Learning with understanding

Learning with understandingThe story about Benny

Learning with understanding

Learning with understanding

Learning with understandingBenny's rules and other examples build up to: •Criticism of Programmed learning

•Lack of focus on developing concepts and understanding can make students develop inexpedient and wrong understanding/rules

•Students create their own understanding no matter what

•Exaggerated focus on rules in math can make the students focus on remembering rules instead of understanding rules as meaningful generalizations  

Learning with understandingIt is not enough that students learn to master rules and procedures.They must understand in order to know when and how to apply.

This has the implication that mathematics must appear as a coherent body of concepts and methods instead of isolated rules to remember.

Learning with understanding5 characteristics of learning with understanding (Carpenter and Lehrer 1999). The student

1. constructs relations between what is learnt and the new to be learnt

2. develops, expands and utilizes/applies mathematical knowledge

3. deliberately reflects on and inquires new concepts and methods instead of unconscious adaption

4. expresses subject understanding in speech, writing, drawing, diagram and symbols

5. makes the subject matter her/his own. So students consider knowledge as something quite different from something you have just been told.

Learning with understandingChildren without algorithms

Learning with understandingBehaviorism (Skinner)

Simple model: Stimulus Response

Positivistic through it’s intention of basing psychology directly upon observable facts avoiding models for mental phenomena which are not observable

The idea of learning as a direct response to stimulus (teaching) implicates that you can test teaching methods like you test medicine, which is of course nonsense

Problem: You cannot consider a teaching learning process through such a simple model.

CognitionPiaget got interested in children’s patterns of systematic thinking. Through a comprehensive research program based on semiclinical interviews of many children Piaget and his co-workers found some regularities in children’s thinking including systematic differences between different age groups.

This result is well known as his stage theory and central concepts like assimilation and akkomodation.Source: ATHERTON J S (2010) Learning and Teaching; Assimilation and Accommodation [On-line] UK: Available: http://www.learningandteaching.info/learning/assimacc.htmAccessed: 9 January 2011

Learning with understandingStage  Characterised by 

Sensori-motor  (Birth-2 yrs) 

Differentiates self from objects  Recognises self as agent of action and begins to act intentionally: e.g. pulls a string to set

mobile in motion or shakes a rattle to make a noise 

Achieves object permanence: realises that things continue to exist even when no longer present to the sense (pace Bishop Berkeley) 

Pre-operational  (2-7 years) 

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 

Concrete operational  (7-11 years) 

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. 

Formal operational  (11 years and up) 

Can think logically about abstract propositions and test hypotheses systematically  Becomes concerned with the hypothetical, the future, and ideological problems 

Stages of Cognitive Development

CognitionAdaptation  What it says: adapting to the world through assimilation and accommodation Assimilation  The process by which a person takes material into their mind from the environment,

which may mean changing the evidence of their senses to make it fit. The environment almost fits the existing schemes

Accommodation  The difference made to one's mind or concepts by the process of assimilation.  Note that assimilation and accommodation go together: you can't have one without the

other. The environment dos not fit – schemes must be adjusted or restructured 

Classification  The ability to group objects together on the basis of common features. Class Inclusion  The understanding, more advanced than simple classification, that some classes or sets

of objects are also sub-sets of a larger class. (E.g. there is a class of objects called dogs. There is also a class called animals. But all dogs are also animals, so the class of

animals includes that of dogs) 

Conservation  The realisation that objects or sets of objects stay the same even when they are changed about or made to look different. 

Operation  The process of working something out in your head. Young children (in the sensorimotor and pre-operational stages) have to act, and try things out in the real world, to work

things out (like count on fingers): older children and adults can do more in their heads. 

Schema (or scheme) 

The representation in the mind of a set of perceptions, ideas, and/or actions, which go together. 

Stage  A period in a child's development in which he or she is capable of understanding some things but not others 

Some of Piaget's Key Ideas

CognitionSvovlsyreMikkel: 30*10 = 20*20 (skema fra +)

Cognition - ConstructivismBased on Piaget it becomes a common perception that-Knowledge and learning are matters for the individual-Knowledge is in the mind of the individual

Constructivism (von Glasersfeld) phrase it like this:

• knowledge is not transmitted and received passively, but actively built by each individual

• understanding is not a question of discovering an objectively existing world but reorganizing your experiencehttp://video.google.com/videoplay?docid=-7841878207694220233#

Cognition - ConstructivismBased on the idea that each individual constructs his/her own knowledge one can ask: What do we know?

Cognition - Constructivism

Experiences

Mind: Concept content

Refer to

External influencesymbolises

A model of Vygotsky’s theory of language and concept development (Høines)

Expressions

Language - Vygotsky

ExperiencesCounting to 7, writing 7, hearing 7

Mind: Concept content on 7

Codes which are used freely carrying concept content are called first order language – below there are 2 parallel first order languages

Expressions”seven”

Language - Vygotsky

ExperiencesCounting to 7, hearing 7

Mind: Concept content on 7

A new content expression introduced needs some translation via a well known expression – 7 is second order language

Expressions”seven”7 ?

Language - Vygotsky

ExperiencesCounting to 7, hearing 7

Mind: Concept content on 7

Working and training makes 7 a first order language – no need for translation

Expressions”seven”7

Language - VygotskyAccording to Vygotsky concept content and language about it develop simultaneously – knowledge cannot be speechless

First order language:You express spontaneously using first order language and think and learn by means of first order language.

Second order language: