were used in the analysis (Ericsson & Simon, 1980). The analysis proceeded in the following phases. First, the protocols were segmented into episodes of solution activity where the solver has appeared to change his/her goals and purpose. Transitions involve solution activity and transition points between episodes (Schoenfeld, 1985). An episode consists of a set of the solver's individual actions, oriented towards a goal or purpose. Transitions involve solution activity where the solver has appeared to change his/her goals and purposes. According to Schoenfeld (1985), organizing the solver's solution activity in terms of episodes and transitions helps the researcher identify solution activity where the solver may be making conceptual progress towards a solution. Each episode was documented by the set of verbal comments made by the solver, indicating both their goal as well as their actions initiated to achieve the goal. Transitions between episodes were indicated. Second, the study was particularly concerned with shifts in the solver's problem posing and problem-solving actions; hence, each episode was examined to identify instances of problem formulation or re-formulation, problem solving, and solution verification. Finally, our review of the research on mathematical problem solving suggested a need to document both quantitative and qualitative products of solution activity. Hence, with a view towards relating the results of the study to existing research in mathematical problem solving, the protocols of each solver were examined to identify the following aspects of the solver's solution activity: (1) the solver's reasoning activity that helps him/her generate their goals, purposes and conjectures; (2) the formal mathematical relationships explored by the solver; (3) the affective and metacognitive comments made by the solver; and (4) the number of examples generated by the solver.

Results

The results of the study are discussed in the following sections. The students demonstrated an interesting range of problem-solving actions as they completed the Billiard Ball task. We have organized the results so that both quantitative and qualitative aspects of the students' solution activity can be discussed. We will begin with a summary of the quantitative aspects of the solution activity of the solvers. Then we will focus on the qualitative aspects of the students' solution activity by discussing the various exploration strategies demonstrated by each solver to better explain how individual differences in their actions led to the results.

Explored Mathematics and General Quantitative Counts

Both students explored an interesting range of formal mathematical relationships. Explored mathematics refers to any relational statements and generalizations made by the subjects, involving at least two variables and tested with at least two examples. Table 1 shows the explored mathematics of solvers SI and S2 as inferred by the researchers.

An examination of the relationships in Table 1 indicates that the students explored a great variety of mathematical relationships, thus suggesting the richness of the microworld SNOOK as a problem posing and problem-solving environment. Student S1 constructed more mathematical relationships than S2 (S1 constructed 12 mathematical relationships and S2 constructed 8 mathematical relationships). There was some overlap in the students' explorations: both subjects generated situations to examine mathematical relationships 1 and 7. Specifically, each student started their explorations by analyzing the case of tables with square dimensions (relationship 1). And both students generated mathematical relationship #7, which is among the most sophisticated relationships in their explorations.

As each solver progressed in their explorations, S1 appeared to construct more sophisticated relationships than S2. For example, S1 spent more time than S2 focusing on the table dimensions and how they influenced the path of the ball (relationships 4-6), and then was able to generalize the dimensions and explore more sophisticated relationships (relationships 14, 15, 18). We examined other aspects of the solver's solution activity to determine if there were other indicators that would help us explain some of the differences in the solvers' problem-solving actions. Each solver spent about 45 minutes exploring the various situations they generated. S1 had 15 episodes and S2 had 14 episodes within approximately 45 minutes.

Goals and Purposes in Mathematical Exploration

Tables 2 and 3 provide summaries of the solution activity of each solver including the number of episodes, the type of goal involved, the number of examples generated, the number of questions posed, the number of conjectures made, the number of example features observed, and the number of relationships between or among examples observed in each of the episodes making up their explorations.

The solvers' purpose within each of their individual explorations was to generate new and potentially useful information, the consideration of which might yield conjectures or conclusions. Within these explorations, solvers were observed to generate qualitatively different kinds of goals to orient their actions. Specifically, S1 and S2 formulated two types of goals in their explorations: primitive and refined. Primitive-goal episodes occurred when the solvers generated a trial to merely try out a preliminary idea, with no stated purpose other than to generate the actual results to aid their further exploration. For example, in the first several episodes, S1 just wanted to generate several examples in order to "see what would happen." In contrast, in one of his early episodes, S2 worked to achieve a more purposeful goal, to see it he could get the ball into a different corner; hence, this was an example of a refined-goal episode because S2 initiated activity with a well-defined goal in mind. In general, whenever S1 or S2 tried to determine specific relationships between variables 1, then the goal was classified as a refined goal. Another example of a refined goal is when S1 tried to find the number of hits for tables of dimension n by (n + 1), where n is any positive integer.

