Implementation of critical thinking in Grade 3 mathematics classrooms : a case study

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(1)COPYRIGHT AND CITATION CONSIDERATIONS FOR THIS THESIS/ DISSERTATION. o Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. o NonCommercial — You may not use the material for commercial purposes.. o ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.. How to cite this thesis Surname, Initial(s). (2012) Title of the thesis or dissertation. PhD. (Chemistry)/ M.Sc. (Physics)/ M.A. (Philosophy)/M.Com. (Finance) etc. [Unpublished]: University of Johannesburg. Retrieved from: https://ujcontent.uj.ac.za/vital/access/manager/Index?site_name=Research%20Output (Accessed: Date)..

(2) I. IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY By Albert Garcer A Minor Dissertation submitted in fulfillment of the degree Master of Education in Curriculum Policy Evaluation (Course Work) Faculty of Education At the University of Johannesburg. Supervisor: Professor Shireen Motala Co – Supervisor: Professor Kakoma Luneta. April 2018.

(3) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 1. ABSTRACT This research case-study was motivated by the need to explore the implementation of critical thinking, as well as the instructional implications of promoting critical thinking in a foundation phase classroom, by determining whether Grade 3 math teachers prioritize the promotion of effective critical thinking skills in the subject of mathematics. The following research question guided the research study: “What effective approaches do foundation phase teachers employ, as a teaching strategy, to promote effective critical thinking skills in a Grade 3 mathematics classroom”. To address the main question, the following sub-questions were posed: . What is Grade 3 math teachers’ understanding of critical thinking?. . How do Grade 3 math teachers foster critical thinking skills in their math classrooms?. . How does learners’ prior content knowledge impact influence their critical thinking skills?. The case study made use of a qualitative research design in the form of a case-study to answer the above questions. The research sites were two primary schools in Gauteng, South Africa. The sampled participants for this study were two Grade 3 classes from which 18 learners were purposively selected. The data collection instruments that were used included interviews, field notes and learners’ written assessments. The learner assessments were conducted prior to the learner interviews. The results of learners’ assessments and interviews were assessed and compared to determine whether teachers’ implementation of their various critical thinking strategies was effective and promoted critical thinking skills within learners, when they took the assessments. Learners solved problems and explained the rationale behind how they solved the word problems in more detail during the interview. Learners could give verbal accounts of their methods during the interview. Both the assessments and interviews made way for the researcher to rate learners’ critical thinking skills using the Learners’ Interview and Assessment Analysis Grid (LIAAG). The study found that there were three main factors that influenced the effectiveness of teachers’ approaches towards implementing and promoting critical thinking as a teaching strategy, in a mathematics classroom. These were namely: teachers’ understanding of critical thinking skills; teaching methods; and learners prior content knowledge. These factors manifested through a reflection of high levels and stages of the Learning Framework in Numbers (LFIN) as well as Stages of Early Arithmetic Learning (SEAL), in the learners’ problem-solving strategies..

(4) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 2. This case study found that neither teacher was fully conversant with the concept of critical thinking so that they thought they were applying critical thinking but according to its full definition and interpretation were not. The teachers did not have effective measures to fully implement critical thinking and promote higher order thinking skills within learners. Both teachers failed to determine the objectives or set expectations on what behaviours they would like to see at the end of each lesson. Consequently, by failing to set expectations, the teachers deprived learners of an important opportunity to develop critical thinking skills, which are emphasised by this study as a vital step in the process. Recommendations for future investigation are also provided. Keywords: Critical thinking, Learning Framework in Numbers (LFIN), Stages of Early Arithmetic Learning (SEAL)..

(5) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 3. DECLARATION OF ORIGINALITY UNIVERSITY OF JOHANNESBURG. Full names of student: Albert Kabwe Garcer Student number: 201474595 I, Albert Garcer, declare that the content of this thesis represents my own unaided work and that the thesis has not previously been submitted for academic examination towards any qualification. Furthermore, it represents my own opinions and not necessarily those of the University of Johannesburg.. ________________________ Signature of student. ________________________ Signature of supervisor. ________________________ Signature of co-supervisor. _________________ Date. _________________ Date. _________________ Date.

(6) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 4. TABLE OF CONTENT IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY.............................................................................................. 0 ABSTRACT ................................................................................................................... 1 DECLARATION OF ORIGINALITY ........................................................................ 3 CHAPTER 1: BACKGROUND OF THE STUDY .......................................................... 10 1.1. INTRODUCTION ............................................................................................ 10. 1.2 BACKGROUND AND SIGNIFICANCE OF THE STUDY ............................. 10 1.3 PURPOSE OF THE STUDY................................................................................ 11 1.4 CRITICAL THINKING AND PROBLEM SOLVING ..................................... 12 1.5 RESEARCH METHODOLOGY AND DESIGN............................................... 13 1.6 LIMITATIONS OF THE STUDY ....................................................................... 14 1.7 OUTLINE OF THE CASE STUDY .................................................................... 15 CHAPTER 2: LITERATURE REVIEW ...................................................................... 17 2.1 INTRODUCTION................................................................................................. 17 2.2 THEORETICAL FRAMEWORK ...................................................................... 17 2.2.1 Social constructivism ..................................................................................... 18 2.2.2 Zone of Proximal Development (ZPD) ......................................................... 19 2.2.3 Scaffolding ...................................................................................................... 20 2.3. PROBLEM SOLVING IN MATH ................................................................. 21.

(7) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 2.3.1. 5. Problem solving ........................................................................................... 21. 2.3.2 Types of problem-solving............................................................................... 21 2.3.3 Examples of problem types ........................................................................... 22 2.3.4 Invented critical thinking skills in problem-solving ................................... 23 2.4 MATHEMATICAL PRIOR KNOWLEDGE ..................................................... 23 2.4.1 Learning Framework in Number ............................................................. 2423 2.5. CRITICAL THINKING ................................................................................. 27. 2.5.1 Active and Passive learning ........................................................................... 28 2.5.2 Singapore math .............................................................................................. 28 2.6 A MODEL TO. PROGRESS. LEARNERS TOWARDS THINKING. CRITICALLY ......................................................................................................................... 29 2.6.1 Step 1: Determining the Learning objectives .............................................. 29 2.6.2 Step 2: Teach through questioning. .............................................................. 31 2.6.3 Step 3: Practice before you assess. ................................................................ 32 2.6.4 Step 4: Reviewing, refining, and improving. ............................................... 33 2.6.5 Step 5: Providing feedback and assessing. ................................................... 34 2.7 CONCLUSION ..................................................................................................... 34 CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY ............................... 36 3.1. INTRODUCTION........................................................................................... 36.

