Efficacy beliefs, problem posing, and mathematics achievement
Aristoklis A. Nicolaou
Perceived self-efficacy beliefs have been found to be a strong predictor of mathematical performance while problem posing is considered fundamental in mathematical learning. In this study we examined the relation among efficacy in problem posing, problem-posing ability, and mathematics achievement. Quantitative data were collected from 176 fifth and sixth grade students, and interview data from six students selected on the basis of hierarchical cluster analysis. Students’ perceived efficacy to construct problems was found to be a strong predictor of the respective performance as well as of the general mathematics achievement. A strong correlation was also found between ability in problem posing and general mathematics performance. The students constructed problems of greater variety and complexity on the basis of informal tasks rather than on the basis of formal tasks. Significant differences were found in problem posing ability, between fifth and sixth grade students. The findings provide support to earlier studies indicating the predictive power of context-specific efficacy beliefs. Implications are drawn about strategies for enhancing students’ efficacy beliefs and problem-posing ability.
Theoretical Background and Aims
Research on mathematics teaching and learning has recently focused on affective variables, which were found to play an essential role that influences behavior and learning (Bandura, 1997). The affective domain is a complex structural system consisting of four main components: emotions, attitudes, beliefs, and values (Goldin, 2002). Beliefs can be defined as one’s subjective knowledge, theories, and conceptions and include whatever one considers as true knowledge, although he or she cannot provide convincing evidence to support it (Pehkonen, 2001). Self-beliefs can be described as one’s beliefs regarding personal characteristics and abilities and include dimensions such as self-concept, self-efficacy, and self-esteem. Self-efficacy can be defined as “one’s belief that he/she is able to organize and apply plans in order to achieve a certain task” (Bandura, 1997, p. 3). This study focuses on self-efficacy of primary students with respect to problem posing.
Self-efficacy is a task-specific construct and there is a correspondence between self-efficacy beliefs and the criterial task being assessed; in contrast, self-concept is the sense of ability with respect to more global goals (Pajares, 2000; Bandura, 1986), while self-esteem is a measure of feeling proud about a certain trait, in comparison with others (Klassen, 2004; Bong & Skaalvik, 2003). The task-specificity of efficacy beliefs implies that related studies are more illuminating when they refer to certain tasks, such as problem posing; the predictive power of self-efficacy is in this case maximized (Pajares & Schunk, 2002). On the other hand, the level of specificity could not be unlimited; as Lent and Hackett (1987) have rightly observed, specificity and precision are often purchased at the expense of practical relevance and validity.
The construct self-efficacy is tightly connected to motivation and plays a prominent role in human development since it directly influences behavior. According to Bandura’s social cognitive theory, every individual possess a system that exerts control on his/her thoughts, emotions and actions. Among the various mechanisms of human agency, none is more central or pervasive than self-efficacy beliefs (Bandura & Locke, 2003; Pajares, 2000).
Research on self-efficacy has recently been accumulated providing among other things notable theoretical advances that reinforce the role attributed to this construct in Bandura’s social cognitive theory. Several studies have indicated a strong correlation between mathematics self-efficacy and mathematics achievement (Klassen, 2004). It was further found that mathematics self-efficacy is a good predictor of mathematics performance irrespective of the indicators of performance (Pajares, 1996; Bandura, 1986) and regardless of any other variables (Bandura & Locke, 2003; Pajares & Graham, 1999). It was found that mathematics self-efficacy is a better predictor of mathematics performance than mathematics anxiety, conceptions for the usefulness of mathematics (Pajares & Miller, 1994), prior involvement in mathematics (Pajares & Miller, 1995), mathematics self-concept and previous mathematics performance (Klassen, 2004; Chemers, Hu, & Garcia, 2001). It is noteworthy that self-efficacy beliefs were even found to be a stronger predictor of performance than general mental ability (Pajares & Kranzler, 1995).
Self-efficacy beliefs have already been studied in relation to a lot of aspects of mathematics learning, such as arithmetical operations, problem solving and problem posing. Pajares and Miller (1994) asserted that efficacy in problem solving had a casual effect on students’ performance. Research findings support the view that high achieving in mathematics students have higher and more accurate efficacy beliefs (Chemers et al., 2001; Pajares & Kranzler, 1995; Zimmerman, Bandura, & Martinez-Pons, 1992). Efficacy beliefs towards a certain task are accurate when they correspond to what the person can actually accomplish.
The development of problem posing competency is generally recognized as an important goal of mathematics teaching and learning; it lies at the heart of mathematical activity (Crespo, 2003; English, 1997b; Brown Walter, 1993). Reformed mathematics education adopted the view that knowing mathematics is identified as “doing” mathematics and learning mathematics is equivalent to constructing meaning for oneself and the ability to handle non-routine problems. In this context, problem posing comprises a primary factor that contributes to enhancing students’ ability to solve mathematical problems. Moreover, from a teaching perspective, problem posing reveals much about the understandings, skills and attitudes the problem poser brings to a given situation and thus becomes a powerful assessment tool (English, 1997a). Not surprisingly, reports such as those produced by the National Council of Teachers of Mathematics (NCTM, 1989, 2000) call for increased emphasis on problem posing activities in the mathematics classroom.