Both students started their solution activity with episodes that involved primitive goals. Student S1's first eight episodes involved primitive goals while only S2's first three episodes involved primitive goals.

For example, while it took S1 about 20 minutes to finally get clear ideas what he wanted to find out, it took S2 about 5 minutes to find out what specifically he wanted to explore. Remaining episodes for both S1 and S2 involved refined-goal explorations. This suggests that once the solvers became familiar with the environment through observation and exploration of the situation, their further exploration became more directed and systematic.

S1 took much longer than S2 to explore the problem in primitive-goal cases (20 minutes versus 5 minutes for S2) before developing more sophisticated goals to frame his actions. This result, together with the result that S1 appeared to construct more mathematically sophisticated relationships in his explorations, is consistent with the problem-solving research finding that able and competent problem-solvers spend increased time planning and reflecting before they develop solution activity (Schoenfeld, 1985; Schoenfeld, 1992).

There are other results that help to shed light on the differences in the solvers' explorations. In particular, while each student generated a similar number of questions to explore (18 for S1 versus 22 for S2), student S1 only needed to generate half as many examples as S2 in developing his explorations (35 versus 71). Hence, if we use the solvers' generated number of examples and conjectures as measures of problem posing and solving respectfully, it would appear that S2 is the more prolific problem poser. However, S1 appeared to put his problem posing to better use in his explorations as indicated by the mathematical relationships that he explored: the relationships that he explored were more generalized than those explored by S2. We interpreted this result to suggest that S1 appeared to develop more abstract problem formulations and re-formulations throughout the course of his explorations.

While we inferred that S1 appeared to develop more sophisticated mathematical relationships than S2, it should note that they took a similar procedure to explore the Billiard Ball task once their solution activity became goal refined. Specifically, within refined-goal episodes, each of S1 and S2 formulated goals to investigate a well-defined problem, initiated appropriate activity to solve the problem, and then verified his/her results. Whenever they solved a particular problem that they had formulated, their solution of the problem often suggested to them new problems to formulate, reflect on, and solve. This continual pushing forward of problem formulation followed by problem solution and verification constituted a problem posing-solving chain of mathematical explorations for the solvers. In addition, both subjects frequently used "what-if' or "what-if-not" approaches to formulate their problems. For example, after S2 investigated the pattern for number of hits in odd X 12 tables, he turned to examine the pattern for number of hits in the even X 12 tables by asking himself, "Um, what should we try here? Uh, hmm, what if we try it with even numbers [by 12 tables] this time?" Similarly, using the "what-if-not" approach, he turned to examine the pattern when table dimensions were odd X 13.

Figure 2 shows S2's sequential goals in his refined-goal episodes and suggests how his "what-if' and "what-if-not" problem posing strategies played a prominent role in his evolving problem-solving actions. In general, both S1 and S2 consistently used a "what-if-not" approach to specify the problems and frame their mathematical explorations. In these cases, the questions help them formulate and refine their goals and purposes (e.g., What if the height is not an odd number?).

During the interviews, S1 posed a total of 18 questions while S2 posed 22 questions. In addition to posing questions which subsequently served as goals of their solution activity, both S1 and S2 posed other kinds of questions to help monitor their on-going actions. For example, in the transition of the episodes 4 and 5, S2 asked himself that "Um, what else?" This question served the purpose of both monitoring his prior actions as well as prompting his future actions (i.e., asking him to examine a new case different from that of the previous episode). In 6 instances for S1 and 5 instances for S2, they posed such monitoring questions during the transition between solution episodes.

[FIGURE 2 OMITTED]

Mathematical Exploration Strategies

In addition to the quantitative results reported in the previous section, there were indications that the solvers used qualitatively different exploration strategies in the course of their mathematical explorations. Specifically, we observed that the solvers employed different reasoning strategies in their solution activity. Student S1 often generated hypotheses to frame and suggest potential courses of actions for his explorations. We refer to these as hypothesis-driven explorations. Working within hypotheses enabled S1 to make some major conceptual advances in his explorations. In contrast, Student S2 very seldom was able to reason beyond the actual results of a specific case or trial. We refer to these as data-driven explorations. For example, we already noted how solver S1 needed to generate fewer examples than S2 to aid his explorations (35 versus 71); this was because he was able to generate some powerful inferences from his results to serve as working hypotheses with respect to mathematical relationships between variables. He proceeded to generate other examples that were highly focused to test and verify his evolving hypotheses. In contrast, S2 needed to generate many examples, most of which were closely related in structure to his prior examples, in order to determine mathematical relationships. He was seldom able to extend his ideas beyond the results at hand. As a result, he appeared to induce lower level mathematical relationships than did S1. Each of these strategies, hypothesis-driven and data-driven, is illustrated and described in the following sections.

