Metacognition and Meta-Affect in Young Students: Does It Make a Difference in Mathematical Problem Solving?
by Meirav Tzohar-Rozen & Bracha Kramarski - 2017
Mathematical problem solving is one of the most valuable aspects of mathematics education and the most difficult for elementary school students. Cognitive and metacognitive difficulties in this area cause students to develop negative attitudes and emotions as affective reactions, hampering their efforts and achievements. These metacognitive and meta-affective reactions are fundamental aspects of self-regulated learning (SRL), a non-innate process that requires systematic, explicit student training. This study investigated the impact of two self-regulation programs among young students (Grade 5)—metacognition (n = 64) and meta-affect (n = 54) versus a control group (n =53)—on enhancing achievements in mathematical verbal problem solving and a novel transfer task, as well as metacognitive and meta-affective regulation processes of a focus group during a thinking-aloud solution. Mixed methods indicate that students who participated in the metacognitive and meta-affective intervention programs presented similar but higher achievements than the control group. Additionally, during the thinking-aloud solution, students from each group broadly implemented the self-regulation processes they were trained in, while consistently referring to all the self-regulation phases. The current study makes an important contribution to practical implications for students with diverse abilities.
Research studies have indicated that students experience difficulties in solving mathematical problems (Organisation for Economic Co-operation and Development [OECD], 2014; Verschaffel, Greer, & De Corte, 2000). These difficulties are manifested in understanding math-problem texts, perceiving alternative ways of solving mathematical problems, and expressing confidence when calculating and verifying solutions (Desoete, Roeyers, & De Clercq, 2003; Schoenfeld, 1992). Moreover, according to certain researchers (OECD, 2014; Schoenfeld, 1992), these difficulties are sometimes caused by inactivation of knowledge, arising from a lack of metacognitive skills that are required to control, monitor, and reflect on the solution processes rather than from lack of mathematical knowledge.
A lack of metacognition causes students to develop negative emotions, such as severe anxiety, frustration, and despair toward mathematics, resulting in inhibition of learning and achievement (Efklides, 2011; Kramarski, Weiss, & Kololshi-Minsker, 2010). According to meta-analysis studies, the formation of negative emotions and attitudes might hold far-reaching consequences, such as avoidance of mathematics in adulthood (Hembree, 1990). These symptoms appear as early as elementary school and reach their peak in the fifth and sixth grades (Pekrun, Goetz, Frenzel, Petra, & Perry, 2011).
To cope with these difficulties, metacognitive intervention programs in mathematics, mostly for the adult student, have been developed (Kramarski & Mevarech, 2003; Mevarech & Kramarski, 1997, 2014). We believe there is significant importance in providing an answer to the young students metacognitive and meta-affective difficulties and in allowing him or her to better cope with verbal problems. Self-regulation in learning is known to be one of the more effective ways to cope with metacognitive difficulties and negative feelings.
SELF-REGULATED LEARNING (SRL)
Self-regulation in learning is perceived as critical for meaningful learning and lifelong success (Boekaerts, 1999; OECD, 2014). Through the use of SRL, students can better determine their own learning goals by attempting to monitor, regulate, and control them while guided by the goals and contextual features of the learning environment (Pintrich, 2000). In theory, SRL is structured on both metacognitive and meta-affective regulation processes (Zimmerman, 2000).
As a foundational construct of self-regulation, metacognition enables students to plan and allocate learning resources, monitor their own knowledge and skill levels, and evaluate their personal learning levels at different phases throughout the learning process (Pintrich, 2000; Zimmerman, 2000). Planning required choosing appropriate strategies and allocating resources that affect performance; monitoring is the students awareness of their own understanding and task performance quality; and evaluation estimates the outcomes and processes of SRL. The purpose of metacognitive regulation is to increase learning competence by providing students with systematic and explicit guidance throughout the process of thinking and reflection on their tasks (Schraw, Crippen, & Hartlay, 2006). Over the years, programs providing metacognitive support for solving mathematical problems were mostly developed for the adult student (Mevarech & Kramarski, 2014; Schoenfeld, 1992). Prior research has indicated that providing students with metacognitive support for solving these problems may empower their achievements and improve their self-regulation skills.