Problem posing can be defined as the generation of new problems and mathematical questions, as well as the reformulation of problems within the process of solving a given problem, when a solver restates or recreates a given problem in some way or other to make it more accessible for solution. Many researchers have reported a positive relation among problem posing ability and mathematics achievement (English, 1998; Leung & Silver, 1997) as well as among problem posing and problem solving ability (English, 1998; Silver & Cai, 1996; Cai, 1998); however, Silver and Mamona (1989) found no clear link between problem posing and problem solving abilities in their work with middle school mathematics teachers. Yet, Brown and Walter (1993) argue that problem posing ability is closely linked to the ability of solving problems; Silver and Cai (1993) found a strong positive relationship between middle school students’ problem solving and problem posing abilities and Ellerton (1986) reported results showing a clear link between mathematical competence and ability in problem posing, meaning that mathematically “able” students are also able to generate problems.
English (1998) has found that children were able to pose a broader range and more complex problems on the basis of informal rather than of formal contexts. By informal context she means that formal symbolism is missing, e.g. when the stimulus provided to students is a verbal mathematical situation or a picture. In formal contexts, e.g. posing a problem from a number sentence, children were confined to the basic change, group-part-part-whole problems. Similar findings have been reported about preservice teachers by Philippou, Charalambous, and Christou (2001).
The quality of mathematical problems is determined on the basis of semantic structure and the number of operations involved in the problem i.e., the number and the kind of relations and the number of operations that are required to solve the problem (English, 1998). Types of problems such as compare/equalize, problems that involve rate, proportion, conditional problems are considered to have higher semantic-structural complexity than problems that involve change, grouping and division problems. The diversity and the complexity of the problems are indicators of the quality of the problems posed.
Despite its importance, problem posing has not yet received analogous attention from the mathematics education community. Indeed, we know relatively little about children’s ability to construct their own problems in formal and informal contexts or about the extent to which these abilities are linked to mathematical competence (English, 1998). Silver and Cai (1996) argue that only a few researchers have examined the mathematics problems posed by children and research has tended to rely on small numbers of subjects and to provide a rather superficial analysis of the posed problems. Furthermore, we are aware of no studies investigating the efficacy beliefs of primary school students towards problem posing and the relation among this construct and the ability to generate problems. An understanding of the diversity and complexity of the problems posed by students could serve as an indication of students’ capabilities in solving similar problems and could provide an insight into students’ thought(s). On the other hand, a possible relationship among efficacy and ability in problem posing would enrich our knowledge about the connection among affective and cognitive factors, with obvious implications in teacher education and teaching.
The purpose of this study was to explore the relationships between elementary school students’ efficacy beliefs in problem posing, their problem posing ability, and their achievement in mathematics. Specifically, the aims of the present study were: (a) to measure efficacy in problem posing and ability in problem posing of fifth and sixth grade students and examine for possible differences by grade, (b) to look for possible relationships between any pair of the following variables: efficacy in problem posing, ability in problem posing, and mathematics achievement, and examine whether efficacy in problem posing could predict ability in problem posing and mathematics achievement, (c) to examine the quality of the problems posed by students in each of the four tasks and the relative difficulty met by students in constructing problems from each of those tasks.
We used questionnaire data collected from 176 fifth and sixth grade students and interview data from a sub-sample. The sample was selected on the basis of purposeful cluster sampling; four urban and rural schools from the three major districts of Cyprus were first selected and then a fifth grade class and a sixth grade class were randomly selected from each of those schools. The sample comprised of a total of 87 fifth-grade and 89 sixth-grade students from eight classes.
A four-part questionnaire, measuring efficacy beliefs in problem posing and ability in problem posing was developed on the basis of earlier studies (English, 1997b, 1998; Philippou et al., 2001). The first three parts measured efficacy beliefs about problem posing and the fourth one measured ability in problem posing. Specifically, in the first part, students were asked to read the following four tasks and state their sense of certainty to pose problems based on each of them, without attempting to pose any problem. Each of the four tasks asked students to pose a problem (a) from a stimulus picture, (b) that should end in a specific question, (c) that could be solved by the division 3/4, and (d) from a given number pattern, respectively (see Figure 1). The second part comprised of five cartoon-type pictures and statements explaining the situation presented by each picture; the students were asked to select the picture that best expressed their efficacy in problem posing (see Figure 2). The third part consisted of 14 five-point Likert type items, reflecting efficacy in problem posing (see Table 1). The fourth part consisted of four tasks similar to those in the first part and the students were asked to pose problems based on those tasks (see Table 2).
The questionnaires were piloted on 26 sixth grade students to detect possible weaknesses or shortcomings. After some minor language improvements and a change in the seventh statement of the third part, the questionnaires were administered to the sample subjects by the first author. The students were instructed to proceed to the fourth part only after they had finished the first three parts; they were given 60 minutes to complete the questionnaire.