(8) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 6. 3.2. RESEARCH DESIGN .................................................................................... 36. 3.3. RESEARCH PLAN ........................................................................................ 37. 3.3.1 Pilot Study....................................................................................................... 38 3.3.2 Data plan ......................................................................................................... 38 3.4. DATA COLLECTION INSTRUMENTS...................................................... 39. 3.4.1 Teacher interviews ..................................................................................... 4039 3.4.2 Lesson observations ....................................................................................... 40 3.4.3 Learner Assessments ...................................................................................... 40 3.4.4. Learner interviews ...................................................................................... 41. 3.4.5. Audio recordings ..................................................................................... 4241. 3.4.6. Field notes .................................................................................................... 42. 3.5. SITE AND SAMPLING ................................................................................. 42. 3.5.1. Site selection ................................................................................................ 42. 3.5.2. Selection of participants ............................................................................. 43. 3.6 DATA ANALYSIS ................................................................................................. 43 3.6.1. Description ............................................................................................... 4443. 3.6.2 Sense making .................................................................................................. 44 3.6.3. Interpretation .............................................................................................. 44. 3.6.4. Drawing implications .................................................................................. 44.

(9) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 7. 3.7 VALIDITY, TRUSTWORTHINESS AND RELIABILITY .......................... 4544 3.8 ETHICAL CONSIDERATION ........................................................................... 45 3.9 CONCLUSION ..................................................................................................... 46 CHAPTER 4: FINDINGS AND DATA ANALYSIS ................................................... 47 4.1 INTRODUCTION................................................................................................. 47 4.2 TEACHER INTERVIEWS .................................................................................. 47 Step 1. Determining the learning objectives. ........................................................ 48 Step 2: Teaching through questioning ................................................................... 48 Step 3: Practicing before they assess ..................................................................... 48 Step 4: Reviewing, refining, and improving ......................................................... 48 Step 5: Providing feedback and assessing learning ............................................. 49 4.3 TEACHERS LESSON OBSERVATIONS .......................................................... 49 4.4 LEARNER ASSESSMENTS ............................................................................... 51 Problem 1 ................................................................................................................. 52 Problem 2 ............................................................................................................. 5453 Problem 3 ................................................................................................................. 56 Problem 4 ............................................................................................................. 5857 Problem 5 ................................................................................................................. 58 4.5 LEARNER INTERVIEWS .................................................................................. 60.

(10) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 8. 4.6 CONCLUSION ..................................................................................................... 63 Outline of instruments used: .................................................................................. 64 CHAPTER 5: DISCUSSION OF FINDINGS .............................................................. 65 5.1 SUMMARY OF THE RESEARCH PROCESS ................................................. 65 5.2 DISCUSSION OF KEY FINDINGS .................................................................... 66 5.2.1 Teachers’ understanding of critical thinking .................................................... 66 5.2.2 Teaching Methods ...................................................................................... 6768 5.2.3 Learner’s prior content knowledge .............................................................. 70 5.3 CONCLUSION ..................................................................................................... 71 5.4 RECOMMENDATIONS ...................................................................................... 71 REFERENCES ...................................................................................................................... 73 APPENDIX ...................................................................................................................... 80 APPENDIX A: MATHEMATICAL PROBLEMS ................................................... 80 APPENDIX B: CRITICAL THINKING INTERVIEW QUESTIONS FOR TEACHERS............................................................................................................................. 86 APPENDIX C: CLASSROOM OBSERVATION RUBRIC ................................... 88 APPENDIX D: TEACHER LESSON OBSERVATION AND LEARNERS LFIN SCORES................................................................................................................................... 93 Table 4.3.1 Teacher lesson observation comparison Step 1 scores ..................... 93.

(11) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 9. Table 4.3.2 Teacher lesson observation comparison Step 2 scores ..................... 94 Table 4.3.3 Teacher lesson observation comparison Step 3 scores ..................... 95 Table 4.3.4 Teacher lesson observation comparison Step 4 scores ..................... 96 Table 4.3.5 Teacher lesson observation comparison Step 5 scores ..................... 96 Table 4.4: Summary of LFIN levels and number of learners for each problem in the Assessment ..................................................................................................................... 97 APPENDIX E: SUMMARY OF LEARNERS’ INTERVIEWS .............................. 99 Table 4.5.1 Teacher 1, summary of learners’ interviews and assessment analysis Grid ...................................................................................................................................... 99 Table 4.5.2 Teacher 2, summary of learners’ interviews and assessment analysis Grid ...................................................................................................................................... 99 APPENDIX F: LEARNERS INTERVIEW AND ASSESSMENT ANALYSIS GRID ................................................................................................................................................. 101 APPENDIX G: LEARNERS’ INTERVIEW QUESTIONS .................................. 105 APPENDIX H: FULL TEACHER INTERVIEW .................................................. 106 APPENDIX I: CONSENT LETTER TO PRINCIPALS ....................................... 109 APPENDIX J: CONSENT LETTER TO PARENTS ............................................ 111.

(12) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 10. CHAPTER 1: BACKGROUND OF THE STUDY 1.1 INTRODUCTION Critical thinking has been part of South Africa’s education policies at all levels, yet the competency-based Curriculum Assessment Policy Statement (CAPS) which aims to produce learners with problem-solving and critical thinking skills has not been properly implemented (Howie, 2007). South Africa’s change of curriculum in the past shows that it moved from the Outcome Based Education (OBE) introduced in 1998 to be replaced by the Revised National Curriculum Statement (RNCS) then later by the National Curriculum Statement (NCS) (Chisholm, 2015). Today the Curriculum Assessment Policy Statement (CAPS) is being implemented, still with a focus on developing mental processes to promote logic and critical thinking skills (Department of Basic Education, 2011). When Curriculum 2005 was introduced, it was received with hostility by certain media companies who criticized low literacy and numeracy levels in the Foundation Phase level (FP), as well as poor grade 12 pass rates (Chisholm, 2015). One of the outcomes within Curriculum 2005 included learners having the ability to “identify, solve problems as well as to make decisions using critical and creative thinking” (Department of Basic Education, 2011). The OBE Outcomes-based curriculum was an attempt then, to make sure that critical outcomes were being acted upon, implemented and operationalized within classrooms across the country. Critics (Jansen, 1998) anticipated that the new curriculum would be ineffective and predicted that it would fail because the implementation of the policy was driven by political forces who had no knowledge of what was happening in the classroom. The first chapter is an outline of the background and rationale for this case study. The research question and sub-questions are outlined, and the methodological and theoretical orientations of the study are presented. In addition, the significance and limitations of the study are set out. 1.2 BACKGROUND AND SIGNIFICANCE OF THE STUDY The current curriculum, namely the Curriculum Assessment Policy Statement (CAPS), was announced in 2012 aiming to “…improve the quality of teaching and learning in our schools” (Department of Basic Education, 2011). The Department of Basic Education allocated five teaching hours, per week, for mathematics in the current curriculum document for grades R, 1, 2, and 3. The mathematics subject has been, and is, taught with the aim “to develop mental processes.

