MENTAL PROJECTIONS IN MATHEMATICAL PROBLEM SOLVING: THE ROLES PLAYED BY ABDUCTIVE INFERENCE AND SCHEMES OF ACTION IN THE EVOLUTION OF MATHEMATICAL KNOWLEDGE

Victor V. Cifarelli

Department of Mathematics

The University of North Carolina at Charlotte

vvcifare@email.uncc.edu

Combining aspects of Piaget’s scheme theory and Peirce’ theory of abduction, this paper examines the novel problem solving actions of a college student. The analysis documents and explains the important role of abductive inference in the solver’s novel solution activity.

Abstraction and abduction describe creative processes in mathematical problem solving. The solver’s ability to abstract mathematical relationships from their problem solving actions enables them to create mathematically powerful ideas (Schoenfeld, 1985). The process of abduction, as described by Charles Saunders Peirce, wherein explanatory hypotheses are generated and tested, enables solvers to reflect on and scrutinize their potential solution activity and make conjectures about its usefulness (Anderson, 1995; Cifarelli, 1998; Fann, 1970; Mason 1995).

The purpose of this paper is to demonstrate how a focus on the abductive reasoning activities of solvers enhances and extends contemporary constructivist analyses of problem solving. The first part of the paper provides a brief overview of the theories of Peirce and Piaget, focusing on how each explains the construction of new knowledge as involving acts of problem solving. The second part of the paper focuses on a Piagetian study of problem solving previously conducted by the author. Through re-examination of selected episodes of solution activity, the revised analysis demonstrates the prominent role that abduction plays in problem solving activity and shows how the Peircean analysis enhances and extends the Piagetian analysis.

Using Peircian and Piagetian Perspectives to Study Problem Solving

Peirce and Piaget each had high regard for problem solving activity and its role in the evolution of knowledge. Peirce’s focus on the importance of logical reasoning that individuals use to explain unexpected or surprising facts, and Piaget’s focus on how a learner’s thinking proceeds in the face of cognitive perturbation, suggests they each saw a fundamental connection between problem solving and learning: an individual solving a problem engages in learning activity and has constructed new knowledge.

Both Peirce and Piaget viewed the construction of new knowledge as involving dynamic, creative activity (Table 1). Peirce identified abduction (the generation of plausible hypotheses to account for surprising facts) as the process that introduces new ideas into the reasoner’s actions.

Table 1: Problem Solving-Based Explanations of New Knowledge

	Peirce	Piaget
Constructs	abductive inference	reflective abstraction
key processes	hypothesis generation and testing: plausible hypotheses are self-generated and tested to explain surprising results	through resolution of perturbations, learners construct and reconstruct their knowledge at increasingly abstract levels
new knowledge	hypotheses evolve through intertwining of induction, deduction, and further abductions	learners develop structure in their problem solving activity (schemes of action)
growth of awareness	explanatory hypotheses	anticipation

In contrast to Peirce’s views, Piaget maintained that learners organize their sensori-temporal actions into mental structures, or schemes, which can be evoked to aid the learner’s interpretive acts when problems are encountered. Schemes become operative as they are generalized and extended. Piaget explained the development of new knowledge in terms of reflective abstraction as the primary process that explains the re‑organization of action as schemes are revised.

While the differences above involve different emphases regarding what counts as meaningful problem solving, they also indicate that Peirce and Piaget held different ideas about how the learner mentally projects their ideas through time and space. The following sections analyze the solution activity of a college student named Marie. By examining critical junctures of her solution activity using a Peircian lens, the revised analysis will help clarify these differences.

Table 2: Sample of Algebra Word Problems Used in the Study

TASK 1: Solve the Two Lakes Problem

The surface of Clear Lake is 35 feet above the surface of Blue Lake. Clear Lake is twice as deep as Blue Lake. The bottom of Clear Lake is 12 feet above the bottom of Blue Lake. How deep are the two lakes?

TASK 2: Solve a Similar Problem Which Contains Superfluous Information

The northern edge of the city of Brownsburg is 200 miles north of the northern edge of Greenville. The distance between the southern edges is 218 miles. Greenville is three times as long, north to south as Brownsburg. A line drawn due north through the city center of Greenville falls 10 miles east of the city center of Brownsburg. How many miles in length is each city, north to south?

TASK 3: Solve a Similar Problem Which Contains Insufficient Information

An oil storage drum is mounted on a stand. A water storage drum is mounted on a stand that is 8 feet taller than the oil drum stand. The water level is 15 feet above the oil level. What is the depth of the oil in the drum? Of the water?

TASK 4: Solve a Similar Problem In Which the Question is Omitted

An office building and an adjacent hotel each have a mirrored glass facade on the upper portions. The hotel is 50 feet shorter than the office building. The bottom of the glass facade on the hotel extends 15 feet below the bottom of the facade on the office building. The height of the facade on the office building is twice that on the hotel.

