Unit 2 Theories of teaching and learning mathematics

Unit 2Theories of teaching and learning mathematics

Department of Mathematics/ICT - KMCE 1

Objectives

By the end of the unit, the student will be able to;

• explain theories of learning

• key characteristics of learning

• explain the philosophical perspective of Behaviorists, cognitivists and Constructivists learning.

• discuss Jean Piaget and cognitive stages of development, Jerome Bruner and instructional phases, Z.P. Dienes and multiple embodiment and variability principles, and Richard Skemp and concept formation.


How will you explain theories of learning to a friend who is not doing this course?


Definitions of a theory

• A theory is a supposition or system of ideas intended to explain something, especially one based on general principles independent of the thing to be explain.

• A theory is a set of principles on which the practice of an activity is based.

• A theory is an idea used to account for a situation or justify a course of action.

• Learning theory therefore describes how students receive, process and retain knowledge during learning.


What are some of the key characteristics of learning?


Key characteristics of learning

• Learning involves change

• All learning involves activities

• Learning requires interaction

• Learning is a lifelong process

• Learning occurs randomly throughout life

• Learning involves problem solving

• Learning is a process of acquiring experience


Outline the main tenets of Behaviourist theory of learning


The main tenets of Behaviourist theory of learning

• Learning is a connection between stimulus and response / response and reinforcement

• Change in behavior as result of an individual response to stimulus that occur in the environment

• Learning is acquisition of a new behaviour through conditioning

• Rewards and punishment

• Edward L. Thorndike, B. F.Skinner, Pavlov and Robert Gagne are the main proponents of behaviourist

• The experiment by Pavlov which conditioned a dog to salivate to sound of bell(classical conditioning)


How will you define reinforcement from the behaviourist point of view?

• Reinforcement is an event which increases the frequency of the response that follows it.


Robert Gagne• Behavioural response of the learner follows some form of instruction.

• Arranging the conditions to bring about the most effective learning of intellectual skills, cognitive strategies, verbal information, motor skills, and attitudes (i.e. five categories of capabilities).

• Teaching and learning should be very specific or goal directed. It must also be based on task analysis.


How do we apply behaviourist learning theory in mathematics classroom?


We can apply behaviourist learning theory in mathematics classroom by:

• Reinforcing the correct responses and discourage wrong responses.

• Controlling the stimuli, choose the correct response and providing the

appropriate reward.

• Providing feedback to learners immediately.

• Using marks, prizes and praise as different reinforcements.

• Arranging lessons into series of graded steps(i.e. programmed learning).

• Providing opportunities for practice(i.e. drill and practice).Department of Mathematics/ICT - KMCE 12

Gestalt psychology

• Gestalt is German word literally means configuration or pattern or form.

• Gestalt psychology is a school of thought that looks at the human mind and behaviour as a whole rather than attempting to break up into small parts.

• They perceived the whole of anything to be greater than any of its parts.


Cognitive /developmentalists Theory of Learning

• Principles that underlie cognitivists learning theory:

1. Learning consists of changes in mental constructs and processes; learning takes place when the learner mentally reorganizes his/her inner world of concepts.

2. Learning is a personal experience and teachers only need to serve as a facilitator.

3. Learning is something that happens as a result of thinking.

4. The child cannot learn the same content as adults and even among children there are individual differences to consider.


Cont.

• For meaningful learning to take place, new information needs to fit in with existing cognitive structures.

• Learning is a process in the learner is actively involved.


Effective application of cognitivism requires that the teacher needs to:

• Know about the students’ previous learning.

• Assist the student in developing meaning by providing puzzles and rules for the students to work with.

• Provide structures or help the students to create a structure to which they can add a new information.

• Employ the inductive-deductive approach of teaching.


Jean Piaget

• Jean Piaget was a Swiss psychologist who has provided numerous insights

into the development of human intelligence, ranging from the random

responses of the young infant to the highly complex mental operations

inherent in adult abstract reasoning.

• Piagetian theory indicates that in learning, children pass through

developmental stages and that the use of active methods which gives scope

to spontaneous research by the child helps him rediscover or reconstruct

what is to be learned “not simply imparted to him”.Department of Mathematics/ICT - KMCE 17

Concepts of assimilation, accommodation and equilibrium

• Assimilation involve fitting new situations into existing psychological

framework.

• Accommodation involves modification of behavior. In accommodation, the

individual modifies or expand those internal structures to conform to the new

information.

• Equilibrium is the balance between assimilation and accommodation.


Developmental stages according to Piaget• Sensory motor(0-2 years of age): This stage is where children begins to use

imitation, memory and thought. The child moves from reflex actions to goal-directed activity.

• Pre-operational (2-7 years) : This is the stage where children gradually develop language and the ability to think in symbolic form.

• Concrete operational(7-11 years): This is the stage where children are able to solve concrete(hands-on) problems in logical fashion.