(13) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 11. that enhance logical and critical thinking, accuracy and problem solving” (Department of Basic Education, 2011). Yet, despite including critical thinking in the curriculum, research shows that learners exposed to teaching intended to promote critical thinking skills are limited to those registered in higher level education and that this group excludes a clear majority of high school students as well as virtually all primary school learners (Solorzano & Ornelas, 2004). Furthermore, research within foundation phase mathematics education and particularly critical thinking is very limited. Evidence indicates a high number of parents are enrolling learners in extra mathematics classes due to poor teaching methods in public schools (The Centre for Development and Enterprise, 2013). In addition to deficiencies regarding critical thinking in the FP phase in mathematics classrooms which research indicates (Howie, 2007), this trend provides meaningful motivation for this case study. This case study stands behind the notion that meeting the current CAPS objectives requires a skilled educator making intentional, informed decisions to teach and implement critical thinking (Department of Basic Education, 2011). 1.3 PURPOSE OF THE STUDY The case study was therefore motivated by a need to explore the implementation of, as well as the instructional implications of promoting, critical thinking in a foundation phase classroom. This purpose is founded on a valid concern that exists as to whether an increased emphasis on critical thinking strategies in the classroom would lead to meeting one of the current objectives which aims to “develop mental processes that enhance logical and critical thinking, accuracy and problem solving” (Department of Basic Education, 2011). Critical thinking and problem solving are vital in any professional field, and applications include understanding symbols, variables, equations and logic. This study recognized the significance of mathematical principles, and the ability to think critically, to any field (Giannakopoulos & Luneta, 2015). It is further motivated by the need to explore the implementation of critical thinking, as well as the instructional implications to promote critical thinking in a foundation phase classroom, by determining whether Grade 3 math teachers prioritize the promotion of effective critical thinking skills in the subject of mathematics. Therefore, an answer to the following research question will be sought through the conduct of the proposed research:.

(14) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 12. What effective approaches do foundation phase teachers employ, as a teaching strategy, to promote effective critical thinking skills in a grade 3 mathematics classroom. The aim of the research suggests several objectives, as well sub-questions needed to address the main question and investigate the effectiveness of teachers’ approaches and practices in the Grade 3 mathematics class: ●. What is Grade 3 Math teachers’ understanding of critical thinking?. ●. How do Grade 3 Math teachers foster critical thinking skills in their math classrooms?. ●. How does learners’ prior content knowledge influence their critical thinking skills?. 1.4 CRITICAL THINKING AND PROBLEM SOLVING Critical thinking is known to be a skill vital in the preparation of learners for tertiary education and the workforce (Bellanca, 2011). It is understood as the mental process that mainly functions in the decision-making process, and organizing of information, to resolve problems (Fisher, 2011). This means that critical thinking, together with problem solving, are vital skills that enable learners to use knowledge, facts, and data to effectively solve educational and life problems. According to Allen and Cowdery (2014), “it is vital to give learners chances to use their minds to apply the knowledge through various ways of thinking, such as critical thinking.” Inculcating critical thinking in a classroom requires a philosophical shift from mere drilling, cramming and practicing content to actual problem solving-based learning, from isolating a subject’s content to integrating various subjects and/or disciplines; a shift in focus from output of learning to understanding the importance of the process (Savery, 2009). This case study, therefore, is founded on the belief that one cannot separate critical thinking from problem solving skills, and that these can be taught, practiced, and transferred through specific teacher instructions (Marina & Halpern, 2011). Mandernach (2006) found that despite their ability to teach content in depth and analyze their subject contents critically, educators mostly struggle to effectively foster critical thinking in young learners due to the partial critical thinking skills of foundation phase learners and their “…lack of background or content knowledge needed to engage in a task” (Lai, 2011). Yet, empirical research (Lai, 2011) shows that individuals start to develop critical thinking skills at a young age. Lai (2011) goes on to show that prior content knowledge is necessary but not enough to enable learners to.

(15) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 13. think critically in a subject. The research therefore agrees with Lai (2011) who states that “Critical thinking involves both cognitive skills and dispositions, which are attitudes or habits of mind, include open and fair-mindedness; inquisitiveness; flexibility; a propensity to seek reason, a desire to be well informed, and a respect for and willingness to entertain diverse viewpoints”. This case study therefore focused on both learners’ background knowledge in the mathematics subject, as well as their use of the content knowledge to solve mathematical word problems. 1.5 RESEARCH METHODOLOGY AND DESIGN The case study focused on trying to understand the Grade 3 math teachers’ practices on critical thinking and how they promote critical thinking within their lessons by making use of a case study qualitative research design. A case study can be defined as a detailed examination of a project’s uniqueness and complexity, from multiple perspectives (Simons, 2009). Rule and John (2011) similarly describes it as being systemic, and a thorough enquiry of a topic with the aim to generate knowledge in its context. This case study will be explored through interviews, teacher lesson observations, and learner assessments in interviews; all of which together will provide a detailed and in-depth method of data collection as derived from Petersen’s research on writing and mathematical problem solving in Grade 3 (Petersen, McAuliffe, & Vermeulen, 2017). The context was to observe the development of critical thinking as a support measure, as well as a fundamental learning technique, provided by teachers for learners to solve mathematical problems. A qualitative approach was used, being the most appropriate, as it stresses what Denzin and Lincoln (2011) describes as “the socially constructed nature of reality” which attempts to make sense of phenomena and connotations. Qualitative research is concerned with an understanding of: a complex phenomenon by examining it in its totality in context (Apple, 2004). The theory of constructivism was also used as an umbrella theory to which various other theories can be linked. The constructivism theory of learning holds that knowledge is acquired by building on previous knowledge, and that “…every learner constructs his/her ideas, as opposed to receiving them” (Selley, 2013). Siemon et al. (2013, p. 81) describe problem-solving, as indicated in CAPS, as “non-routine problems, higher order understanding and the ability to break a problem down into its component parts”. The investigation employed a research design similar to that of Petersen (Petersen, McAuliffe, & Vermeulen, 2017) which:.