Throughout the course of learning, students experience achievement emotions derived from achievement activities or achievement outcomes (Pekrun et al., 2011(. Achievement emotions can be grouped by standards and degree of implied activation. In terms of standards, positive emotions such as pleasant enjoyment can be distinguished from negative emotions such as unpleasant anxiety. These emotions are critically important for students learning and performance (Pekrun, Goetz, Daniels, Stupnisky, & Perry, 2010; Schutz & Pekrun, 2007), particularly in problem solving (Gross, Sheppes, & Urry, 2011; Isen, 2001). To effectively deal with negative emotions, it is important that the student identify the negative emotion by the meta-affect process. Meta-affect relates to the thinking about affect and uses this information to guide thought and action. Following the emotion identification (meta-affect), it is important that the student know how to cope with the negative emotion. This process is known as affect regulation, a relatively new domain of research (Gross, 2013) that examines the manner in which emotions are regulated, either before or after they occur (DMello, Strain, Olney, & Graesser, 2013).
Once students experience an emotion they are aware of, the question of its regulation arises. The assumed goal of emotion regulation is to downregulate negative emotions while upregulating positive ones. To cope with negative emotions, researchers suggested developing affect regulation processes among students. Explicit affective regulation refers to students conscious monitoring of their own affective responses and conscious decisions, and its effect on their thinking and learning (Gross, 2013; Kramarski et al., 2010; Malmivuori, 2006).
Despite these recommendations, the majority of studies dealing with the emotional regulation aspect focused mainly on the relationship between positive and negative emotions and on spontaneous and achievement self-regulation processes (Efklides, 2011; DMello et al., 2013), and almost no meta-affective regulation programs for mathematics have been developed. On the other hand, SRL programs mostly focus on metacognitive regulation (e.g., Kramarski & Mevarech, 2003; Schoenfeld, 1992).
The need for these two regulation aspects (metacognition and meta-affect) in mathematics poses new challenges for self-regulation programs, specifically during problem solving by young students, who cope with difficulties in understanding the task, adapting strategies for solving the task, and elevating their confidence when calculating and verifying solutions (Kramarski et al., 2010). Although most self-regulation programs in mathematics relate to metacognitive aspects, only several programs address affective needs during mathematical problem solving (Mevarech & Kramarski, 2014). Therefore, a comparison of the two kinds of programs (metacognition and meta-affect) among young students is crucial.
THEORETICAL FOUNDATION OF THE INTERVENTION
The theoretical foundation of the intervention program in the current study was based on central elements of Pintrichs model (2000), adapted to the young student. Pintrich (2000) described a four-phase model of self-regulated learning. Forethought, planning, and activationStudents create plans for their learning sessions and activate relevant knowledge (hereinafter, planning). At this phase, we addressed three elements: goal setting, prior content knowledge activation, and metacognitive knowledge or emotion activation. MonitoringStudents monitor various aspects of the self, the task, and other contextual conditions. Judgments made during the active monitoring processes in the second phase are relayed onto the third phase, control. ControlStudents attempt to regulate aspects of the self, the task, or the context that is perceived to hinder the progress of learning goals. Our study adjoined these two phases that will be referred to hereinafter as monitoring and control. Reaction and reflectionStudents engage in reaction and reflection on the self, the task, and the context (hereinafter, reflection).
Pintrich stated that although these four phases suggest a time-ordered sequence, students can simultaneously engage in monitoring, control, and reaction during a learning task. The information gathered from these three phases can be used to update goals and plans created during the first phase. Each of the phases refers to both the metacognitive and meta-affective aspects in self-regulation processes. The details of the phases and primary elements according to Pintrichs model are presented in Table 1.
