A survey was conducted to investigate the emotions, attitudes, and beliefs of 5^th- and 6^th-graders in the context of mathematical problem solving. The study was carried out in elementary schools in Cyprus using a 34-item questionnaire and a sample size of 277 students. Findings revealed high levels of self-efficacy beliefs, pleasure, and control, and low levels of phobia and anxiety during problem solving tasks. Well-known conceptions on mathematical problem solving, such as “only clever students can solve mathematical problems”, “learning mathematics is mostly memorization” and “a mathematical problem must be solved in 10 minutes or less otherwise it cannot be solved” were confirmed. The participants on average expressed disagreement to the commonly held belief that “mathematics problems have only one solution”. The subscales measuring emotions and attitudes – self-efficacy, phobia-anxiety, pleasure, control and problem type preferences – were run through factor analysis. All five subscales were all positively correlated with each other to a medium to high degree. No gender or grade differences on the subscales were identified apart from a difference on self-efficacy in favor of 5^th-graders. This study demonstrates some descriptive findings about various affective factors, which in turn influence mathematical problem solving. We propose that other methodologies must be employed in future studies on this topic.

Emotions, attitudes, and beliefs are components of the affective domain and are critical for the understanding of students’ behavior in mathematics.

Emotions are considered an individual’s reactions to events, agents and objects. Their particular nature is determined by the way the individual understands the situation causing them. Attitudes refer to a person’s tendency to react consistently toward certain events, people or institutions, objects or curriculum subjects positively or negatively (Philippou & Christou, 2001). A student’s previous experience in mathematics may lead to, for example, a stressful or an enthusiastic approach toward the subject. Schoenfeld (1985) defined a personal system of beliefs for mathematics as the way one perceives the world of mathematics: the set of factors that influence the behavior of a person toward him-/herself, the environment, and the subject matter. These beliefs have some bearing on mathematical problem solving (MPS) performance (Lerch, 2004; Silver, 1985).

MPS is an important activity and purpose in mathematics teaching and learning; students’ MPS proficiency reflects their knowledge of mathematics and level of creative and critical thinking. Research has substantiated that affective variables such as emotions, beliefs and attitudes have a powerful influence on student behavior in MPS (Schoenfeld, 1992).

The purpose of this paper is to examine ways in which the affective domain relates to MPS. In particular, we pose the following research questions: What are the descriptive characteristics of the emotions, beliefs and attitudes in MPS of 5^th and 6^th graders based on their questionnaire responses and what are the relationships between these variables? Are there any group differences between males and females and between 5^th- and 6^th-graders on their responses on emotions, attitudes, and beliefs related to MPS?

 

We collected data from 277 5^th- and 6^th-grade students through a 34-item questionnaire. The sample consisted of 161 boys and 116 girls; 126 students were 5^th-grades and 151 6^th-graders from urban, suburban and rural schools from the western region of Cyprus. 

Most of the questionnaire 5-point, Likert-type items were used in previous research on the affective domain and MPS. We added items related to the beliefs of students based on previous research (Schoenfeld, 1992; Silver, 1985). The questionnaire included statements on: phobia - anxiety (7 items), students’ happiness and pleasure in MPS (4), self-efficacy beliefs (5), control in MPS (5) attitudes in “routine” and “non routine” types of tasks (6), beliefs related to MPS (7).

According to students’ responses they feel happy when solving mathematical problems and have a rather high level of self-efficacy beliefs on such tasks (a mean higher than 3 is interpreted as a positive reaction to the statements; Table 1). On average, they don’t feel stress or phobia when solving mathematical problems. They believe they can control MPS procedures; they are in the position to notice when their efforts are productive or not during MPS. Lastly, they do not seem to have an apparent preference toward routine versus non-routine tasks, since the mean on the “problem type” category is very close to 3.

Items on beliefs were examined separately because of their diverse content. Students held deep-seated convictions such as “only clever students can solve mathematical problems”, “learning mathematics is mostly memorization” and “a mathematical problem must be solved in 10 minutes or less otherwise it cannot be solved” (Lerch, 2004; Schoenfeld, 1992; Silver, 1985). In contrast to other research findings, the students in the sample expressed disagreement to the statement “mathematical problems have only one solution”.

The questionnaire had a high level of internal consistency (alpha=0.89). Excluding the beliefs items, we conducted factor analysis to examine the structure of the questionnaire. The factors were slightly revised according to the factor analysis results. As expected, most of items grouped in five factors, which could be interpreted as self–efficacy, phobia-anxiety, pleasure, control and problem type.

            Pearson correlation coefficients between the five factors ranged from 0.331 to 0.716 (Table 1). Notably high coefficients (> 0.6) were found between self-efficacy beliefs, pleasure, and control. Students who are confident in their ability to tackle and solve mathematical problems feel comfortable and in control in MPS activities.

Independent samples t-tests were carried out to compare gender and grade groups (Table 2). The only significant difference in all comparisons was the higher score of 5^th-graders on MPS self-efficacy compared to 6^th-graders; the latter seemed to be more pragmatic about their MPS proficiency.

Research has established the importance of the affective domain in the efficiency of students in MPS and has focused further into the study of specific categories of emotions, attitudes and beliefs and how they relate to students’ MPS behavior. Our paper describes a survey of student affective reactions to statements on MPS and reports results on specific subscales of emotions, attitudes, and beliefs. Findings revealed high levels of self-efficacy, pleasure, and control, and low levels of phobia toward problem solving tasks. Common beliefs on problem solving were confirmed with one exception (“mathematics problems have only one solution”). We found that there were medium or high positive relationships between the responses to the various subscales, but overall there were no significant differences between gender and grade groups.

There are limitations to the interpretations of self-responses to survey questionnaires. In a follow-up study we complement these findings with results from observations and interviews with students on the same research questions. Multiple methodological approaches are needed in the study of the affective domain to examine the multifaceted nature of the topic and to triangulate the findings.

 

Lerch, C. M. (2004). Control decisions and personal beliefs: their effect on solving mathematical problems. Journal of Mathematical Behavior,23,21-36.

Philippou, G., & Christou, K. (2001). Affective factors and Mathematics learning (in Greek). Athens:Atrapos.

Schoenfeld, A. H. (1992). Learning and think mathematically: Problem-solving, metacognition, and sense making in mathematics. In D.A.Grows(Ed.), Handbook of research in mathematics teaching and learning,(pp.334-368). New York:Macmillan.

Silver, E. A. (1985). Research on Teaching Mathematical Problem Solving: Some Underrepresented Themes and Needed Directions. In E.A.Silver(Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives,(pp.247-266). Hillsdale:Lawrence Erlbaum.