(16) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 14. a. administered a pilot assessment to record the level of learners in the subject b. interviewed teachers to record their understanding of critical thinking. c. Teaching was then observed to identify the various ways in which critical thinking was promoted and learners were enabled to solve mathematical problems. d. Learners’ assessment scripts were analyzed to identify the various problem-solving skills that they used when solving a mathematical problem. This was used as an indication of how effective the teachers’ implementation of critical thinking was (Giannakopoulos & Luneta, 2015). e. Interviews with learners were conducted with the aim for learners to explain their thinking when solving the mathematical problems. 1.6 LIMITATIONS OF THE STUDY Due to limited funding, the case study was confined to two schools in the Midrand region of Gauteng, and the sample of the study was therefore small. Eighteen learners were purposively selected from two Grade 3 classes (Nine per class), and two teachers. The small sample limited the study resulting in the inability to apply its results to a broader population. A large sample may provide a broader perspective on how Grade 3 math teachers can promote critical thinking within their classrooms. Flyvberg (2006, p. 241) highlights that the advantages of a larger sample size include the facilitation of a widespread viewpoint. In most cases, a combination of quantitative and qualitative methods improved representation (Flyvberg, 2006). It is the viewpoint of the researcher that quantitative methods can enable researchers to firstly determine the extent to which math teachers understand critical thinking. Subsequently, qualitative investigations are useful to gain more profound insight. As one of the principals of a school in the case study, the researcher acknowledges and considers the potential bias to the research process (Rajendran, 2001). As the principal of one of the selected schools/Grade 3 classes, the researcher was close to the teachers who took part in the research and had “insider knowledge” of where to find outstanding practices within the classroom (Hammersley & Atkinson, 2007). This bias could have affected the sampling selection of the eighteen learners and the data analysis process (Morrell & Carroll, 2010). The validity of the data was ensured, however, by using multiple data collection instruments such as audio-recordings of teacher and.

(17) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 15. learners’ discussions and interviews. The validity of the data helped to secure an objective thesis report. The normal school program had an impact on data collection envisaged prior to the pilot study. Data could not be collected as planned. As a result, data collection was shortened to accommodate the assessment program of the school. The data was collected towards the beginning of term 3 due to the inability to collect data in terms 2 and 4 since schools are preparing for examinations. This resulted in time constraints and there was not sufficient time to conduct a full pilot study. It is acknowledged that there was some subjectivity when the data was interpreted, as per the nature of qualitative research. When data is interpreted subjectively, this tends to provoke the aspect of bias in the study, which could be considered as a limitation. According to Flyvberg (2006, p. 219), case studies are subjective and allow researchers to interpret their own data, and therefore the validity of the case study may be questionable. Case studies present real-life experiences and multiple case studies are essential in developing a subtle nuanced perspective of reality (Flyvberg, 2006). The mathematical problems used in the study were differentiated according to the expected number ranges of the three mathematical ability groups. The contexts, however, were identical for the problems across the ability groups. For some of the mathematical problems, it appeared that the number range was too low and did not present enough of a challenge for learners. This was evident in all the mathematical ability groups. At other times, some of the learners from the above average ability group were not sufficiently challenged by the problem. It appeared that either the context of the problem was too simple, or the number range was not suitable. Learners were not adequately encouraged to develop strategies that encourage a higher cognitive function: the solution and/or strategy may have been obvious. On other occasions, the context of the problem proved too perplexing for the below average learners. In addition, most learners found aspects of reading and language difficult. The context of the problems may have caused learners to have difficulty identifying the mathematical content within them. 1.7 OUTLINE OF THE CASE STUDY The case study is reported and outlined as follows: Chapter one presents the background and rationale for this case study, with its purpose. It outlines an overview and importance of critical thinking and problem-solving in mathematics, as well as.

(18) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 16. an overview of the methodology and significance of the study. Limitations of the study are also mentioned. Chapter two outlines the literature review relevant to the study, as well as the theoretical framework. It begins by acknowledging the importance of Vygotsky’s theory of social constructivism in teaching and learning by highlighting the Zone of Proximal Development (ZPD) and scaffolding (Petersen, McAuliffe, & Vermeulen, 2017). Critical thinking and problem solving are also discussed followed by the frameworks of problem-solving strategies with reference to the 5-Step model (Duron, Limbach, & Waugh, 2006) to move learners toward critical thinking that can be employed in the mathematics classroom to support problem-solving thinking strategies. Chapter three presents the case study as qualitative research as well as the research plan which describes the data collection plan. Subsequently the site and sample are discussed. The data collection instruments include learners’ written assessments, audio-recordings of ability group discussions, field notes and interviews with three learners, per class, selected from two Grade 3 math classes, per school. The process of data analysis is also explained. Chapter 4 presents the findings of this study, as well as the research results of teachers’ understanding of critical thinking, teacher lesson observations, analysis of learners’ assessments, as well as learner interviews. Chapter five extracts themes from the data gathered in Chapter 4 which are discussed to answer the sub questions and the main research question. Lastly, this case study makes recommendations, and possible areas of further research are highlighted..

(19) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 17. CHAPTER 2: LITERATURE REVIEW 2.1 INTRODUCTION The purpose of the case study was to investigate the methods that foundation phase teachers employ, as a teaching strategy, to promote effective critical thinking skills in a Grade 3 mathematics classroom. Previous results in the sample schools show that learners often struggle to solve word problems because they require a deeper conceptual understanding of mathematical ideas. The literature review discusses theories of learning and schools of thought in mathematics that relate to the research question. This chapter outlines the theoretical framework that supports the case-study, Vygotsky’s social constructivist theory, scaffolding the Zone of Proximal Development (ZPD) and other scaffolding concepts relating to the study. In addition, the chapter reviews literature on critical thinking frameworks pertaining to this investigation, including levels of problem-solving strategies with reference to the 5-Step model (Duron, Limbach, & Waugh, Critical Thinking Framework For Any Discipline, 2006) to move learners toward critical thinking that can be employed in the mathematics classroom to support problem-solving thinking strategies.. 2.2 THEORETICAL FRAMEWORK The research study emanates from the belief that learners rely on support provided by the teacher and peers for them to think critically when solving mathematical problems, and will, therefore, focus on identifying the various methods and practices that Foundation Phase (FP) teachers use to promote effective critical thinking skills within the classroom. The ability to think is a process that comes naturally to all, including learners, but thinking alone can be uninformed, prejudiced and possibly biased; showing that excellence in thinking must be cultivated (Duron, Limbach, & Waugh, 2006). Critical thinking does not develop by learners merely memorizing content, but by means of a praxis (Daniel & Auriac, 2011). The ability of learners to evaluate and analyze information, formulate their own questions and ideas clearly, assess and gather relevant information to solve math problems, develops and broadens their critical thinking skills. Constructivism is, therefore, used in this case study as an umbrella theory to which various theories can be linked. Constructivism is “a learning theory describing the process of knowledge construction” (Major & Mangope, 2012). It is based on the notion that through encouraging learners to actively contribute.