The Ward’s method of hierarchical cluster analysis was then used on the quantitative data for the selection of subjects for the interviews. Clustering was based on the following variables: ability in problem posing, efficacy in problem posing and complexity of the problems posed. The analysis revealed that students could be clustered into six distinct groups. The six-group solution was selected because the Agglomeration scale showed a fairly large increase in the value of the distance measure from a six-cluster to a five-cluster solution (Norusis, 1993). The means and standard deviations for the aforementioned variables (ability in problem posing, efficacy in problem posing and complexity of the problems posed) were then calculated for each of the six groups (see Table 3).
Table 3 shows that the students of G1 had high efficacy beliefs in problem posing, high ability in problem posing, and constructed problems of higher complexity. [G.sub.2] students showed high ability in problem posing, relatively moderate to high efficacy beliefs in problem posing as well as with respect to complexity of the problems posed. The students in [G.sub.3] and [G.sub.4] had moderate scores in all the three variables, whereas [G.sub.5] had moderate to low scores and [G.sub.6] had the lowest scores in all three constructs. Thus, [G.sub.1] and [G.sub.2] can be considered as the relatively “high score” groups, [G.sub.3] and [G.sub.4] are the “moderate score” and [G.sub.5] and [G.sub.6] are the relatively “low score” groups. From each cluster group a student was randomly selected for interviews, in such a way that the six students came from six different classes and three of them were boys and three were girls.
The interviews were semi-structured encouraging the interviewees to answer the same questions and give explanations and clarifications where necessary. The interviews were conducted by the first author at the child’s school and were tape-recorded. No time limit was set, and the interviewer was prepared to provide clarifications whenever a student seemed unable to understand something of the issue. The interviews focused on nine key issues aimed at eliciting students’ attitudes towards mathematics, efficacy beliefs in problem posing, ability in problem posing from formal and informal tasks and the quality of problems posed in either task. The tasks used were a picture (informal task) and an operation (formal task) (see Interview Guide-Appendix A).
[FIGURE 2 OMITTED]
For the analysis of the interviews, students’ responses were classified according to main issues and compared between them. Moreover, the responses in some questions, (e.g. questions 6 and 7) that examined efficacy in problem posing and ability in problem posing respectively, were compared to the responses on the respective questionnaire items and the profile of the cluster group in which the student belonged.
Appendix A: The Interview Guide
1. How do you like mathematics? You think it is an easy subject? You get high marks?
2. Have you got any experience in making mathematical problems of your own? How often do you spend time on doing this in your class?
3. In the case you have constructed some problems, what was the stimulus? Did your teacher give you instructions and helped in that task? Was there a relevant activity in your mathematics textbook?
4. When facing a task to propose a problem of your own, how did you like the idea? Did you feel comfortable? Was it an easy task?
5. What did you like more: solving or constructing problems? Why?
6. What did you like more, to construct a problem in relation to a stimulus picture or a problem that could be solved by performing an operation, such as the multiplication 34×13? Why? Which of the two do you think is easier?
Do you think you are able to construct problems in relation to the following tasks (tell me how certain do you feel about doing so)? Explain the reasons why yes or why not.
a) Construct a problem in relation to the following picture.
b) Construct a problem that could be solved by the operation 2/3.
7. Now try to really construct a problem in each one of the following cases, and describe me the way you proceed to make it.
(i) in relation to the above picture (provide hints if necessary).
(ii) that could be solved by performing the operation 7×6 (provide hints if necessary).
8. Which of the two tasks was easier for you and why?
9. Which one of the problems you have constructed do you like more and why? (e.g. because it was more complex?)
Appendix B: A sample of the problems posed by students in each task.
“There were 50 balloons in the party. 15 balloons were red and 25 were yellow. How many balloons were blue?” (G1)
“Mrs Maria has her birthday today, she becomes 50 years old. Her brother is 27 years younger and her cousin is 14 years older. What is the age of her brother and what is the age of her cousin?” (G5)
“The woman in the picture is organizing a birthday party. She invited twenty friends but three of them could not come to the party. If each friend brought her two presents, how many presents would she take?” (G3)
“Mr. Nicos has got a rectangular field. He planted 30 lemon trees across one side, with a distance of 2.5m from each other. Across the other side, he planted 20 apple trees, with a distance of 2m from each other. What is the perimeter of the field?” (G2)
“My grandfather has got a rectangular plot of land. Its width is 50m and its length is twice as long. What is the perimeter of the field?” (G6)
“There are two apples to be fairly shared among three children. How many apples will each child get?” (G2)
“Thomas has two chocolate bars and he wants to divide them equally among his three friends. Which portion of the chocolate bar will each of his friends receive?” (G4)
a) 2, 6, 18, 54, 162, 486, 1458
b) On the first day, two persons declared participation in a competition, on the second day six persons joined them, 18 on the third day, 54 on the fourth day, 162 on the fifth day, 486 on the sixth day, and 1458 on the seventh day. If this pattern continues for another three days, how many persons will declare participation altogether?” (G4)
a) 2, 6, 18, 54, 162, 486, 1458, 4374
b) The first day I had two marbles, the second day I had six, the third day 18, the fourth day 54, the fifth day 162, the sixth day 486, the seventh day 1458, and the eighth day 4374. By how many times did the number of marbles increased?” (G6).
