Examining College of Education Students’ Metacognitive Consciousness in Solving Problems on Geometric Theorem ()
1. Background of the Study
Besides the development of professional competence, transformations of the Colleges of Education in Ghana over the years were also geared towards improving the mathematics competence in both content and pedagogical knowledge. The programs rolled out over the years include, post middle certificate teachers training programme, phased out in 1992 and replaced with a post-secondary certificate teachers programme, which was also replaced with a Diploma awarding programme dubbed IN-IN-OUT programme with an introduction of a full academic year of internship. Currently a drastically reformed programme leading to the award of a four-year bachelor of education certificate with an enhanced exposure to practicum was implemented in 2018.
To improve content knowledge, the First-Year College Courses in mathematics are foundation courses meant to bridge the gaps in college students’ High School mathematics (University of Cape Coast, 2020). To build a solid foundation in mathematics as a prerequisite, a more improved strategy of learning mathematics needs to be adopted by students in the Colleges of Education. To effectively achieve this drive, it is very crucial that more efficient strategies are adopted in teaching and learning in the Colleges of Education, considering that, products of the Colleges would also be involved in teaching of learners at the Primary and Junior High Schools. Metacognition should therefore be given significant consideration as a strategy for effective teaching and learning.
According to Tansyani (2016), consciousness in significant levels of metacognition is closely related to student’s ability to develop and use improved learning process by designing enhanced models, monitoring, evaluating and reflecting on the process of learning. Ulfa, Sistiana, and Setiono (2018), cite a number of authors (Fauzi, 2013; Panchu, Bahuleyan, Seethalakshmi, & Thomas, 2016) who emphasized the need for good metacognition during learning and demanded great efforts from teachers and lecturers to empower students with metacognitive abilities. The argument from these authors presupposes that student with good metacognitive awareness will learn better and faster relative to those who do not have good metacognitive awareness. Cognitive awareness in effect would enhance learning abilities through reflection, self-regulation and self-evaluation.
Teachers in Colleges of Education are expected to equip student teachers with knowledge in both content and pedagogy to be able to teach competently as Basic School Teachers. It is therefore very critical that college students are exposed to comprehensive instructional methodologies which would strengthen their competencies and enable them to teach children mathematics effectively and as well enable them to use metacognitive skills in learning mathematics concepts and to solve mathematics problems. Schraw, Crippen and Hartley (2006), report that teaching students how to learn allows them to develop the ability of transferring skills which is essential for them throughout the period of furthering their education and career. This affirms the essence for teachers to equip their students with metacognitive skills to effectively learn mathematics concepts.
“Learner’s use of metacognitive skills and strategies, such as task analysis planning monitoring checking and reflection, self and group monitoring skills, reading and writing skills, self-regulation (SR) and self-assessment (SA) helped them in mathematics problem-solving. The learners could also solve problems more easily through group discussions and thinking about their own thinking” (Tachie, 2019). Despite the significant overlap of metacognition with cognition, Mevarech and Kramarski (2014), however illustrates some differences between the two with an example. For example, remembering your Automated Teller Machine (ATM) card Personal Identification Number (PIN) is cognition but metacognition is being aware of the strategies by which you can recall the PIN.
Schofield (2012) concluded in a study that successful learning requires metacognitive capabilities. Metacognition is about becoming aware of what you know as well as knowing the gaps in what you know. She narrates that when students were interviewed, several of them noted that they often completed a task without really knowing if they understood it.
Metacognitive strategies refer to methods used to help students understand the way they learn; in other words, it means processes designed for students to “think” about their “thinking” (Inclusive Schools Network, 2015). Classroom practices considered as metacognitive strategies include; asking questions, self-reflection, self-quizzing, self-recording, modeling, directly teaching learners these strategies, availability and accessibility of mentors, solving problems with colleagues, think aloud, self-explanation, keeping learning journals and providing opportunities for making errors (Drew, 2020; Marilyn, 2015; Kim, Park, & Baek, 2009). Du Toit and Kotze (2009), also reviewed literature and enumerated a number of metacognitive strategies suggested by the authors. Among the strategies were:
1) Planning strategy: teachers draw learners’ attention to the rules and steps at the beginning of solving a task.
2) Generating questions: Du Toit and Kotze (2009) writes that Ratner (1991), sees generating questions as a metacognitive strategy and considers questioning of given information and assumptions as a vital component of intelligence.
3) Choosing consciously: through teacher support learners would be able to identify underlying relationship between their actions and the results of their actions.
4) Setting and pursuing goals: evaluating the way of thinking and acting, learners look back on their learning activity and identify feasible (practicable) strategies, exciting situations, obstacles and distasteful experiences.
5) Additional strategies they listed were, identifying the difficulty of a task, paraphrasing, elaborating and reflecting on ideas of learners, problem-solving activities, “thinking-aloud”, journal-keeping, cooperative learning and modelling.
Most Pre-Tertiary Curricula might contain some activities and processes which are metacognitive strategies but do not directly classify them as metacognitive strategies. However, early childhood curriculum directly lists metacognitive processes which is captured as “learning to learn” as one of the five key principles which must pervade every domain of learning at that level (New Zealand Ministry of Education, 2007). Metacognitive training programmes introduces individuals to a set of rules and effective strategies to learn concepts in specific subject areas like English, Mathematics and Science (Flavell, 1987). A wide range of metacognitive strategies that can be adopted include, error-detecting, effort and allocation, elaborating, self-quizzing, self-explanation, developing visual representation of ideas visually, activating prior knowledge, double reading of difficult questions and reviewing experiences and concepts learnt. Metacognitive strategies support learners to effectively manage and settle on main thoughts when confronted with conflicting thoughts (Lin, 2001). These metacognitive strategies stealthily find their way into students approaches of learning or tackling problems on GT. These approaches might have been adopted consciously through teacher coaching or unconsciously during interaction in class, peer interaction or through self-strategizing.
Khan (2020) and Pintrich and colleagues (1991) have established that, there was no significant difference in awareness, knowledge and control of cognitive process between students in the traditional learning environment and online sessions. In a similar comparison, Khan noted that there was very minimal effect of the online sessions compared to the traditional sessions in terms of achievement and metacognitive awareness though students in the online sessions developed higher enthusiasm.
Results in a study conducted by Rezvan et al. (2006) suggest that metacognitive training increased the academic achievement of the group that received treatment. Their study also showed that metacognitive training also raised the happiness score-average of the same treatment group. Abdellah (2015) concluded in another study, that metacognitive skills play a significant role in student learning and recommend the inclusion of metacognition courses in Colleges of Education Teacher Training Programs. Schunk (2012), also agrees to the assertions that metacognition improves achievement but draws attention to the sticky point that, even after metacognition awareness, learners would not automatically adopt it when they encounter a task. These then put the responsibility on tutors to expose learners to numerous task that would engage metacognitive strategies to a point, when learners on a regular basis consciously and unconsciously use these strategies in GT task.
Pintrich and De Groot (1990) attest that, though academic achievement is not solely dependent on metacognitive strategies, it plays an essential role in students learning progression. Metacognition is self-motivating and further improves a student’s desire towards learning and academic achievement. It follows also that students who are capable of motivating themselves are equally able to engage in metacognitive strategies such as planning, assessment and evaluating how well they are accomplishing a learning task.
Cognitive strategy instruction comprises of the cognitive and metacognitive strategies which promotes and results in learning and higher achievements, where the complexity of the strategies varies as a function of task difficulty (Montague, Krawec, Enders, & Dietz, 2014). Coughlin and Montague (2011), add that instructional approaches that use cognitive strategies have been shown to increase academic achievements. In addition to this, it is more rewarding to students who have mathematics learning disabilities after their use of cognitive strategies that support mathematics problem solving proficiency.
Gurat and Medula (2016) enumerate the factors that affect failure or success of problem-solvers to include metacognitive strategy knowledge. During mathematics problem-solving, metacognitive strategy knowledge emerges in a three-phase cyclic model. The model comprises preparatory metacognitive strategy knowledge, production metacognitive strategy knowledge and evaluation metacognitive strategy knowledge. The problem-solver initiates the process of solving the task by planning metacognitive strategies that would be employed in solving the problem. The second phase involves designing and execution of the metacognitive strategy planned the first stage and eventually the going over the process and solution obtained to determine its success. This spiral into another cycle if the solution obtained is not satisfactory.
1.1. Problem Statement
Learners’ challenges and aversions to mathematics might not necessarily be a result of bad teacher attitudes and strategies employed in the mathematics classroom but probably the approach students adopt towards learning and solving mathematics problems. A study carried out in a traditional classroom showed that students’ achievement is largely related to a range of learning strategies (Weinstein & Palmer, 2002).
This study was borne out of the interest to probe and determine the metacognitive strategies Trainee Teachers adopt in learning Geometric Theorems which is considered by some teachers and most students as one of the difficult topics. Domingo (2016), mentions the topics that secondary school teachers encounter challenges in teaching to include Geometric Theorem. Geometric concept is one of the topics perceived to be difficult. This perception is the result of poor teacher instructional strategies, student-to-student interaction or misinformation on Geometry and the inability to use metacognitive approaches when they encounter problems on Geometry.
1.2. Objectives of the Study
The objectives of the study were to examine:
1) Whether College of Education students evaluate their abilities and vulnerability in Geometric Theorem problems during or after instructional sessions.
2) Whether College of Education students adopt metacognitive strategies when they encounter problems on Geometric Theorem.
1.3. Research Questions
The study sought to probe the problem through the following research questions:
1) Are College of Education students conscious of their cognitive abilities and vulnerability when solving problems on Geometric Theorem?
2) Do College of Education students adopt metacognitive strategies in solving problems on Geometric Theorem?
2. Methodology and Participants
The study adopted a largely quantitative research approach using a collective case study design which is in tune with the purpose. For collective or multiple case study, two or more groups are studied concurrently or simultaneously by researchers in their desire to generate a clearer view of the particular issue of interest (Crowe, Creswell, & Robertson, 2011). For collective or multiple case study, the groups selection is carefully done to allow for the merit of comparison across several groups which further allows for generalizing of theory (Yin, 2009).
The accessible population was level 100 students of Bagabaga College of Education and Tamale College of Education. These Colleges had a First-Year student population of 1396 comprising 696 from Bagabaga College and 700 from Tamale College.
Purposive sampling was used to select these two Colleges from nine Colleges in the Northern Zone of the country, whilst stratified sampling procedure was used to select the sample from the two Colleges. The participants in the population possessed similar characteristics. The basis for the use of purposive sampling to select Bagabaga College of Education and Tamale College of Education was based on the conviction that the two Colleges give fair representation of Teacher Education programmes in the country since the two Colleges combined, run all the various Teacher Education programs pursued in Colleges of Education in the country. Questionnaire was employed to collect the data which was adapted from a standard tool for measuring the subscales under metacognition. This instrument was so selected because it was designed by Schraw and Dennison (1994) to measure the subscales that were intended to be measured in this study. The study equally employed descriptive statistics for the analysis which were presented in tabular form.
3. Discussions
3.1. Research Question One
What extent are College of Education students aware of their cognitive abilities and vulnerability when solving problems on Geometric Theorem?
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
Research Question One sought to find out the level of awareness of respondents in their use of metacognitive strategies. The results in Table 1 show a Mean score, ranging from 3.21 to 3.41, indicating a remarkable level of awareness of procedural knowledge. Pimentel (2019) states the interpretation of a Mean value of 3.28 to 4.00 as lot or strongly agree. This shows that, respondents are conscious of strategies they have used in the past which can be rewarding in new GT task they encounter. The overall mean score from Table 1 (M = 3.158) suggests remarkable levels of consciousness of procedural knowledge among the College of Education students. This is confirmed by a very low standard deviation of 0.205. Procedural knowledge is a component of knowledge about cognition (Flavel, 1979). For the last item, I restate the GT problems encountered, the Mean score of (M = 2.7; SD = 1.044), suggests that it is not a procedure frequently adopted by students in tackling GT task they encounter.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
Table 2 contains items on declarative knowledge and constitutes the second sub-scale under knowledge about cognition of the metacognitive awareness instrument. The overall mean score (M = 3.142) indicates that respondents largely
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Table 1. Items on procedural knowledge.
Source: Field Survey (2021).
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Table 2. Items on declarative knowledge.
Source: Field Survey (2021).
agreed to the items under the subscale confirming College of Education students’ consciousness of this aspect of knowledge about cognition. Panda (2017) noted that, metacognitive awareness of learners improves in studying from early ages in school and develop remarkably in students at the College level. “I know what type of information is most important when solving a problem on GT” (M = 2.99, SD = 0.881) and “I learn more because I am interested in GT topics” (M = 2.82, SD = 0.916) were the only items that recorded mean values of less than three. The SD values (SD = 0.881, SD = 0.916) show that the responses were further away from the Mean values relative to items with mean scores greater than three.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
Results in Table 3 give a high indication that information management strategies are adopted by the students (overall M = 3.252). The individual items each recorded Mean values ranging from M = 3.00 to M = 3.44. This suggests consciousness or use of strategies such as “focusing on important information on GT” (M = 3.44 SD = 0.727), “creating own examples to make information more meaningful” (M = 3.27 SD = 0.829), “representing GT problems diagrammatically” (M = 3.23 SD = 0.808) and “translating new information on GT problems into their own words” (M = 3.33, SD = 0.780). These affirm the assertions made by Victor (2004) cited in Belet and Guven (2011) representing GT problems diagrammatically and translating new information on GT problems into their own words are metacognitive strategies that complement the constructivist theories of learning.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
The results in Table 4 give strong evidence of debugging strategies. That is, students make attempts to correct their mistakes or address challenges they encounter
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Table 3. Items on information management strategies.
Source: Field Survey (2021).
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Table 4. Items on debugging strategies.
Source: Field Survey (2021).
when they encounter problems on GT. From the results in the Table 4, students gave strong indication that, they stop and re-examine a question if they get confused (M = 3.61, SD = 0.615). The Standard deviation (0.615) confirms that a large level of responses to the item concentrated around the mean (M = 3.61). Similarly, the results show a strong agreement that respondents pause and go over their work when they get stuck; I stop and re-examine a question if I get confused (M = 3.44, SD = 0.669). From the open-ended items, students indicated that they always go over their work to effect corrections on mistakes made. The overall mean score (M = 3.370) reflects students’ strength at the use of debugging strategies in learning or solving GT task.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
Items in Table 5 sought to determine whether students design a plan to tackle GT problems. The overall mean (M = 3.309) reveals that respondents are conscious of the need to design a plan as metacognitive strategy to tackle GT problems. Respondents strongly agreed to all the planning strategies mentioned in this subscale; students pacing themselves to work within time (M = 3.21, SD = 0.820), setting specific goals before executing a task (M = 3.16, SD = 0.839), considering or identifying the materials needed for the task (M = 3.21, SD = 0.810), considering several approaches to solving the problem (M = 3.39, SD = 0.716). The standard deviation values of less than one suggests that most of the responses are closer to the mean scores, thus, more positive affirmation to the consciousness of the need to design a plan as metacognitive strategy to tackle GT problems.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
Results in Table 6 give an overall mean of 3.231, generally showing that respondents monitor the strategies they adopt in solving metacognitive strategies. The results further reveal a high mean value (M = 3.33, SD = 0.714) to the response on whether they ask themselves how well they are doing or learning something new on geometric theorem. From Table 6 both the mean of means and overall standard deviations (M = 3.231 and SD = 0.095), the results show
Source: Field Survey (2021).
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Table 6. Items on monitoring comprehension.
Source: Field Survey (2021).
that respondents strongly accepted that they pause regularly to check their understanding when they get stuck on GT problems.
Mean ranges: 1.00 - 1.75 = Not much; 1.76 - 2.51 = A little; 2.52 - 3.27 = Some; 3.28 - 4.00 = A lot
There was a remarkable level of evaluation of metacognitive strategies by students and this is supported by a Mean value of 3.150 and a small standard deviation of 0.098, as captured in Table 7. Mean score of responses to the individual items in the table range from 3.06 to 3.32 which is a further confirmation of respondents’ consciousness and use of evaluation as a strategy to tackle GT task.
3.2. Research Question Two
What metacognitive strategies do College of Education students adopt in solving problems on Geometric Theorem?
From the results in Table 8, the overall mean score was computed as 2.86 which is evidence of the fact that some metacognitive strategies are rarely used
Source: Field Survey (2021).
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Table 8. Respondent’s use of metacognitive strategies in solving GT problems.
Source: Field Survey (2021).
by respondents. From Table 8, I resort to solving GT problems with a colleague or a team (M = 3.07, SD = 0.974), recall formulae to use (M = 3.09; SD = 0.891), Self-reflection (M = 3.13, SD = 0.898), Self-questioning (M = 3.06; SD = 0.948) were very often used by respondents. The frequent use of rote or recall of rules by students to solve GT problems was confirmed by Ozkan and Kesen (2008) but was blamed on educational systems that students went through.
4. Responses to the Open-Ended Items
The open-ended items offered respondents the opportunity to list both ordinary and innovative strategies they use to solve GT questions. The third and fourth open-ended items sought to determine the thoughts, feelings and emotions of respondents when they encounter difficult task. Respondents listed the following as strategies they adopt when they encounter GT problems:
1) Use of formula and the properties of geometric shapes
2) Reflection, recall and analysis of the problem
3) Breaking problems into smaller parts and approach it step by step
4) Use of pictures and diagrams to solve GT problems
5) Consider alternative strategies and select one
6) Change strategies when if one or the first one fails
7) Seek help from colleagues when they encounter difficult problems
8) Review previous topics/knowledge to see if old information relates to the new problems
9) Going over their work to check for lapses
Regarding their emotions when they encounter difficult questions, the common responses were: feeling of sadness, depression, disappointment and inadequacy in the learning of concepts. These strategies conform to the metacognitive strategies listed by many researchers (Inclusive Schools Network, 2015; Resourceaholic, 2014; Drew, 2020; Marilyn, 2015; Kim, Park, & Baek, 2009; Belet & Guven, 2011).
5. Conclusion
The research was borne out of the interest to determine the metacognitive consciousness of College of Education students in the strategies they adopt in solving problems on GT. It was found that, College of Education students were highly conscious of the use of metacognitive strategies when they encounter GT problems. This conforms to the assertions of Panda (2017) who conducted similar studies into metacognitive awareness of college students in Haryana state, India and observed similar findings. Hence, being conscious of metacognitive strategies help improve students’ understanding and performance.
Recommendations
This study recommends College of Education students be made to adopt metacognitive strategies when they encounter problems in Geometry and other content courses. To raise the consciousness of trainee teachers the College of Education curriculum should deliberately mention metacognition and as well highlight activities in the curriculum that are metacognitive strategies.