(20) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 18. in the lesson being taught, as well as their learning through social interactions, they learn to confront, develop and construct their own knowledge (Bhowmik, 2015). Major et al (2012) also mention that constructivism is a concept of learning with no prescriptive practice, but rather which informs both the methodology and pedagogy. Furthermore, constructivism can be categorized into two types, namely radical and social constructivism. Radical constructivism theory asserts that each individual constructs reality for him or herself. This means that learner’s experiences are subjective, and are filtered through the prism of individual perceptions, biases, and experiences (Shashidhar, 2011). The brain simply organizes experienced knowledge into reality. Conversely, social constructivism theory emphasizes the role of culture and its context in developing learners and their shared interpretations of reality (Cooperstein & Kocevar-Weidinger, 2004). Although both theories share the idea that reality is constructed, this case study makes use of social constructivism due to the researcher’s belief that this construction does not exist before its social invention. The shared belief is that knowledge is a product of a learning social process, and that social patterns shape learners’ meaning by the assumptions encapsulated in language (Shashidhar, 2011).. 2.2.1 Social constructivism Piaget, Bruner, Vygotsky, and Bandura among others, defined social constructivism as active learning that involves much more than encouraging class participation, encouraging and presenting hands-on activity, or having learners move around the room (Cooperstein & Kocevar-Weidinger, 2004). Social constructivism learning states that the learning experience follows the action instead of preceding it. Learners engage in activities, from the start of the lesson, where they develop critical thinking skills, instead of the traditional methods of learners copying what teachers say, “word for word” as a learning approach. According to Piaget, learners “construct increasingly complex ‘maps’ of their world to organize, understand and adapt to it” (Ackermann, 2001). The learner can progress from a concrete, preoperational stage to an intellectual and abstract stage according to Piaget’s developmental stages. Piaget’s theory is more concerned with the physical aspects of cognitive development in the construction of knowledge than interaction and culture. Whereas Vygotsky’s theory of social constructivism provides a theoretical underpinning that focuses on the role of parents, teachers, and peer community in constructing knowledge (Liu & Matthews, 2005)..

(21) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 19. Vygotsky’s theoretical system highlights as a central concept the role of social collectivity in individual development (Liu & Matthews, 2005). Like Piaget, Vygotsky also thought that learning is progressive and constructive, except for “spontaneous" and "scientific" concepts (Fosnot, 1996). Vygotsky reasoned that scientific concepts stem from a structured classroom instruction activity that imposes on the learner’s formal abstractions as well as logically defined concepts compared to those constructed spontaneously (Fosnot, 1996). He argued that scientific concepts are culturally agreed upon, and much more formalized. This research takes the view that social constructivist theory supports interaction and collaborative learning, and further agrees with the view that constructivism is an active formation of knowledge through learners’ construction and their social interaction.. 2.2.2 Zone of Proximal Development (ZPD) The theory of the Zone of Proximal Development states that learners learn through collaboration rather than as individuals, and through working with more skilled peers, teachers and/or parents, they can “…internalize new concepts and skills” (Shabani, Khatib, & Ebadi, 2010). This casestudy adopts the definition of ZPD as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (Vygotsky, 1987). Fosnot (1996) understood this as the ability of a learner to learn through effective teaching assistance, and that this zone is more pedagogically important than the belief that a learner can learn without assistance. This means that ZPD describes a learners’ intellectual level of development and how he/she can acquire new knowledge with the help of more skilled peers, parents, and/or teachers’ facilitation (Shabani, Khatib, & Ebadi, 2010). Learning within the ZPD makes use of the knowledge the learner already possesses as the foundation on which to construct prospective knowledge. In the ZPD, teaching represents the means through which development is advanced (Fosnot, 1996). Vygotsky’s theory of ZPD and social constructivism is believed to underpin the theory of scaffolding, discussed in the next section (Shabani, Khatib, & Ebadi, 2010)..

(22) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 20. 2.2.3 Scaffolding In educational research (Verenikina, 2003), scaffolding is referred to as “a way of operationalising Vygotsky’s concept of working in the zone of proximal development” (Well, 2000). In the ZPD, learners work together with peers who are more capable than those who cannot work independently due to the difficulty level of a task (Verenikina, 2003). The more capable learner, and/or the educator; peer or parent, supports the learning process by individualizing the process of problem solving (Wass, Harland, & Mercer, 2011). The social interaction between the learner and the more skilled individual is vital because together learners can develop a mutual understanding of the task and what is required to be achieved, thereby owning the development process together. In a mathematic classroom setting, the teacher is held responsible to develop and structure small steps and interactions based on what learners are already capable of performing on their own. The teacher will then, continuously, provide support until the learner can solve the problems independently.. The school in this study makes use of Singapore math as their math curriculum. Singapore math is a national mathematics curriculum method used from Grade R to Grade 6 in Singapore which enables learners to solve mathematical problems in an abstract way by using numbers and symbols. Research by Hofer (2015) involved evaluating a teacher’s impact on learner’s education, by introducing the Singapore bar model in a Grade 1 classroom and its effect on their ability to solve math problems. Hofer (2015) found that using the bar model to solve word problems enabled learners to link addition and subtraction concepts. Some learners could engage in higher order thinking and apply their knowledge identifying patterns to deepen their understanding. However, learners needed concrete experience to support the pictorial representations by means of manipulatives and scaffolding questions for them to profit from the model. Scaffolding can be described as the ability to help and enable a learner to acquire new knowledge and achieve control over a new function and/or learning concept under the guidance of a teacher, parent, or a more skilled peer (Maybin, Mercer, & Stierer, 1992). This study recognizes that scaffolding is a difficult concept to define (Granott, 2005), therefore, it examines the various support structures that can help learners develop their critical thinking skills. Constructivism conceives of learning as constructed individually and co-constructed socially by learners built on their prior knowledge, and interpretation of the world they live in (Liu &.

(23) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 21. Matthews, 2005). Rather, this study supports the view that the teacher’s instruction provides learners with experiences that can be interpreted and facilitates the learners’ individual and independent knowledge construction.. 2.3. PROBLEM SOLVING IN MATH. This section reviews different perspectives of mathematical problems. Solving problems in mathematics is explained as well as the use of word problems as a type of problem-solving exercise. The role of previous knowledge and conceptual development is elaborated upon: both relate to mathematical problem-solving. Various types of word problems are dealt with as they are presented in a mathematics lesson. 2.3.1 Problem solving Problem-solving may be defined as the ability to solve real-life problems that encourage the use of skills such as prediction and analysis. Kilpatrick (2001) believes that solving mathematical problems is the foundation of maths as a subject and should provide teachers with the ability to assess learners’ performances on various objectives within the framework. Problem solving enables learners to go through the ZPD by enabling them to take an active role, trying to solve a mathematical problem through collaboration (Hoven & Garelick, 2007). This study assumes that it is the teacher’s role to expose learners to different methods of solving problems through scaffolding, understanding important math concepts, analysing possible alternatives to the solution, and informing possible consequences that can be encountered when dealing with different types of word problems. This puts a focus on the process which promotes critical thinking (Hermanowicz, 1961). The ability to generate ideas further enhances the understanding that problem-solving requires higher order, critical thinking because solutions are, by definition, not immediately observable. The process of problem-solving may require learners to work through various possible solutions to solve problems. 2.3.2 Types of problem-solving Murray et al (Murray, Olivier, & Human, 1998) states that “learning occurs when students grapple with problems for which they have no routine methods.” They go on to say that there are different.

