Mathematical discovery: a covariance analysis.

Abstract

A covariance analysis of constructs related to the process of mathematical discovery is presented. The constructs are derived from Polya's ideas on problem solving and include sophistication, traits that predict affect, and emotion. The relative importance of these constructs is investigated in specific problem solving situations. How the model is refined and validated is discussed. The analysis provides a foundation for exploring the nature of mathematical thinking during problem solving.

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The importance of affect and emotion during mathematical discovery and problem solving has long been recognized (Dreger & Aiken, 1957; Higbee & Thomas, 1999). Emotion can organize, focus, disrupt, distract or energize problem solving (Abella & Heslin, 1989), and the influence of emotion can be immediate or delayed (Lazarus, 1991). Further, emotion has a representational function in problem solving (Bebellis & Goldin, 1997) and forms enduring affective pathways that contribute to individuals' mathematical power (Goldin, 2000).

   If the student had no opportunity in school to familiarize himself
   with the varying emotions of the struggle for the solution, his
   mathematical education failed in the most vital point. (Polya, 1985,
   p. 94).

This paper develops a model of the process of mathematical discovery. The model is built from Polya's ideas on mathematical discovery and problem solving, and augmented with Bloom's (1956) concept of sophistication, Anderson's (1981) traits that predict affect, and Mandler's (1989) theory of emotion during problem solving. After identifying variables involved in the mathematical discovery process, the model is refined and validated to provide a foundation for exploring the nature of mathematical thinking during problem solving (Schoenfeld, 2000).

In influential work on the cognitive activities that occur during mathematical problem solving and discovery, Polya (1981, 1985) describes ongoing cognitive evaluations that lead to emotion. Polya contends that every step in problem solving is accompanied by the partially correlated evaluations of relevancy, proximity, and quality. These evaluations are judgements about the problem solving process and performance. Relevancy is an estimate of either how important the problem is to the problem solver, or how important a particular idea or guess is in obtaining the solution. Proximity is an estimate of how close or far away the solution is. Described by Polya, but not specifically named as such, quality is an estimate of how good an idea or guess is, or how good the problem solving performance is.

Polya claims that more sophisticated people experience more differentiated emotion during problem solving. Polya does not define sophistication but in the context of problem solving, Polya is probably referring to individuals who have greater experience and expertise in problem solving, who perform better at problem solving, and who have more acute evaluations of their problem solving process (Bloom, 1956). Thus, sophistication can be defined and measured in terms of problem solving performance and in terms of variables that predict affective behavior.

Anderson (1981) identifies five traits correlated with emotion which influence learning and testing. The five traits are academic motivation, academic self-esteem, mathematics anxiety, interests in mathematics, and locus of control. The importance of these traits are tested. Polya's model of mathematical discovery is purely cognitive, yet several emotion theories contend that emotion has a physiological component (Lazarus, 1991; Mandler, 1989). Polya's model is easily extended to include a physiological component using Mandler's (1975, 1989) theory of emotion during problem solving. Central to Mandler's theory is that emotion arises from interruptions of thoughts. An ongoing evaluation process interprets an interruption as positive or negative depending on the incongruity between what was expected and the perception of the actual event. Mandler (1989) shows that physiological activity is a direct consequence of incongruity and interruption. Mandler claims that emotion tone (whether agreeable or disagreeable) depends on evaluations whereas emotion intensity depends on the degree of physiological activity. Thus, cognitive interruptions during problem solving can lead directly to emotion.

Validation of the Mathematical Discovery Model

To investigate the extent to which the variables described above are related to mathematical discovery, data were collected on constructs that measure Polya's evaluations, Bloom's sophistication, Anderson's affect, and Mandler's interruptions and physiology.

Subjects

The subjects participating in this experiment attended a public university in the Northeast and were representative of undergraduates at public universities across the country. The first and third quartiles of SAT/ACT scores of students attending the university were 950 and 1150 respectively. Further, 17% were minorities, 13% were from other states, 3% were international, and 18% were more than 25 years old. Two-hundred-nine subjects participated in the study. The sample size was selected to be consistent with common guidelines for factor stability. Each subject was available for one hour.

Research Instruments

Three instruments developed by the authors were used to measure evaluations, affect traits, emotion, and problem solving expertise, and sophistication. The Emotion Questionnaire (Allen & Carifio, 1999a) is a 32 item semantic differential. Emotion and affect traits were measured by five to eight item subscales that were highly reliable with internal consistency estimates between .84 and .96. The Mathematics Affect Trait Questionnaire (Allen & Carifio, 1999b) measured traits that influence learning, testing, and affect as identified by Anderson (1981). Each trait was measured by a 5-item subscale with internal consistency estimates between .27 and .80. Because these subscales contained high unique-variance items, they were concurrently valid measures. The Math Problem Set consisted of one easy problem and one difficult problem. These two "Polya Problems" (Allen & Carifio, 1999c) were randomly selected from a validated set of twenty-four problems. One problem was a traditional algebra word problem. The second problem was a novel, unconventional problem designed to challenge students' problem solving skills.