S1's Hypothesis-Driven Exploration

The following excerpts illustrate S1's hypothetical reasoning activity that constituted his mathematical explorations. At this point of the interview, S1 has already found that if the table was square-shaped, then the ball would hit in the upper right corner of the table.

  Let's try, ah, try something like, um. Let's try something where it's
  close to the square but not exactly. Height equals 10, width equals 9.
  So, if I'm right, since it went straight to the pocket whenever these
  sides are equal. This one's slightly unequal, it's going to hit the
  pocket and bounce around a little bit so (generates 10 x 9 table). Oh,
  boy. Yup, there you go.

The comments of the solver indicate his hypothesis that having a table of unequal dimensions (in this case 10 x 9) would result in the ball bouncing around the table and not landing in the upper right corner as it did for the case of a square-shaped table. He generated a table of dimension 10 x 9 to test his hypothesis. The solver reflected on the results as he continued to explore tables of unequal dimensions.

  The distance is getting greater between them. It's going to end up in
  this pocket (points to top left corner). So, let's see, so whenever,
  so that does make a difference if you elongated it, depending on the
  number that you pick the ball's going to hit, you know, that distance
  further down the side of a little bit, so to speak. So, you have this
  figure, I think that, then whenever you have it when it's a little bit
  shorter, a little bit longer. Whichever way you want to look at it
  rather than go straight to there, it's going to go like this. So it's
  gonna hit the difference, away from that pocket. So, let's try this.
  Let's try ten and eight. So, this is going to be, side X, X-2
  (generates 10 x 8 table). There you go.

With his comments, the solver hypothesized that the ball would hit somewhere short of the upper right corner, along the top bank. In the following excerpt, the solver further developed his hypothesis about tables with unequal dimensions by examining the case of a 10 x 5 table.

  It's the same thing. This distance increased. So, when you have a
  perfect square, it goes to here (points to corner). When you have a
  rectangle that's close to a perfect square, it'll hit there. Then when
  you have one's that's got, got 2 subtracted from the width, it's going
  to go further away. So, then if you were to take half this number and
  hit, I would think. Don't quote me on this but is should hit that
  center. It should hit dead center and probably go in this pocket. So,
  let's try 10 and 5 (generates 10 x 4 table). It should, yep, see right
  there, bounce around the pocket, bounce around the pocket.

The solver reflected on his results in the case of a 10 x 4 table. At this point, he realized that there were similarities between the 10 x 4 case and the case of a table of dimension 2 x 5, a case that he had generated and explored earlier in the interview. This led S1 to develop a more generalized mathematical relationship between cases: if a table's dimensions are a multiple of another table's dimensions, then the two tables have the same number of hits and the ball will land in the same pocket.

  So, if you've got ten and four, with this 10 and this one's 4. So,
  let's see here. That would be the same as 2 and the 5. So, if I take
  and shrink the table down, I'll get the same figure. Yeah, one is 4,
  10. I'll get this same figure. So whenever this 4 and 10, I got this
  diamond shaped figure something like that. I should get that same
  figure here. Um, it's just showing that, you know, given a certain
  number. If there is a relationship between those at all. Like these,
  these here, given these numbers, this would be the same as saying,
  this side times two. So this would be the same as 2Xand 2 Y So what
  I basically did is said that, that's that, then I'll divide by 2 and
  see why X and Y are. And it should be the same figure because these
  are really what I would call the critical numbers, that mean
  something. I mean, you could take these numbers, these base numbers
  and like, as long as you multiply them by any given number, both sides
  equally, you should have the same figure. So, let's try 5 and 2
  (generates 5 x 2 table). There it is so, just to prove a point, you
  can take, uh, this equal 50 multiply it by a side that's ten, this
  equal 20 and you get the same figure only it'll be in bigger
  perspective (generates 50 x 20 table). So there's that.

The solver asserted that the dimensions of a 5 x 2 table represented " critical numbers" in the sense that he knew that all tables with dimensions that are multiples of the 5 x 2 case behave in the same way (i.e., they have the same number of hits and land in the same pocket). In this way, he had formed a generalized mathematical relationship.

S2's Data-Driven Exploration

The following excerpts illustrate S2's data driven activity that constituted his mathematical explorations. At this point of the interview, S2 had found a general pattern for cases of tables having dimensions A x 12, where A is an odd number.

In the following excerpt, S2 explored cases for tables having dimensions B x 12, where B is an even number. In the course of his activity, he constructed a chart and recorded the results of the different cases he generated to aid his search patterns.

  Uh, hmm. What if we tried it with even numbers this time? Make it two
  by 12 (generates 2 x 12 table). Oh, that's gonna, now we had seven
  hits (generates 4 by 12 table). So, four hits (generates 6 x 12
  table). Um, okay six and 12, we get, three hits. (solver laughs) So
  there's a weird pattern here. (generate 8 x 12). And, eight and 12,
  get five hits. Hmmm. Let's see, something differently. I don't see a
  pattern (generates 10 by 12 table). Um, 11 hits, oh boy. On 12, it
  just be one. 14 (generates 14 x 12 table) It's hard to see if there's
  even a pattern, that's a problem. We have 13 then. Oh, boy. Uh, I
  don't see anything here (looks at his work on the paper). Um, seven,
  four, three, five, 11, one, 13. Boy, that's not right. Um, there
  doesn't seem to be any pattern in here. Oh, boy, um, no, I don't know.
  Five plus six equals... Okay, let's try 16 x 12 and see if there can
  be any further (generates 16 x 12 table). It's only seven hits. Oh,
  maybe this is like repeating things so if I were to guess the next
  one, if we had like 18 by 12, I'd guess 4, I'm gonna guess four
  (generates 18 x 12 table). Let's see if I'm right here. Yeah, oh
  there's five hits. That just throws it all off. Um, okay. Boy. 20 by
  12 (generates 20 x 12 table). Okay, we have eight hits. Oh, gosh. All
  right. Seven, okay we had five and eight. Um, let's try 20, we just
  tried it, I'm sorry. Okay, 22 by 12 (generates 22 x 12 table). Okay,
  we had 17. So, 17, no, that's right. I don't see anything.

While S2 generated a sequential set of trials and results, he was not able to develop generalized mathematical relationships from his solution activity.

As indicated in Tables 1 and 2, S1 spent longer time than S2 working on primitive goals (about 20 minutes for S1 as compared to 5 minutes for S2). The fact that S1 spent much longer time than S2 working on primitive goals might be due to their different exploration strategies. S2 relied heavily on generating many examples. In particular, S2 generated 71 examples, which is more than twice as many as that was generated by S1 (35). Although S2 systematically generated examples, his results were "blind" to him in the sense that he seldom derived or inferred patterns from his results; consequently he seldom generated novel hypotheses about mathematical relationships. In contrast, after careful observations of a few examples, S1 made explicit hypotheses about patterns of examples. In most cases, these hypotheses helped pave the way for S1 to determine novel mathematical relationships.

The solvers' self-generated examples

Examples generated by the solvers aided their explorations in three different ways: 1) to get familiar with the SNOOK environment; 2) to make and test conjectures; and 3) to solve the problems they formulated. Even though both solvers used examples to become familiar with SNOOK at the beginning of the interview, the solvers differed in the ways they used examples as their explorations progressed because of using different exploration strategies. S1 used examples to aid his generation of novel conjectures and then generated additional conjectures to test the viability of his conjectures. In contrast, S2 used examples to find patterns involving the number of hits and seldom generated novel conjectures from the examples.

Table 4 shows the mean number of examples' features and interrelationships between or among examples observed by S1 and S2 during their exploration episodes. Example features refer to intrinsic characteristics and properties of tables having specific dimensions. For example, in 2 x 8 table, there are the following features: 8 = 23, the ball landed in bottom-right pocket; there are 5 hits; the ball passed 8 grids; etc. The interrelationships between and among examples refer to similarities and differences of features across two or more examples. S1 had a larger mean number of observed examples' features than S2, but their mean numbers of observed interrelationships are very close. Both S1 and S2 had larger mean numbers of observed examples' features in the primitive goal exploration episodes than they had for their refined goal exploration episodes. For the observations of interrelations, a difference between S1 and S2 was noted. Specifically, S1 had a larger mean number of observed interrelationships in the primitive-goal exploration episodes than he had in his refined goal exploration episodes. However, S2 had a larger mean number of observed interrelationships in the refined-goal exploration episodes than he had for his primitive goal exploration episodes.

Even though S1's exploration strategy was primarily hypothesis-driven and thus appeared to indicate that he was operating at an abstract level of reasoning, his observations of example features did not always prove useful to him in confirming his conjectures. For example, in episode 11, he investigated the following conjecture: Given a table with specified dimensions, if either one of the dimensions of the table is squared and the other remains constant, the ball in the figure with the squared dimension will land in the same pocket as it does in the figure with the given dimensions. He generated tables of dimensions 5 x 2, 25 x 2 and 5 x 4 and found that the ball landed in the same pocket each time. He conjectured that trying other examples of the same kind would have the same result. When he generated another example to confirm his conjecture (i.e., tables of dimensions 4 x 7, 16 x 7, and 4 x 49), he made several feature observations of the examples he generated, the majority of which were unrelated to confirming his conjecture. Below is a part of episode 11 which shows S1 made several irrelevant observations of generated examples' features.

  So, let's try something. What's 4 and 7 look like,..., there we go, 4
  and 7 right here (generates 4x 7 table). Okay, so when this table
  here, started in here, ended in there, then if you notice all these
  are perfectly the same size, right there (points to upper-left
  pocket). And, these triangles right here, excluding the ones on the
  sides. So these two are equal. So, it's basically, looks like a bunch
  of slanted rectangles on the side.

While the solver's observations about the triangles were irrelevant, he continued his activity of checking out his conjecture by generating tables of dimensions 4 x 49 and 16 x 7.

  So, now let's take numbers, were four and seven, I believe right? So,
  it's 4 and 7. So, let's take this number squared. Let's square 7, see
  if you get the same thing. This was, so we're going to take that and
  see which pocket it goes into and then we'll square 4 and see if it
  goes into the same pocket.--So you got 4 and, uh, 49, okay, humongus
  table again (generates 4 x 49). It's going to do this for a little
  while. Actually there's an interesting thing, ignoring the figure,
  notice that when it started out, it was sort of like a frequency, like
  a Sin (x) ray, like in trigonometry? You know this? And since this
  number was different and it didn't hit in the pocket, every time it
  missed the pocket, it sort of phased shift. So, it's like, here's your
  graph, here's where the phase shift is and moved it over versus this.
  So you get that effect. So there might be some ways. Uh, huh. Now,
  where is that ball landing? The way the ball lands in this pocket over
  here (points to upper-left corner). So, this is where it landed. So,
  this is where it landed. So, now let's try this. Let's try 16 and 7.
  See if it's the same as previously. It shouldn't be but let's see
  (generates 16 x 7). So it does. So, there's a definite relationship
  there, it lands on the same side.

By characterizing the exploration strategies of S1 and S2 as hypothesis-driven and data-driven, we do not mean to imply every episode followed these descriptions. For example, it appeared that S1 discovered mathematics #7 by accident.

  So, then, let's see what else can you explore here? Let's try taking
  less than that. If you take less than, if you've got, if you take more
  than half of that away, it's going to hit off the pocket again and
  make some figure based upon, you know, the size of it. Or it's going
  to go, like that or something or it's going to bounce around the
  pockets a little bit. So, if I try 10 and 4 (generates 10 x 4 table).
  It should, yep, see right there, bounce around the pocket, bounce
  around the pocket. So, if you've got ten and four, with this 10 and
  this one's 4 So, let's see here. That would be the same as 2 and the
  5. So, if I take this and shrink the table down, I'll get the same
  figure. Yeah, one is 4, 10. I'll get this same figure. So whenever
  this is 4 and 10, I got this diamond shaped figure, something like
  that. I should get that same figure here. Um, it's just showing that,
  you know, given a certain number. If there is a relationship between
  those at all. Like these, these here, given these numbers, this would
  be the same as saying, this side times two. So this would be the same
  as 2X and 2Y So what I basically did is said that, that's that, then
  I'll divide by 2 and see why X and Y are. And it should be the same
  figure because these are really what I would call the critical
  numbers, that mean something. I mean, you could take these numbers,
  these base numbers and like, as long as you multiply them by any given
  number, both side equally, you should have the same figure. So, let's
  try 5 and 2 (generates 5 x 2 table). There it is so, just to prove a
  point, you can take, uh, this equal 50 multiply it by a side that's
  ten, this equal 20 and you get the same figure only it'll be in bigger
  perspective (generates 50 x 20 table). So there s that.

Similarly, S2's solution activity included episodes where his seemingly random and unorganized approach resulted in him determining novel relationships. For example, his discovery of explored mathematics #7 was made in the transition of primitive-goal phase to the refined-goal phase.

  Um, let's try getting some other patterns. Let's strike up some
  patterns that the program's creating. Um, as we saw we had a, like a
  two by two, it just was straight line. Let's try, like a two by six,
  two by three (generates 2 x 3). It's just, all right, no round, no
  square. Um, why not a two by four (generates 2 x 4 table). Okay, it's
  interesting that if you has a, uh, an even number and an odd number,
  the ball always goes backwards. You know, it'll go right, then to the
  left. Well, if, like we had four by six, it will see [backward]
  (generates 4 x 6 table). Yes, it does go back. Oh, that was because in
  the two by three, this is a multiple of that, so we get the same
  pattern, just on a larger scale. So we get the same thing if we had
  like, um, eight by 12 (generates 8 x 12 table). We'd get the exact
  same pattern. Yeah, just more, same number of hits, same everything.
  So that's kind of interesting.

Discussion

Since the study was explorative in nature and reported on the mathematical actions of only two students, we need to exercise great care in trying to generalize the results to larger student populations. For example, while the Billiard Ball Task provided an environment for students to investigate open-ended problems, it will be interesting to see if other studies that make use of other problem-solving tasks can achieve similar findings. Nevertheless, we believe that the results do extend our knowledge of problem solving in open-ended or ill-structured problem situations.

Transformation of ill-structured into well defined problems

According to Reitman (1964), ill-structured problems differ from well structured problems by the fact that they have open constraints, or problem conditions, that suggest to the solver a wide range of possible interpretations; these open constraints require the solver to formulate (or re-formulate) the problem statement before problem-solving actions can be pursued.

Reitman described the solution of ill-structured problems as a process through which the learner "fills" the open constraints, and thus begins to focus on a particular formulation of the problem from which solutions may be determined. Voss & Post (1988) described the process as a transformation process, wherein solvers proceed from their initial interpretations of the task to transform the problem into terms that are more meaningful to them and from which they can develop appropriate goals of action. Achieving these goals then becomes the focus of the solvers' activity. In this way, ill-structured problems are transformed into well-structured problems.

The results of the study help to clarify this complex transformation. Both S1 and S2 demonstrated activity of a transformational nature in their mathematical explorations. They both were able to begin with initial analyses of simple cases of the problem (primitive goal-driven activity) and progress to more sophisticated analyses of mathematical relationships in their explorations (refined-goal driven). In this way, their initial explorations helped them to reformulate the ill-structured problem in terms that aided their development of subsequent solution activity, and thus made possible their exploration of some sophisticated mathematical relationships. Hence, the solvers' transition from primitive-goal directed to refined-goal directed activity would appear to be a major transition in the transformation process described by Reitman (1964).

Evolution of exploration processes

The results of the study indicate the complexity of the processes that constitute mathematical exploration. Mathematical exploration can be characterized as a recursive process in which solvers determine goals and formulate problems, initiate activities to achieve their goals (and hence, solve their problems), and reflect upon their solution activity to formulate new problems as needed. We hypothesized this process to involve a complex interaction of the solvers' cognitive and metacognitive processes, where processes of problem formulation and problem solving intermingle to provide cognitive "nourishment" to one another, which feed back to the solvers' evolving interpretations of the problem situation. Their revised interpretations may then suggest to them novel possibilities that serve as source material for new problems for them to pose and solve.

The results help extend our knowledge of the ways solvers develop of structure knowledge from their explorations. First, we reported that the solvers demonstrated problem posing-solving chains in their mathematical explorations, a recursive process within which both solvers made frequent use of "what-if" or "what-if-not" approaches to formulate their problems. These results are compatible with results from the literature on how solvers structure and organize their experiences as they develop their mathematical understandings (Kieran & Pirie, 1991) and on how solvers incorporate problem posing strategies into their solution activity (Brown and Walter, 1993; Silver, 1994; Silver and Cai, 1996).

But we may also consider structure in terms of the abstractness of the solvers' mathematical explorations (Hershowitz, Schwarz, & Dreyfus, 2001). According to Hershowitz, Schwarz and Dreyfus (2001), abstraction of one's mathematical activity involves the epistemic actions of constructing, recognizing, and building--with.

Solvers S1 and S2 appeared to demonstrate such constructive activity as they developed their mathematical explorations. We reported that the students pursued different solution routes and demonstrated different exploration strategies in their problem solving and we believe that these differences add to our knowledge about how solvers make conceptual advances in their actions, from primitive to more abstract understandings. Specifically, S1 chose to initiate many trials to reflect on before he began to make significant progress. In contrast, S2 formulated more highly refined goals very early in the interview. In so doing, S2 demonstrated a facility for problem posing that would seem to suggest that he would make more significant conceptual advances in his explorations than student S1. However this did not turn out to be the case. While both students constructed and explored the case of the table's dimensions being increased or decreased proportionally (relationship #7), S1 appeared to extend and evolve his knowledge to consider more abstract situations (e.g., #12, 13, 14) than did S2. It is possible that this difference is related to the fact that S1 adopted a provisional reasoning approach in his hypothesis-driven explorations; as a result, S1 appeared to explore more abstract relationships because he examined a greater variety of general cases than S2. In contrast, S2's use of data-driven exploration strategies appeared to limit somewhat the scope of his explorations; as a result, S2 explored mathematical relationships that were based on low-level inductions of his prior results. Future studies are needed to verify these results.

In explaining some of the qualitative performance differences between S1 and S2, we used the classifications of hypothesis-driven versus data-driven to discuss and explain differences between them in the conceptual advances they made. We believe that these results may be viewed as reflecting different conceptual orientations that problem solvers adopt and sustain in their solution activity. In particular, student S2's use of data-driven strategy indicated that he operated within a fairly closed conceptual framework, always needing concrete examples on which to base his actions and determine new relationships. His novel relationships were seldom much more advanced conceptually than his prior ideas--he always had a sense that he knew where he was going. In contrast, we inferred that student S1's dominant use of a hypothesis-driven strategy was indicative of a more open conceptual orientation, in which S1 saw the usefulness of projecting his ideas into new situations, even if he could not be sure of where the results would lead him. These results would appear to be compatible with the two levels of generalized activity proposed by Krutetski; (1976):

  We shall consider the ability to generalize mathematical material ...
  on two levels: (1) as a person's ability to see something general and
  known to him in what is particular and concrete (subsuming a
  particular case under a known general concept), and (2) the ability to
  see something general and still unknown to him in what is isolated and
  particular (to deduce the general from particular cases, to form a
  concept) (Krutetskii, 1976, pp. 237).

Summary and Future Directions

While we do not view the results as exhaustive, we believe that they help explain how mathematical knowledge evolves in the course of ongoing mathematical activity, when solvers have the opportunity to generate their own problems to solve. While the study was exploratory in nature, we believe that the results contribute to research on mathematical problem solving in the following ways. First, the hypothesis-driven and data-driven strategies that were identified appear to be useful classifications for analyzing the mathematical explorations of solvers. Future studies with more subjects and different problem tasks could determine if there are any other major categories of strategies that differ from these examples, or if there are sub-categories of either of these classifications.

Second, students S1's hypothesis-driven activity suggests an important way that solvers open up their reflective gaze to consider and assess new ideas for their usefulness while in the process of solving problems. In this way, the solver's self-generated actions provided new learning opportunities to extend and deepen his understanding of the formal mathematical relationships underlying the task. This characterization of the solver's problems solving as linked to mathematics learning is consistent with the recommendations of McGuinn & Boote (2003) who found that problem solving that is closely connected to learning contributes to the solvers' development of autonomous problem-solving actions and, hence, should be an instructional goal for mathematics teachers at all levels.

Third, the contrast between the exploration strategies demonstrated by student S1 (hypothesis-driven) and S2 (data-driven) suggests that we need to be careful as we incorporate more open-ended tasks into the mathematics curriculum. While we encourage all mathematics educators to rethink the tasks they give to their students and provide more open-ended problems where appropriate, these results suggest that our focus should be on students' mathematical thinking and learning and on helping them to open up and explore their mathematical situations. Student S1's hypothesis-driven explorations were purposeful and motivated with the goal of developing deeper understandings of the mathematical relationships underlying the task. In contrast, S2's data-driven explorations more closely approximated a random trial-and-error approach. In other words, the mere inclusion of an open-ended task did not stimulate student S2 to undertake mathematically sophisticated explorations; if this were a classroom situation, an intervention by the teacher with a helpful hint might have helped S2 to consider some appropriate new problems to explore that would extend and deepen his understanding of the task.

Our future work in this area will focus on testing the hypothesis about the processes of mathematical exploration from a wider context, especially in the context of classroom. While the current study explained how solvers' cognitive actions can inform their evolving understanding, enabling them to actively seek out new problems to solve, more explanation is needed concerning the ways their cognitive actions influence their strategic and managerial actions. In particular, studies are needed to characterize the complex interaction of the solvers' cognitive and metacognitive processes in mathematical exploration.

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Jinfa Cai, University of Delaware

Vic Cifarelli, University of North Carolina at Charlotte

(1) Several distinctive correlation variables include the height of the tables (HT), the width of the tables (WT), the number of hits (NH), the location of the final pocket (FP), the number of squares passed through by the ball on its path (NS), and the number of regions formed by the trace of the path of the ball (NR).

Table 1 Explored Mathematics

                                                                  S1  S2

 1) In perfect square tables, the ball will land in upper-right   X   X
    comer
 2) It is impossible to keep the ball from not going to any           X
    pocket
 3) The figure in a table with odd numbers as dimensions is more  X
    complex than that with even numbers as dimensions
 4) The width and the height of a table determines the figure     X
 5) A relationship between the width and the height of a table    X
    determines the figure
 6) If width = 1/2 height, or height = 1/2 width in a table, the  X
    ball will hit directly the dead center of the opposite side
    and land in the far pocket on the same side
 7) Given a table dimensions, if you increase or decrease the     X   X
    dimensions in the same proportion, the figure with the given
    dimensions is the same as that with increased or decreased
    table dimensions except for scale
 8) In the cases of a table odd number by 12 tables, the balls    X
    all land in the same pocket
 9) Given a table's dimensions, if you square either one of the   X
    dimensions and keep the other constant, the ball in the
    figure with squared dimensions will land in the same pocket
    as it does in the figure with the given dimensions
10) There is a pattern for number of hits in tables of dimension      X
    (odd number) by 12
11) There is a pattern for number of hits in tables of dimension      X
    (odd number) by 13
12) When one of the dimensions is 1, it is a specific case of     X
    relationship 7)
13) In 1 by n tables, the number of hits = 1 + n                      X
14) In n by (n+1) tables, the number of hits = 2n + 1             X
15) Given a table dimensions, if you cube either one of the       X
    dimensions, keep the other one constant, the ball in the
    figure with cubed dimensions will land in the same pocket as
    it does in the figure with the given dimensions
16) For some tables with specific dimensions such as 6 by 4 and       X
    6 by 8, the number of hits
17) Given a table dimensions, if you reduce them into two             X
    mutually prime numbers, then the number of the hits = sum of
    the two prime numbers
18) In a figure with n and n-1 dimensions, there is not a         X
    certain rule to predict the pocket in which the ball will
    land.
Total:                                                            12  8

Table 2 Some Results for S1

          Goal                 Posed                   Example
Episodes  Type       Examples  Questions  Conjectures  Features

 1        Primitive   1         1          0             4
 2        Primitive   1         2          1             1
 3        Primitive   1         1          0             4
 4        Primitive   1         1          0            15
 5        Primitive   1         1          2             4
 6        Primitive   1         1          0             8
 7        Primitive   3         2          1             8
 8        Primitive   1         0          0             5
 9        Refined     3         0          4            12
10        Refined     3         1          4             4
11        Refined     1         2          2             8
12        Refined    11         5          4            28
13        Refined     2         1          1             3
14        Refined     3         0          2             4
15        Refined     2         0          2             4
Total                35        18         23           112

          Number of
Episodes  Relationships

 1         0
 2         0
 3         3
 4         2
 5        10
 6         3
 7         5
 8         0
 9         3
10         4
11         0
12         7
13         0
14         0
15         6
Total     43

Table 3 Some Results for S2

          Goal                 Posed                   Example
Episodes  Type       Examples  Questions  Conjectures  Features

 1        Primitive   2         0          1            7
 2        Primitive   2         1          0            4
 3        Primitive   3         2          2            5
 4        Refined     4         1          2            6
 5        Refined     9         6          3            9
 6        Refined    12         1          2           10
 7        Refined     4         0          2            4
 8        Refined     5         3          2            6
 9        Refined     4         3          2            5
10        Refined    10         1          3           11
11        Refined     7         1          2            6
12        Refined     1         0          2            1
13        Refined     5         2          3           21
14        Refined     3         1          1            3
Total                71        22         27           98

          Number of
Episodes  Relationships

 1         0
 2         1
 3         3
 4         5
 5        15
 6         3
 7         1
 8         1
 9        14
10        20
11         5
12        11
13         8
14         3
Total     90

Table 4 Mean Number of Example Features and Interrelationships Observed
by S1 and S2

    Mean Number of Examples  Mean Number of Inter-relationships
    PGE  RGE  AAE            PGE  RGE  AAE

S1  4.9  2.5  3.2            2.3  0.8  1.2
S2  2.0  1.3  1.4            0.7  1.4  1.3

PGE: Mean number of example features observed in primitive-goal episodes
RGE: Mean number of example features observed in refined-goal episodes
AAE: Mean number of example features observed in across all episodes
PGE: Mean number of interrelationships observed in primitive-goal
episodes
RGE: Mean number of interrelationships observed in refined-goal episodes
AAE: Mean number of interrelationships observed in across all episodes