• Formal operation(11-15 years): this is the stage where children are able to solve abstract problems in logical fashion.


Conservation of numbers and volume

Conservation of numbers: This experiment was to see if a child realizes that the number of element in a set remains unchanged even as the set is physically rearranged.

• Conservation of volume: This experiment was to determine whether the child understands that volume is unchanged under a certain type of physical rearrangement.

• Conservation of length has to do with the child being able to recognize that the length of an object remains the same no matter the different positions or inclinations


Jerome Bruner

• Brunner was interested in the general nature of cognition(i.e. Conceptual development).

• Bruner was of the view that any subject can be taught effectively in some intellectually honest form to any child at any stage of development(Bruner,1966).


The three sequential phases of learning by Bruner

• Concrete which involves hands on activity.

• Semi concrete which involves the use of visual medium.

• Abstract which involves the use of symbols.


The three phase/levels of acquiring sequential thinking by Bruner

• Enactive level which involves hands on or direct experience: The student is engaged in first hand manipulating, constructing, or arranging of real-life objects. In other words, students interact directly with the concrete materials.

• Iconic level or imaginary phase: This phase is characterized by the use of visual medium which is dominated by visuals and perceptual organizations such as films, pictures, diagrams: It is semiconcrete in nature.

• Symbolic level involves the students being able to use abstract symbols to represent reality.


Implication of Bruner’s work in teaching and learning

• Textbooks can hardly provide enactive experiences but they are exclusively iconic and symbolic containing pictures of things(physical objects and situational problems or tasks) and the symbols associated to the things are not the things themselves.

• Mathematics programmes that are dominated by textbooks are inadvertently creating a mismatch between the nature of the learner’s needs and the mode in which content is to be learned by students.

• A mathematics programme that does not make use of the environment to develop mathematical concepts eliminates the three levels of representing mathematical ideas.


Cont.

• Textbook activities need to be supplemented with real-world experiences. Mathematics programmes should include more manipulatives and more experiences in applying mathematical ideas in the real world.

• Manipulatives aids help learners to move from concrete situations and problems to abstract ideas.


Zoltan P. Dienes

• Dienes’ theory combines some ideas of Piaget and Bruner. He describes how mathematics is to be learnt and how it should be taught (both prescriptive and descriptive).

• Every learning should start with applications which the learner can actually experience and then progress to formal mathematical summary.


Dienes’ six-stage theory of learning mathematics

• Stage 1 :Preliminary free play stage: Children interact with physical materials within the environment. Different embodiments provide exposures to the same basic concepts but at this stage few commonalities are observed.

• Stage 2: Structural activities stage: Following the free experimenting, some regularity appears in the situation, which can be formulated as rules of the game.

• Stage 3: Intuitive abstraction activities or comparing stage: At this stage, patterns, commonalities and regularities are observed and abstracted across the models.


Cont.

• Stage 4: Representation of activities stage: This involves the use of images and pictures to provide representations.

• Stage 5: Examination and descriptive activities or symbolization stage: This stage involves the description of the representation in mathematical symbols and checking or verifying results.

• Stage 6:Axiomatic Formulation of activities or formulation stage: This where the fundamental rules and properties are recognized as structures of the systems.


Implication of Dienes’ work

• The whole class or large group lesson would be greatly deemphasized in order to accommodate individual differences in ability and interests. Individual and small group activities will benefit students learning.

• The role of the teacher would include exposition as well as being a facilitator where students’ role would be expanded to assume a greater degree of responsibility for their own learning.

• The newly defined learning environment would create new demands for additional sources of information and direction such as a learning laboratory containing materials such as computers.


Constructivism learning theory

• Constructivism states that learning is an active process in which learners construct new ideas or concepts based on current /past experience.


Application of the constructivist learning theory

• Teachers should tailor teaching strategies to student responses and encourage students to analyze, interpret, and predict information.

• Teacher and learners can negotiate meanings of actions and words as they interact.

• Learners must be encouraged to verbalize mathematical thinking, to explain and justify mathematical solutions and to resolve complicating points of view.

• Learners are to be allowed to assess themselves and have regular dialogue with teachers. They should feel free to comment why they are have not achieved a particular learning goal.


The teacher in a constructivist teaching and learning environment should:

• Orchestrate discussion among learners.

• Encourage learners to verbalize the mathematics they are constructing when doing activities.

• Encourage learners to make connections between different aspects of mathematics.

• Encourage learners to explain and justify their solutions.


The learner in a constructivist teaching and learning environment should:

• Discuss work with peers and teacher.

• Pose and solve own problems, work in groups and generate mathematical problems.

• Verbalize the mathematics they are constructing when doing activities.

• Make connections between different aspects of mathematics.


Unit 2 Theories of teaching and learning mathematics

Documents

Transcript of Unit 2 Theories of teaching and learning mathematics