LEARNING THEORIES AND TEACHING METHODS WITH SPECIAL REFERENCE TO ACTIVE LEARNING

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CHAPTER THREE: EMPIRICAL EVIDENCE OF INFLUENCES OF/ON ACTIVE LEARNING OF MATHEMATICS AT UNIVERSITY

INTRODUCTION

In the previous chapter (Chapter two), some learning theories and their associated teaching methods were explained. This chapter presents empirical evidence of the results of active learning of mathematics at universities. Furthermore, this chapter explores factors such as the attitudes of lecturers and students, the training of lecturers, support from academic department heads and deans, class size, and instructional material that affect the implementation of an active learning approach of mathematics at university.

THE EFFECTS OF USING ACTIVE LEARNING METHODS IN MATHEMATICS LEARNING

The goal of this section is to understand the effects of active learning methods in mathematics learning at university. There has been much work done on the effect of active learning and teaching methods on student cognitive learning. The relationship between active learning/student-centred teaching methods on student learning has consistently shown a positive effect (of such approaches) on students’ cognitive and affective outcomes. Particular attention is given to the effects of active learning on students’ cognition, motivations, their attention to and emotional response to learning, and the value they attach to learning. Investigations of the effect of the teaching approach, particularly an active learning/student-centred teaching approach on students’ cognitive and affective learning has consistently shown positive effects. In general, active learning methods are more effective than lecturer-centred/traditional learning and teaching methods for achieving a variety of learning outcomes (Timmermans & Van Lieshout, 2003:11-16). The cognitive and affective effects of active learning are discussed in the next two sections.