Table 1. Self-Regulation Phases (Planning, Monitoring and Control, and Reflection), Elements and Self Question
We focused on this model from several reasons: (a) this model is circular and dynamic, allowing the student to develop throughout the learning process, and (b) the model refers explicitly to both metacognitive and meta-affective aspects and includes several phases in each one: planning, monitoring, and reflection. Based on this, structurally uniform intervention programs have been developed for each aspect and adapted to the young student.
Besides Pintrichs theoretical framework, the interventions focused on two key factors, explicit training and asking self-questions, identified as effective in metacognitive regulation intervention. These factors were extended in the current study to include meta-affect regulation.
EXPLICIT TRAINING AND ASKING SELF-QUESTIONS
In explicit training, the teacher explains the learning strategy to the students, stressing its meaning and importance (Veenman, Van Hout-Wolters, & Afflerbach, 2006). Learning explicit strategies is most effective for students (Perels, Gürtler, & Schmitz, 2005), given that the skills required for self-regulation are not innate and cannot be acquired naturally (Dignath & Büttner, 2008; Gyurak, Gross, & Etkin, 2011; Kramarski et al., 2010; Perels et al., 2005). In the current study, students were exposed to explicit learning in metacognitive regulation and meta-affective regulation, in accordance with the self-regulation processes of Pintrichs model (see the method section). This process was conducted by self-questioning.
Research in metacognitive self-questioning and mathematical problem solving suggested that self-questioning techniques might improve students achievements (King, 1992; Mevarech & Karmarski, 1997, 2014; Schoenfeld, 1992). The self-questioning exercise was meant to increase students awareness of their own understanding by encouraging them to practice thinking of questions to ask themselves before, during, and after problem solving. Self-questioning permits reappraisal also in affective-regulation, during which the student analyzes his or her feelings regarding the situation, consciously coping with his or her feelings, thus changing his or her emotional reaction from negative to positive (Webb, Miles, & Sheeran, 2012).
In the current study, the self-questioning strategy was extended to the meta-affect component of self-regulation and applied to mathematical problem solving in two intervention program groups. One program was oriented by metacognitive regulation and delivered to the MC (metacognitive) group and the other by meta-affect regulation and delivered to the MA (meta-affective) group. Both approaches were based on self-questions and oriented toward Pintrichs (2000) cyclical model for young students, in the context of mathematical problem solving, while examining its efficacy in comparison with a control group.
The study addressed the following two questions:
Q1. Which group (MC or MA), if any, would be more efficient in improving mathematical problem solving compared with a control group?
Q2. Would the gains of both groups (MC and MA) in problem-solving practice in class be transferred to a novel task that demands new skills?
Finally, because the literature that discusses suitable assessment methods for self-regulation perceives it as a dynamic and complex process within an authentic setting (Azevedo, 2015; Zimmerman, 2008), the current study included additional process analysis and case illustrations to examine MC and MA sequential pattern data in 2 participants (one from each group), resulting from their thinking aloud novel problem solving. Comparing the process data shed light on patterns of MC versus MA self-regulation.
The sample comprised 170 fifth graders (boys and girls) who attended six middle socioeconomic schools that were randomly selected from one district and randomly assigned to each group: MC group (n = 63), MA group (n = 54), and control group (n = 53).
The study included two stages with regard to the sample: the macro stage170 students who participated in the various intervention programs and were examined in problem solving and on the novel transfer task; and the focus stage—30 students from the overall sample (10 students randomly selected from each group) who solved a mathematical problem aloud. Preintervention testing found no significant differences between groups in terms of gender: χ² = 1.72, df = 2, p > 0.05, or mathematical and linguistic level, F(6,310) = .86, p > .05.
MC, MA SELF-REGULATION GROUP INTERVENTION VERSUS CONTROL GROUP
Pintrichs (2000) theory of self-regulated learning adapted for the young student embedded with self-question prompts provided the theoretical framework for two interventions that compared the metacognitive and meta-affective components of SRL. To adapt the model to the young student, we have divided it into three stages: planning, monitoring and control, and reflection.