Procedure

Subjects met in a classroom in groups of 25 for one hour. The experimental procedures were read to the subjects who then spent fifteen minutes answering the Mathematics Affect Trait Questionnaire. Students then spent thirty five minutes trying to solve two math problems in a randomly presented order. To sample moment-to-moment changes in evaluation and emotion, subjects completed a total of six Emotion Questionnaires while trying to solve the two problems. Each questionnaire took about two minutes to complete. In all, 152 students worked on the two problems and completed all items on the Math Affect Trait Questionnaire and six Emotion Questionnaires. Solutions to the two problems were assigned correctness scores by the first-named author using a holistic scoring rubric. Prior to analyzing the data, the responses were sorted into high, medium, and low score-level groups.

Construct Validation

To test Polya's claim that emotion in problem solving arises from three types of evaluation, responses to the Emotion Questionnaire were factor analyzed. With an eigenvalue cutoff set at 1.0, the N=1082 items clustered into exactly three factors that corresponded very closely to the evaluations proposed by Polya. Factor I reflected the quality of one's work in solving a problem and explained 55% of the item variance. Factor II reflected how relevant the problem was to the problem solver and explained 8% of the variance. Factor III reflected the perceived proximity to the problem's solution and explained 6 percent item variance. Polya's conjectures about three types of evaluation during problem solving were confirmed by the empirical factor structure. The three factors, however, were not of equal importance. The quality factor contributed almost 7 times as much in explaining the total variance as did the relevancy factor and over 9 times as much as the proximity factor contributed.

As with conjectures and prototypical theories, factors are typically identified but not quantitatively weighted relative to their strength or "percentage of variance" they explain. Polya's conjectures and model are no different. The point of this empirical research, however, was to not only confirm or deny Polya's model but also to weight the factors in the model as has been done here. Although quality is the major factor by an order of 9, proximity (6%) and relevancy (8%) also account for as much variance as, for example, method in the typical learning experience. Moreover, the 14% of variance accounted for by proximity and relevancy combined is more than many trait variables contribute in learning and performance experiments. A nonorthogonal factoring of evaluation, perceived physiology, and emotion data revealed that quality and proximity are closely aligned with emotion. Physiological activity and relevancy appeared as separate factors but accounted for only 4 and 3 percent of the total variance respectively.

Problem Correctness, Evaluation, and Emotion

High scoring subjects had emotion scores on the order of one standard deviation above their low scoring counterparts. Moreover, perceived physiology values were found to increase as math interest and academic self-esteem increased, and as mathematics anxiety decreased. An examination of means revealed that subjects with low problem correctness score totals bad slightly negative evaluations and emotion, and mid-scale physiological activity. Those with high problem correctness score totals had positive evaluations and emotion, and high physiological activity. Thus, subjects with both problems essentially correct had more positive evaluations, positive emotion, and more physiological activity than subjects with both problems essentially incorrect. Medium and low scoring subjects had roughly the same subscale responses. Thus, sophistication, as measured by math interest, self esteem, lack of anxiety, and problem correctness, is an important predictor of emotion in problem solving.

Sophistication and Emotion

It was observed that as sophistication increases, all three evaluation variables and emotion tended to increase. The physiology variable, however, showed very little change supporting Polya's decision to not include physiological activity in his model. With increasing sophistication, highly significant increases in the evaluation variables, emotion, and perceived physiology were observed. The analyses showed significant increases in emotion between both low versus medium, and medium versus high trait levels. Similar results were found for all of Polya's evaluation variables. These results support Polya's claim that more sophisticated people have more differentiated emotions.

Evaluations and Emotion

High correlations were found between the evaluation variables and emotion (.73 to .94). Though no causation is established, these results are consistent with Polya's claim that the three forms of evaluation lead to emotion. Regression analysis was used to determine how well relevancy, proximity, quality, and perceived physiology predict emotion. Linear regression equations were found for both the easy and difficult problems that explained 93 percent of the variance in emotion. Further, regression equation coefficients indicated that relevancy and quality are roughly twice as important in Polya's theory as is proximity. To investigate emotion dynamics, a repeated measures MANOVA test with two within-subject factors was used. The two within-subject factors were time-of-measurement (beginning, middle, and end) and problem difficulty (easy and difficult). The analysis showed a significant difference in relevancy, proximity, quality, and emotion (the dependent variables) across time-of-measurement and problem difficulty. Post hoc tests showed that proximity, and quality were influenced by both problem difficulty and time-of-measurement. Further, emotion was influenced by time-of-measurement, and relevancy was influenced by problem difficulty. There were no evaluation or emotion interactions with time and problem difficulty. The following results were also observed. First, evaluations of relevancy are dynamic and ongoing from problem to problem but stable within a given problem. This is consistent with Polya's claims. Second, proximity evaluations change with time and with problem difficulty. This is also consistent with Polya's model. Third, both quality evaluations and emotion change with time but are fairly stable from problem to problem. Again, this is consistent with Polya's model.