Cognitive effects of active learning on the student

The goal of an active teaching and learning process in mathematics education at university level should be to establish a learning culture that promotes intelligent learning and deep understanding of mathematical concepts. The knowledge of the learned mathematical concepts could in such a case be called conceptual knowledge (Setati & Adler, 2000:263-265). In short, the before mentioned researchers (Setati & Adler, 2000:263-265) found that conceptual knowledge and relational understanding is generated and intelligent learning occurs when the student is given a chance to actively create rich structures of cognitive connections within and between mathematical concepts. Hence, the teaching procedures and interpersonal relations in the mathematics classroom should provide a framework that enhances such a creative and active learning culture. The research calls this type of framework an active learning approach.
Active learning is an approach that is extremely effective in maintaining students’ information processing, developing skills, attitude and interest. The responsibility for learning is focused on the students. Most importantly, to be actively involved, students must engage in such higher-order mathematical thinking tasks as analysis, synthesis and evaluation. Students are involved in acquiring information and interpreting or transforming it. To do all this, time must be provided within the curriculum. Norman and Schmidt (2000:727–728) argue that optimal student participation in the teaching and learning process is imperative to ensure that the students are able to effectively practice self-regulated learning methods. Research underlying the active learning/student-centred approach confirms that learning is nonlinear, recursive, continuous, complex, relational and natural in humans (Burbach, Matkin & Fritz, 2004: 482-493). An active learning approach which is based on constructivist theory helps students absorb knowledge and make connections in their mind, understanding not just what they learn, but how they learn (Benek-Rivera & Mathews, 2004:104).
Active teaching and learning methods offer opportunities for interaction between lecturers and students, amongst the students themselves, as well as between students and the materials (Schaeffer, Epting, Zinn & Buskist, 2003:135-136). Students are expected to become active learners who can demonstrate what they know and do by applying their knowledge and skills to real problem-solving situations. According to O’Sullivan and Copper (2003:449) if students are not actively involved in their mathematics learning, they will less likely to construct personal meanings or retain the lesson. Rather they simply memorise answers to the questions that will appear on tests. Research has shown that active learning is an exceptionally effective teaching and learning approach (Chou, 2004:18-21). Doerr and Lesh (2003:19-21) assert that in mathematics education students cover more material, retain the information longer, and enjoy the class more through active learning methods compared to lecturer-centred learning and teaching methods (such as lectures). McNair (2000:560-565) notes that the application of active learning methods (cooperative, inquiry, discussion, discovery and problem-based learning methods) in mathematics education help students to make connections to and apply mathematical knowledge in the real world. Boyer (2002:49-51) also argues that the use of an active learning approach in the classroom enables students to apply mathematical concepts and to foster meaningful learning.
A study conducted on mathematics education shows that an inquiry-based learning and teaching process is superior to lecturer-centred/traditional instruction for cognitive learning, which includes conceptual and subject learning, reasoning ability and creativity, as well as for non-cognitive learning including manipulative skills and attitudes (Aleven & Koedinger, 2002:171-176). Stead (2005:122-128) also asserts that an inquiry-based method is likely to be more effective than lecturer-centred teaching methods in helping students gain understanding of concrete observable phenomena. He recommends the planning of activities around questions that students can answer directly via investigation and activities oriented towards concrete concepts. He also places an emphasis on use of materials for which students have the prerequisite skills and on activities that involve situations familiar to students. In addition, lecturers need to pose a sufficient level of challenge to help students develop better thinking skills.
As regards cognitive effects, cooperative learning has proven itself to be superior to other methods (Harton, Richardson, Barreras, Rockloff & Latané, 2002:13-14). Evidence to this claim is in fact abundant and there is little disputing that the active learning/student-centred approach which encourages active, collaborative and constructivist learning improves students’ learning in more ways than one. For example, a very important outcome of active learning/student-centred approach, which is often less noticeable, is its effect on students’ approach to learning. Working together with fellow students, solving problems together and talking through material together has other benefits as well (Johnson & Johnson, 1999:68-70): student participation, lecturer encouragement, and active learning/student-student interaction positively relate to improved critical thinking. These different activities confirm other research and theories stressing the importance of active learning, motivation and feedback in thinking skills as well as other skills. This confirms that discussions are superior to lectures in improving thinking and problem solving in mathematics education (Johnson & Johnson, 1999:78-80). Students who collaboratively work with peers in active learning situations are able to identify solutions to problems, develop negotiation and mediation skills, distribute cognitive responsibilities amongst members and externalise thinking through explaining ideas to peers (Tan, 2005:38).
The students’ ability to perform logical operations as described by Piaget in active learning in mathematics education is manifested in their ability to solve word problems involving those logical operations. Evidence of attainment of thought processes at Piaget’s levels of intellectual development can be gathered through investigation of their problem-solving skills. An indicator of the acquisition of the problem-solving skill is the ability to articulate one’s problem-solving solutions and reason these out adequately. For instance, one significant finding came from a study by Duch et al., (2001:3-11). They investigated the levels of cognitive achievement of university freshmen using the test of logical operations. Interviews were conducted to investigate the adequacy of their reasoning patterns. This study showed the relation between the levels of cognitive skills achievement of university freshmen and their formal reasoning patterns. A significant relationship was drawn between reasoning abilities and cognitive skill achievement in mathematics. This study showed that 61% of the university freshmen were at the concrete level. This study also revealed that as an individual goes through the four successive cognitive levels of performance, expertise on reasoning develops progressively. This study further provided evidence that there are certain logical operations that are not fully developed even at the university level. An investigation in this study showed that more than 50% of the university students have inadequate understanding of the concept of ratio and proportion as they exhibited ambiguous reasoning patterns during the interview (Duch et al., 2001:3-11). Active learning by using problem-solving skills can facilitate problem solving at more abstract levels.