The two interventions and the control group were implemented by six math teachers (two teachers for each group) in their scheduled lessons on verbal problem solving. The teachers of all three groups were similar in their mathematical professional expertise. The length and structure of the intervention were identical in all groups. Following are the differences and similarities between groups during the intervention.
Stage A: Teachers Guidance Meeting (3 Hours)
The research assistance guided all math teachers (separately, in accordance with the research group) in the following topics. The teachers of all three study groups were guided on deeper understanding of serial problems (from the Ministry of Education collection) and their characteristics. The teachers from the MC and MA groups were also guided on the phases of self-regulation (planning, monitoring, and reflection) with their elements and the use of self-questioning prompts for metacognition/ meta-affect activation (respectively for MC and MA groups) according to Table 1.
Stage B: Intervention Program (1-Hour Session Twice a Week for 5 Weeks, Total of 10 Hours)
Teachers introduced the students to the self-regulation model in its metacognitive/meta-affective aspect (for MC and MA groups, respectively) and discussed its importance for the learning process. The teachers exposed the students to raising metacognitive/meta-affective questions (for MC and MA, respectively) for the problem-solving process (see Table 1); students solved serial problems while applying metacognitive/meta-affective regulation processes (for MC and MA groups, respectively). At the end of each session, a discussion was held regarding the contribution of metacognitive/meta-affective regulation (respectively). In the control group, teachers introduced the students to the problem-solving unit and discussed the solution process.
To keep on the fidelity of the intervention, an assistant researcher (blind to the aims of the study) observed every second of the lesson and provided feedback to the teacher according to the goals of the study group.
A mixed methods approach was usedquantitative and qualitative analysesfor assessing research variables.
Verbal Problem Solving (n = 170)
Two problems were processed from the Grade 5 standardized exam developed by the Ministry of Education (2004) and administered before and after the intervention (after the numbers and context were changed). Each question consisted of two sections; each section required a response and an explanation of the solution path (strategy).
Scoring: A score of 1 was given for a correct answer and a score of 0 for an incorrect answer. For evaluating the scoring of the solution strategy of the student, an indicator consisting of three grades (2-0) was constructed. A higher score indicates a higher level of the strategy. An interjudgment reliability (0.95) between two researchers was found.
Novel Transfer Task (n = 170)
After the intervention, students were asked to solve a novel transfer problem based on Kramarski and Mevarechs study (2003). This problem included a graph interpretation and required high cognitive reasoning to make comparisons, provide explanations, and draw conclusionsskills that were not required previously. Scoring: A score of 1 was given for a correct answer, and a score of 0 was given for an incorrect answer. For scoring the students explanations, an indicator consisting of three grades (2-0) was constructed. The higher the score, the higher the precision level of the explanation. An interjudgment reliability (0.95) between two researchers was found.
Thinking Aloud Protocols
The week after the intervention and the verbal problem solving tests, a focus group of 30 students (10 from each group) was selected randomly from all three groups. Each student solved a verbal problem of similar level to the transfer task, only in a different context. The goals of the aloud solution were to compare between the frequencies of students (n = 30) using SRL phases and incidence of its elements in different groups.
The thinking aloud protocols were transcribed, and two experts discussed them to reach a full agreement on the SRL phases and their elements for each group according to Strauss and Corbins (1990) approach. Both experts were highly knowledgeable in self-regulation processes and were not aware of the goals of the study. An interjudgment reliability (0.91) between the two experts was found.
Verbal Problem Solving (n = 170)
A one-way analysis of variance (ANOVA) in the problem-solving achievements before the intervention indicated no significant difference between the groups, F(2,167) = .78, p < .05. Furthermore, a 3 × 2 ANOVA analysis with repeated measures to examine the differences between the measurement before and after the intervention indicated a significant time effect between the measurement before and after the intervention, F(1,168) = .18.92 p < .001, η2 = .79. In addition, significant interaction of group × time was found F(2,164) = 12.38, p < .001, η2 = .13 (see Table 2; Figure 1).
Figure 1. Means of the Mathematical Problem Solving Test by Time and Group
Table 2. Means, Standard Deviations, and F Values of the Problem Solving by Time and Group
*** p < .001
Pairwise comparison by Bonferroni analyses for each group separately indicated significant differences in all groups. However, the MC group, F(2,167) = 14.1, p < .001, η2 = .38, and the MA group, F(1,162) = .219.82, p < .001, η2 = .57, exhibited a greater difference than the control group, F(1,162) = .71.22, p < .001, η2 = .31. Finally, no significant differences were found between the MC and MA groups (Figure 1).
Novel Transfer Task (n = 170)
A one-way ANOVA indicated a significant difference between all three groups, F(2,167) = 26.11, p < .001, η2 = .24. A follow-up Bonferroni analysis indicated that the achievements of the MC (M = 1.91, SD = .99) and MA (M = 1.56, SD = .86) groups were greater than those of the control group (M = .83, SD = .42). However, no significant difference was found between the MC and MA groups.
SELF-REGULATION PROCESSES DURING AN ALOUD SOLUTION FOCUS GROUP (n = 30)
First, we examined the frequency between the metacognitive and meta-affective elements without differentiating between the phases of self-regulation (planning, monitoring and control, reflection) in each of the study groups. We found that the MC group used 74.7% (68 elements) from the total metacognitive elements used by the three groups but did not use any meta-affective elements at all. The MA group alone used 60 (100%) meta-affective elements and 16.6% (15 elements) of the metacognitive elements even though this group was not given a specific intervention regarding the metacognitive component. The control group made the least use of metacognitive elements: 8.7% (eight elements). To deepen the self-regulation analysis, we analyzed the metacognitive and meta-affective regulation elements in each study group separately.
The Metacognitive Regulation
A. Distribution of the students who used metacognitive elements in the various regulation phases by groups. Figure 2 indicates that in the MC group, 100% of the students used metacognitive elements across the three phases of self-regulation (planning, monitoring and control, and reflection). In the MA group, 80% of the students used metacognitive elements in the planning phase, and 70% of the students used them in the monitoring and control phase. The lowest frequency of students was found in the control group, where only 50% of the students used metacognitive elements in the planning phase.
Figure 2. Distribution of the Students Who Used Metacognitive Elements in the Various Regulation Phases by Group
Regarding the planning phase, a difference in the distribution of the students who used metacognitive elements was detected. A significant difference was found in c² analyses for comparison between the students, c² = 11.60, df =2, p < .01. In the follow-up c² analyses conducted for comparison between pairs of groups, a significant difference was found between the MC group and the control group, c² = 6.68, df = 1, p < .01.
Regarding the monitoring and control phase, similar to the planning phase, the c² analyses found a significant difference between the groups, c² = 30.43, df = 2, p < .001. Similar to the planning phase, significant differences between the MC group and the control group were found here as well, c² = 20.00, df = 1, p < .001. Furthermore, a significant difference was found between the MA group and the control group, c² = 10.77, df = 1, p < .001.
Regarding the reflection phase, slightly diverse differences were found. As can be seen in Figure 2, whereas all the students of the MC group used the reflection elements, none of the students in the MA and the control group used this element. The c² analyses found significant differences between the MC group and the MA and control groups, c² = 20.00, df = 2, p < .001. Examples of the elements can be seen later on.
B. Distribution of the metacognitive elements in the various regulation phases by groups. The aloud solutions indicated that all three groups used metacognitive elements with significantly different frequency. The MC group made the most extensive use of metacognitive elements (71 elements): planning phase 31%; monitoring and control phase 52%; reflection phase 17%. The MA group followed (15 elements): planning phase 53%; monitoring and control phase 47%. The control group made the least use of metacognitive elements (8 elements): planning phase (100%). According to Figure 3, the aloud solutions indicated that the MC group made the most consistent use of metacognitive elements during all the self-regulation phases. The highest frequency of elements was found in the monitoring and control phase, followed by the planning phase and, finally, the reflection phase. The MA group used these elements only in the planning phase and in the monitoring and control phase. The control group made the least consistent use of metacognitive elements and used it in the planning phase only.