(24) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 22. kinds of problems in mathematics and thus mathematic tasks can have different kinds of problems, namely: “… Investigations, projects, traditional story sums, real-life problems, abstract problems, puzzles, etc”. For this paper, the researcher adopts the notion that a word problem in math can be defined as a multi-step operation whereby a learner is given mathematical values, with known and unknown quantities, with the expectation to solve/find the value of an unknown quantity (Charles, 2011). Charles (2011) outlines two ways in which teachers teach problem solving, namely the key word approach and the problem-solving step approach, before suggesting the bar model method as a means to solving mathematical word problems. The relevance of the bar model technique is important because the sample schools make use of Singapore Math which primarily uses this method as its means throughout the curriculum. According to Charles (2011), teachers ought to: “1. Model the bar diagram on a regular basis, not just in special lessons but frequently when word problems are encountered; 2. Discuss the structure of bar diagrams and connect them to quantities in the word problem and to operation meanings; 3. Use bar diagrams to focus on the structure of a word problem, not surface features like key words; 4. And lastly, encourage students to use bar diagrams to help them understand and solve problems.” 2.3.3 Examples of problem types The case study used different types of word problems to stimulate and develop learners’ problemsolving skills and their ability to solve problems through critical thinking. The word problems focused on basic mathematical operations such as adding, subtracting, multiplying and dividing; using whole numbers. Naudé and Meier (2004, p. 105) in (Petersen, McAuliffe, & Vermeulen, 2017) identifies three problem types that relate to the basic operations such as problems that involve adding and subtracting; repeated addition as a means to conceptualise multiplication; as well as grouping and sharing as a means to conceptualise division. Combining addition, subtraction, multiplication and division in the study ensured that the mathematical problems were presented in a way that enabled learners to use either operation as strategies (i.e. the problems were inverse operations). Some learners may have used addition as a strategy while other learners may have used subtraction to.

(25) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 23. solve the same problem. This duality also applied to the multiplication and division problems with learners able to use either strategy to solve a word problem. This enabled problem-solving to have multiple paths to a solution. 2.3.4 Invented critical thinking skills in problem-solving Fülöp (2015, p. 49) defines a strategy as the thinking aspect of problem-solving that is invented and flexible. She goes on to say that it is “an overarching idea involving arranging or combining what is otherwise discrete and independent with a particular end in view”. Critical thinking strategy involves making decisions while the doing aspect (methods and algorithms) entails implementing the decisions made. In addition, Campbell, Rowan, and Suarez, (1998, p. 49) states that, when learners invent their own strategies, they enhance their learning. About a project, Campbell et al. (1998, p. 49) found that “students often solved problems by inventing algorithms based on their interpretations of the problems, their understanding of arithmetic operations, and their representation of numerical relationships”. When learners are encouraged to explain their invented and personal strategies (Campbell, Rowan, & Suarez, 1998, p. 50), they display the ability to arrive at a solution, demonstrating the conceptual, procedural knowledge and critical thinking skills needed in the process. Murphy (2006, p. 219) adds that, when learners use their own critical thinking strategies, they rely on established mathematical ideas such as commutativity and associativity while they develop their mathematical reasoning abilities leading to critical thinking skills.. 2.4 MATHEMATICAL PRIOR KNOWLEDGE According to Rumiati and Wright (Rumiati & Wright, 2010), one needs to first understand if learners have prior number knowledge as an important part of teaching. This case study makes use of the works of Wright, Martland and Stafford (2006) and Wright, Martland, Stafford and Stanger (2006), as mentioned in Peterson et al (2017) as fundamental in understanding learners’ levels of conceptual knowledge and critical thinking skills used in tackling mathematical problems..

(26) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 24. 2.4.1 Learning Framework in Number The Learning Framework in Numbers (LFIN) is an instrument used to assess a learner’s number knowledge, as well as early arithmetic (Wright, 2013). It maps out numeracy knowledge based on four domains: “…the ability to count; forward and backward number word sequences; and lastly the ability to identify numbers” (Wright, 2013). LFIN includes the areas of learning numbers such as Stages of Early Arithmetic Learning (SEAL); which will be most applicable in this case-study, with an addition of early multiplication and division (Petersen, McAuliffe, & Vermeulen, 2017). These aspects of SEAL are most applicable to this case study because they relate to skills used by the sampled Grade 3 learners in solving the word problems provided in the learner assessments. Most learners in the sampled Grade 3 math classes were at a stage of their number learning where they coped well with number words, as well as numerals. Other aspects of the LFIN, such as structuring number strands and conceptual place value are more applicable to the number learning required in lower grades, because they were not areas of focus in defining and understanding the LFIN within this case study. 2.4.1.1 SEAL (Stages of Early Arithmetical Learning) SEAL is a model that outlines a learner’s early arithmetic learning, by observing the child while he/she solves the word and or mathematical problem (Rumiati & Wright, 2010). Table 2.1 shows the SEAL model, with its stages used to determine a learner’s level, tracking the ability of a child moving from counting items toward a more sophisticated approach of non-counting-by-one. Petersen et al (2017) states that “learners progress to at least stage 3 of the SEAL, and generally begin to develop base-ten arithmetical strategies.” Shown in table 2.2, the Base Ten Arithmetical strategies outline the levels and descriptions of whether a learner can count by ten or not ( (Rumiati & Wright, 2010).. Table 2.1: The Model for Stages of Early Arithmetic Learning (SEAL) Stage. Name of stage. Definition. 0. Emergent. Unable to count Objects.. counting.

(27) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 1. Perceptual. 25. Ability to only count objects that they can see.. counting. 2. Figurative. Ability to count objects that they cannot see, starting from one.. counting. 3. 4. Initial. number Ability to count on instead of starting from one, as well as solving. sequence. addition or missing addition tasks.. Intermediate. Ability to use the most effective count-down-from and count-. number sequence down-to strategy. 5. Facial. number Ability to use non-counting-by-one strategies, such as doubles,. sequence.. add through ten, compensation, etc.. Adapted from Rumiati et al (2010), and Wright et al (2006, p. 9) in (Petersen, McAuliffe, & Vermeulen, 2017). Table 2.2: The Model for development of base-ten arithmetical strategies Level Name of Level. Definition. 1. Initial Concept of Ten. Cannot perceive ten as a unit made up of ten ones.. 2. Intermediate. Concept Can see ten as a unit that is made up of ten ones.. of Ten. 3. Facile Concept of Ten. Ability to solve addition and subtraction tasks that involve tens and ones.. Adapted from Rumiati et al (2010), and Wright et al (2006, p. 9) in (Petersen, McAuliffe, & Vermeulen, 2017) 2.4.1.3 Early multiplication and division In addition to the CAPS curriculum’s expectation of learners to have a conceptual understanding of place value, addition, and subtraction, it also introduces learners to early multiplication and.