The problems constructed by participants in the fourth part of the questionnaire were scored as follows: in the first two tasks, one point was given for each mathematical problem constructed, in the third task two points were given for the construction of a problem and in the fourth task, one point was given for correct completion of the number pattern and one point for constructing a problem. This marking scheme guaranteed equal weight of students’ ability to construct problems from each of the four tasks (a range from zero to two points in each task). The average score in the four tasks determined the ability in problem posing.
The subjects’ achievement in mathematics was drawn on the teachers’ grades for the school year 2002-2003. Though these grades were given on a scale from 0 to 20, in the analysis we used the ordinal place of students by class, to cater for possible variance in the judgment of individual teacher.
To determine the overall measure of efficacy in problem posing, we recoded those items of the third part of the questionnaire that were negatively stated, and then the average score of the statements of the first three parts of the questionnaire was calculated. In other words, the mean value of efficacy beliefs in problem posing was drawn on the basis of three complementary sources.
As regards the quality of a problem, we adopted the criteria proposed by English (1997b, 1998), i.e., the semantic-structure and the operational complexity of the problem. In order to determine semantic-structure complexity, the problems were classified as basic or complex (English, 1997b). As basic problems we consider elementary change situations for addition and substraction, grouping problems for addition, equivalent set problems for multiplication and partitive and quotitive problems for division. We classify compare and equalize problems for addition and subtraction, scalar or multiplicative comparison problems, and cartesian product (combinatorial) as complex problems (English, 1997b); problems involving rate, proportion and conditional situations were also treated as complex. Basic problems were given one point, whereas complex problems were given two points. The number of operations in a problem, which was the second dimension of complexity was taken into account cumulatively; for instance in the case of a two-step problem involving one change and one comparison situation, 1+2=3 points were given, while a three-point problem involving repeated comparison, was assigned 2+2+2 or 3×2=6 points.
Pearson’s r correlation coefficient was used to examine for relationship between efficacy in problem posing and ability in problem posing. Spearman’s p correlation coefficient was used to test for possible connection between efficacy beliefs in problem posing and mathematics achievement. Linear regression analysis was performed to examine the possibility of efficacy in problem posing to predict the ability in problem posing and mathematics achievement. Finally, we used independent samples t-test to examine differences between fifth and sixth grade students regarding efficacy in problem posing and problem posing ability, the quality of the problems posed in the various tasks and check for possible differences among the tasks with respect to the above mentioned variables.
The findings of the study are organized according to the three research questions drawing upon quantitative and qualitative data.
1. Efficacy in problem posing, ability in problem posing, and differences by grade
In general, the students’ efficacy in problem posing was found to be of a satisfactory level. This is evident from the responses of the questionnaire (Mean=3.58 out of a maximum of 5) and the interviews where all the interviewees expressed confidence in their ability to construct problems. On the contrary, the students’ actual ability in problem posing was found to be at a moderate level; the overall mean response was found to be 1.15 out of a maximum of 2. That was in line with children’s responses in the interviews, where four children were able to construct problems in both tasks without any help, while the other two faced great difficulties and the interviewer had to provide considerable hints in both tasks.
An independent samples t-test was performed to examine whether there were significant differences in the level of efficacy beliefs in problem posting and ability in problem posing between fifth and sixth grade students. The analysis showed that sixth graders outperformed the fifth graders in problem posing in three out of the four tasks of the fourth part of the questionnaire, namely in the second (Mea[n.sub.sixth]=1.250, Mea[n.sub.fifth]=0.896, t=2.205, p=0.029), the third (Mea[n.sub.sixth]=1.039, Mea[n.sub.fifth]=0.569, t=3.379, p=0.001) and the fourth task (Mea[n.sub.sixth]=1.273, Mea[n.sub.fifth]=0.856, t=3.673, p=0.000). Sixth grade students were also found to feel more efficacious to pose problems only in the second task of the first part of the questionnaire (Mea[n.sub.sixth]=3.539, Mea[n.sub.fifth]=3.161, t=2.223, p=0.028). No significant differences were found concerning efficacy beliefs in the other problem posing tasks.
2. Efficacy in problem posing, ability in problem posing, and mathematics achievement
A significant correlation was found between efficacy about and ability in problem posing, r=0.48, p=0.001. Similarly, in the interviews, children with high efficacy in problem posing were able to construct problems without any support, whereas low efficacy children either were unable to construct problems or constructed problems after support was provided. For instance, [S.sub.2] a high efficacy student who expressed confidence in his ability to construct mathematical problems that could be solved by the operation 2/3, said:
“I think it is very easy. I feel quite sure. I believe that 2/3 is a
simple operation and I would have no difficulty to construct problems
that could be solved by this operation”.
In line with his confidence, he was later able to construct the following two problems that could be solved by the multiplication 7×6, as required:
P1: “A group of children consists of seven children and each child has six marbles. How many marbles do the children altogether have?”
P2: “Seven pizzas were cut into six pieces each. How many pieces of pizzas were there?”
On the contrary, [S.sub.5] who felt uncertain about her ability to construct problems that could be solved by the operation 2/3, was later on unable to pose any problem that could be solved by the multiplication 7×6. Indeed, she didn’t manage to pose any problem on her own, despite considerable assistance provided by the interviewer.
In order to examine whether efficacy in problem posing could predict ability in problem posing, a linear regression analysis was performed. The analysis showed that efficacy in problem posing was a good predictor of students’ ability in problem posing (R=0.480, F=52.097, p=0.000). This was also confirmed from qualitative analysis; the level of children’s perceived efficacy in problem posing expressed before they tried to pose problems matched well their actual success or failure to construct problems in each of the two tasks. These findings are in line with previous results, where a strong correlation was found between efficacy with respect to a certain task and actual achievement in that task, and also that efficacy predicted actual achievement in that task (Bandura, 1997; Pajares, 1996; Pajares & Miller, 1994). The present results are also in line with earlier findings by Philippou et al. (2001) who found positive correlation among pre-service teachers’ efficacy beliefs in problem posing and their ability in problem posing.
Efficacy in problem posing was positively correlated with general mathematics achievement ([rho]=0.431, p=0.001). Additionally, it was found that efficacy beliefs in problem posing could predict mathematics achievement fairly well (R=0.427, F=38.339, p=0.000). These results are in agreement and confirm the findings of previous studies indicating that mathematics efficacy beliefs were correlated and could predict mathematics achievement (Pajares & Graham, 1999; Zimmerman et al., 1992). However, the correlation between efficacy in problem posing and mathematics achievement was lower than the correlation between efficacy in problem posing and ability in problem posing. Moreover, efficacy beliefs in problem posing could predict mathematics achievement in a lower degree than they could predict problem posing ability. On the other side, a strong positive relationship ([rho]=0.566, p=0.000) was found between the ability in problem posing and mathematics achievement. This finding confirms the findings of previous studies (English, 1998; Leung & Silver, 1997; Ellerton, 1986).
3. Difficulty of the tasks and quality of the problems posed in each task
Dependent samples t-test was used to test for the relative difficulty in problem posing between the four tasks. It was found that constructing a problem from a stimulus picture was the easiest task, followed by posing a problem that should end in a specific question and constructing a problem from a given pattern; the latter two were found to be of the same difficulty. Constructing a problem that could be solved by performing the division operation 2/3 was found to be the most difficult task. Quantitative results agree with previous results of studies in Cyprus (Philippou et al., 2001) and other countries (English, 1998) indicating that students faced more difficulties when asked to pose a problem in formal rather than informal contexts.
The quantitative results, however, did not seem to be in complete agreement with the results of the interviews concerning problem-posing difficulty in formal and informal contexts. Children’s responses in questions 7 and 8 cannot lead to clear conclusions concerning which task (formal or informal) is more difficulty. The children were first asked to construct a problem from a stimulus picture (different from the picture in the questionnaire) and construct a problem and that could be solved by the multiplication 7×6 (instead of the division 2/3), and then judge which of the two tasks was more difficulty. The students’ responses when asked to construct a problem (question 7) indicate that the formal task was harder, but they expressed variable views about the relative difficulty of the tasks (question 8). One student ([S.sub.1]) said both tasks were of equal difficulty, three students ([S.sub.3], [S.sub.4] and [S.sub.6]) perceived the formal task of constructing a problem that could be solved by the multiplication 7×6 as harder, and two students ([S.sub.2], [S.sub.5]) thought that the informal task was more difficult. The students who thought that constructing a problem in an informal task is harder, belonged to groups with relatively high efficacy beliefs, high ability in problem posing and high complexity of the problems posed, as well as from groups with relatively low efficacy beliefs in problem posing, ability in problem posing and complexity of the problems posed. Among the three students that supported the opposite view, two belonged to the “moderate score” groups, whereas one of them belonged to the “low score” group. Hence, the students from the “moderate score” groups found the formal task harder, while we do not have evidence to support which of the two tasks students belonging in the “high score” or the “low score” groups consider as more difficult.
A typical explanation provided by the children who considered problem posing from a stimulus picture as more difficult follows:
[S.sub.2]: “This is because when asked to construct a problem that
could be solved by an operation, the operation gives you the data
whereas the picture does not provide you with any data; you have to
construct a problem entirely on your own. In the case of the
picture, the task is harder, because you are not given any clue, as
in the case of the operation where you are given some data; in the
picture problem there are no data; it takes more thinking.”
An indicative explanation provided by children who considered the multiplication 7×6 more difficult was given by S4:
[S.sub.4]: “This is because I was searching for things in
correspondence to number 7, things in correspondence to number 6 in
order to write a problem. In the case of the picture, some elements
were given and I constructed a problem.”