Figure 3. Frequency of the Metacognitive Elements in the Various Regulation Phases by Group
Note: Each of the groups elements were calculated separately for each phase so that the total of elements amounted to 100%.
To deepen and characterize the differences between the groups, we calculated the distribution of the metacognitive elements in each of the self-regulation phases (planning, monitoring and control, reflection) in the different groups. The percentages were calculated for each group separately with respect to each of the phases (see Table 3).
Table 3 indicates that all three groups used metacognitive elements in different frequencies, whereas the MC group made the most consistent use of these elements. As can be recalled, the MA group and the control group were not trained on metacognitive regulation. An interesting finding was that students in the MA group made more consistent use of the metacognitive regulation than in the control group.
Table 3. Distribution of the Metacognitive Elements in the Various Regulation Phases by Group
Note: Percentages in each group were calculated by a division of the amount of each element by the total number of elements in each phase, multiplied by 100.
In the planning phase, all groups used the goal setting element in the highest frequency of all other elements in this phase. Relating to the monitoring and control phase, in the MC group, all students used both monitoring and control elements in an integrated manner that complemented one another (in 100% of the elements), as they learned during the intervention. For instance, they examined whether the strategy they chose was effective and, if not, they chose a different one. The students also checked their responses in the course of learning. The MA group used this phase as well. However, an interesting finding was that the students did not integrate between monitoring and control; rather, they referred to monitoring or to control separately (50% monitoring; 50% control). Relating to the reflection phase, only the metacognitive group made use of it.
The Meta-Affective Regulation
A. Distribution of the students who used meta-affect elements in the various regulation phases by groups. Findings indicated that all the students in the MA group used meta-affect elements in all three phases, whereas in the MC and control groups, none of the students used the meta-affective elements. In the c² analyses, this difference was found to be significant c² = 40.00, df =2, p < .001. Regarding the reflection phase, 50% of the students in the MA group used meta-affect elements. In the other groups, none of the students used meta-affect elements in this phase. These differences were supported by c² analysis, c² = 17.14, df =2, p < .001.
B. Distribution of the meta-affect elements in the various regulation phases by groups. To examine the consistency of the meta-affective regulation, we examined the manner in which the students implement the meta-affective regulation elements during the various phases (planning, monitoring and control, reflection). Each of the groups was calculated separately so that the total of each groups elements amounted to 100%. The aloud solutions indicated (Figure 4) that the MA group used 60 meta-affective elements consistently during all the self-regulation phases: the planning phase (25%), monitoring and control phase (60%), and reflection phase (15%).
Figure 4. Frequency of the Meta-Affects Elements in the Various Regulation Phases in the MA Group
Note: The elements of the MA group were calculated separately for each phase so that the total of elements amounted to 100%.
In conclusion, unlike the metacognitive regulation, which was used also by students in the MA group who were not trained for it, in the meta-affect aspect, only the MA group trained for meta-affective regulation made high-quality use of it. It is also interesting to see that students in each of the groups implemented the self-regulation processes in accordance with the component they was trained for, whereas in each of the groups, the meaningful phase was the monitoring and control phase, then the planning phase, and, finally, the reflection phase.
The following illustrations of the MC group and MA group represent a sample of the self-regulation processes sequence at all the phases in each of the groups, indicating the process of a student of each group. These illustrations emphasize the self-regulation process sequence at all of its phases (planning, monitoring and control, reflection) as well as the timing with regard to each of the phases during the solution. The duration of the total solution in each of the groups was approximately 20 minutes.
SELF-REGULATION PROCESSES DURING AN ALOUD SOLUTION MC GROUP
Time: approximately 20 minutes
The illustration above indicates that the student used only metacognitive elements during the three phases. It was found that during the first 6 minutes, the student focused on the metacognitive planning phase (30% of the solution duration) while referring to the elements: goal setting; prior content knowledge activation; and metacognitive knowledge activation.
Then, for a duration of 12 minutes, the student focused on the monitoring and control phase (60% of the solution duration).
