This section provides an overview of the theories that assist mathematics teachers to effectively teach mathematics. Generally, theories help teachers to plan instruction, communicate the instruction, assess students’ performance, evaluate the teaching process and keep professional ethics (Machaba, 2013). In simple terms, a theory on teaching endeavours to give an explanation to teachers of how human beings acquire new knowledge and in so doing it assists them to understand the complex process of learning. Illeris (2004) defined learning as a process that involves emotional, cognitive and environmental influences and experiences for gaining, enhancing or making changes in a person’s values, skills, knowledge, behaviour and views. According to Hill (2002), theories of teaching and learning have two main functions. They provide teachers with the vocabulary to use in giving instruction and a conceptual framework to interpret the examples of learning that the teachers observe. The theories also suggest possible solutions to practical problems faced by the teachers and their learners. However, these theories may not provide the teachers with the exact solutions to the problems but they could direct the teachers towards the possible solutions.
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Theories that are used in teaching mathematics can be categorised into three categories, namely behaviourism, cognitive and constructivism (Machaba, 2013).
2.2.1 Behaviourism learning theory
According to Kelly (2012), behaviourists deal with observable aspects of the learning process. They believe that individuals change their behaviour due to a relationship between stimuli and responses. Bush (2006) stated that according to behaviourists, not all learning outcomes are worthy studying. Only those that can be observed and measured are worthy scientific inquiry. That is the reason why teachers who follow the behavioural learning theory frame their lesson objectives in behavioural and measurable terms. Bacanli (2012) explained that behaviourists view human brain as a black box. One cannot see the contents of the box.
However, to a behaviourist it is not important to know what is inside the box. What matters is what gets into the box and what comes out of it. Therefore, behaviourists are not keen to study the human brains, but they are interested in studying the information that the human brains receive and the reaction of the individual after receiving the information.
According to behaviourists, learning begins when a learner responds to a cue or a stimulus and learning is judged by a change in behaviour (Kelly, 2012). The change in behaviour is predictable and controllable (Weegar & Pacis, 2012). That means a teacher can easily predict a students’ reaction to an instruction. The teacher can also control how the students respond to an instruction. Kelly stated that, according to behaviourism, the learning process can be reinforced so that the desired outcome can be maintained. For instance, getting a good grade can serve to reinforce good studying behaviour by a student. Kelly went on to state that when teachers reward or punish students, they apply the principles of behaviourism. Rewarding students for good performance reinforces hard work done by students. Punishing students for doing wrong things serves to make sure that the students desist from repeating the same action.
Weegar and Pacis (2012) reported that, according to behaviourism, the learning environment can affect the learning process. Some environments lead to poor performance by student whilst some encourage students to work hard. According to Weegar and Pacis, all students have potential to perform well provided the learning environment is conducive to effective learning.
However, as stated by Kelly (2012), behavioural instruction does not train students to be creative thinkers or to be competent problem solvers. It teaches students to follow rules,
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formulae and algorithms without making any changes and without bothering to prove them.
According to Kelly, examples of behaviourism teaching strategies include drilling or rote teaching, repetitive practice and provision of incentives or verbal comments like ‘good work’.
As stated by Kelly, behaviourism theory was founded by Skinner and Watson.
2.2.2 Cognitive learning theory
According to Machaba (2013), cognitive theories endeavour to explain brain based learning.
Unlike behaviourists, cognitivists believe that individuals process new information or new experiences rather than simply responding to cues. They believe that behaviour change can be noticed on individuals after a learning process, but the change in behaviour is a result of processes that take place inside the individual’s brains. When the learner receives new knowledge, the learner relates it to the knowledge already existing in the learner’s brains. The new knowledge is processed and included in the existing knowledge structure in the learner’s brains. According to Kelly, the founder of the cognitive learning theory is Jean Piaget.
According to cognitive theory, learning is defined as change in what the learner knows and keeps in his or her memory and not just a change in behaviour. The cognitive theory explains how the human brain works. It recognises that mental processes are affected by both intrinsic and extrinsic factors. Learning difficulties are a result of ineffective mental processes (Sincero, 2019),
Sincero (2019) reported that cognitive learning theory can be divided into two more specific theories which are social cognitive theory and cognitive behavioural theory. Social cognitive theory deals with three variables or factors that are interrelated. The variables are behavioural factors, environmental factors and the personal factors. Environmental factors are extrinsic factors while personal factors are intrinsic. Sincero explained that a combination of the three factors affects the way a learner thinks and behaves. Interaction between personal factors and environmental factors modifies the beliefs, ideas and cognitive competences of a person.
Interaction between personal factors and behavioural factors affects the way a person behaves. When environmental factors and behavioural factors interact, the way a person behaves is also affected. The interaction can also result in change in the environment.