Most of the problems constructed by the children were one-step (85% of the total) and of low complexity. The low complexity of the problems constructed is revealed by the mean value for complexity (Mean=1.50 with a maximum of 5.23). In general, children were confined to specific kinds of simple problems (basic change, part-part-whole problems). Children constructed problems of a greater variety in the first task, when the stimulus was a picture, than in any of the other three tasks. This is evident from the kinds of one-step, two-step, three-step and multi-step problems constructed in the first task. In particular, children were able to construct more diverse one-step problems in the first task, including a significant percent of compare/equalize and other types of problems. In contrast, the one-step problems posed in the third task (formal context), were confined to only one kind of problems (division allocation problems). Children also constructed more one-step, two-step, three-step, and multi-step problems in the first task than in any of the other three tasks. Quantitative analysis concerning differences between the four tasks as regards the complexity of the problems constructed, showed that children constructed more complex problems in the first task, followed by the second task, the fourth task, while the least complex problems were posed in the third task. The results seem to be in agreement with the findings of previous studies, indicating that children were able to pose more complex problems in formal contexts (English, 1998; Philippou et al., 2001).
The analysis of the interviews, however, has demonstrated somewhat different results as compared to the findings of quantitative analysis, regarding the children’s sense on which task they had constructed more complex problems. Indeed, the children expressed contrasting views on this issue; it seems that at their age, they are not ready to appraise their problems in this respect. Specifically, [S.sub.4] held the view he has constructed problems of roughly the same complexity, [S.sub.5] felt simply unable to make her mind, [S.sub.1] and [S.sub.2] responded that they had constructed more complex problems in the informal task, whereas [S.sub.3] and [S.sub.6] asserted that they had constructed more complex problems in the formal task. The children, who claimed that they constructed more complex problems from the picture-task, explained that the operation-task does not provide opportunities for posing composite problems. Specifically, [S.sub.1] said:
[S.sub.1]: “I believe that I posed more complex problems on the
basis of the picture. This is because the operation 7×6 limits the
possibility of posing a greater diversity of more complex problems.
On the contrary, the picture-task provides much more chances; you
can construct a problem that could be solved by any operation you
want e.g. 4×3, 2/2 etc.”
On the other side, the children considering that more complex problems can be constructed in formal tasks support their opinion mentioning that posing problems in formal tasks demand more time to think. It is noteworthy that after a conversation with the researcher, [S.sub.2] admitted that the content of a task (what the picture presents, what is the operation and the numbers involved) and not only the kind of the task affects complexity.
Finally, the results of the questionnaire are highly reliable. There is a high positive correlation between answers in statements with a similar content and a negative moderate correlation between statements with contradictory content. Gronbach’s Alpha coefficient was also found to be high (a=0.818). The validity of the questionnaire (criterion related validity) is satisfactory, since in most cases children’s answers in the questionnaire were consistent with their answers in the interviews.
Given that ability in problem posing is influenced by related previous experiences, the problem posing performance of the participants in this study could be considered as satisfactory. According to the interviews, the subjects’ prior experience in problem posing was limited; they were given this opportunity to work on problem posing tasks only a few times each month. Comparing the level of students’ perceived efficacy in problem posing and actual performance in problem posing, we conclude that students have in general overestimated their competence. This finding is in agreement with the results of previous studies that revealed a tendency of the majority of students to be overoptimistic about their abilities to undertake a certain mathematical task (Pajares & Miller, 1994). This gap among the level of efficacy beliefs and the actual ability in problem posing can be attributed to social and cultural factors that encourage this tendency among Cypriot students. One of the regular tendencies of Cypriot parents is to let encourage their offspring to believe that they are capable to accomplish much more than they actually can.
The question concerns the magnitude of this residual. Bandura (1986) argued that one’s successful functioning with respect to a certain task is best served by reasonable accurate efficacy appraisals, although the most functional efficacy judgements are those that slightly exceed what one can actually accomplish, for this overestimation may serve a motive to increase effort and persistence. This overconfidence should not be excessive; one should guard against putting the bar too high and face the danger of ultimate disappointments. On the other hand, unrealistically low math efficacy perceptions, rather then the lack of capability or skill, may in part be accountable for avoidance of math related courses and failure in a mathematics task (Chemers et al., 2001; Pajares, 1996).
Efficacy in problem posing was found to be positively correlated with and a good predictor of the ability in problem posing. This is in agreement and reinforces the results of other studies that lead to the same conclusions (Bandura, 1997; Pajares & Miller, 1995; Pajares, 1996). The findings of the present study are also consistent with the results reported by Philippou et. al. (2001) in Cyprus; they have found that prospective teachers’ efficacy beliefs in problem posing were significantly related to their problem posing ability.
The construct efficacy in problem posing was also positively linked and a good predictor of mathematics achievement; it could predict mathematics performance fairly well. These findings again confirm similar results reported in other studies (e.g., Klassen, 2004; Bandura & Locke, 2003; Pajares & Graham, 1999). Both the correlation between efficacy in problem posing and ability in problem posing as well as the predictive power of efficacy in problem posing and ability in problem posing were higher than the correlation between the respective link among efficacy in problem posing and mathematics achievement. These results seem to justify Bandura’s demand (1986) for specificity, claiming that efficacy beliefs with a respect to a certain task are strongly correlated and are best predictors of achievement at the same task. The aforementioned results are also consistent with Pajares & Miller (1994) position that efficacy beliefs should be measured with respect to a certain task in order to better predict achievement in that task.
