presentation math 09

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WEEK 2 : TEACHING AND LEARNING OF

BASIC MATHEMATICS SKILLS

Y



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



Pavlov

Thorndike

Skinner

Watson

Albert Bandura



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



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.



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



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.



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.



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.



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



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



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 asÂ past (i.e. traditional behaviorism),

promised (imagined incentives)and vicarious (seeing and

recalling the reinf orced model)

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