Therefore in the cognitive theory perspective, for effective learning to take place, positive personal characteristics, appropriate behaviour and a conducive environment are necessary and mandatory.
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According to Sincero (2019), from the social cognitive theory’s viewpoint some basic cognitive aspects are noticeable in people. The aspects include observational learning, reproduction, self-efficacy, emotional coping, self-regulatory capability. Observational learning means learning from others by simply observing how they operate. Reproduction involves putting a learner in an environment that encourages the learner to repeat certain behaviour. Such an environment should provide all the required materials to enable the learner to retain new knowledge or skill for a long time. Self-efficacy involves putting a new skill or new knowledge into practice so that it can be retained for a long time. Emotional coping refers to development of coping mechanisms that enable the learner to learn effectively in environments that are not favourable. Self-regulatory capability refers to a person’s ability to control his or her behaviour even in situations that maybe difficult for the person to control himself or herself, for instance, in situations that are tempting.
Cognitive social theory involves the use of cognition to make predictions or to determine the behaviour of an individual ( Sincero, 2019). Cognition simply means knowledge. According to the cognitive social theory, individuals develop self concepts about their behaviour. Self concepts refer to what individuals think about themselves. An individual can think positively or negatively about himself or herself. The self concepts control the behaviour of the individual. They can be controlled by the environment in which the individual lives.
Cognitive triad is used in cognitive social theory to explain human behaviour. Cognitive triad comprises thoughts about oneself, the world and one’s future. The thoughts can be negative, for instance thought like ‘I am a fool’. Such thoughts control the behaviour of a person, as explained by cognitive social theory.
Cognitive theory is applicable in many situations in teaching and learning. Examples of situations where ideas from cognitive theory are applied in mathematics teaching include situations in which teachers use the following strategies in teaching: problem solving, discussions, analogies and examples from real life situations.
2.2.3 Constructivism learning theory
Constructivists use ideas developed by theorists like Piaget, Vygostky and Bruner. According to Machaba (2013), constructivism theories take learning as a process in which the learners actively build new knowledge and concepts. It is based on the belief that individuals build their own perspectives of the world through their own experiences and internal processes.
According to constructivists, learning is unique and it differs from individual to individual
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depending on how the individual interprets the world. According to McLeod (2019), individuals make their own meanings and interpretations of the world. The meanings and the interpretations are a result of an interaction between the individual’s prior and new knowledge and experiences. As explained by McLeod, teachers and their students share authority and knowledge. Although the teacher acts as a facilitator in the process in which the students learn, the teacher also learns from the students. McLeod also suggested that a teacher who teaches according to the constructivism theory should make use of small groups which consists of students who are heterogeneous. The reason for using small groups is to make sure that each member of the group gets an opportunity to participate actively in group activities. Heterogeneous groups ensure that students with different abilities and who interprets the world differently mix and share their perspectives of the world.
To a constructivist, learning outcomes are not predictable. Therefore a teacher who follows a constructivist approach to learning cannot tell the outcome of his or her teaching process before delivering the lesson. When teaching mathematics, constructivism theory can be applied in the following situations among others: brainstorming, problem solving, guided discovery learning and collaborative learning.
Constructivists define learning as how the individual learner interprets and creates own meaning of the new experiences that the individual learner receives. The learning process starts with the learner making some hypotheses. The learners are actively involved in the learning process, in which they construct knowledge depending on their prior and new knowledge and experiences. The role of the teacher is to facilitate learning and to negotiate meaning with the learners and not to dictate an interpretation to the students (Driscoll, 2015).
The teacher gives help to the students whenever it is needed, especially in the early stages of learning a concept.
According to McLeod (2019), constructivism theory has the following five principles:
Knowledge cannot be passively absorbed
This forms the central idea in constructivism theory. The theorists believe that knowledge cannot be transferred from one person to the other. Self discoveries are important in order to gain an understanding of concepts.
Learning is a process
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Learners need to be actively involved in the learning process. The learner can be actively involved in the learning process by carrying out experiments, making inquiries and through problem solving. An expert, for example a teacher, should only serve to guide the learner.
Knowledge is socially constructed
According to constructivists, a learner acquires knowledge through interaction. The learner interacts with peers and experts in the area under study. Constructivists believe that the environment in which the learner lives affects the meanings that the learner attaches to the world (Vygostky, 1978).
Knowledge is personal
Learners can be taught by the same teacher using the same teaching strategies and under the same environment, but the learners understand the concepts learnt differently.
Learning takes place in the mind of the learner
Individuals develop their own mental models of the world around them. The process takes place in the individual’s mind.
The theories of learning discussed in this section provide a framework for various studies on teaching and learning. They form the basis or the foundation of the theory on learning styles.
The learning styles theory borrows ideas from the three categories of the theories of learning discussed in this section.