The significant positive relationship that was found between the ability in problem posing and mathematics achievement could have been expected. It is consistent with the findings reported in previous studies (Leung & Silver, 1997; English, 1998). Specifically, English (1998) found that students with high achievement in mathematics were better able to generate problems, and Ellerton (1986) argued that more able students in mathematics constructed more complex problems than the less able students.
The differences found in favor of sixth grade students support the developmental nature of these competencies. It can be interpreted as the outcome of physical maturation, supported by additional experience and teaching, and further practicing in both general mathematics and problem posing.
Quantitative analysis showed that children faced more difficulties to construct problems in formal rather than in informal contexts, a result consistent with the results of other studies (English, 1998; Philippou et al., 2001), which can be attributed to the disconnection between students’ informal, intuitive mathematical knowledge and formal “school math.” The lack of connection between school mathematics and real life experiences prevents children from recognizing formal symbolisms as representing verbal mathematical problems in formal tasks. On the contrary, it could be argued that informal tasks provide more opportunities for advanced problem posing than formal tasks (Silver & Cai, 1996; English, 1998).
Constructing a problem that should end in a given question and constructing a problem from a given number pattern were found to be of about the same difficulty; somewhere between the less difficult task of constructing a problem from a stimulus picture and the more difficult task of constructing a problem that could be solved by the division 2/3. It seems that both tasks lie somewhere between formal and informal tasks, each providing students with able opportunities to think and construct problems. For example, students in the second task can construct problems that involve figures, concepts such as length, area and end up with a specific question.
Students constructed a greater variety and more complex problems in the first task (informal context) than in the other three tasks. These findings are in line with the results of previous studies (English, 1998; Philippou et al., 2001). For instance, English (1998) in exploring the complexity of the problems posed by 8-year-olds found out that in informal contexts, children posed a great variety of problems including the more complex equalize/compare problems, while in formal contexts children were confined to a limited range of the less complex group/change problems. Similarly, pre-service teachers constructed a greater variety of problems in informal than in formal contexts (Philippou et al., 2001).
Quantitative results are not in complete agreement with qualitative results concerning the problem-posing difficulty and the variety and complexity of the problems posed in formal and informal contexts. The differences can be attributed to the small number of interviews that prevented drawing solid conclusions, and probably even more to the dependence of the findings from the interviews on the individual characteristics of the subjects. For example [S.sub.6], a boy that comes from the “low score” group systematically presented higher efficacy beliefs and problem posing ability compared to the profile of the group he belonged. In this particular case that might well be due to the fact that girls in general tend to understate their perceived efficacy compared to boys; in this case gender may be responsible for the unexpected results (Seegers & Boekarts, 1996; Philippou et al., 2001). Further research with a bigger sample might lead to more secure conclusions. In a future study it might be profitable to develop a questionnaire with more problem posing tasks, of a wider variety and more interviews; it should be noticed that the content of the task also affects the complexity of the problems posed. This is in accord with Wiest’s (2002) view as regards problem solving ability based on her findings that problem context aspects such as readability, verbal structure, and story concepts affect problem solving ability. It seems, however, that we still know little about the cognitive processes that take place during problem posing activity and the reasons why students face more difficulties in certain tasks.
The results of the present study reinforce the importance of efficacy beliefs in mathematics. Bandura (1997) argued that the uncertainty for one’s capability can overflow the results of his/her efforts, even if he/she had the best competence in a certain task. Thus, teachers must work towards the development of efficacy beliefs and must pay attention to students’ self perceptions with respect to certain tasks, since these beliefs can be an indicator of future achievement in the specific task (Bandura & Locke, 2003). Hackett and Betz (1989) asserted that teachers should pay as much attention to students’ perceptions of competence in mathematics as to actual competence. Therefore it is argued that one’s behavior is more influenced by his/her beliefs that by his/her knowledge or ability.
Given that efficacy beliefs towards problem posing can predict ability in problem posing, teachers must develop ways to enhance efficacy beliefs, particularly in the case they are relatively low and do not match the child’s ability in problem posing. Efficacy beliefs in problem posing can be developed by providing children with activities in which they can succeed. Certainly, this does not imply that all problem posing activities should be easy, but that it would be better if teaching starts with easier activities (problem posing in informal contexts) gradually inserting more difficult activities, such as problem posing in formal contexts.
In addition, efficacy in problem posing is very important due to the emphasis that is lately attributed to problem posing. Problem posing importance is shown by its positive link to mathematics achievement. Philippou et. al (2001) mentioned that pre-service teachers valued problem posing as the ultimate goal of mathematics learning. Learning how to construct mathematical tasks is considered one of the challenges of learning and teaching mathematics (Crespo, 2003). Students’ moderate ability in problem posing and the current limited opportunities given to children to get involved in problem posing activities, make the need for further inclusion of more and varied such activities in teachers’ repertorie imperative. Moreover, according to Lowrie (2002) since most problem solving experiences presented in classroom contexts do not have connections to real-world experiences, it is understandable that students tend to pose traditional problems that simply are variation of those found in textbooks (Hernandez & Socas, 2001). Thus teachers have to offer children a great variety of problems to solve; problems that provide young children with the opportunity to engage in more diverse and flexible thinking and the re-examination of the problem posing tasks cited in mathematics textbooks is also necessary; a greater diversity of problem posing tasks is required.