(28) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 26. division (Department of Basic Education, 2011). Furthermore, according LFIN, learners develop through five levels when learning multiplication and division, as shown in table 2.3.. Table 2.3: Model for early multiplication and division levels (Wright, Martland, Stafford, & Stanger, 2006, p. 14) Stage. Name of stage. Definition. 1. Initial Grouping. Ability to count by one and divide objects into groups of a given size.. 2. Perceptual counting. Ability to use multiplication counting methods to count objects in that they can see, organized in equal groups.. multiples. 3. 4. Figurative. Ability to use multiplication counting methods to count objects. composite. that are organized in equal groups in the case where the objects. grouping. cannot be seen.. Repeated. Ability to count composite units in repetitive addition or. abstract. subtraction.. composite grouping. 5. Multiplication. Ability to see the number in each group and the number of groups. and division as as a composite unit. They are able instantly to derive the basic operations. facts of multiplication and division.. Adapted from Wright et al (2006) in (Petersen, McAuliffe, & Vermeulen, 2017). According to Peterson et al (2017) Grade 3 learners can be expected to have a conceptual understanding of place value. They go on to say that “…Learners who have appropriately developed their understanding in this area make use of it as part of their strategies when thinking.

(29) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 27. critically to solve mathematical problems”. The next section of this chapter addresses Critical thinking in mathematics.. 2.5. CRITICAL THINKING. Known as “…the mental process that is organized and plays a role in the decision-making process to resolve problems” (Fisher, 2011), critical thinking analyses and assesses a way of thought with the intention to improve it. There are various definitions of what critical thinking is. It can also be described as the skill to decide what to accept as truth and what not to (Norris, 1985). Harris and Hodges (1995) defined it as a critical evaluation process to judge the impact and/or value of a text based on its quality examination. While Elder and Paul (1995) proposed the notion that it can be understood as a “…thinkers’ ability to take charge of their own thinking”. The concept of critical thinking was first expounded by Bloom (1956), who recognized six levels in the cognitive sphere, with each one linking to a level of cognitive skills. This taxonomy offers a straightforward method of classifying instructional tasks as they progress from easy to difficult. While the lower order of thought requires less thinking ability, higher order thinking would require more complex thinking. These are as follows: Lower order thinking: . Knowledge: To be able to remember and recite information.. . Comprehension: When one can relate and organize previously learned information.. . Application: To be able to remember information based on rules or principles.. Higher order thinking: . Analysis: To be able to analyse information and its functions.. . Synthesis: To be able to put information together to construct a new and/or original idea.. . Evaluation: To be able to value and judge knowledge based on information.. This case study adopts the definition of critical thinking as the ability to analyse and evaluate information, with the aim to apply cognitive skills. This definition points out the importance of a learner’s intellectual processes when learning which can develop through a learner’s environment. This links to active learning because it would require learners to take an active role in their “…perceptions, thoughts, beliefs, attitudes, and values in constructions, acquisitions, retrieval, or forgetting” (Schunk, 2008). In addition, this definition highlights the mental importance of.

(30) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 28. learners’ thinking abilities and development such as analysing, interpreting, inducing, and evaluating of information. These abilities are recognized as key foundations for critical thinking (Adler, 2000). 2.5.1 Active and Passive learning Critical thinking enables learners to “…raise vital questions and problems, formulate them clearly, gather and assess relevant information, use abstract ideas, think open-mindedly, and communicate effectively with others” (Duron, Limbach, & Waugh, 2006). Learners’ ability to critically think is very unlikely to increase within a lecture format lesson. Introducing new objectives sequentially instead of critically, can cause learners to memorize the content being taught rather than understand it, since a lecture format type of lesson only facilitates the rushed covering of large amounts of content (Bonwell & Eison, 1991). This instructional format places learners in a passive role, instead of an active environment with teachers doing all the speaking, thinking and enquiring. Active learning, on the other hand, encourages learners to critically think, and enables both the teacher and the learner to enjoy the subject. While it is useful for learners to be exposed to a subject’s content knowledge, learners will be unable to understand the content being taught, and see its meaning, unless the activities in the lesson are hands-on and promote active learning / participation (Duron, Limbach, & Waugh, 2006). This will be investigated in the study. 2.5.2 Singapore math The sample schools makes use of the Scholastics PRIME curriculum. This mathematics curriculum focuses on problem solving. It follows “…the approach used by the Singapore Ministry of Education that turned Singapore from a low-performing math nation into a high-performing one by making use of the Bar model technique to solve mathematical problems and a focus on word problems” (Hofer, 2015). The Singapore Math curriculum emphasises the importance of problem solving skills and conceptual understanding (Hoven & Garelick, 2007). It emphasizes essential math skills recommended rather covering several topics in depth. Learners are taught how to problem solve, making use of a bar model method. This has enabled the Singapore learners to excel in international math exams, year in and year out (Hoven & Garelick, 2007). The first time that learners come across the bar model is in Grade 3, and they are required to apply it to solve easy, one-step, mathematic word problems. They then apply this same.

(31) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 29. model to harder, multistep word problems in Grades 4, 5, and 6. When learners get to Grade 7, they have a solid foundation to step into algebra. The bar model enables learners to solve mathematical problems and represent these problems symbolically (Hoven & Garelick, 2007). Bar modelling is a simple “draw a picture mathematics problem-solving” strategy (Mahoney, 2012), and because Singapore makes use of this one method unwaveringly from one grade to the next, learners become familiar with what is expected of them (Mahoney, 2012). Learners find this model useful when solving problems involving part-whole calculations, rates of change, and comparisons (Hoven & Garelick, 2007) because it graphically communicates knowledge that learners already know and shows them how to use that knowledge to solve math problems. This visual method can be used to model any type of word problem, from multiplying and/or dividing, to addition and/or subtracting. The Ministry of Education in Singapore describes this method of teaching as being like the c-p-a (concrete-pictorial-abstract) teaching technique found in their primary math curriculum (Mahoney, 2012). Learners in Grades R to 3 progress from working with concrete materials, on to the abstract mathematical concepts. Bar models are used as a bridging step between learning with manipulatives, tangible objects and the math abstract concepts. Helping learners to better understand how to solve math problems is important and improving their math performance has a clear implication in enabling learners better to understand the mathematical problem, and thus think critically. The next section will discuss the 5-step critical thinking model (Duron, Limbach, & Waugh, 2006) that will be used throughout the research. 2.6 A MODEL TO PROGRESS LEARNERS TOWARDS THINKING CRITICALLY This interdisciplinary model is made up of five steps namely: “Determining the learning objectives; teaching through questioning; practicing before assessing; reviewing, refining and improving; and lastly providing feedback and assessing the content learned” (Duron, Limbach, & Waugh, 2006). 2.6.1 Step 1: Determining the Learning objectives When looking at the importance of an objective in providing the ground knowledge that will be used as the foundation of other objectives, it is vital for a teacher to highlight the core purpose of an objective which will determine the expected behaviour that learners ought to display at the end of a teaching session. The highlighted learning objectives and their main purpose, the pre-planned.

(32) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 30. activities linked to Bloom’s taxonomy of higher level thinking (Bloom, 1956), as well as the assessments of the lesson will result in critical thinking amongst learners (Duron, Limbach, & Waugh, 2006). The objectives should include the behaviours that are appropriate and linked to chosen thinking levels of Bloom’s taxonomy (Duron, Limbach, & Waugh, 2006). Bloom’s taxonomy was developed in the 1950’s as a model for teachers to plan educational and cognitive objectives. Its aim was to offer a hierarchy for the cognitive sphere learning objectives through six groups namely, the ability to know, to comprehend information, to apply knowledge, to analyse, synthesise, and evaluate information (Bloom, 1956), with the last four requiring higher order thinking processes related to critical thinking. Within the knowledge category, examples of questions that teachers can ask learners to answer are to describe, state, list, who and what. While the comprehension category would ask learners to answer questions that demonstrate understanding of the content and information being taught. Such questions could include summarizing, explaining, paraphrasing, comparing, and contrasting. Application is the last of Bloom’s lower order thinking categories and would ask learners to demonstrate abilities to use evidence, skills and models in new situations. Such enquiries would include and require learners to apply, solve, show, construct, and discover. Higher order thinking of Bloom’s taxonomy includes analysis, which requires learners to demonstrate abilities to classify information, theories and concepts into different parts. For example, learners might be expected to demonstrate skills in analysing, examining, differentiating and categorizing information. Synthesising would require learners to demonstrate abilities that link information from different areas to make an original idea. Learners at this stage would be able to use, manipulate and combine information. The evaluation category entails learners being able to critique information based on reasonable arguments. Math problems on this level may expect learners to evaluate, predict, assess, suggest and criticize objectives being taught. A teacher determining an objective would identify the specific behaviours he/she would want to see at the end of a lesson, introduce and ensure that learners practice those behaviours during the lesson, and complete the lesson by ensuring that learners are able to exhibit the required behaviours. For example, if the objective of a lesson to be taught is that “Learners will be able to classify shapes between two dimensional and three-dimensional shape”, to classify is a doing word, and, therefore, a verb. It is also a behaviour that can be recognized at the analysis level in Bloom’s taxonomy. The different shape groups would then be taught to learners through.

(33) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 31. questioning to develop the learners’ understanding, by asking questions such as: What are the different types of shapes; how can you identify a two-dimensional shape compared to a threedimensional shape? and define a three/two-dimensional shape. Once learners grasp and understand the different shapes, they can, for instance, be placed in groups, and asked to go around the class/school to identify objects, linking them to a shape. Each group can then name the object and what shape it is in a discussion with the whole class. Finally, in the last part of the lesson, learners can be individually questioned to classify different objects by shape category.. 2.6.2 Step 2: Teach through questioning. For teachers to actively foster understanding and stimulate critical thinking within learners, they must pose arguments, ask questions, and critique learners’ evidence. This is a vital teaching and learning procedure because, when teachers focus on activities, experiences, or interventions clearly linked to the lesson objectives, they grow and involve more learners in the lesson, enabling deeper and more critical thought throughout the lesson (Smart & Csapo, 2007). Such activities would include direct activities/experience with a concept, indirect activities that simulate learner experiences and enable learners to relate to situations and reflect through interactive questioning discussions that promote critical thinking. Examples of such questions would be: What clue can you find that tells you how to solve the equation? How would you solve the problem differently? How would you adapt to…? Although there are various methods that can influence learners’ critical thinking, a teacher’s questions have the most effect (Clasen & Bonk, 1990) because teaching through questioning fosters the thinking skills of learners. These questions can be differentiated in various ways such as using the convergent and divergent types of questions. Convergent questions differ from divergent questions because they seek just one correct response, whereas divergent questioning seeks a wide range of correct responses. Within Bloom’s taxonomy, Convergent questioning applies to lower order thinking of knowledge and application. As mentioned above, lower order thinking allows for answers to questions such as “Define a right angle,”. For teachers to encourage learner participation, teachers must actively plan and become highly skilled questioners. This would mean teachers would have to ask brief and specific questions, rephrase questions, prompt explanations to answers from learners, use a variety of techniques to redirect responses from one.

(34) THE IMPLEMENTATION OF CRITICAL THINKING IN GRADE 3 MATHEMATICS CLASSROOMS: A CASE STUDY. 32. learner to the other, spread questions to different learners in the classroom, provide feedback and reinforcement objectives without repeating answers. For learners to experience higher level questioning once they learn a new concept, Elder and Paul (1995) proposes that “…students learn math by asking questions about math…”, and for them to take the content seriously, teachers must ask genuine questions which seek to inspire critical thinking in the classroom. Teaching through questioning and active learning enables teachers to establish learners’ prior knowledge and later extend beyond that by developing new understandings. This can also be used to stimulate teacher-learner interaction and for learners’ opinions and answers to be challenged by their peers. For teachers to effectively teach through questioning, they would have to pre-plan and prepare questions developed for the appropriate level to accomplish the lesson objectives and consider how each of their questions can support learners to think critically. 2.6.3 Step 3: Practice before you assess. “Practice is necessary to master any skill…” (Duron, Limbach, & Waugh, 2006); and education systems have taken a major shift to make learning more active by “…adding experiential learning and opportunities for reflective dialog, to enhance learners’ overall learning experience” (Duron, Limbach, & Waugh, 2006). Active learning can be described as involving learners in the teaching and learning process with activities designed to prompt them to think about what they are doing (Duron, Limbach, & Waugh, 2006). This concept of learning indicates that learners retain knowledge and learn more when they obtain it by actively participating in the lesson rather than just listening (Duron, Limbach, & Waugh, 2006). This therefore means that teachers must give learners the chance to practice the content, skills and concepts taught, as well as the behaviours and attitudes that teachers have set out to evaluate towards the end of a lesson. Duron et. al (2006), suggests that teachers can follow two main principles that ought to be well-thought-out when choosing a learning activity to enable learners to take responsibility for their individual knowledge and for teachers to promote an engaging learning environment. Teachers should clearly define what is expected of learners, and carefully track learners’ involvements. In addition, they might choose activities from the following techniques of active learning (Duron, Limbach, & Waugh, 2006):.

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