Learning theories in

mathematics

Outcome

1. Explain some learning theories that can be applied in Mathematics teaching; by using examples, demonstrate the application of these learning theories and show how you would adapt your teaching to suite the level of development of the learners.

introduction

• How do learners learn Mathematics?

• How can learners be assisted to gain an insight in the structure of Mathematics?

Behaviourism

• between 1920s to 1950

• Ivan Pavlov (1849-1936)

• John B. Watson (1878-1958)

• concerned with observable behaviour

• Observable (i.e. external) behaviour can be objectively and scientifically measured.

• Behaviour is the result of stimulus – response

• Later learning depends on earlier concepts being grasped

• Sometimes learners need to drop back a level to reinforce what they have learnt before.

• Behaviourism emphasises the effect of punishments and rewards in learning.

• Positive rewards for correct answers and disapproval for wrong answers result in learners practising until they can perform the steps automatically.

• The emphasis is on developing Mathematics skills (drill and practice) and less on concepts and strategies.

• A weakness is that learners are often unable to transfer their skills to alternative contexts.

Constructivism

• Emphasizes the building or construction that occurs in peoples’ minds when they learn.

• Prior learning is a key component: build on what the learner knows.

LEARNING THEORIES

(a) Hilgard:

• Learners’ capacity varies with age

• Motivation to learning makes the fixing of the learning material easier

• Intensive motivation (anxiety, tension) distracts the attention from the task

• Success and reward – more beneficial outcomes than failure and punishment

• Intrinsic motivation is better than extrinsic motivation

• Success experiences lead to an ability to tolerate failures

• Setting own goals – realistic formulation of aims

• Personal history – influence on reaction towards educator

• Active participation rather than passive acceptance

• Meaningful assignments and responsibilities are easier to learn than meaningless contents

• Exercise leads to automatic response

• Learning is supported by knowledge of success and failure

• Transfer is supported by discovery and experience

• Spaced-out reviewing helps to fix contents

Cognitive constructivismPiaget (1896-1980)

• three fundamental processes, which contributed to the child’s cognitive development., namely assimilation, accommodation, and equilibrium.

• Assimilation involved the incorporation of new events into pre-existing cognitive structures.

• Accommodation is the adjustment involved in the formation of new mental structures needed to accommodate new information.

• Accommodation is the adjustment involved in the formation of new mental structures needed to accommodate new information.

• When a child experienced a new event, disequilibrium set in until he was able to assimilate and accommodate the new information and thus attain equilibrium.

4 phases of development

(1) Sensor-motoric phase (0-24 months)

• Use of reflexes

• First habits and primary circular reactions

• Coordination of vision and understanding

• Coordination of secondary schemes and application to new situations

• Differentiation of action schemes and solving of some problems by deduction

(2) Pre-conceptual phase (2-7 years)

• Appearance of symbolic function and the beginning of internalised actions accompanied by imaging

• Learns to use language and to represent objects by images and words 

• Thinking is still egocentric: has difficulty taking the viewpoint of others 

• Classifies objects by a single feature: e.g. groups together all the red blocks regardless of shape or all the square blocks regardless of colour 

(3) Concrete operational phase (7-11 years)

• Imaging organisations based on either static configuration or on assimilation

• Imaging regulations are stressed

• Can think logically about objects and events 

• Achieves conservation of number (age 6), mass (age 7), and weight (age 9) 

• Classifies objects according to several features and can order them in series along a single dimension such as size. 

(4)Formal operational phase (11 years and up)

• Can think logically about abstract propositions and test hypotheses systematically. 

• Becomes concerned with the hypothetical, the future, and ideological problems 

• Hypothetic-deductive logics and combined calculations (11-14 years)

• Structuring and grouping of 4 transformations (14 years - …)

(c) Van Hiele3 levels (niveaux) of how a child learns Mathematics:

(1) Base niveau

• Concretely – there is a visual, tangible differentiation of concepts

(2) First niveau

• Verbal description of concepts (intuitive descriptive definitions). Qualities are tabulated. Generalisation takes place.

(3) Second niveau

• Exploration of logical relations between qualities, e.g. 6 = 2 x 3 because 6 = 3 + 3

Achievement of niveaus by means of the learning process – not by biological ripening

Following phases during learning progression from one niveau to the next:

(a) Information: The child becomes familiar with the context of the specific field of study.

(b) Bound orientation: The learner comes into contact with the most important combinations of the relations net to be formed, for example, two rows of three should be read as “three together with three more”, or as “twice three”, or seen from a different viewpoint, as “three times two”, etc.