In conclusion, the findings of the present study suggest that developing efficacy beliefs in problem posing should be an integral part of mathematics teaching and learning. It has been verified once again, this time with primary students, that efficacy beliefs constitute an important component of motivation and behavior; the correlations found among the efficacy in problem posing, ability in problem posing and mathematics achievement suggest a possible focus for further research.
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Aristoklis A. Nicolaou & George N. Philippou
Department of Education University of Cyprus, Nicosia
Figure 1: Efficacy in problem posing
The following task refers to your perceived ability to construct
problems in various situations. Please indicate your degree of certainty
on a scale from one to five, “1-not at all certain, 5-very much certain”
(DO NOT TRY TO CONSTRUCT ANY PROBLEMS NOW).
1. Construct two different mathematical problems in 1 2 3 4 5
relation to the following picture.
2. Construct two problems that should end with the 1 2 3 4 5
following question: “What is the area of the field?”
3. Construct a problem that could be solved by 1 2 3 4 5
performing the operation 3/4.
4. Construct a mathematical problem from the following 1 2 3 4 5
number pattern: 2, 4, 8, 16, 32, 64, 128 …
1 = not at all certain, 2 = rather certain, 3 = quite certain 4 = much
certain, 5 = very much certain
Table 1: Efficacy questionnaire
The statements that follow refer to the construction of mathematical
problems and the numbers correspond to the level of agreement, 1-not at
all certain, 5-very much certain. In each statement circle the number
that reflects better your own beliefs.
1. I can easily construct mathematical problems. 1 2 3 4 5
2. I believe that I am one of the best students of my 1 2 3 4 5
class in mathematical problem posing.
3. My teacher believes that I am able to construct 1 2 3 4 5
4. I do not feel comfortable when asked to pose a 1 2 3 4 5
5. I believe that I can pose problems without any 1 2 3 4 5
6. Posing problems is more difficult than solving 1 2 3 4 5
7. In the case I cannot construct a mathematical 1 2 3 4 5
problem in 5 minutes, I give up.
8. I think that the construction of mathematical 1 2 3 4 5
problems is a difficult task.
9. I prefer solving than constructing mathematical 1 2 3 4 5
10. I believe that someone who can easily solve problems 1 2 3 4 5
is able to construct problems as well.
11. I found it very difficult when asked to construct 1 2 3 4 5
12. I need much assistance in order to construct 1 2 3 4 5
13. Someone who is good in mathematics should easily 1 2 3 4 5
construct mathematical problems.
14. Someone who is good in constructing problems is also 1 2 3 4 5
able to solve problems.
1 = not at all certain, 2 = rather certain, 3 = quite certain, 4 = much
certain, 5 = very much certain
Table 2: The problem posing tasks
In this part you must think and construct problems following the
1. Construct two different mathematical problems in relation to the
following picture (the picture was the same as in the first part of
2. Construct two problems that should end with the following question:
“What is the perimeter of the field?”
3. Construct a problem that could be solved by performing the operation
4. You are presented with the following numbers in the specified order:
2, 6, 18, 54, 162,… There is a constant relation between any two
successive numbers, i.e. 2 is related to 6 in the same manner that 6
is related to 18.
a) Continue the number pattern.
b) Construct a mathematical problem from the following number pattern:
2, 6, 18, 54, 162,…
Table 3: Means and standard deviations for the cluster groups in
efficacy beliefs in problem posing, ability in problem posing and
complexity of the problems posed.
in problem Ability in problem Complexity of the
posing (1) posing (2) problems posed (3)
Groups [bar.X] SD [bar.X] SD [bar.X] SD
Group 1 (N=20) 4.033(H) 0.510 1.515(H) 0.394 2.250(H) 1.274
Group 2 (N=31) 3.843(MH) 0.554 1.561(H) 0.399 2.006(MH) 0.724
Group 3 (N=27) 3.620(M) 0.435 1.127(M) 0.416 1.546(M) 0.897
Group 4 (N=35) 3.583(M) 0.496 1.246(M) 0.503 1.563(M) 0.792
Group 5 (N=36) 3.438(ML) 0.646 0.896(ML) 0.525 1.074(ML) 0.835
Group 6 (N=25) 3.061(L) 0.517 0.635(L) 0.535 0.818(L) 0.655
(1) Maximum score = 5, Minimum = 1, (2) Maximum score = 2, Minimum = 0,
(3) Maximum score = 5.23, Minimum = 0, L: Relatively low score, ML:
Relatively moderate-low score, M: Moderate score, MH: Relatively
moderate-high score, H: Relatively high score, [G.sub.1] (H, H, H),[G.sub.2] (MH, H, MH), [G.sub.3] (M, M, M), G4 (M, M, M), [G.sub.5] (ML,
ML, ML), [G.sub.6] (L, L, L)
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