The reflection phase lasted 2 minutes (10% of the solution duration).
SELF-REGULATION PROCESSES DURING AN ALOUD SOLUTION MA GROUP
Time: Approximately 20 minutes
The illustration above indicates that the student used mostly meta-affective elements during all the self-regulation phases and few metacognitive elements. During the first 4 minutes, the student focused on the planning phase (20% of the solution duration). This phase consisted of both the meta-affective component and the metacognitive component. In this phase, the student referred to the elements: prior content knowledge activation and primary emotion activation.
Following the planning phase, for a duration of 14 minutes (70% of the solution duration), the student focused on the monitoring and control phase (60% of the solution duration), while referring to the meta-affective component and somewhat to the metacognitive component.
The reflection phase lasted 2 minutes (10% of the total solution duration), during which the student described his feelings toward the task:
The current study examined the contribution of two self-regulation intervention programsmetacognitive and meta-affectiveto achievements in mathematical verbal problem solving, a novel transfer task, and self-regulation processes among young students. Each group was trained directly and explicitly in self-regulation processes but with a different focus, compared with a control group. The findings indicate that the MC and MA groups presented similarly high achievements in verbal problem solving and in the novel transfer task compared with the control group. This finding strengthens and broadens previous findings in studies conducted in the field of self-regulation, indicating that explicit guidance of metacognitive regulation in mathematics through self-questioning improves achievements among the adult student (Karmarski & Mevarech, 2003; Schoenfeld, 1992; Veenman et al., 2006
The current study proposes a mental intervention model for metacognitive self-regulation in verbal problem solving, explicitly adapted to the young student. The model focuses the student on high metacognitive processes throughout the solution phases (planning, monitoring and control, and reflection). This focus enables successful coping with the complexity of the verbal problem that demands multiple metacognitive processes, such as the use of existing knowledge (facts, definitions, and competencies) and selection of problem-solving strategies (analysis), thus leading to improved achievements.
Regarding meta-affective regulation, previous studies indicated that negative emotions exercise a negative effect on the learning process. According to these studies, negative emotions instigate feelings of alienation from school and from the learning process (Artino, 2009; Gross, 2013; Kramarski et al., 2010), beginning as early as the fifth and sixth grades (Pekrun et al., 2010). Nonetheless, majority of studies dealing with self-regulation focused mainly on the relation between negative and positive emotions, spontaneous self-regulation processes, and achievements (DMello et al., 2013; Efklides 2011; Malmivuori, 2006).
Given that there are almost no intervention programs in this field, the current study amplifies the importance of the meta-affective process to the students achievements in mathematics at a young age and offers an intervention model for meta-affective self-regulation, which assists the young student in coping with negative emotions during problem solving. The focused and intensive training enabled the students to reach a state of calmness that induced a reduction in negative emotions. Thus, when the students identified a negative emotion while working on a task, they dealt with it by using the emotion-management strategy they had practiced earlier, which presumably contributed to improvement in achievements.
Regarding the interplay between the metacognitive and meta-affect components, it appears as though both MC and MA groups similarly improved their achievements in tasks before and after the intervention. This finding is supported by previous studies that found a connection between metacognitive and meta-affective processes among the adult student, meaning that grooming one aspect of self-regulation affects the entire process, leading to an improvement in achievements (Hoffman, 2010; Mevarech & Kramarski, 2014). The current study broadened these findings by emphasizing the interplay between the self-regulation processes (metacognitive and meta-affective) among the young student, leading to improved achievements. Interestingly, no differences were found for the novel transfer task between the MC and MA groups. So far, it was found that metacognitive regulation contributes to the students assignment ability by offering deep and general scaffolding to coping with tasks (e.g., Kramarski, Weiss, & Sharon, 2013; Mevarech & Kramarski, 2014). The current study expands this knowledge and indicates that meta-affective regulation also enables the young student to efficiently cope with the novel transfer task. Each of the programs can assist students to holistically perceive the mathematical steps they are required to take, productively access and interact with content, as well as think about the deeper concepts and structures of disciplinary relations. Moreover, this finding allows educators to adjust the self-regulation process to the students learning style (metacognitive or meta-affective), because both components lead to a similar improvement in achievements.
