presentation math 09
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WEEK 2 : TEACHING AND LEARNING OF
BASIC MATHEMATICS SKILLS
Y
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That science, or class of sciences, which treats of the exact relationsexisting between quantities or magnitudes, and of the methods bywhich, in accordance with these relations, quantities sought arededucible from other quantities known or supposed; the science of spatial and quantitative relations.
http://www.brainyquote.com/words/ma/mathematics
188408.html
math·e·mat·ics (mth'-mt'ks)n. (used with a sing. verb)The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.
Mathematics is the study of quality, structure, space, and change
WIKIPEDIA
The study of numbers, equations, functions, and geometric shapes(see geometry ) and their relationships. Some branches of mathematics are characterized by use of strict proofs basedon axioms. Some of its major subdivisions are arithmetic, algebra ,geometry, and calculus
SCIENCE DICTIONARY
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Pavlov
Thorndike
Skinner
Watson
Albert Bandura
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BEHAVIORISM THEORIES
PAVLOV (1849-1936)
For most people, the name "Pavlov" rings a bell. The
Russian physiologist is best known for his work in
classical conditioning or stimulus substitution.
Pavlov's most famous experiment involved food, a dog
and a bell
Pavlov¶s Experiment
ringing the bell caused no response f rom the dog.
Placing f ood in f ront of the
dog initiated salivation.
BEFORE
the bell was rung a f ew seconds bef ore
the dog was presented with f ood.
DURING
the ringing of the bell alone produced salivation
AFTER
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Stimulus and Response Items of Pavlov's Experiment
Food Unconditioned Stimulus
Salivation Unconditioned Response (natural, not learned)
Bell Conditioned Stimulus
Salivation Conditioned Response (to bell)
Other Observations Made by Pavlov~ Stimulus Generalization : Once the dog has learned to salivate at the sound of the bell, it will
salivate at other similar sounds.
~ Extinction: If you stop pairing the bell with the food, salivation will eventually cease in response to
the bell.
~ Spontaneous Recovery: Extinguished responses can be "recovered" after an elapsed time, but
will soon extinguish again if the dog is not presented with food.~ Discrimination: The dog could learn to discriminate between similar bells (stimuli) and discern
which bell would result in the presentation of food and which would not.
~ Higher-Order Conditioning: Once the dog has been conditioned to associate the bell with food,
another unconditioned stimulus, such as a light may be flashed at the same time that the bell is
rang. Eventually the dog will salivate at the flash of the light without the sound of the bell.
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Thorndike (1874 - 1949)His theory, Connectionism, stated that learning was
the formation of a connection between stimulus andresponse.
~ The "law of effect" stated that when a connection between a stimulus and
response is positively rewarded it will be strengthened and when it is negatively
rewarded it will be weakened. Thorndike later revised this "law" when he found that
negative reward, (punishment) did not necessarily weaken bonds, and that someseemingly pleasurable consequences do not necessarily motivate performance.
~ The "law of exercise" held that the more an S-R (stimulus response) bond is
practiced the stronger it will become. As with the law of effect, the law of exercise
also had to be updated when Thorndike found that practice without feedback does
not necessarily enhance performance.
~ The "law of readiness" : because of the structure of the nervous system, certain
conduction units, in a given situation, are more predisposed
to conduct than others
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Watson (1878 - 1958)
Watson believed that humans are born with a few
reflexes and the emotional reactions of love and rage.All other behavior is established through stimulus-
response associations through conditioning.
Watson's Experiment
~ Watson demonstrated classical conditioning in an experiment involving a young
child (Albert) and a white rat.~ Originally, Albert was unafraid of the rat; but Watson created a sudden loud noise
whenever Albert touched the rat.
~ Because Albert was frightened by the loud noise, he soon became conditioned to
fear and avoid the rat. The fear was generalized to other small animals.
~ Watson then "extinguished" the fear by presenting the rat withoutthe loud noise.
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Skinner (1904 ± 1990)
Skinner's Operant Conditioning Mechanisms
POSITIVEREINFORCEMENT
OR REWARD
Responses thatare rewarded arelikely to berepeated
NEGATIVEREINFORCEMENT
Responses thatallow escape frompainful or undesirablesituations arelikely to berepeated
EXTINCTION ORNON-
REINFORCEMENT
Responses thatare not reinforcedare not likely to berepeated.
PUNISHMENT
Responses thatbring painful or undesirableconsequences willbe suppressed,but may reappear if reinforcement
contingencieschange.
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Skinner and Behavioral Shaping
If placed in a cage ananimal may take a verylong time to figure out
that pressing a lever willproduce food.
To accomplish suchbehavior successiveapproximations of thebehavior are rewardeduntil the animal learns
the association betweenthe lever and the food
reward.
To begin shaping, theanimal may be rewardedfor simply turning in thedirection of the lever,
then for moving towardthe lever, for brushingagainst the lever, and
finally for pawing thelever.
Behavioral chainingoccurs when asuccession of stepsneed to be learned.
The animal wouldmaster each step insequence until the entire
sequence is learned.
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Reinf orcement Schedules
Once the desired behavioral response is accomplished, reinforcement doesnot have to be 100%; in fact it can be maintained more successfully through
what Skinner referred to as partial reinforcement schedules. Partial
reinforcement schedules include interval schedules and ratio schedules.
the target response is reinf orced af ter a f ixed amount of time has passed since the
last reinf orcement
Fixed Interval Schedules
similar to f ixed interval schedules, but the amount of time that must pass between
reinf orcement varies
Variable Interval Schedules
a f ixed number of correct responses must
occur bef ore reinf orcement may recur.
Fixed Ratio Schedules
the number of correct repetitions of the correct response f or reinf orcement varies.
Variable Ratio Schedules
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People can learn by observing the
behavior is of others and theoutcomes of those behaviors
Learning can occur a change inbehavior
Cognition plays a role in learning
Can be considered a bridge or atransition between behavioristlearning theories and cognitive
learning theories
General Principles Of Social Learning Theories
Albert Bandura
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Necessar y Conditions For Eff ective Modeling
ATTENTION ² various f actors increase or decrease the amount of
attention paid. Includes distinctiveness, aff ective valence,
prevalence, complexity, f unctional value. One¶s characteristics (e.g. sensor y capacities, arousal level,
perceptual set, past reinf orcement)aff ect attention.
RETENTION ² remembering what you paid attention to. Includes symbolic coding, mental images, cognitive
organization, symbolic rehearsal, motor rehearsal
REPRODUCTION ² reproducing the image. Including physical
capabilities, and self-observation of reproduction.
MOTIVATION ² having a good
reason to imitate. Includes motives such as past (i.e. traditional behaviorism),
promised (imagined incentives)and vicarious (seeing and
recalling the reinf orced model)
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