TASK 9: Make Up a Problem Which has a Similar Solution Method

Marie’s Problem Solving

Marie was interviewed as she solved a set of nine similar algebra word problems (Table 2). These tasks were designed by Yackel (1984) to induce problematic situations across a range of similar mathematical situations.

The Piagetian analysis of Marie’s activity is summarized as follows. She was inferred to have constructed a conceptual structure while solving Task 1. While solving Tasks 2-9, Marie’s sense of problem “sameness” (Lobato, 1996) evolved to the extent that she could begin to reflect on and anticipate results of potential solution activity prior to carrying it out with paper‑and‑pencil. This development of her solution activity was interpreted as she having constructed a conceptual scheme that enabled her to see each successive task as “the same” and act accordingly to solve the problems (Figure 1). Her growing awareness of the efficacy of her solution activity was characterized as increasing levels of abstraction of the scheme (Table 3).

Figure 1: Marie’s Evolving Scheme

Tasks 1-2		Tasks 3-9
Solves the target task and follow-up		Solves variations of original task
	Emerging Structure
	evolving awareness of solution activity
Primitive Structure		Abstract Structure
Solver needs to carry out solution activity with paper and pencil		Solver can reflect on potential activity and “see” results

Table 3: Marie’s Solution Activity as Levels of Abstraction

Level of Activity	Characterization	Examples
Abstraction	Solver can coordinate potential actions and “run through” potential solution activity in thought and operate on its results.	Solver can “see” or anticipate results of potential activity (and draw inferences) prior to carrying out solution activity with paper-and-pencil.
Re-Presentation	Solver can coordinate prior actions and “run through” prior solution activity in thought.	Solver can “see” or anticipate potential difficulties in new problem situations.
Perceptual Expression	Solver uses diagram to aid reflection.	Solver can reason from diagrams to anticipate potential problems.
Recognition	Solver sees the relevance of using previously constructed solution activity to solve new problems	Solver recognizes usefulness of diagrammatic analysis used to solve Task 1 to solve Tasks 2-9

A Revised Analysis of Marie’s Solution Activity

Marie’s solution activity while solving Tasks 1 & 2 will be summarized. Her subsequent solution activity in Tasks 3 & 4 will illustrate and explain the gradual generation of novelty into her evolving solution activity in terms of abductive reasoning.

Marie’s solution to Task 1. Marie’s solution activity for Task 1 was by no means routine. She initially interpreted the task about the two lakes as an “algebra word problem in two variables” and generated a system of several equations, no two of which were consistent (Figure 2). When she realized that this approach did not lead to a solution, she constructed a side-by-side diagram of the lakes, and translated relevant lengths from the diagram to a vertical axis, which served as a reference aid in constructing relationships. This solution activity eventually led to a correct solution (Figure 3).

After solving Task 1, Marie interpreted Task 2 as similar, remarking that “the first thing that strikes me is that this problem is a lot like the first one” and constructing a diagram similar to that she constructed to solve Task 1 (Figure 4). While her anticipation indicated some sense of similarity between solution of the current task and Task 1, it did not allow her to see and address the potentially problematic information. It was only after she carried out her solution activity that she realized the potentially problematic information:

Marie: This information seems to have nothing to do with the problem. So, I’ll just consider all of the other relationships first.

Marie went on to construct a correct solution for Task 2. Marie attempted to solve Task 3 in the same way as she solved Tasks 1 & 2. However, she soon found herself faced with a problematic situation she had not anticipated, which galvanized her with a sense of excitement and wonderment, as she looked to explain the new problem she now faced (Figure 5):

Marie: I am going to draw a picture. Here is my oil stand. And we have a water storage 8 feet taller. And here’s level water. And here’s the oil level. (reflection) So, solve it ... the same way. (She smiles, then displays a facial expression suggesting sudden puzzlement) Impossible!! It strikes me suddenly that there might not be enough information to solve this problem. I suspect I’m going to need to know the height of one of these things (points to containers). I don’t know though, I am going to go over, all the way through.

Marie’s anticipation that “the same way” would not work was followed by her abduction that the problem might not contain enough information, later refined to the hypothesis that she needed more information about the heights of the unknowns. While her hypothesis contained uncertainty, it helped to organize and structure her subsequent solution activity as she explored and tested its plausibility as an explanatory device. She spent much time pursuing the elusive information and finally concluded that the problem could not be solved. 

Marie’s solution activity while solving Task 3 suggested a qualitatively different level of inquiry than she demonstrated while solving Tasks 1 and 2. While she could temporarily suspend conditions of the problem to solve Task 2 (by ignoring the extra information), Marie’s sudden experience of surprise while solving Task 3 fueled her desire to generate and entertain novel explanations that were radically different from her prior sense-making actions; this indicated a major opening-up of her conceptual boundaries. She ventured to explore her ideas and convictions with a sense of open-endedness, free in the sense that she no longer was constrained by the conventions she previously operated within. The crucial point here is that her abductive actions opened up conceptual boundaries for potential action, and did not merely suspend the constraining conditions that constituted the problem (as was the case in Task 2).

A second indication that the solver had transformed her actions to a new level of inquiry was the shift in her reflective orientation, whereby she began to formulate goals for action in terms drawing from potential states of the problem. This change of orientation came out of her need to explain a result in “present time” for which there was no room for explanation given her current understandings. With her abduction she generated plausible explanations within the world of future events and imagined action, thereby forging her deliberation over future events that ultimately served to constrain her current actions. She organized her sense‑making actions in terms of future events (specific actions concerning the problem conditions she needed to perform in order to verify that a solution was possible) which then beckoned back to her to make them real. This drawing from the future to chart a present course of action helped the solver make‑sense of her current problem and paved the way for her to pose new problems.

In what ways did Marie’s abductions help to evolve her solution activity while solving later tasks? A partial answer to this question is that she became more cautious in her activity, spending increased time reflecting on her potential activity. However, her reflections on potential solution activity continued to exhibit hypothetical qualities that led to novel conjectures. For example, while solving Task 4, Marie quickly noticed the omission of a question from the problem statement yet was able to hypothesize potential problems for her to solve from the information.

Marie: There’s no question! (Long reflection here) ... The things they could ask for are things like ... (HYPOTHESIS) ... the height of one of the buildings but ... (ANTICIPATION) ... there's not enough information to get that....  The only thing we have information about is ... (HYPOTHESIS) ... Ah, the relative heights of the two facades. So, if I were ... if somebody wanted me to solve any problem, that's probably what they're asking for. 

Marie’s hypothetical statements about potential problems that could be solved were provisional in the sense that they lent themselves to further scrutiny, and plausible since, based on her current understandings, these were problems that could conceivably be solved. Marie’s anticipation following her first hypothesis indicated she had deduced from her hypothesis the necessary conditions of the problem, and had performed a mental “run through” of the imagined action of trying to solve the problem, the result of which she rejected her hypothesis. Similarly, she explored the plausibility of her second hypothesis, concluding that it made more sense to her that the problem of finding the heights of the two facades was a problem that could be solved.

Unlike her solution activity in Tasks 2-3, where her solution activity involved making explicit comparison to Task 1, here Marie employed hypothetical states of new and future problems in initiating her solution activity. Her anticipations were now connected to specific hypotheses. The solver demonstrated this highly abstract activity prior to constructing her diagrams. She constructed a solution to the problem, utilizing diagrams to construct relationships, in much the same way as she solved earlier tasks (Figure 6).

Marie: Okay. Let’s see if there is anything here that will at least give me information. Okay, the hotel is 50 feet shorter than the office building. So we have distance here which is 50. The facade of the hotel extends 15 feet below the facade of the office building. That distance would be 15. The height of the facade on the office building is twice that on the hotel. (Long reflection here) So I call this distance X, this distance here is 2X. All right! And then I can say that X minus ... I’m trying to find a relationship between these two. And I know that ... X minus 15 plus 50 is going to equal 2X. So, 35 equals one X. So that would indicate that the facade on the hotel is 35 feet. On the office building is 70 feet.

Discussion

Abductions as Motivating Orientations for Future Actions. As Marie elaborated and extrapolated her hypotheses, her reflective scope widened in the sense that she could ‘see’ among many options for action and determine those which aided her progress towards solving her problem. In this way, those future events beckoned to Marie for her to actualize specific trials, the results of which provided feedback for her evolving hypotheses. This reflective phenomena of formulating explanations in terms of future action is an aspect of abductive reasoning that involves the ability to coordinate images of action with one’s evolving goals and purposes. The philosopher Bertrand de Jouvenel explained how images of action are projected and ‘stored’ into the future:

“Our actions seek to validate appealing images and invalidate repugnant images. But where do we store these images? For example, I “see myself” visiting China, yet I know I have never been there …. There is not room for the image in the past or present, but there is room for it in the future. Time future is the domain able to receive as “possibles” those representations which elsewhere would be “false”. And from the future in which we now place them, these possibles “beckon” to us to make them real.” (de Jouvenel, 1967, p. 27)

The Role of Anticipations. The original Piagetian analysis explained Marie’s growth of awareness of in terms of the process of anticipation: “anticipation is nothing other than application of the scheme to a new situation before it actually happens” (Piaget, 1971, p. 195). As Marie solved variations of the original problem, she developed awareness of the structure of her solution activity, enabling her to anticipate results of potential solution activity prior to carrying it out with paper‑and‑pencil.

By considering aspects of Marie’s solution activity in terms of abductive reasoning, her anticipations were seen to be connected to her evolving hypotheses, and hence took on greater impact -- they were constituted within Marie’s hypothesis‑elaborating and hypothesis‑testing activities, and thus helped her confer degrees of clarification and certainty in her on‑going reasoning. In this way, problem solving for Marie was less about resolving problematic situations by revising her current scheme, but more about making her hypotheses work for her.

References

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