Furthermore, Siciliano (2001:12) argues that, when students participate or share in an active learning method in the mathematics lecture room, this serves to appropriate the purpose that actuates it; students become familiar with its methods and mathematical contents, acquire needed cognition and skills, and are saturated with its emotional spirit. Contemporary approaches to active learning/student-centred methods build on these early foundations but place greater emphasis on the context of theory-practice integration, learning communities, and implementation of a wide range of active learning in mathematics education (Gupta, 2005:48-50). In the present context of societal change and significant educational reform, active learning provides a flexible and multifaceted approach to meet the diverse needs and circumstances of students in university. Therefore, the lecturer’s role in choosing worthwhile problems and mathematical tasks is crucial in implementing active learning approach. By analysing and adapting a problem, anticipating the mathematical ideas that can be brought out by working on the problem, and anticipating students’ questions, a lecturer can decide if particular problems will help to further the mathematical goals for the class in active learning. There are many problems that are interesting and fun but that may not lead to the development of the mathematical ideas that are important for a class at a particular time. Choosing problems wisely and using and adapting problems from instructional materials, is the difficult part of using active learning methods in teaching mathematics (Blumberg, 2007:11-125; NCTM, 2000:53).
Furthermore, lecturers with an active learning/student-centred approach assist the students in learning content focusing on thinking skills. By asking questions and providing access (to what?), they interpret, organise and transfer knowledge which is important to solve authentic problems in the content areas being studied, and in daily life as well. Thus, an important role of the lecturer is choosing learning problems and situations that make students actively involved and stimulate interest in understanding how mathematics is applied in real-world situations (Holton, 2001:41). The development of a supportive classroom environment can also serve to enhance student motivation for learning mathematics (Boyer, 2002:48-50; Hines, 2002:275-278).
Students may transfer the skills that they acquire through active learning to other learning tasks. When students engage in active learning when set problem-based learning tasks, several steps are followed. These steps are: meet the problem, define the problem, gather facts about the problem, hypothesise solutions to the problem, research the problem, rephrase the problem, generate alternatives and advocate solutions to the problem (Angeli, 2002:9-15). Many of these steps align with standards in several disciplines apart for mathematics (NCTM, 2000:42). Moreover, a lecturer in a university may use a problem-based learning task to encourage students to investigate standard deviation while a lecturer in another university may use the same problem-based learning task to encourage students to investigate correlation (Normala & Maimunah, 2004:16).
The findings of a study conducted by Wilson et al. (2005: 83-91) asserts that students who frequently solve problems related to mathematics topics and discuss practical problems using active learning methods (cooperative, inquiry, discussion, discovery and problem-based learning methods) tend to score higher mathematics test scores than other students. It is shown that the results of students whose lecturers frequently requested them to do mathematics problems during typical lessons achieved better than students whose lecturers discussed and completed the given activities themselves. Thus, the frequent use of an active learning approach is significantly related to high mathematics test scores.
Further, central to the goals of active learning like cooperative learning methods in mathematics education is the enhancement of achievement, problem-solving skills, attitudes and inculcation of values. Several research findings asserted the positive effect of cooperative learning on academic achievement and problem-solving skills. For instance, the study conducted by McConnell (2005:35-38) shows that the experimental group that applied the active learning method (cooperative learning) significantly achieved better results in mathematics and problem-solving skills than the control group that was instructed in the lecturer-centred, traditional lecture manner. McConnell also finds that students instructed in the active learning method had a favourable response towards group work. Other researchers also report findings that assert the achievement benefits of cooperative learning (Siciliano, 2001:15-18).
Several research findings assert that as an inquiry-based learning method, active learning has a positive effect on students’ achievement. For example, Hill, Rowan and Ball (2005:398-402) find that an inquiry learning method improved the academic achievement and critical thinking skills of students. An analysis of the results of 81 experimental studies on thousands of students shows that the inquiry learning method produces significant positive gains for academic achievement, student perceptions, process skills and analytic abilities (Steinberg, Empson & Carpenter, 2004:252-261).
However, the results of the different studies differ. For example, Healey and Roberts (2004:31–34) note that: (a) the effects on academic achievement (i.e., reading, writing, mathematics) of a lecturer-centred educational approach was generally found to be more effective than the active learning/student-centred approach; (b) research that compared active learning and lecturer-centred approaches found an interaction with mathematics class such that lecturer-centred approach was particularly beneficial for lower achievement students. The data on the effect of the two approaches were either equivocal or non-existent for middle-class students. Interestingly, some data suggest that the active learning approach may have a negative effect on the achievement level of low-achieving students who are unable to engage in the desired behaviours required by this approach; (c) the advantages of individualised learning (i.e., different pace for different students, choice of what and how to learn about a topic, and learning style differences) have not found empirical support. This finding was particularly true for lower achieving students; (d) learning by groups and by lecturer-led instruction leads to higher achievement; (e) the methods (derived from the work of Thorndike and Skinner) which have the greatest positive effect on achievement use cues, engagement, corrective feedback, and reinforcement and are more likely to occur in a lecturer directed context;
(f) while there is a paucity of data on the comparative effects of lecturer-centred and student-centred approaches at the different levels.

Affective effects of active learning on the student

The affective effects of active learning on the student include psychological notions as such as feelings, emotions, moods, interest, motivation and values. Active learning also impacts on behavioural changes, students’ self concept, self-esteem and social interactions in the learning environment.
Various studies on these emotional effects have been done highlighting different aspects:
Shen, Leon, Callaghan and Shen (2007:274) state that:
It has become evident that effective teaching is not a question of putting information across a group of students. Rather, it is more of a question of initiating behavioral change in every student… Indeed, it has become clear that students learn in dynamic social learning environments in which the various interactions continuously influence each other, thereby changing the leaning situation itself as well as their own appraisal of the situation. Theories of learning that focus exclusively on information processing cannot grasp this complexity.
Researchers such as Schaeffer et al. (2003:133-136) and Shen et al. (2007:267-278) support an active learning approach. This method considers the contextual information of the students and the learning setting, and generates appropriate responses to the student, based on their emotional state, cognitive abilities and learning goals. This method can also be used to customise the interaction between the student and the active learning process, to predict students’ responses to behaviour, and their interactions with the active learning process.
Zan and Martino (2007:165-168) point out that emotions could provide feedback to lecturers in the classroom and to fellow students in team work. An active learning approach enables lecturers to recognise the emotional states of their students and respond in ways that positively affect students’ learning. Lecturers can provide a solution for problems via real-time feedback to students’ emotional states. This is valuable because emotion plays an important role in interaction, involvement and development. Hence, the lecturer should be aware of the students’ emotional states while organising group discussion so as to enhance the information flow within the group by smoothing the emotion flow.
Webb et al. (2002:15-18) agree that active learning has emotional effects on learners. The use of an active learning method, such as cooperative learning, increases students’ motivation for working on mathematical proofs and thus resulted in improved achievement. According to Webb et al. (2002:13-20), students who have been taught by this active learning method developed interactional (communications) skills, such as how to ask for help and help each other. Such skills had resulted in positive outcomes like an increase in intrinsic motivation, love for the university and improved self-esteem (Savin-Baden & Wilkie, 2004:18-21).
Effective implementation of active learning approaches also has other positive effects in mathematics education. In this regard, the study conducted by Bush (2005:122-
134) on active learning involving 966 students and using Jigsaw structures, found that active learning in cooperative learning experiences inculcated values such as independency, love and cleanliness. Gutstein (2003:63-72) from the results of his investigation conducted using Jigsaw as a model and including 180 sample students from eight universities concluded that the values of self-dependence, rational thinking, love and hard working are prominently inculcated by an active, cooperative learning method. It was also found that this method enhances students’ mathematical skills and achievement, and promotes enquiry learning.

The effects of active learning on the lecturer

Learning by doing is not an innovation in mathematics education; however, there is great emphasis on active learning techniques today. It is recognised as an important teaching and learning method and the findings of several researches have shown the effectiveness of this approach. Lecturers’ attitudes towards teaching-learning approach influence their teaching behaviours and the selection of methods. Lecturers with positive attitudes to active learning implement active learning/student-centred methods to provide opportunities to their students to participate in the teaching-learning process. This approach involves all aspects of the lecturers’ personalities – their opinions, attitudes, cognitions, feelings, and insights – to be involved in the mathematics teaching and learning process. It is the only paradigm explicitly aimed at the personal, social, and cognitive growth of the facilitator of learning (the lecturer) and provides satisfaction from the interaction with their students.
In an active learning approach, the role of the lecturer is to facilitate (rather than lead), or to coach students’ personal learning via a guided discovery approach (Dewey & Meyer, 2000:268, Tan, 2005:39). Emphasis therefore is on interpersonal values. This makes the lecturer aware of his/her own values.

Summary

To sum up, the findings of different studies indicate that:
Students become more actively engaged in solving mathematical problems when an active learning method like cooperative learning is employed in mathematics education. Thus, reluctant students, those who previously did not do their work, begin to participate in the problem-solving process.
In active learning students move from a competitive to a cooperative stance. That is, they begin to share their problem-solving skills while solving the problems and discussing their answers to mathematical problems rather than competing for the correct answer. They also learn different ways of solving problems in general and specific mathematical terms.
The classroom lecturer becomes more aware of students’ abilities when they work in small groups. Some students who do not normally participate in whole group activities are actively involved in small group work (Fink, 2002:16; Gupta, 2005:26-39; Zehr, 2004:55-56).
Active learning methods such as collaborative learning and cooperative learning enhance the accountability of students for their learning. Problem-based learning enhances students’ retention and ability to apply material (Prince, 2004:223-231: Zehr, 2004:55-56).
As an active learning method, problem-based learning has a robust positive effect on students’ skill development and understanding the interconnections among concepts (Gijbels, Dochy, Van den Bossche & Segers, 2005:46-58), and on deep conceptual understanding, ability to apply appropriate meta-cognitive and reasoning strategies (Novick & Bassok, 2005:335-342),. Problem-based learning has also been shown to promote self-directed learning and the adoption of a deep (meaning-oriented) method to learning, as opposed to a superficial (memorisation-based) approach (Blumberg, 2007:11-125; Kroesbergen, Van Luit & Maas, 2004:233-251).
Classrooms designed for an active learning approach include more hands-on activities and provide opportunities for students to discuss their solutions with each other; teaching incorporates problems based on realistic situations (Benek-Rivera & Mathews, 2004:104). There is a positive correlation between the amount of time students spend on discussion of learning activities and high mathematics test scores (Alexander 2002:36-37; Dixon in Kroesbergen et al., 2004: 233-251).

TABLE OF CONTENTS
CHAPTER ONE INTRODUCTION AND OVERVIEW
1.1 INTRODUCTION AND RATIONALE FOR THE STUDY
1.2 PROBLEM STATEMENT AND RESEARCH QUESTIONS
1.3 AIMS OF THE RESEARCH
1.4 SIGNIFICANCE OF THE STUDY
1.5 DEFINITION OF CONCEPTS
1.6 RESEARCH DESIGN AND METHODOLOGY
1.7 THE DIVISION OF CHAPTERS
1.8 SUMMARY
CHAPTER TWO THEORETICAL FRAMEWORK: LEARNING THEORIES AND TEACHING METHODS WITH SPECIAL REFERENCE TO ACTIVE LEARNING
2.1 INTRODUCTION
2.2 THEORIES OF LEARNING
2.3 TEACHING METHODS THAT INFLUENCE ACTIVE LEARNING
2.4 ADVANTAGES AND DISADVANTAGES OF ACTIVE LEARNING
2.5 SUMMARY
CHAPTER THREE EMPIRICAL EVIDENCE OF INFLUENCES OF/ON ACTIVE LEARNING OF MATHEMATICS AT UNIVERSITY
3.1 INTRODUCTION
3.2 THE EFFECTS OF USING ACTIVE LEARNING METHODS IN MATHEMATICS LEARNING
3.3 FACTORS AFFECTING THE IMPLEMENTATION OF ACTIVE LEARNING IN MATHEMATICS EDUCATION
3.4 SUMMARY
CHAPTER FOUR RESEARCH DESIGN AND METHODOLOGY
4.1 INTRODUCTION
4.2 RESEARCH PROBLEM
4.3 RESEARCH DESIGN AND METHODOLOGY
4.4 RESEARCH METHODS
4.6 SUMMARY
CHAPTER FIVE RESULTS AND DISCUSSION
5.1 INTRODUCTION
5.2 ANALYSIS OF BIOGRAPHICAL DATA
5.3 ACTIVE LEARNING/STUDENT-CENTERED APPROACHES AND INFLUENCING FACTORS IN MATHEMATICS EDUCATION
5.4 DISCUSSION OF RESULTS
5.5 SUMMARY
CHAPTER SIX CONCLUSIONS, RECOMMENDATIONS AND LIMITATIONS
6.1 INTRODUCTION
6.2 CONCLUSIONS
6.3 RECOMMENDATIONS
6.4 LIMITATIONS OF THE STUDY
6.5 SUMMARY
LIST OF REFERENCES
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