1. Psychology for Learning Examining the way people learn and the implications for learning mathematics Dr Karen Skilling

2. Why is this important for teaching? Learning is defined as any process in living organisms that leads to permanent capacity change and not solely due to biological maturation or ageing (Illeris, 2007) Learning is influenced by extensive and complicated processes Conditions for learning are also influenced by contextual, environmental, social and affective factors some of which are internal and others external to the learner Your approach to pedagogy and the teaching practices you use means that you will be influencing your students learning and must also take account of individual learners needs

3. How do we learn Changes in performance – valuing repetitive practice and feedback Development of conceptual understanding – reflecting, cognitive conflict and discussion Socialisation process – where the learner is an apprentice

4. Theories about learning Developed from different political and philosophical orientations Most are not theories in the strictest sense of the word but ‘perspectives’ that offer alternate ways of discussing teaching and learning

5. Theories about learning Descriptive not prescriptive – not recipes Not theories of teaching but do provide alternative orientations and approaches to teaching practice and curriculum design

6. Shifts in theories of learning Behaviourism – dominant until 1970’s Assumes a stimulus-response mechanism for learning Thorndike (1922)proposed: Law of exercise – responses to a situation become associated with it Law of effect – responses followed by satisfaction were likely to recur and less likely if associated with discomfort

7. Neo-Behaviourism Emphasised making behavioral objectives for learning explicit Gagné defined objectives as observable outcomes and learning involved practicing performance until it became fluent before moving to more complex performances Programmed learning and schemes e.g. textbooks, work cards – small and imitative steps – simple to complex

8. Constructivism From the 1960’s a student-centred focus emerged Strongly influenced by the work of Piaget Children construct individual conceptual frameworks – ‘discovery learning’ (Bruner) Socio and socio cultural constructivism – role of teacher and significant others (Vygotsky)

9. Influence of Psychology Cognitive psychology - literally means “knowing”. Cognitive psychology is the study of human mental processes and their role in thinking, feeling, and behaving. Developmental psychology - scientific study of how and why human beings change over the course of their life. Educational psychology - concerned with the scientific study of human learning.

10. What does the cognitive science approach mean for mathematics? Cognitive – internal reps of learner’s knowledge Mathematical – structural mathematics representations Symbolic – external representations of a notation Explanatory – models and theories that account for cognitive structures and processes Computational – such as hypothetical representations in computer programs based on observed human behaviours. Quality based on number AND strength of connections

11. Jean Piaget (1896-1980) Swiss biologist and psychologist -asked to develop a standardised reasoning test using pupils in a school in Paris Piaget became interested in the patterns emerging from students’ incorrect responses Then undertook his own research to consider the thinking processes behind these

12. Jean Piaget Using tasks with physical materials and questioning Piaget investigated how children think and learn The responses to the intellectual tasks were interpreted as reflecting children’s level of reasoning Noticed that children of like age groups responded in similar ways and were markedly different from adults responses and expectations

13. Theoretical Concepts The learner constructs theories by the processes of assimilation and accommodation Assimilation – absorption of new ideas into our frameworks Accommodation- modifications made to cognitive structures as a result of ‘fitting’ new ideas into an existing framework

14. Theoretical Concepts Processes of assimilation and accommodation operate simultaneously to achieve intellectual balance Cognitive Conflict/perturbations Equilibrium is sought – change/extend

15. Theoretical Concepts Reversibility (reversible cognitive processes) Seriation studies – ordering Transitivity studies-handling two way relationships e.g. A>B, B>C

16. Stages of development StageAge RangeCharacteristics Prelogical stages Sensori-motorBirth- 2yearsCo-ordination of physical actions Preoperational2-7 yearsAbility to represent action through thought and language; prelogical Advanced logical thinking stages Concrete operational 7-11 yearsLogical thinking but limited to physical reality Formal operational 11-15 yearsLogical thinking, abstract and unlimited

17. Conservation Tasks Test a child's ability to see that some properties are conserved after an object undergoes physical transformation Piaget proposed that an inability to conserve is due to a weakness in the way children think during the pre-operational stage (age4-5) At the stage the focus is on the static characteristics of objects instead of focusing on the change

18. Conservation relevant to Mathematics of… Number Length Liquid amount Solid amount Area Weight Displacement volume

19. Conservation Tasks www.youtube.com/watch?v=gnArvcWaH6I

20. Issues at the Concrete-Operational Stage (Thornton, 1982)

21. Lev Vygotsky (1896-1934) Russian psychologist Interested in the relationship between thinking and learning language – underlying focus of social constructivism is conversation Acquisition emphasises social interaction and less on stages of behaviour Learning is about internalising the language and actions of others

22. Lev Vygotsky Language and cognition emerge and develop about the same time but are distinct processes When considering development both what the child can do already and the child’s potential The distance between these he termed the ‘zone of proximal development’ (ZPD)

23. Lev Vygotsky Vygotsky believed that learning was mediated by cultural and metacognitive tools With the aid of stimuli designed by teachers, the learner reacts and uses speech and symbols to organise thoughts and actions Speech is internal- egocentric-inner Speech provokes metacognition – self control

24. Internalisation Process Primitive Stage – do activities that do not require verbal operations Practical Intelligence – having a grasp of language forms and practical problem-solving ability but not linked External Means – use language/tools to help internal problem solving Thinking becomes internal and language/tools are for reflection and development

25. Implications for Mathematics Discussion is an important element for internalising mathematical ideas Children learn effectively from each other Mathematics should be incorporated into meaningful tasks E.g. teaching the theorem that the sum of 3 inner angles of a triangle would present manipulations with students discussing/manipulating leading to geometric proof

26. Piaget and Vygotsky Born in the same year/new of and critiqued each others work Piaget emphasised the individuals construction of knowledge and his approach is based on individual schemes Vygotsky emphasised internalisation based on the social relations between individuals and then within individuals

27. Compatibility? Some criticism of waiting for ‘readiness’ – stage theories Some elements about the mechanisms for learning are - may influence teaching design E.g. – perturbations may be helpful for promoting accommodation E.g. – encouraging discussion and reflection rather than individual methods

28. Jerome Bruner (1915-present) Jerome Bruner – American psychologist (cognitive and later applied to education) Social constructivist - emphasising the importance of language and shared understandings to construct knowledge and meaning Interested in how needs, motivations and expectations influence perceptions i.e. not passive

29. Jerome Bruner It was, we thought, an all-out effort to establish meaning as the central concept of psychology…it focused upon the symbolic activities that human beings employed in construction and in making sense not only of the world, but of themselves (Bruner, 1990, p.2)

30. Scaffolding and Spiral Curriculum Scaffolding – helpful interactions that assist the child in taking on new learning Spiral curriculum – children capable of abstract thought at all ages with increasingly complex ideas (e.g. in enactive mode then later modes)

31. Conceptions of the mind Imitation – apprenticeship like Didactic exposure Children as thinkers Children as knowledgeable

32. Three modes of representation Enactive – doing things/action based Iconic – images and sense of things Symbolic – using agreed symbols Overlapping and entwined but not hierarchical

33. Implications for Teaching Mathematics children's 'readiness' to learn is not linked to age development of language is important to concept formation adults are important in structuring and supporting children's developing ideas new concepts (regardless of the age) should be taught enactively, then iconically and, finally, symbolically as ways of capturing experiences in the memory it is important to include practical activities and discussion as an integral part of mathematics Do you think the concept of a 'spiral curriculum' is reflected in the National Curriculum?

34. Richard Skemp(1919-1995 Richard Skemp – mathematician who later studied psychology and drew on both to explain how mathematics is learnt Believed that children could learn intelligently from an early age, producing a complete curriculum framework for age 5–11 known as Structured Activities in Intelligent Learning.

35. Richard Skemp http://www.skemp.org.uk

36. Schema Concepts Conceptual Structures Schema – indispensable tools for the acquisition of further knowledge - efficient -preparation of mental tools to use again - consolidating earlier content of the schema -reduces cognitive load

37. Instrumental Understanding A mechanical, rote kind of learning Is usually easier to understand Rewards more immediate and apparent Can often get the right answer more quickly However learning a ‘method that works’ is an impoverished kind of learning and leads to weak understandings of the nature of mathematics

38. Relational Understanding Ability to deduce specific rules or procedures from more general mathematical relationships More adaptable to new tasks Easier to remember An effective goal in itself- self motivating Relational schemas are organic – allow for development

39. Two Kinds of Learning Instrumental and relational learning - both are important for mathematics e.g. see Watson pp.34-36 When might different kinds of learning take precedence? What do you observe takes place more often – instrumental or relational learning?

40. Activities Compare: Piaget and Vygotsky (group 1) Piaget and Bruner (group 2) Bruner and Vygotsky (group 3) Skemp and Bruner (group 4) Skemp and Vygotsky (group 5) Skemp and Piaget (group 6) Time line and key contributions (group 7)

41. References Bruner, J (1960) The process of education. Harvard University Press, Cambridge Bruner, J (1990) Acts of meaning. Cambridge, MA. Harvard University Press Daniels, H (Eds)(2005) An introduction to Vygotsky. Routledge, London Labinowicz, E (1980) The Piaget primer. Addison-Wesley Publishing, Menlo Park Skemp, R (1987) The psychology of learning mathematics. Routledge, New York Smith, L., Dockrell, J., & Tomlinson, P (Eds)(1997) Piaget, Vygotsky and beyond. Routledge, London Swan, M (2006) Collaborative learning in mathematics. NRDC, London Thornton, M.C (1982) Piaget and mathematics students (Paper from the ADAPT Program) Watson, A (2006) Raising Achievement in Secondary Mathematics. New York, Open University Press. Wertsch, M (1985)Vygotsky and the social formation of mind. Harvard University Press, Cambridge