SELF-REGULATION PROCESSES DURING AN ALOUD THINKING SOLUTION (FOCUS GROUP)
Real-time analysis of self-regulation processes indicated that students in the MC and MA groups used self-regulation processes and broadly focused self-regulation processes while referring to all self-regulation processes (planning, monitoring and control, reflection) and the various elements (see Table 1). Furthermore, students from both groups timed the regulation processes similarly, so that the majority of regulation processes focused on monitoring and control, in terms of both frequency and duration. This process is most meaningful because it keeps the students alert to monitor and control cognitive strategies and emotion management while performing the task, thus optimally coping with problem solving. This finding is supported by previous studies indicating the importance of monitoring and control in the mathematical context (Schoenfeld, 1992). Analysis indicated that the control group made the least use of self-regulation processes. This finding is in line with previous studies indicating that self-regulation is not spontaneous and must be taught explicitly (Dignath & Büttner, 2008; Kramarski et al., 2010).
When taking into consideration the findings on improvement of achievements in mathematics in the MC and MA groups, these analyses emphasize the efficiency, importance, and effectiveness of the intervention programs despite the different focus. It appears that students from the MC group used metacognitive elements throughout the self-regulation phases, which contributed to their coping with the problem, understanding and organizing it, as well as dealing with difficulties. Students from the MA group used meta-affective regulation throughout the self-regulation phases, while focusing on coping with negative emotions. They identified negative and positive emotions and consciously dealt with negative feelings. This analysis sheds light on the importance and efficiency of the meta-affective regulation and supports previous studies presented in the context of the relation between negative emotions and achievements.
Although not trained in it, students from the MA group and control group made partial use of metacognitive regulation. This finding can be explained by the exposure of all groups to self-awareness, in a different focus. Although self-awareness in the MA group and control group were not aimed toward the metacognitive component explicitly, this awareness led to a field-to-field assignment in these groups as well. This finding is supported by other studies that found a connection between grooming one component and its effect on the other (Karmarski et al., 2010; Sansone & Thoman, 2005; Wolters, 2003). However, it should be noted that the use of metacognitive processes in the MA and control groups was considerably meager as compared with the MC group, a finding that strengthens the importance of explicit regulation. On the other hand, no assignment to the meta-affective aspect was found: The MC and control groups made no use of meta-affective regulation. This finding strengthens our hypothesis that the awareness of meta-affective regulation should be broadened and taught to the students explicitly.
CONTRIBUTION OF THE STUDY AND FURTHER RESEARCH
The current study suggests a theoretical, methodological, and practical contribution. Theoretically, it deepens our understanding of the importance of supporting self-regulation in metacognition and meta-affect with regard to students problem-solving achievements and self-regulation processes. The present study was based on Pintrichs (2000) self-regulation model, focusing on the metacognitive and affective aspects of self-regulation in adults. The study extended this model to the affective aspect of self-regulation while adapting it to the young student. Methodologically, the study suggests principles for real-time analysis of self-regulation processes, in coherence with the current widespread perception that real-time analysis of self-regulation processes is meaningful for understanding SRL (Azevedo, 2015). In practical terms, the study developed two unique self-regulation interventions for the young student, based on one theoretical model (Pintrich, 2000): a MC regulation intervention and a MA regulation intervention. As noted, in light of the lack of MA intervention programs, the study suggests an effective program tailored to young students. Both interventions can be useful for teacher trainings as well as training seminars on the subject of self-regulation approaches. These interventions can also be adapted to adults and young students alike.
The study examined the effect of the MC and MA interventions only in relation to mathematical verbal problem solving and one thinking-aloud problem. These effects were examined immediately following the completion of the study. The interventions efficacy could therefore be examined in other subjects and using a variety of SRL assessment methods. Longitudinal studies can be useful for determining the long-term impact of the different programs.
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