Cognition and Emotion

In developing the covariance model of mathematical discovery presented here, one particular set of findings is highly suggestive and may be extraordinarily important. The nonorthogonal factorization of all Emotion Questionnaire items showed that perceived physiology items accounted for only four percent of the total variance in Emotion Questionnaire responses. Because emotion currently is viewed as having both cognitive and physiological ("in the body") components that interact with each other (Mandler, 1989; Lazarus, 1991), this result implies that perceived physiology is a weak contributor to the intensity and quality of emotion reported by subjects. Such a result contradicts both the implicit view of behaviorists and the neurological view of emotion which contend that physiology accounts for "9/10ths" of the variance in emotion and that cognition is a minor "after effect." Implicitly, the cognitive view of emotion (particularly in a therapeutic context) holds that cognition accounts for the major portion of variance in emotion and that physiology is a far less powerful contributor for most people in most contexts. Polya's contention and argument is that more sophisticated people have more intense and more highly differentiated emotions. This argument follows logically as an implication of both the cognitive view and the findings of the factor analysis.

Polya did not explicitly state this view or its logic, and was to some degree trying to account for the qualitative differences in emotion across levels of human development and education. But the cognitive view is inherent in Polya's model and throughout his writings and the results reported here strongly support this view. That cognition may be four or more times as important as physiology in emotion is a finding that needs to be researched further because of its importance and far reaching implications. Not even Mandler, a respected emotion theorist in mainstream academic psychology, has addressed the relative weights and importance of cognition and physiology in his two factor model of emotion. Polya not only implicitly addressed this question but also provides a more differentiated model of cognition in emotion. This differentiated model is supported by the results presented here.

Mathematics Teaching, Learning and Emotion

It is commonly believed that the main effects of emotion in problem solving are disruptive and distracting, and that negative emotions interfere with and diminish performance. These effects and outcomes, however, are not supported by the empirical evidence presented here. Instead, these results clearly indicate that positive emotions energize, organize, focus, and improve performance. Furthermore, negative emotions provide highly valuable problem solving information for problem solvers in general and for sophisticated problem solvers in particular. The results of this study have immediate implications for the wide-spread "feel good" approaches used to teach mathematics during the last decade. In particular, these teaching approaches were misdirected in their over-generalized and undifferentiated views about emotion and its role in problem solving and learning. Cognitive dissonance and emotional conflict, within appropriate limits, have very positive psychological and learning functions and are not inimical factors to be eliminated from the learning process. Allowing students to have both positive and negative emotions as an integral part of mathematics learning is not necessary demeaning nor detrimental. Instead, according to these results, a varied-emotion approach generates valuable meta- cognitive evaluation information as well as stimulating, energizing, organizing, and focusing effects from the concomitant emotion.

The key to success in problem solving, according to the results presented here, comes when students consider the cognitive evaluations generated during problem solving to be personally important and then when they constructively utilize the differentiated emotions that occur as a result. Provided that the importance of cognitive evaluations and concomitant emotions has been taught as part of a general model of problem solving practice, the use of authentic, challenging, and intrinsically relevant problems, where students have a personal stake in the outcome, may be particularly beneficial in helping students become better problem solvers. Lastly, the results indicate that less sophisticated problem solvers would gain valuable problem solving experience from having more sophisticated problem solvers act as role models and peer mentors in cooperative learning situations. This mentoring approach might be most effective, from a long-term benefit perspective, when applied in middle school contexts where students encounter elementary but real Polya Problems for the first time.

References

Allen, B. D. & Carifio, J. (1999a). The Development and Validation of an Emotion Questionnaire for the Investigation of Affect During Mathematical Problem Solving. Springfield, VA: ERIC Document Reproduction Service No. ED434037.

Allen, B. D. & Carifio, J. (1999b). The Development and Validation of a Math Affect Trait Questionnaire for the Investigation of Affect During Mathematical Problem Solving. Springfield, VA: ERIC Document Reproduction Service No. ED434038.

Allen, B. D. & Carifio, J. (1999c). A Problem Set for the Investigation of Mathematical Problem Solving. Springfield, VA: ERIC Document Reproduction Service No. ED434039.

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Bradford D. Allen, Florida Institute of Technology

James Carifio, University of Massachusetts-Lowell

Allen, Ed.D. is Assistant Professor of Mathematical Sciences and Mathematics Education. Carifio, Ph.D. is Professor of Education.