(c) Explicitising: Here the language of the field of study is learnt, and the learner can pronounce it.

(d) Free orientation: The learner learns to find his/her way in the relations net using the available combinations, for example, he/she learns what he/she can do with multiplication and division and when he/she should use it.

(e) Integration: A synopsis of the various ways of thinking. General rules are drawn up and own actions are reconsidered.

Social constructivism

• Vygotsky (1896-1934)

• Emphasizes the importance of social interactions in the learning process.

• Vygotsky and Activity Theory

• Emphasizes the collaborative nature of learning.

• Learner must be actively involved in the learning process.

• Zone of proximal development (ZPD) which is defined as the “distance between the actual development 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.

• This zone is seen as the gap between the actual development level and the potential level a learner can reach.

• The way this zone can be crossed is through mediation by a more competent peer.  

• “more knowledgeable other,” (MKO). The MKO in a community of practice might be a teacher who represents a ‘keystone’ species (master teacher). The role of a keystone species as mediator is that of providing collaborative dialogue and scaffolding to assist other in their development.

• Motivation according to this theory is seen as both extrinsic and intrinsic and teachers learn through team work and gain knowledge as they develop by way of social interactions with peers.

Reuven Feuerstein

• born 21 August 1921 in Botoşani, Romania

• Disagreed with Piaget who considered cognitive development as a result of maturation and experience with objects.

• Human interaction is essential and calls it mediated learning experience (MLE).

• Renowned for his theory of intelligence which states “it is not ‘fixed’, but rather modifiable”.

• “Intelligence is not a static structure, but an open, dynamic system that can continue to develop throughout life”.

• Intelligence can be modified through mediated interventions.

• These interlocked practices provide educators with the skills and tools to systematically develop students’ cognitive functions and operations to build meta-cognition.

• The theories of Structural Cognitive Modifiability and Mediated Learning Experience and a definition of intelligence as "the capacity of the individual to use previous experience in his adaptation to new situations."

• A belief that all learners can learn, irrespective of age or stage of development

• Mediated learning experience

• Dynamic assessment

• A cognitive development programme, known as instrumental enrichment.

• He believes that human cognitive structures can be modified and that all individuals can change the way they think and learn at any age or stage of development.

• He stressed a social and cultural origin of cognitive development or intelligent behaviour.

• Cultural beliefs, customs and habits are passed on, or mediated, by the older to the younger generation.

• He suggested two ways of acquiring cognitive processes:

1. By direct learning through interaction with the stimuli from the environment

2. By mediated learning experience (MLE) via interaction with a human mediator.

Howard Gardner

• (1943- )

What is a learning style?

Learning styles

• Sensory pathways through which learners prefer to receive information.

• It is a biologically and developmentally imposed set of characteristics that explain why the same lesson, readings, interactions, classroom settings and teachers affect individuals differently.

three categories

1. Perceptual modalities that define biologically based reactions to our physical environment and represent the way we most efficiently adopt data.

2. Information processing distinguishes between the way an individual senses, thinks, solves problems and remembers information. (Refer to De Bono’s hats).

3. Personality patterns focus on attention, emotion and values. These reflect the way an individual will react, feel about and value different situations.

perceptual modalities: learning styles are the visual, auditory, kinaesthetic and tactile.

(a) Visual learners prefer seeing what they are learning. As teachers we must bring a wide variety of visuals and colour to our classrooms: posters, charts, models, pictures, slides, etc.

(b) Auditory learners prefer spoken messages and learn by listening and verbalizing. Some individuals may even need to hear their own voices as they construct mental dialogues. In these cases we should provide opportunities for oral presentations, debates, discussions, radios, videos, films, question-and-answer sessions, and so on.

(c) Kinaesthetic learners want to sense the position and movement of what they are working on. We can accommodate these learners best in field trips, simulation games, role playing sessions, and other activities encouraging movements.

(d) Tactile learners want to touch and feel learning materials. By including laboratory activities, field trips and other forms of practical work, allow this group to handle and manipulate equipment, materials, and objects we accommodate these learners best.

Assessment

1. Discuss behaviourism, cognitive constructivism and social constructivism as learning theories and give examples how these learning theories are applied in Mathematics education.

2. Discuss perceptual modalities as a learning style category and the implications it have on your teaching of Mathematics?

School experience learning guides

Study material division

D1 Lab K05

5-22 February 2014